diffn_util.cc

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// Copyright 2010-2025 Google LLC
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
 
#include "ortools/sat/diffn_util.h"
 
#include <stddef.h>
 
#include <algorithm>
#include <array>
#include <cmath>
#include <cstddef>
#include <cstdint>
#include <limits>
#include <optional>
#include <ostream>
#include <sstream>
#include <string>
#include <string_view>
#include <tuple>
#include <type_traits>
#include <utility>
#include <vector>
 
#include "absl/algorithm/container.h"
#include "absl/container/btree_set.h"
#include "absl/container/flat_hash_map.h"
#include "absl/container/flat_hash_set.h"
#include "absl/container/inlined_vector.h"
#include "absl/log/check.h"
#include "absl/random/bit_gen_ref.h"
#include "absl/types/optional.h"
#include "absl/types/span.h"
#include "ortools/base/logging.h"
#include "ortools/base/stl_util.h"
#include "ortools/base/strong_vector.h"
#include "ortools/graph/connected_components.h"
#include "ortools/graph/strongly_connected_components.h"
#include "ortools/sat/integer_base.h"
#include "ortools/sat/util.h"
#include "ortools/util/fixed_shape_binary_tree.h"
#include "ortools/util/integer_pq.h"
#include "ortools/util/strong_integers.h"
 
namespace operations_research {
namespace sat {
 
bool Rectangle::IsDisjoint(const Rectangle& other) const {
  return x_min >= other.x_max || other.x_min >= x_max || y_min >= other.y_max ||
         other.y_min >= y_max;
}
 
absl::InlinedVector<Rectangle, 4> Rectangle::RegionDifference(
    const Rectangle& other) const {
  const Rectangle intersect = Intersect(other);
  if (intersect.SizeX() == 0) {
    return {*this};
  }
 
  //-------------------
  //|   |    4    |   |
  //|   |---------|   |
  //| 1 |  other  | 2 |
  //|   |---------|   |
  //|   |    3    |   |
  //-------------------
  absl::InlinedVector<Rectangle, 4> result;
  if (x_min < intersect.x_min) {
    // Piece 1
    result.push_back({.x_min = x_min,
                      .x_max = intersect.x_min,
                      .y_min = y_min,
                      .y_max = y_max});
  }
  if (x_max > intersect.x_max) {
    // Piece 2
    result.push_back({.x_min = intersect.x_max,
                      .x_max = x_max,
                      .y_min = y_min,
                      .y_max = y_max});
  }
  if (y_min < intersect.y_min) {
    // Piece 3
    result.push_back({.x_min = intersect.x_min,
                      .x_max = intersect.x_max,
                      .y_min = y_min,
                      .y_max = intersect.y_min});
  }
  if (y_max > intersect.y_max) {
    // Piece 4
    result.push_back({.x_min = intersect.x_min,
                      .x_max = intersect.x_max,
                      .y_min = intersect.y_max,
                      .y_max = y_max});
  }
 
  return result;
}
 
CompactVectorVector<int> GetOverlappingRectangleComponents(
    absl::Span<const Rectangle> rectangles) {
  if (rectangles.empty()) return {};
 
  std::vector<std::pair<int, int>> intersections =
      FindPartialRectangleIntersections(rectangles);
  const int num_intersections = intersections.size();
  intersections.reserve(num_intersections * 2 + 1);
  for (int i = 0; i < num_intersections; ++i) {
    intersections.push_back({intersections[i].second, intersections[i].first});
  }
 
  CompactVectorVector<int> view;
  view.ResetFromPairs(intersections, /*minimum_num_nodes=*/rectangles.size());
  CompactVectorVector<int> components;
  FindStronglyConnectedComponents(static_cast<int>(rectangles.size()), view,
                                  &components);
  CompactVectorVector<int> result;
  for (int i = 0; i < components.size(); ++i) {
    absl::Span<const int> component = components[i];
    result.Add({});
    for (const int r : component) {
      result.AppendToLastVector(r);
    }
  }
  return result;
}
 
bool ReportEnergyConflict(Rectangle bounding_box, absl::Span<const int> boxes,
                          SchedulingConstraintHelper* x,
                          SchedulingConstraintHelper* y) {
  x->ClearReason();
  y->ClearReason();
  IntegerValue total_energy(0);
  for (const int b : boxes) {
    const IntegerValue x_min = x->ShiftedStartMin(b);
    const IntegerValue x_max = x->ShiftedEndMax(b);
    if (x_min < bounding_box.x_min || x_max > bounding_box.x_max) continue;
    const IntegerValue y_min = y->ShiftedStartMin(b);
    const IntegerValue y_max = y->ShiftedEndMax(b);
    if (y_min < bounding_box.y_min || y_max > bounding_box.y_max) continue;
 
    x->AddEnergyMinInIntervalReason(b, bounding_box.x_min, bounding_box.x_max);
    y->AddEnergyMinInIntervalReason(b, bounding_box.y_min, bounding_box.y_max);
 
    x->AddPresenceReason(b);
    y->AddPresenceReason(b);
 
    total_energy += x->SizeMin(b) * y->SizeMin(b);
 
    // We abort early if a subset of boxes is enough.
    // TODO(user): Also relax the box if possible.
    if (total_energy > bounding_box.Area()) break;
  }
 
  CHECK_GT(total_energy, bounding_box.Area());
  x->ImportOtherReasons(*y);
  return x->ReportConflict();
}
 
bool BoxesAreInEnergyConflict(absl::Span<const Rectangle> rectangles,
                              absl::Span<const IntegerValue> energies,
                              absl::Span<const int> boxes,
                              Rectangle* conflict) {
  // First consider all relevant intervals along the x axis.
  std::vector<IntegerValue> x_starts;
  std::vector<TaskTime> boxes_by_increasing_x_max;
  for (const int b : boxes) {
    x_starts.push_back(rectangles[b].x_min);
    boxes_by_increasing_x_max.push_back({b, rectangles[b].x_max});
  }
  gtl::STLSortAndRemoveDuplicates(&x_starts);
  std::sort(boxes_by_increasing_x_max.begin(), boxes_by_increasing_x_max.end());
 
  std::vector<IntegerValue> y_starts;
  std::vector<IntegerValue> energy_sum;
  std::vector<TaskTime> boxes_by_increasing_y_max;
 
  std::vector<std::vector<int>> stripes(x_starts.size());
  for (int i = 0; i < boxes_by_increasing_x_max.size(); ++i) {
    const int b = boxes_by_increasing_x_max[i].task_index;
    const IntegerValue x_min = rectangles[b].x_min;
    const IntegerValue x_max = rectangles[b].x_max;
    for (int j = 0; j < x_starts.size(); ++j) {
      if (x_starts[j] > x_min) break;
      stripes[j].push_back(b);
 
      // Redo the same on the y coordinate for the current x interval
      // which is [starts[j], x_max].
      y_starts.clear();
      boxes_by_increasing_y_max.clear();
      for (const int b : stripes[j]) {
        y_starts.push_back(rectangles[b].y_min);
        boxes_by_increasing_y_max.push_back({b, rectangles[b].y_max});
      }
      gtl::STLSortAndRemoveDuplicates(&y_starts);
      std::sort(boxes_by_increasing_y_max.begin(),
                boxes_by_increasing_y_max.end());
 
      const IntegerValue x_size = x_max - x_starts[j];
      energy_sum.assign(y_starts.size(), IntegerValue(0));
      for (int i = 0; i < boxes_by_increasing_y_max.size(); ++i) {
        const int b = boxes_by_increasing_y_max[i].task_index;
        const IntegerValue y_min = rectangles[b].y_min;
        const IntegerValue y_max = rectangles[b].y_max;
        for (int j = 0; j < y_starts.size(); ++j) {
          if (y_starts[j] > y_min) break;
          energy_sum[j] += energies[b];
          if (energy_sum[j] > x_size * (y_max - y_starts[j])) {
            if (conflict != nullptr) {
              *conflict = rectangles[b];
              for (int k = 0; k < i; ++k) {
                const int task_index = boxes_by_increasing_y_max[k].task_index;
                if (rectangles[task_index].y_min >= y_starts[j]) {
                  conflict->GrowToInclude(rectangles[task_index]);
                }
              }
            }
            return true;
          }
        }
      }
    }
  }
  return false;
}
 
bool AnalyzeIntervals(bool transpose, absl::Span<const int> local_boxes,
                      absl::Span<const Rectangle> rectangles,
                      absl::Span<const IntegerValue> rectangle_energies,
                      IntegerValue* x_threshold, IntegerValue* y_threshold,
                      Rectangle* conflict) {
  // First, we compute the possible x_min values (removing duplicates).
  // We also sort the relevant tasks by their x_max.
  //
  // TODO(user): If the number of unique x_max is smaller than the number of
  // unique x_min, it is better to do it the other way around.
  std::vector<IntegerValue> starts;
  std::vector<TaskTime> task_by_increasing_x_max;
  for (const int t : local_boxes) {
    const IntegerValue x_min =
        transpose ? rectangles[t].y_min : rectangles[t].x_min;
    const IntegerValue x_max =
        transpose ? rectangles[t].y_max : rectangles[t].x_max;
    starts.push_back(x_min);
    task_by_increasing_x_max.push_back({t, x_max});
  }
  gtl::STLSortAndRemoveDuplicates(&starts);
 
  // Note that for the same end_max, the order change our heuristic to
  // evaluate the max_conflict_height.
  std::sort(task_by_increasing_x_max.begin(), task_by_increasing_x_max.end());
 
  // The maximum y dimension of a bounding area for which there is a potential
  // conflict.
  IntegerValue max_conflict_height(0);
 
  // This is currently only used for logging.
  absl::flat_hash_set<std::pair<IntegerValue, IntegerValue>> stripes;
 
  // All quantities at index j correspond to the interval [starts[j], x_max].
  std::vector<IntegerValue> energies(starts.size(), IntegerValue(0));
  std::vector<IntegerValue> y_mins(starts.size(), kMaxIntegerValue);
  std::vector<IntegerValue> y_maxs(starts.size(), -kMaxIntegerValue);
  std::vector<IntegerValue> energy_at_max_y(starts.size(), IntegerValue(0));
  std::vector<IntegerValue> energy_at_min_y(starts.size(), IntegerValue(0));
 
  // Sentinel.
  starts.push_back(kMaxIntegerValue);
 
  // Iterate over all boxes by increasing x_max values.
  int first_j = 0;
  const IntegerValue threshold = transpose ? *y_threshold : *x_threshold;
  for (int i = 0; i < task_by_increasing_x_max.size(); ++i) {
    const int t = task_by_increasing_x_max[i].task_index;
 
    const IntegerValue energy = rectangle_energies[t];
    IntegerValue x_min = rectangles[t].x_min;
    IntegerValue x_max = rectangles[t].x_max;
    IntegerValue y_min = rectangles[t].y_min;
    IntegerValue y_max = rectangles[t].y_max;
    if (transpose) {
      std::swap(x_min, y_min);
      std::swap(x_max, y_max);
    }
 
    // Add this box contribution to all the [starts[j], x_max] intervals.
    while (first_j + 1 < starts.size() && x_max - starts[first_j] > threshold) {
      ++first_j;
    }
    for (int j = first_j; starts[j] <= x_min; ++j) {
      const IntegerValue old_energy_at_max = energy_at_max_y[j];
      const IntegerValue old_energy_at_min = energy_at_min_y[j];
 
      energies[j] += energy;
 
      const bool is_disjoint = y_min >= y_maxs[j] || y_max <= y_mins[j];
 
      if (y_min <= y_mins[j]) {
        if (y_min < y_mins[j]) {
          y_mins[j] = y_min;
          energy_at_min_y[j] = energy;
        } else {
          energy_at_min_y[j] += energy;
        }
      }
 
      if (y_max >= y_maxs[j]) {
        if (y_max > y_maxs[j]) {
          y_maxs[j] = y_max;
          energy_at_max_y[j] = energy;
        } else {
          energy_at_max_y[j] += energy;
        }
      }
 
      // If the new box is disjoint in y from the ones added so far, there
      // cannot be a new conflict involving this box, so we skip until we add
      // new boxes.
      if (is_disjoint) continue;
 
      const IntegerValue width = x_max - starts[j];
      IntegerValue conflict_height = CeilRatio(energies[j], width) - 1;
      if (y_max - y_min > conflict_height) continue;
      if (conflict_height >= y_maxs[j] - y_mins[j]) {
        // We have a conflict.
        if (conflict != nullptr) {
          *conflict = rectangles[t];
          for (int k = 0; k < i; ++k) {
            const int task_index = task_by_increasing_x_max[k].task_index;
            const IntegerValue task_x_min = transpose
                                                ? rectangles[task_index].y_min
                                                : rectangles[task_index].x_min;
            if (task_x_min < starts[j]) continue;
            conflict->GrowToInclude(rectangles[task_index]);
          }
        }
        return false;
      }
 
      // Because we currently do not have a conflict involving the new box, the
      // only way to have one is to remove enough energy to reduce the y domain.
      IntegerValue can_remove = std::min(old_energy_at_min, old_energy_at_max);
      if (old_energy_at_min < old_energy_at_max) {
        if (y_maxs[j] - y_min >=
            CeilRatio(energies[j] - old_energy_at_min, width)) {
          // In this case, we need to remove at least old_energy_at_max to have
          // a conflict.
          can_remove = old_energy_at_max;
        }
      } else if (old_energy_at_max < old_energy_at_min) {
        if (y_max - y_mins[j] >=
            CeilRatio(energies[j] - old_energy_at_max, width)) {
          can_remove = old_energy_at_min;
        }
      }
      conflict_height = CeilRatio(energies[j] - can_remove, width) - 1;
 
      // If the new box height is above the conflict_height, do not count
      // it now. We only need to consider conflict involving the new box.
      if (y_max - y_min > conflict_height) continue;
 
      if (VLOG_IS_ON(2)) stripes.insert({starts[j], x_max});
      max_conflict_height = std::max(max_conflict_height, conflict_height);
    }
  }
 
  VLOG(2) << " num_starts: " << starts.size() - 1 << "/" << local_boxes.size()
          << " conflict_height: " << max_conflict_height
          << " num_stripes:" << stripes.size() << " (<= " << threshold << ")";
 
  if (transpose) {
    *x_threshold = std::min(*x_threshold, max_conflict_height);
  } else {
    *y_threshold = std::min(*y_threshold, max_conflict_height);
  }
  return true;
}
 
absl::Span<int> FilterBoxesAndRandomize(
    absl::Span<const Rectangle> cached_rectangles, absl::Span<int> boxes,
    IntegerValue threshold_x, IntegerValue threshold_y,
    absl::BitGenRef random) {
  size_t new_size = 0;
  for (const int b : boxes) {
    const Rectangle& dim = cached_rectangles[b];
    if (dim.x_max - dim.x_min > threshold_x) continue;
    if (dim.y_max - dim.y_min > threshold_y) continue;
    boxes[new_size++] = b;
  }
  if (new_size == 0) return {};
  std::shuffle(&boxes[0], &boxes[0] + new_size, random);
  return {&boxes[0], new_size};
}
 
absl::Span<int> FilterBoxesThatAreTooLarge(
    absl::Span<const Rectangle> cached_rectangles,
    absl::Span<const IntegerValue> energies, absl::Span<int> boxes) {
  // Sort the boxes by increasing area.
  std::sort(boxes.begin(), boxes.end(), [cached_rectangles](int a, int b) {
    return cached_rectangles[a].Area() < cached_rectangles[b].Area();
  });
 
  IntegerValue total_energy(0);
  for (const int box : boxes) total_energy += energies[box];
 
  // Remove all the large boxes until we have one with area smaller than the
  // energy of the boxes below.
  int new_size = boxes.size();
  while (new_size > 0 &&
         cached_rectangles[boxes[new_size - 1]].Area() >= total_energy) {
    --new_size;
    total_energy -= energies[boxes[new_size]];
  }
  return boxes.subspan(0, new_size);
}
 
void ConstructOverlappingSets(absl::Span<IndexedInterval> intervals,
                              CompactVectorVector<int>* result,
                              absl::Span<const int> order) {
  result->clear();
  DCHECK(std::is_sorted(intervals.begin(), intervals.end(),
                        IndexedInterval::ComparatorByStart()));
  IntegerValue min_end_in_set = kMaxIntegerValue;
 
  // We do a line sweep. The "current" subset crossing the "line" at
  // (time, time + 1) will be in (*intervals)[start_index, end_index) at the end
  // of the loop block.
  int start_index = 0;
  const int size = intervals.size();
  for (int end_index = 0; end_index < size;) {
    const IntegerValue time = intervals[end_index].start;
 
    // First, if there is some deletion, we will push the "old" set to the
    // result before updating it. Otherwise, we will have a superset later, so
    // we just continue for now.
    if (min_end_in_set <= time) {
      // Push the current set to result first if its size is > 1.
      if (start_index + 1 < end_index) {
        result->Add({});
        for (int i = start_index; i < end_index; ++i) {
          result->AppendToLastVector(intervals[i].index);
        }
      }
 
      // Update the set. Note that we keep the order.
      min_end_in_set = kMaxIntegerValue;
      int new_start = end_index;
      for (int i = end_index; --i >= start_index;) {
        if (intervals[i].end > time) {
          min_end_in_set = std::min(min_end_in_set, intervals[i].end);
          intervals[--new_start] = intervals[i];
        }
      }
      start_index = new_start;
    }
 
    // Add all the new intervals starting exactly at "time".
    // Note that we always add at least one here.
    const int old_end = end_index;
    while (end_index < size && intervals[end_index].start == time) {
      min_end_in_set = std::min(min_end_in_set, intervals[end_index].end);
      ++end_index;
    }
 
    // If order is not empty, make sure we maintain the order.
    // TODO(user): we could only do that when we push a new set.
    if (!order.empty() && end_index > old_end) {
      std::sort(intervals.data() + old_end, intervals.data() + end_index,
                [order](const IndexedInterval& a, const IndexedInterval& b) {
                  return order[a.index] < order[b.index];
                });
      std::inplace_merge(
          intervals.data() + start_index, intervals.data() + old_end,
          intervals.data() + end_index,
          [order](const IndexedInterval& a, const IndexedInterval& b) {
            return order[a.index] < order[b.index];
          });
    }
  }
 
  // Push final set.
  if (start_index + 1 < size) {
    result->Add({});
    for (int i = start_index; i < size; ++i) {
      result->AppendToLastVector(intervals[i].index);
    }
  }
}
 
void GetOverlappingIntervalComponents(
    std::vector<IndexedInterval>* intervals,
    std::vector<std::vector<int>>* components) {
  components->clear();
  if (intervals->empty()) return;
  if (intervals->size() == 1) {
    components->push_back({intervals->front().index});
    return;
  }
 
  // For correctness, ComparatorByStart is enough, but in unit tests we want to
  // verify this function against another implementation, and fully defined
  // sorting with tie-breaking makes that much easier.
  // If that becomes a performance bottleneck:
  // - One may want to sort the list outside of this function, and simply
  //   have this function DCHECK that it's sorted by start.
  // - One may use stable_sort() with ComparatorByStart().
  std::sort(intervals->begin(), intervals->end(),
            IndexedInterval::ComparatorByStartThenEndThenIndex());
 
  IntegerValue end_max_so_far = (*intervals)[0].end;
  components->push_back({(*intervals)[0].index});
  for (int i = 1; i < intervals->size(); ++i) {
    const IndexedInterval& interval = (*intervals)[i];
    if (interval.start >= end_max_so_far) {
      components->push_back({interval.index});
    } else {
      components->back().push_back(interval.index);
    }
    end_max_so_far = std::max(end_max_so_far, interval.end);
  }
}
 
std::vector<int> GetIntervalArticulationPoints(
    std::vector<IndexedInterval>* intervals) {
  std::vector<int> articulation_points;
  if (intervals->size() < 3) return articulation_points;  // Empty.
  if (DEBUG_MODE) {
    for (const IndexedInterval& interval : *intervals) {
      DCHECK_LT(interval.start, interval.end);
    }
  }
 
  std::sort(intervals->begin(), intervals->end(),
            IndexedInterval::ComparatorByStart());
 
  IntegerValue end_max_so_far = (*intervals)[0].end;
  int index_of_max = 0;
  IntegerValue prev_end_max = kMinIntegerValue;  // Initialized as a sentinel.
  for (int i = 1; i < intervals->size(); ++i) {
    const IndexedInterval& interval = (*intervals)[i];
    if (interval.start >= end_max_so_far) {
      // New connected component.
      end_max_so_far = interval.end;
      index_of_max = i;
      prev_end_max = kMinIntegerValue;
      continue;
    }
    // Still the same connected component. Was the previous "max" an
    // articulation point ?
    if (prev_end_max != kMinIntegerValue && interval.start >= prev_end_max) {
      // We might be re-inserting the same articulation point: guard against it.
      if (articulation_points.empty() ||
          articulation_points.back() != index_of_max) {
        articulation_points.push_back(index_of_max);
      }
    }
    // Update the max end.
    if (interval.end > end_max_so_far) {
      prev_end_max = end_max_so_far;
      end_max_so_far = interval.end;
      index_of_max = i;
    } else if (interval.end > prev_end_max) {
      prev_end_max = interval.end;
    }
  }
  // Convert articulation point indices to IndexedInterval.index.
  for (int& index : articulation_points) index = (*intervals)[index].index;
  return articulation_points;
}
 
namespace {
bool IsZeroOrPowerOfTwo(int value) { return (value & (value - 1)) == 0; }
 
void AppendPairwiseRestriction(const ItemWithVariableSize& item1,
                               const ItemWithVariableSize& item2,
                               std::vector<PairwiseRestriction>* result) {
  const int state =
      // box1 can be left of box2.
      (item1.x.end_min <= item2.x.start_max) +
      // box1 can be right of box2.
      2 * (item2.x.end_min <= item1.x.start_max) +
      // box1 can be below box2.
      4 * (item1.y.end_min <= item2.y.start_max) +
      // box1 can be up of box2.
      8 * (item2.y.end_min <= item1.y.start_max);
 
  if (!IsZeroOrPowerOfTwo(state)) {
    return;
  }
 
  switch (state) {
    case 0:
      // Conflict. The two boxes must overlap in both dimensions.
      result->push_back(
          {.first_index = item1.index,
           .second_index = item2.index,
           .type = PairwiseRestriction::PairwiseRestrictionType::CONFLICT});
      break;
    case 1:
      // box2 can only be after box1 on x.
      if (item1.x.end_min > item2.x.start_min ||
          item2.x.start_max < item1.x.end_max) {
        result->push_back({.first_index = item1.index,
                           .second_index = item2.index,
                           .type = PairwiseRestriction::
                               PairwiseRestrictionType::FIRST_LEFT_OF_SECOND});
      }
      break;
    case 2:  // box1 an only be after box2 on x.
      if (item2.x.end_min > item1.x.start_min ||
          item1.x.start_max < item2.x.end_max) {
        result->push_back({.first_index = item1.index,
                           .second_index = item2.index,
                           .type = PairwiseRestriction::
                               PairwiseRestrictionType::FIRST_RIGHT_OF_SECOND});
      }
      break;
    case 4:
      // box2 an only be after box1 on y.
      if (item1.y.end_min > item2.y.start_min ||
          item2.y.start_max < item1.y.end_max) {
        result->push_back({.first_index = item1.index,
                           .second_index = item2.index,
                           .type = PairwiseRestriction::
                               PairwiseRestrictionType::FIRST_BELOW_SECOND});
      }
      break;
    case 8:  // box1 an only be after box2 on y.
      if (item2.y.end_min > item1.y.start_min ||
          item1.y.start_max < item2.y.end_max) {
        result->push_back({.first_index = item1.index,
                           .second_index = item2.index,
                           .type = PairwiseRestriction::
                               PairwiseRestrictionType::FIRST_ABOVE_SECOND});
      }
      break;
    default:
      break;
  }
}
}  // namespace
 
void AppendPairwiseRestrictions(absl::Span<const ItemWithVariableSize> items,
                                std::vector<PairwiseRestriction>* result) {
  for (int i1 = 0; i1 + 1 < items.size(); ++i1) {
    for (int i2 = i1 + 1; i2 < items.size(); ++i2) {
      AppendPairwiseRestriction(items[i1], items[i2], result);
    }
  }
}
 
void AppendPairwiseRestrictions(
    absl::Span<const ItemWithVariableSize> items,
    absl::Span<const ItemWithVariableSize> other_items,
    std::vector<PairwiseRestriction>* result) {
  for (int i1 = 0; i1 < items.size(); ++i1) {
    for (int i2 = 0; i2 < other_items.size(); ++i2) {
      AppendPairwiseRestriction(items[i1], other_items[i2], result);
    }
  }
}
 
void CapacityProfile::Clear() {
  events_.clear();
  num_rectangles_added_ = 0;
}
 
void CapacityProfile::AddRectangle(IntegerValue x_min, IntegerValue x_max,
                                   IntegerValue y_min, IntegerValue y_max) {
  DCHECK_LE(x_min, x_max);
  if (x_min == x_max) return;
 
  events_.push_back(
      StartRectangleEvent(num_rectangles_added_, x_min, y_min, y_max));
  events_.push_back(EndRectangleEvent(num_rectangles_added_, x_max));
  ++num_rectangles_added_;
}
 
void CapacityProfile::AddMandatoryConsumption(IntegerValue x_min,
                                              IntegerValue x_max,
                                              IntegerValue y_height) {
  DCHECK_LE(x_min, x_max);
  if (x_min == x_max) return;
 
  events_.push_back(ChangeMandatoryProfileEvent(x_min, y_height));
  events_.push_back(ChangeMandatoryProfileEvent(x_max, -y_height));
}
 
void CapacityProfile::BuildResidualCapacityProfile(
    std::vector<CapacityProfile::Rectangle>* result) {
  std::sort(events_.begin(), events_.end());
  IntegerPriorityQueue<QueueElement> min_pq(num_rectangles_added_);
  IntegerPriorityQueue<QueueElement> max_pq(num_rectangles_added_);
  IntegerValue mandatory_capacity(0);
 
  result->clear();
 
  result->push_back({kMinIntegerValue, IntegerValue(0)});
 
  for (int i = 0; i < events_.size();) {
    const IntegerValue current_time = events_[i].time;
    for (; i < events_.size(); ++i) {
      const Event& event = events_[i];
      if (event.time != current_time) break;
 
      switch (events_[i].type) {
        case START_RECTANGLE: {
          min_pq.Add({event.index, -event.y_min});
          max_pq.Add({event.index, event.y_max});
          break;
        }
        case END_RECTANGLE: {
          min_pq.Remove(event.index);
          max_pq.Remove(event.index);
          break;
        }
        case CHANGE_MANDATORY_PROFILE: {
          mandatory_capacity += event.y_min;
          break;
        }
      }
    }
 
    DCHECK(!max_pq.IsEmpty() || mandatory_capacity == 0);
    const IntegerValue new_height =
        max_pq.IsEmpty()
            ? IntegerValue(0)
            : max_pq.Top().value + min_pq.Top().value - mandatory_capacity;
    if (new_height != result->back().height) {
      result->push_back({current_time, new_height});
    }
  }
}
 
IntegerValue CapacityProfile::GetBoundingArea() {
  std::sort(events_.begin(), events_.end());
  IntegerPriorityQueue<QueueElement> min_pq(num_rectangles_added_);
  IntegerPriorityQueue<QueueElement> max_pq(num_rectangles_added_);
 
  IntegerValue area(0);
  IntegerValue previous_time = kMinIntegerValue;
  IntegerValue previous_height(0);
 
  for (int i = 0; i < events_.size();) {
    const IntegerValue current_time = events_[i].time;
    for (; i < events_.size(); ++i) {
      const Event& event = events_[i];
      if (event.time != current_time) break;
 
      switch (event.type) {
        case START_RECTANGLE: {
          min_pq.Add({event.index, -event.y_min});
          max_pq.Add({event.index, event.y_max});
          break;
        }
        case END_RECTANGLE: {
          min_pq.Remove(event.index);
          max_pq.Remove(event.index);
          break;
        }
        case CHANGE_MANDATORY_PROFILE: {
          break;
        }
      }
    }
    const IntegerValue new_height =
        max_pq.IsEmpty() ? IntegerValue(0)
                         : max_pq.Top().value + min_pq.Top().value;
    if (previous_height != 0) {
      area += previous_height * (current_time - previous_time);
    }
    previous_time = current_time;
    previous_height = new_height;
  }
  return area;
}
 
IntegerValue Smallest1DIntersection(IntegerValue range_min,
                                    IntegerValue range_max, IntegerValue size,
                                    IntegerValue interval_min,
                                    IntegerValue interval_max) {
  // If the item is on the left of the range, we get the intersection between
  // [range_min, range_min + size] and [interval_min, interval_max].
  const IntegerValue overlap_on_left =
      std::min(range_min + size, interval_max) -
      std::max(range_min, interval_min);
 
  // If the item is on the right of the range, we get the intersection between
  // [range_max - size, range_max] and [interval_min, interval_max].
  const IntegerValue overlap_on_right =
      std::min(range_max, interval_max) -
      std::max(range_max - size, interval_min);
 
  return std::max(IntegerValue(0), std::min(overlap_on_left, overlap_on_right));
}
 
ProbingRectangle::ProbingRectangle(
    const std::vector<RectangleInRange>& intervals)
    : intervals_(intervals) {
  minimum_energy_ = 0;
  if (intervals_.empty()) {
    return;
  }
  interval_points_sorted_by_x_.reserve(intervals_.size() * 4 + 2);
  interval_points_sorted_by_y_.reserve(intervals_.size() * 4 + 2);
 
  Rectangle bounding_box = {.x_min = std::numeric_limits<IntegerValue>::max(),
                            .x_max = std::numeric_limits<IntegerValue>::min(),
                            .y_min = std::numeric_limits<IntegerValue>::max(),
                            .y_max = std::numeric_limits<IntegerValue>::min()};
 
  for (int i = 0; i < intervals_.size(); ++i) {
    const RectangleInRange& interval = intervals_[i];
    minimum_energy_ += interval.x_size * interval.y_size;
 
    bounding_box.x_min =
        std::min(bounding_box.x_min, interval.bounding_area.x_min);
    bounding_box.x_max =
        std::max(bounding_box.x_max, interval.bounding_area.x_max);
    bounding_box.y_min =
        std::min(bounding_box.y_min, interval.bounding_area.y_min);
    bounding_box.y_max =
        std::max(bounding_box.y_max, interval.bounding_area.y_max);
 
    interval_points_sorted_by_x_.push_back({interval.bounding_area.x_min, i});
    interval_points_sorted_by_x_.push_back(
        {interval.bounding_area.x_min + interval.x_size, i});
    interval_points_sorted_by_x_.push_back(
        {interval.bounding_area.x_max - interval.x_size, i});
    interval_points_sorted_by_x_.push_back({interval.bounding_area.x_max, i});
 
    interval_points_sorted_by_y_.push_back({interval.bounding_area.y_min, i});
    interval_points_sorted_by_y_.push_back(
        {interval.bounding_area.y_min + interval.y_size, i});
    interval_points_sorted_by_y_.push_back(
        {interval.bounding_area.y_max - interval.y_size, i});
    interval_points_sorted_by_y_.push_back({interval.bounding_area.y_max, i});
  }
 
  full_energy_ = minimum_energy_;
  // Add four bogus points in the extremities so we can delegate setting up all
  // internal state to Shrink().
  interval_points_sorted_by_x_.push_back({bounding_box.x_min - 1, -1});
  interval_points_sorted_by_x_.push_back({bounding_box.x_max + 1, -1});
  interval_points_sorted_by_y_.push_back({bounding_box.y_min - 1, -1});
  interval_points_sorted_by_y_.push_back({bounding_box.y_max + 1, -1});
 
  auto comparator = [](const IntervalPoint& a, const IntervalPoint& b) {
    return std::tie(a.value, a.index) < std::tie(b.value, b.index);
  };
  gtl::STLSortAndRemoveDuplicates(&interval_points_sorted_by_x_, comparator);
  gtl::STLSortAndRemoveDuplicates(&interval_points_sorted_by_y_, comparator);
 
  grouped_intervals_sorted_by_x_.reserve(interval_points_sorted_by_x_.size());
  grouped_intervals_sorted_by_y_.reserve(interval_points_sorted_by_y_.size());
  int i = 0;
  while (i < interval_points_sorted_by_x_.size()) {
    int idx_begin = i;
    while (i < interval_points_sorted_by_x_.size() &&
           interval_points_sorted_by_x_[i].value ==
               interval_points_sorted_by_x_[idx_begin].value) {
      i++;
    }
    grouped_intervals_sorted_by_x_.push_back(
        {interval_points_sorted_by_x_[idx_begin].value,
         absl::Span<IntervalPoint>(interval_points_sorted_by_x_)
             .subspan(idx_begin, i - idx_begin)});
  }
 
  i = 0;
  while (i < interval_points_sorted_by_y_.size()) {
    int idx_begin = i;
    while (i < interval_points_sorted_by_y_.size() &&
           interval_points_sorted_by_y_[i].value ==
               interval_points_sorted_by_y_[idx_begin].value) {
      i++;
    }
    grouped_intervals_sorted_by_y_.push_back(
        {interval_points_sorted_by_y_[idx_begin].value,
         absl::Span<IntervalPoint>(interval_points_sorted_by_y_)
             .subspan(idx_begin, i - idx_begin)});
  }
 
  Reset();
}
 
void ProbingRectangle::Reset() {
  indexes_[Edge::LEFT] = 0;
  indexes_[Edge::RIGHT] = grouped_intervals_sorted_by_x_.size() - 1;
  indexes_[Edge::BOTTOM] = 0;
  indexes_[Edge::TOP] = grouped_intervals_sorted_by_y_.size() - 1;
 
  next_indexes_[Edge::LEFT] = 1;
  next_indexes_[Edge::RIGHT] = grouped_intervals_sorted_by_x_.size() - 2;
  next_indexes_[Edge::BOTTOM] = 1;
  next_indexes_[Edge::TOP] = grouped_intervals_sorted_by_y_.size() - 2;
 
  minimum_energy_ = full_energy_;
  ranges_touching_both_boundaries_[0].clear();
  ranges_touching_both_boundaries_[1].clear();
 
  for (int i = 0; i < 4; ++i) {
    corner_count_[i] = 0;
    intersect_length_[i] = 0;
    cached_delta_energy_[i] = 0;
  }
 
  // Remove the four bogus points we added.
  Shrink(Edge::LEFT);
  Shrink(Edge::BOTTOM);
  Shrink(Edge::RIGHT);
  Shrink(Edge::TOP);
}
 
Rectangle ProbingRectangle::GetCurrentRectangle() const {
  return {
      .x_min = grouped_intervals_sorted_by_x_[indexes_[Edge::LEFT]].coordinate,
      .x_max = grouped_intervals_sorted_by_x_[indexes_[Edge::RIGHT]].coordinate,
      .y_min =
          grouped_intervals_sorted_by_y_[indexes_[Edge::BOTTOM]].coordinate,
      .y_max = grouped_intervals_sorted_by_y_[indexes_[Edge::TOP]].coordinate};
}
 
namespace {
// Intersects `rectangle` with the largest rectangle that must intersect with
// the range in some way. To visualize this largest rectangle, imagine the four
// possible extreme positions for the item in range (the four corners). This
// rectangle is the one defined by the interior points of each position. This
// don't use IsDisjoint() because it also works when the rectangle would be
// malformed (it's bounding box less than twice the size).
bool CanConsumeEnergy(const Rectangle& rectangle,
                      const RectangleInRange& item) {
  return (rectangle.x_max > item.bounding_area.x_max - item.x_size) &&
         (rectangle.y_max > item.bounding_area.y_max - item.y_size) &&
         (rectangle.x_min < item.bounding_area.x_min + item.x_size) &&
         (rectangle.y_min < item.bounding_area.y_min + item.y_size);
}
 
std::array<bool, 4> GetPossibleEdgeIntersection(const Rectangle& rectangle,
                                                const RectangleInRange& range) {
  std::array<bool, 4> result;
  using Edge = ProbingRectangle::Edge;
  result[Edge::LEFT] = rectangle.x_min >= range.bounding_area.x_min;
  result[Edge::BOTTOM] = rectangle.y_min >= range.bounding_area.y_min;
  result[Edge::RIGHT] = rectangle.x_max <= range.bounding_area.x_max;
  result[Edge::TOP] = rectangle.y_max <= range.bounding_area.y_max;
  return result;
}
 
}  // namespace
 
// NOMUTANTS -- This is a sanity check, it is hard to corrupt the data in an
// unit test to check it will fail.
void ProbingRectangle::ValidateInvariants() const {
  const Rectangle current_rectangle = GetCurrentRectangle();
 
  IntegerValue intersect_length[4] = {0, 0, 0, 0};
  IntegerValue corner_count[4] = {0, 0, 0, 0};
  IntegerValue energy = 0;
  CHECK_LE(next_indexes_[Edge::LEFT], indexes_[Edge::RIGHT]);
  CHECK_LE(next_indexes_[Edge::BOTTOM], indexes_[Edge::TOP]);
  CHECK_GE(next_indexes_[Edge::TOP], indexes_[Edge::BOTTOM]);
  CHECK_GE(next_indexes_[Edge::RIGHT], indexes_[Edge::LEFT]);
 
  for (int interval_idx = 0; interval_idx < intervals_.size(); interval_idx++) {
    const RectangleInRange& range = intervals_[interval_idx];
 
    const Rectangle min_intersect =
        range.GetMinimumIntersection(current_rectangle);
    CHECK_LE(min_intersect.SizeX(), range.x_size);
    CHECK_LE(min_intersect.SizeY(), range.y_size);
    energy += min_intersect.Area();
 
    std::array<bool, 4> touching_boundary = {false, false, false, false};
    CHECK_EQ(CanConsumeEnergy(current_rectangle, range) &&
                 current_rectangle.Area() != 0,
             range.GetMinimumIntersectionArea(current_rectangle) != 0);
    if (CanConsumeEnergy(current_rectangle, range)) {
      touching_boundary = GetPossibleEdgeIntersection(current_rectangle, range);
    }
 
    CHECK_EQ(
        touching_boundary[Edge::LEFT] && touching_boundary[Edge::RIGHT],
        ranges_touching_both_boundaries_[Direction::LEFT_AND_RIGHT].contains(
            interval_idx));
    CHECK_EQ(
        touching_boundary[Edge::TOP] && touching_boundary[Edge::BOTTOM],
        ranges_touching_both_boundaries_[Direction::TOP_AND_BOTTOM].contains(
            interval_idx));
 
    if (touching_boundary[Edge::LEFT] && !touching_boundary[Edge::RIGHT]) {
      intersect_length[Edge::LEFT] += Smallest1DIntersection(
          range.bounding_area.y_min, range.bounding_area.y_max, range.y_size,
          current_rectangle.y_min, current_rectangle.y_max);
    }
 
    if (touching_boundary[Edge::RIGHT] && !touching_boundary[Edge::LEFT]) {
      intersect_length[Edge::RIGHT] += Smallest1DIntersection(
          range.bounding_area.y_min, range.bounding_area.y_max, range.y_size,
          current_rectangle.y_min, current_rectangle.y_max);
    }
 
    if (touching_boundary[Edge::TOP] && !touching_boundary[Edge::BOTTOM]) {
      intersect_length[Edge::TOP] += Smallest1DIntersection(
          range.bounding_area.x_min, range.bounding_area.x_max, range.x_size,
          current_rectangle.x_min, current_rectangle.x_max);
    }
 
    if (touching_boundary[Edge::BOTTOM] && !touching_boundary[Edge::TOP]) {
      intersect_length[Edge::BOTTOM] += Smallest1DIntersection(
          range.bounding_area.x_min, range.bounding_area.x_max, range.x_size,
          current_rectangle.x_min, current_rectangle.x_max);
    }
 
    if ((touching_boundary[Edge::LEFT] && touching_boundary[Edge::RIGHT]) ||
        (touching_boundary[Edge::TOP] && touching_boundary[Edge::BOTTOM])) {
      // We account separately for the problematic items that touches both
      // sides.
      continue;
    }
    if (touching_boundary[Edge::BOTTOM] && touching_boundary[Edge::LEFT]) {
      corner_count[RectangleInRange::Corner::BOTTOM_LEFT]++;
    }
    if (touching_boundary[Edge::BOTTOM] && touching_boundary[Edge::RIGHT]) {
      corner_count[RectangleInRange::Corner::BOTTOM_RIGHT]++;
    }
    if (touching_boundary[Edge::TOP] && touching_boundary[Edge::LEFT]) {
      corner_count[RectangleInRange::Corner::TOP_LEFT]++;
    }
    if (touching_boundary[Edge::TOP] && touching_boundary[Edge::RIGHT]) {
      corner_count[RectangleInRange::Corner::TOP_RIGHT]++;
    }
  }
 
  CHECK_EQ(energy, minimum_energy_);
  for (int i = 0; i < 4; i++) {
    CHECK_EQ(intersect_length[i], intersect_length_[i]);
    CHECK_EQ(corner_count[i], corner_count_[i]);
  }
}
 
namespace {
 
struct EdgeInfo {
  using Edge = ProbingRectangle::Edge;
  using Direction = ProbingRectangle::Direction;
  using Corner = RectangleInRange::Corner;
 
  Edge opposite_edge;
 
  struct OrthogonalInfo {
    Edge edge;
    Corner adjacent_corner;
  };
 
  Direction shrink_direction;
  Direction orthogonal_shrink_direction;
  // Lower coordinate one first (ie., BOTTOM before TOP, LEFT before RIGHT).
  OrthogonalInfo orthogonal_edges[2];
};
 
struct EdgeInfoHolder {
  using Edge = ProbingRectangle::Edge;
  using Direction = ProbingRectangle::Direction;
  using Corner = RectangleInRange::Corner;
 
  static constexpr EdgeInfo kLeft = {
      .opposite_edge = Edge::RIGHT,
      .shrink_direction = Direction::LEFT_AND_RIGHT,
      .orthogonal_shrink_direction = Direction::TOP_AND_BOTTOM,
      .orthogonal_edges = {
          {.edge = Edge::BOTTOM, .adjacent_corner = Corner::BOTTOM_LEFT},
          {.edge = Edge::TOP, .adjacent_corner = Corner::TOP_LEFT}}};
 
  static constexpr EdgeInfo kRight = {
      .opposite_edge = Edge::LEFT,
      .shrink_direction = Direction::LEFT_AND_RIGHT,
      .orthogonal_shrink_direction = Direction::TOP_AND_BOTTOM,
      .orthogonal_edges = {
          {.edge = Edge::BOTTOM, .adjacent_corner = Corner::BOTTOM_RIGHT},
          {.edge = Edge::TOP, .adjacent_corner = Corner::TOP_RIGHT}}};
  static constexpr EdgeInfo kBottom = {
      .opposite_edge = Edge::TOP,
      .shrink_direction = Direction::TOP_AND_BOTTOM,
      .orthogonal_shrink_direction = Direction::LEFT_AND_RIGHT,
      .orthogonal_edges = {
          {.edge = Edge::LEFT, .adjacent_corner = Corner::BOTTOM_LEFT},
          {.edge = Edge::RIGHT, .adjacent_corner = Corner::BOTTOM_RIGHT}}};
  static constexpr EdgeInfo kTop = {
      .opposite_edge = Edge::BOTTOM,
      .shrink_direction = Direction::TOP_AND_BOTTOM,
      .orthogonal_shrink_direction = Direction::LEFT_AND_RIGHT,
      .orthogonal_edges = {
          {.edge = Edge::LEFT, .adjacent_corner = Corner::TOP_LEFT},
          {.edge = Edge::RIGHT, .adjacent_corner = Corner::TOP_RIGHT}}};
};
 
constexpr const EdgeInfo& GetEdgeInfo(ProbingRectangle::Edge edge) {
  using Edge = ProbingRectangle::Edge;
  switch (edge) {
    case Edge::LEFT:
      return EdgeInfoHolder::kLeft;
    case Edge::RIGHT:
      return EdgeInfoHolder::kRight;
    case Edge::BOTTOM:
      return EdgeInfoHolder::kBottom;
    case Edge::TOP:
      return EdgeInfoHolder::kTop;
  }
}
 
IntegerValue GetSmallest1DIntersection(ProbingRectangle::Direction direction,
                                       const RectangleInRange& range,
                                       const Rectangle& rectangle) {
  switch (direction) {
    case ProbingRectangle::Direction::LEFT_AND_RIGHT:
      return Smallest1DIntersection(range.bounding_area.x_min,
                                    range.bounding_area.x_max, range.x_size,
                                    rectangle.x_min, rectangle.x_max);
    case ProbingRectangle::Direction::TOP_AND_BOTTOM:
      return Smallest1DIntersection(range.bounding_area.y_min,
                                    range.bounding_area.y_max, range.y_size,
                                    rectangle.y_min, rectangle.y_max);
  }
}
 
}  // namespace
 
template <ProbingRectangle::Edge edge>
void ProbingRectangle::ShrinkImpl() {
  constexpr EdgeInfo e = GetEdgeInfo(edge);
 
  bool update_next_index[4] = {false, false, false, false};
  update_next_index[edge] = true;
 
  IntegerValue step_1d_size;
  minimum_energy_ -= GetShrinkDeltaEnergy(edge);
  const std::vector<PointsForCoordinate>& sorted_intervals =
      e.shrink_direction == Direction::LEFT_AND_RIGHT
          ? grouped_intervals_sorted_by_x_
          : grouped_intervals_sorted_by_y_;
 
  const Rectangle prev_rectangle = GetCurrentRectangle();
  indexes_[edge] = next_indexes_[edge];
  const Rectangle current_rectangle = GetCurrentRectangle();
 
  switch (edge) {
    case Edge::LEFT:
      step_1d_size = current_rectangle.x_min - prev_rectangle.x_min;
      next_indexes_[edge] =
          std::min(indexes_[edge] + 1, indexes_[e.opposite_edge]);
      next_indexes_[e.opposite_edge] =
          std::max(indexes_[edge], next_indexes_[e.opposite_edge]);
      break;
    case Edge::BOTTOM:
      step_1d_size = current_rectangle.y_min - prev_rectangle.y_min;
      next_indexes_[edge] =
          std::min(indexes_[edge] + 1, indexes_[e.opposite_edge]);
      next_indexes_[e.opposite_edge] =
          std::max(indexes_[edge], next_indexes_[e.opposite_edge]);
      break;
    case Edge::RIGHT:
      step_1d_size = prev_rectangle.x_max - current_rectangle.x_max;
      next_indexes_[edge] =
          std::max(indexes_[edge] - 1, indexes_[e.opposite_edge]);
      next_indexes_[e.opposite_edge] =
          std::min(indexes_[edge], next_indexes_[e.opposite_edge]);
      break;
    case Edge::TOP:
      step_1d_size = prev_rectangle.y_max - current_rectangle.y_max;
      next_indexes_[edge] =
          std::max(indexes_[edge] - 1, indexes_[e.opposite_edge]);
      next_indexes_[e.opposite_edge] =
          std::min(indexes_[edge], next_indexes_[e.opposite_edge]);
      break;
  }
 
  absl::Span<ProbingRectangle::IntervalPoint> items_touching_coordinate =
      sorted_intervals[indexes_[edge]].items_touching_coordinate;
 
  IntegerValue delta_corner_count[4] = {0, 0, 0, 0};
  for (const auto& item : items_touching_coordinate) {
    const RectangleInRange& range = intervals_[item.index];
    if (!CanConsumeEnergy(prev_rectangle, range)) {
      // This item is out of our area of interest, skip.
      continue;
    }
 
    const std::array<bool, 4> touching_boundary_before =
        GetPossibleEdgeIntersection(prev_rectangle, range);
    const std::array<bool, 4> touching_boundary_after =
        CanConsumeEnergy(current_rectangle, range)
            ? GetPossibleEdgeIntersection(current_rectangle, range)
            : std::array<bool, 4>({false, false, false, false});
 
    bool remove_corner[4] = {false, false, false, false};
    auto erase_item = [this, &prev_rectangle, &range, &touching_boundary_before,
                       &remove_corner](Edge edge_to_erase) {
      const EdgeInfo& erase_info = GetEdgeInfo(edge_to_erase);
      intersect_length_[edge_to_erase] -= GetSmallest1DIntersection(
          erase_info.orthogonal_shrink_direction, range, prev_rectangle);
 
      if (touching_boundary_before[erase_info.orthogonal_edges[0].edge] &&
          touching_boundary_before[erase_info.orthogonal_edges[1].edge]) {
        // Ignore touching both corners
        return;
      }
      for (const auto [orthogonal_edge, corner] : erase_info.orthogonal_edges) {
        if (touching_boundary_before[orthogonal_edge]) {
          remove_corner[corner] = true;
        }
      }
    };
 
    if (touching_boundary_after[edge] && !touching_boundary_before[edge]) {
      if (touching_boundary_before[e.opposite_edge]) {
        ranges_touching_both_boundaries_[e.shrink_direction].insert(item.index);
        erase_item(e.opposite_edge);
      } else {
        // Do the opposite of remove_item().
        intersect_length_[edge] += GetSmallest1DIntersection(
            e.orthogonal_shrink_direction, range, prev_rectangle);
        // Update the corner count unless it is touching both.
        if (!touching_boundary_before[e.orthogonal_edges[0].edge] ||
            !touching_boundary_before[e.orthogonal_edges[1].edge]) {
          for (const auto [orthogonal_edge, corner] : e.orthogonal_edges) {
            if (touching_boundary_before[orthogonal_edge]) {
              delta_corner_count[corner]++;
            }
          }
        }
      }
    }
 
    for (int i = 0; i < 4; i++) {
      const Edge edge_to_update = (Edge)i;
      const EdgeInfo& info = GetEdgeInfo(edge_to_update);
      const bool remove_edge = touching_boundary_before[edge_to_update] &&
                               !touching_boundary_after[edge_to_update];
      if (!remove_edge) {
        continue;
      }
 
      update_next_index[edge_to_update] = true;
 
      if (touching_boundary_before[info.opposite_edge]) {
        ranges_touching_both_boundaries_[info.shrink_direction].erase(
            item.index);
      } else {
        erase_item(edge_to_update);
      }
    }
 
    for (int i = 0; i < 4; i++) {
      corner_count_[i] -= remove_corner[i];
    }
  }
 
  // Update the intersection length for items touching both sides.
  for (const int idx : ranges_touching_both_boundaries_[e.shrink_direction]) {
    const RectangleInRange& range = intervals_[idx];
    const std::array<bool, 2> touching_corner =
        (e.shrink_direction == Direction::LEFT_AND_RIGHT)
            ? std::array<bool, 2>(
                  {current_rectangle.y_min >= range.bounding_area.y_min,
                   current_rectangle.y_max <= range.bounding_area.y_max})
            : std::array<bool, 2>(
                  {current_rectangle.x_min >= range.bounding_area.x_min,
                   current_rectangle.x_max <= range.bounding_area.x_max});
    if (touching_corner[0] == touching_corner[1]) {
      // Either it is not touching neither corners (so no length to update) or
      // it is touching both corners, which will be handled by the "both
      // sides" code and should not contribute to intersect_length_.
      continue;
    }
 
    const IntegerValue incr =
        GetSmallest1DIntersection(e.shrink_direction, range, prev_rectangle) -
        GetSmallest1DIntersection(e.shrink_direction, range, current_rectangle);
    for (int i = 0; i < 2; i++) {
      if (touching_corner[i]) {
        intersect_length_[e.orthogonal_edges[i].edge] -= incr;
      }
    }
  }
 
  for (const auto [orthogonal_edge, corner] : e.orthogonal_edges) {
    intersect_length_[orthogonal_edge] -= corner_count_[corner] * step_1d_size;
  }
 
  for (int i = 0; i < 4; i++) {
    corner_count_[i] += delta_corner_count[i];
  }
 
  auto points_consume_energy =
      [this,
       &current_rectangle](absl::Span<ProbingRectangle::IntervalPoint> points) {
        for (const auto& item : points) {
          const RectangleInRange& range = intervals_[item.index];
          if (CanConsumeEnergy(current_rectangle, range)) {
            return true;
          }
        }
        return false;
      };
  if (update_next_index[Edge::LEFT]) {
    for (; next_indexes_[Edge::LEFT] < indexes_[Edge::RIGHT];
         ++next_indexes_[Edge::LEFT]) {
      if (points_consume_energy(
              grouped_intervals_sorted_by_x_[next_indexes_[Edge::LEFT]]
                  .items_touching_coordinate)) {
        break;
      }
    }
  }
  if (update_next_index[Edge::BOTTOM]) {
    for (; next_indexes_[Edge::BOTTOM] < indexes_[Edge::TOP];
         ++next_indexes_[Edge::BOTTOM]) {
      if (points_consume_energy(
              grouped_intervals_sorted_by_y_[next_indexes_[Edge::BOTTOM]]
                  .items_touching_coordinate)) {
        break;
      }
    }
  }
  if (update_next_index[Edge::RIGHT]) {
    for (; next_indexes_[Edge::RIGHT] > indexes_[Edge::LEFT];
         --next_indexes_[Edge::RIGHT]) {
      if (points_consume_energy(
              grouped_intervals_sorted_by_x_[next_indexes_[Edge::RIGHT]]
                  .items_touching_coordinate)) {
        break;
      }
    }
  }
  if (update_next_index[Edge::TOP]) {
    for (; next_indexes_[Edge::TOP] > indexes_[Edge::BOTTOM];
         --next_indexes_[Edge::TOP]) {
      if (points_consume_energy(
              grouped_intervals_sorted_by_y_[next_indexes_[Edge::TOP]]
                  .items_touching_coordinate)) {
        break;
      }
    }
  }
 
  probe_area_ = current_rectangle.Area();
  CacheShrinkDeltaEnergy(0);
  CacheShrinkDeltaEnergy(1);
}
 
void ProbingRectangle::Shrink(Edge edge) {
  switch (edge) {
    case Edge::LEFT:
      ShrinkImpl<Edge::LEFT>();
      return;
    case Edge::BOTTOM:
      ShrinkImpl<Edge::BOTTOM>();
      return;
    case Edge::RIGHT:
      ShrinkImpl<Edge::RIGHT>();
      return;
    case Edge::TOP:
      ShrinkImpl<Edge::TOP>();
      return;
  }
}
 
IntegerValue ProbingRectangle::GetShrinkDeltaArea(Edge edge) const {
  const Rectangle current_rectangle = GetCurrentRectangle();
  const std::vector<PointsForCoordinate>& sorted_intervals =
      (edge == Edge::LEFT || edge == Edge::RIGHT)
          ? grouped_intervals_sorted_by_x_
          : grouped_intervals_sorted_by_y_;
  const IntegerValue coordinate =
      sorted_intervals[next_indexes_[edge]].coordinate;
  switch (edge) {
    case Edge::LEFT:
      return (coordinate - current_rectangle.x_min) * current_rectangle.SizeY();
    case Edge::BOTTOM:
      return (coordinate - current_rectangle.y_min) * current_rectangle.SizeX();
    case Edge::RIGHT:
      return (current_rectangle.x_max - coordinate) * current_rectangle.SizeY();
    case Edge::TOP:
      return (current_rectangle.y_max - coordinate) * current_rectangle.SizeX();
  }
}
 
void ProbingRectangle::CacheShrinkDeltaEnergy(int dimension) {
  const Rectangle current_rectangle = GetCurrentRectangle();
  Rectangle next_rectangle_up = current_rectangle;
  Rectangle next_rectangle_down = current_rectangle;
  IntegerValue step_1d_size_up, step_1d_size_down;
  IntegerValue units_crossed_up, units_crossed_down;
  IntegerValue* delta_energy_up_ptr;
  IntegerValue* delta_energy_down_ptr;
 
  if (dimension == 0) {
    // CanShrink(Edge::RIGHT) and CanShrink(Edge::LEFT) are equivalent
    if (!CanShrink(Edge::LEFT)) {
      cached_delta_energy_[Edge::LEFT] = 0;
      cached_delta_energy_[Edge::RIGHT] = 0;
      return;
    }
 
    next_rectangle_up.x_min =
        grouped_intervals_sorted_by_x_[next_indexes_[Edge::LEFT]].coordinate;
    next_rectangle_down.x_max =
        grouped_intervals_sorted_by_x_[next_indexes_[Edge::RIGHT]].coordinate;
 
    step_1d_size_up = next_rectangle_up.x_min - current_rectangle.x_min;
    step_1d_size_down = current_rectangle.x_max - next_rectangle_down.x_max;
    units_crossed_up = intersect_length_[Edge::LEFT];
    units_crossed_down = intersect_length_[Edge::RIGHT];
    delta_energy_up_ptr = &cached_delta_energy_[Edge::LEFT];
    delta_energy_down_ptr = &cached_delta_energy_[Edge::RIGHT];
  } else {
    if (!CanShrink(Edge::TOP)) {
      cached_delta_energy_[Edge::BOTTOM] = 0;
      cached_delta_energy_[Edge::TOP] = 0;
      return;
    }
 
    next_rectangle_up.y_min =
        grouped_intervals_sorted_by_y_[next_indexes_[Edge::BOTTOM]].coordinate;
    next_rectangle_down.y_max =
        grouped_intervals_sorted_by_y_[next_indexes_[Edge::TOP]].coordinate;
 
    step_1d_size_up = next_rectangle_up.y_min - current_rectangle.y_min;
    step_1d_size_down = current_rectangle.y_max - next_rectangle_down.y_max;
    units_crossed_up = intersect_length_[Edge::BOTTOM];
    units_crossed_down = intersect_length_[Edge::TOP];
    delta_energy_up_ptr = &cached_delta_energy_[Edge::BOTTOM];
    delta_energy_down_ptr = &cached_delta_energy_[Edge::TOP];
  }
  IntegerValue delta_energy_up = 0;
  IntegerValue delta_energy_down = 0;
 
  // Note that the non-deterministic iteration order is fine here.
  for (const int idx : ranges_touching_both_boundaries_[dimension]) {
    const RectangleInRange& range = intervals_[idx];
    const IntegerValue curr_x = Smallest1DIntersection(
        range.bounding_area.x_min, range.bounding_area.x_max, range.x_size,
        current_rectangle.x_min, current_rectangle.x_max);
    const IntegerValue curr_y = Smallest1DIntersection(
        range.bounding_area.y_min, range.bounding_area.y_max, range.y_size,
        current_rectangle.y_min, current_rectangle.y_max);
    const IntegerValue curr = curr_x * curr_y;
    delta_energy_up += curr;
    delta_energy_down += curr;
 
    if (dimension == 0) {
      const IntegerValue up_x = Smallest1DIntersection(
          range.bounding_area.x_min, range.bounding_area.x_max, range.x_size,
          next_rectangle_up.x_min, next_rectangle_up.x_max);
      const IntegerValue down_x = Smallest1DIntersection(
          range.bounding_area.x_min, range.bounding_area.x_max, range.x_size,
          next_rectangle_down.x_min, next_rectangle_down.x_max);
 
      delta_energy_up -= curr_y * up_x;
      delta_energy_down -= curr_y * down_x;
    } else {
      const IntegerValue up_y = Smallest1DIntersection(
          range.bounding_area.y_min, range.bounding_area.y_max, range.y_size,
          next_rectangle_up.y_min, next_rectangle_up.y_max);
      const IntegerValue down_y = Smallest1DIntersection(
          range.bounding_area.y_min, range.bounding_area.y_max, range.y_size,
          next_rectangle_down.y_min, next_rectangle_down.y_max);
 
      delta_energy_up -= curr_x * up_y;
      delta_energy_down -= curr_x * down_y;
    }
  }
  delta_energy_up += units_crossed_up * step_1d_size_up;
  delta_energy_down += units_crossed_down * step_1d_size_down;
  *delta_energy_up_ptr = delta_energy_up;
  *delta_energy_down_ptr = delta_energy_down;
}
 
bool ProbingRectangle::CanShrink(Edge edge) const {
  switch (edge) {
    case Edge::LEFT:
    case Edge::RIGHT:
      return (next_indexes_[Edge::RIGHT] > indexes_[Edge::LEFT]);
    case Edge::BOTTOM:
    case Edge::TOP:
      return (indexes_[Edge::TOP] > next_indexes_[Edge::BOTTOM]);
  }
}
 
namespace {
std::vector<double> GetExpTable() {
  std::vector<double> table(101);
  for (int i = 0; i <= 100; ++i) {
    table[i] = std::exp(-(i - 50) / 5.0);
  }
  return table;
}
 
}  // namespace
 
FindRectanglesResult FindRectanglesWithEnergyConflictMC(
    const std::vector<RectangleInRange>& intervals, absl::BitGenRef random,
    double temperature, double candidate_energy_usage_factor) {
  FindRectanglesResult result;
  ProbingRectangle ranges(intervals);
 
  static const std::vector<double>* cached_probabilities =
      new std::vector<double>(GetExpTable());
 
  const double inv_temp = 1.0 / temperature;
  absl::InlinedVector<ProbingRectangle::Edge, 4> candidates;
  absl::InlinedVector<IntegerValue, 4> energy_deltas;
  absl::InlinedVector<double, 4> weights;
  while (!ranges.IsMinimal()) {
    const IntegerValue rect_area = ranges.GetCurrentRectangleArea();
    const IntegerValue min_energy = ranges.GetMinimumEnergy();
    if (min_energy > rect_area) {
      result.conflicts.push_back(ranges.GetCurrentRectangle());
    } else if (min_energy.value() >
               candidate_energy_usage_factor * rect_area.value()) {
      result.candidates.push_back(ranges.GetCurrentRectangle());
    }
    if (min_energy == 0) {
      break;
    }
    candidates.clear();
    energy_deltas.clear();
 
    for (int border_idx = 0; border_idx < 4; ++border_idx) {
      const ProbingRectangle::Edge border =
          static_cast<ProbingRectangle::Edge>(border_idx);
      if (!ranges.CanShrink(border)) {
        continue;
      }
      candidates.push_back(border);
      const IntegerValue delta_area = ranges.GetShrinkDeltaArea(border);
      const IntegerValue delta_energy = ranges.GetShrinkDeltaEnergy(border);
      energy_deltas.push_back(delta_energy - delta_area);
    }
    const IntegerValue min_energy_delta =
        *std::min_element(energy_deltas.begin(), energy_deltas.end());
    weights.clear();
    for (const IntegerValue delta_slack : energy_deltas) {
      const int64_t table_lookup =
          std::max((int64_t)0,
                   std::min((int64_t)((delta_slack - min_energy_delta).value() *
                                          5 * inv_temp +
                                      50),
                            (int64_t)100));
      weights.push_back((*cached_probabilities)[table_lookup]);
    }
    // Pick a change with a probability proportional to exp(- delta_E / Temp)
    ranges.Shrink(candidates[WeightedPick(weights, random)]);
  }
  if (ranges.GetMinimumEnergy() > ranges.GetCurrentRectangleArea()) {
    result.conflicts.push_back(ranges.GetCurrentRectangle());
  }
  return result;
}
 
std::string RenderDot(std::optional<Rectangle> bb,
                      absl::Span<const Rectangle> solution,
                      std::string_view extra_dot_payload) {
  const std::vector<std::string> colors = {"#0000ff80", "#ee00ee80",
                                           "#ff000080", "#eeee0080",
                                           "#00ff0080", "#00eeee80"};
  std::stringstream ss;
  ss << "digraph {\n";
  ss << "  graph [ bgcolor=lightgray ]\n";
  ss << "  node [style=filled shape=box]\n";
  if (bb.has_value()) {
    ss << "  bb [fillcolor=\"grey\" pos=\"" << 2 * bb->x_min + bb->SizeX()
       << "," << 2 * bb->y_min + bb->SizeY() << "!\" width=" << 2 * bb->SizeX()
       << " height=" << 2 * bb->SizeY() << "]\n";
  }
  for (int i = 0; i < solution.size(); ++i) {
    ss << "  " << i << " [fillcolor=\"" << colors[i % colors.size()]
       << "\" pos=\"" << 2 * solution[i].x_min + solution[i].SizeX() << ","
       << 2 * solution[i].y_min + solution[i].SizeY()
       << "!\" width=" << 2 * solution[i].SizeX()
       << " height=" << 2 * solution[i].SizeY() << "]\n";
  }
  ss << extra_dot_payload;
  ss << "}\n";
  return ss.str();
}
 
std::vector<Rectangle> FindEmptySpaces(
    const Rectangle& bounding_box, std::vector<Rectangle> ocupied_rectangles) {
  // Sorting is not necessary for correctness but makes it faster.
  std::sort(ocupied_rectangles.begin(), ocupied_rectangles.end(),
            [](const Rectangle& a, const Rectangle& b) {
              return std::tuple(a.x_min, -a.x_max, a.y_min) <
                     std::tuple(b.x_min, -b.x_max, b.y_min);
            });
  return PavedRegionDifference({bounding_box}, ocupied_rectangles);
}
 
std::vector<Rectangle> PavedRegionDifference(
    std::vector<Rectangle> original_region,
    absl::Span<const Rectangle> area_to_remove) {
  std::vector<Rectangle> new_area_to_cover;
  new_area_to_cover.reserve(original_region.size());
  for (const Rectangle& rectangle : area_to_remove) {
    new_area_to_cover.clear();
    for (const Rectangle& r : original_region) {
      const auto& new_rectangles = r.RegionDifference(rectangle);
      new_area_to_cover.insert(new_area_to_cover.end(), new_rectangles.begin(),
                               new_rectangles.end());
    }
    original_region.swap(new_area_to_cover);
    if (original_region.empty()) break;
  }
  return original_region;
}
 
// Each node in the tree will hold either a single box that is covering the
// whole interval represented by the node, or, if no such box exists, a superset
// of all the connected components of boxes that are overlapping the interval.
// It is a superset and not the exact set of connected components because we
// don't delete nodes that became stale, as explained in the class comment
// below.
struct BinaryTreeNode {
  // Contains exactly one element if occupying_box_index != -1.
  absl::flat_hash_set<int> connected_components_descendants;
  // Hold the x_max of the box that is currently occupying this node (if any) to
  // know when it is stale.
  int occupying_box_x_max;
  // -1 if not occupied.
  int occupying_box_index = -1;
};
 
// A data structure to store which boxes are overlapping the current sweep line.
// This uses a binary tree in a slight non-standard way: in a typical use of a
// binary tree the actual values are stored in the leaves and the intermediate
// nodes are there just to make finding the right leaf efficient. Here we do the
// opposite: the values are stored as high up in the tree as possible.
// For example, for a tree of size 8 a box that occupies the y interval [0, 7]
// will be stored as a single node at the root. In the same tree, a box that
// occupies [3, 7] will be stored with the nodes representing the [3, 4), [4, 6)
// and [6, 8) intervals. There is no difference on what is stored in the
// intermediate nodes or on the leaves. When the sweep line moves, we don't
// update the existing nodes on the tree. Thus, some nodes will become stale and
// will represent boxes that no longer overlap the sweep line. Those stale nodes
// get removed lazily.
struct SweepLineIntervalTree {
  explicit SweepLineIntervalTree(int max_y, int num_boxes)
      : tree(LeafIndex(max_y + 1)), tree_nodes(tree.StorageSize()) {
    union_find.SetNumberOfNodes(num_boxes);
  }
 
  // Recompute the connected components of a given node, by simply setting it to
  // {self} + left.connected_components + right.connected_components.
  void RecomputeConnectedComponents(TreeNodeIndex idx) {
    BinaryTreeNode& node = tree_nodes[idx];
    if (node.occupying_box_index != -1) {
      node.connected_components_descendants = {
          union_find.FindRoot(node.occupying_box_index)};
      return;
    }
    node.connected_components_descendants.clear();
    if (tree.IsLeaf(idx)) return;
    for (const TreeNodeIndex child_idx :
         {tree.LeftChild(idx), tree.RightChild(idx)}) {
      // The order is non-deterministic, but since this is doing the union of
      // hash sets the result is deterministic.
      for (const int c :
           tree_nodes[child_idx].connected_components_descendants) {
        node.connected_components_descendants.insert(union_find.FindRoot(c));
      }
    }
  }
 
  // We don't have global deletion method in this class, but this method
  // checks if a single interval is fully to the left of the sweep line and
  // removes it if so, also updating its connected components.
  void RemoveNodeIfXMaxLowerOrEqual(TreeNodeIndex idx, int x_threshold) {
    BinaryTreeNode& node = tree_nodes[idx];
    if (node.occupying_box_index == -1) {
      // Node is already empty.
      return;
    }
    if (node.occupying_box_x_max > x_threshold) {
      // Node is still overlapping the sweep line.
      return;
    }
    node.occupying_box_index = -1;
    RecomputeConnectedComponents(idx);
  }
 
  void UpdateChildrenIntersecting(TreeNodeIndex idx, int sweep_line_x_pos,
                                  int component_index,
                                  std::vector<int>* new_connections) {
    if (tree.IsLeaf(idx)) return;
    for (const TreeNodeIndex child_idx :
         {tree.LeftChild(idx), tree.RightChild(idx)}) {
      RemoveNodeIfXMaxLowerOrEqual(child_idx, sweep_line_x_pos);
      BinaryTreeNode& child_node = tree_nodes[child_idx];
      if (child_node.occupying_box_index != -1) {
        if (union_find.AddEdge(child_node.occupying_box_index,
                               component_index)) {
          new_connections->push_back(child_node.occupying_box_index);
        }
        // No need to recurse here: we already connected everything on this
        // branch to the new box.
        continue;
      }
      const bool had_different_component =
          absl::c_any_of(child_node.connected_components_descendants,
                         [this, component_index](const int c) {
                           return !union_find.Connected(c, component_index);
                         });
      // Since everything is intersecting the current box, all descendants
      // must be in one single component.
      child_node.connected_components_descendants = {component_index};
 
      // Only go down on the tree if we have below either:
      //  - a different component to connect.
      //  - a box to remove that is in a different component.
      // In any case, we will visit O(c + d) terminals, where c is the number of
      // components we are connecting and d is the number of boxes that we will
      // delete. Since a box can only be deleted log N times (one per interval
      // it was cut into) and we can only connect O(N) components in total, the
      // amortized cost of a call to UpdateChildrenIntersecting is O((log N)^2).
      if (had_different_component) {
        UpdateChildrenIntersecting(child_idx, sweep_line_x_pos, component_index,
                                   new_connections);
      }
    }
  }
 
  bool UpdateParents(TreeNodeIndex node, int sweep_line_x_pos,
                     int component_index, std::vector<int>* new_connections) {
    if (node == tree.Root()) return false;
    for (TreeNodeIndex parent = tree.Parent(node); parent != tree.Root();
         parent = tree.Parent(parent)) {
      RemoveNodeIfXMaxLowerOrEqual(parent, sweep_line_x_pos);
      BinaryTreeNode& parent_value = tree_nodes[parent];
      if (parent_value.occupying_box_index != -1) {
        if (union_find.AddEdge(parent_value.occupying_box_index,
                               component_index)) {
          new_connections->push_back(parent_value.occupying_box_index);
          return true;
        }
      }
      parent_value.connected_components_descendants.insert(component_index);
    }
    return false;
  }
 
  // Add a new box to the sweep line. This will store it in the tree (split in
  // log N intervals) check if it connects to one or more existing connected
  // components, and for each case it does, add the box that it is overlapping
  // to new_connections.
  void AddInterval(TreeNodeIndex idx, int sweep_line_x_pos, int box_index,
                   int x_max, std::vector<int>* new_connections) {
    RemoveNodeIfXMaxLowerOrEqual(idx, sweep_line_x_pos);
    int cur_box_component = union_find.FindRoot(box_index);
    BinaryTreeNode& node = tree_nodes[idx];
    if (node.occupying_box_index == -1) {
      node.connected_components_descendants = {box_index};
      node.occupying_box_index = box_index;
      node.occupying_box_x_max = x_max;
      const bool had_occupied_parent = UpdateParents(
          idx, sweep_line_x_pos, cur_box_component, new_connections);
      // We can only be connecting children if it is not already connect via
      // something above on the tree.
      if (!had_occupied_parent) {
        UpdateChildrenIntersecting(idx, sweep_line_x_pos, cur_box_component,
                                   new_connections);
      }
    } else {
      // We have already something fully occupying this interval.
      if (union_find.AddEdge(node.occupying_box_index, cur_box_component)) {
        new_connections->push_back(node.occupying_box_index);
        cur_box_component = union_find.FindRoot(cur_box_component);
      }
      node.connected_components_descendants = {cur_box_component};
      if (node.occupying_box_x_max < x_max) {
        // Replace the existing box by the new one.
        node.occupying_box_index = box_index;
        node.occupying_box_x_max = x_max;
      }
    }
  }
 
  FixedShapeBinaryTree tree;
  util_intops::StrongVector<TreeNodeIndex, BinaryTreeNode> tree_nodes;
  DenseConnectedComponentsFinder union_find;
};
 
struct Rectangle32 {
  int x_min;
  int x_max;
  int y_min;
  int y_max;
  int index;
};
 
// Requires that rectangles are sorted by x_min and that sizes on both
// dimensions are > 0.
std::vector<std::pair<int, int>> FindPartialRectangleIntersectionsImpl(
    absl::Span<const Rectangle32> rectangles, int32_t y_max) {
  // We are going to use a sweep line algorithm to find the intersections.
  // First, we sort the rectangles by their x coordinates, then consider a sweep
  // line that goes from the left to the right. See the comment on the
  // SweepLineIntervalTree class for more details about what we store for each
  // line.
  DCHECK(std::is_sorted(rectangles.begin(), rectangles.end(),
                        [](const Rectangle32& a, const Rectangle32& b) {
                          return a.x_min < b.x_min;
                        }));
 
  SweepLineIntervalTree interval_tree(y_max, rectangles.size());
 
  std::vector<TreeNodeIndex> interval_pieces;
  std::vector<int> new_connections;
  std::vector<std::pair<int, int>> arcs;
  for (int rectangle_index = 0; rectangle_index < rectangles.size();
       ++rectangle_index) {
    DCHECK_LT(rectangles[rectangle_index].x_min,
              rectangles[rectangle_index].x_max);
    DCHECK_LT(rectangles[rectangle_index].y_min,
              rectangles[rectangle_index].y_max);
    const int sweep_line_x_pos = rectangles[rectangle_index].x_min;
    const Rectangle32& r = rectangles[rectangle_index];
    interval_pieces.clear();
    interval_tree.tree.PartitionIntervalIntoNodes(
        LeafIndex(r.y_min), LeafIndex(r.y_max - 1), &interval_pieces);
    new_connections.clear();
    for (const TreeNodeIndex& node : interval_pieces) {
      interval_tree.AddInterval(node, sweep_line_x_pos, rectangle_index,
                                r.x_max, &new_connections);
    }
    for (const int new_connection : new_connections) {
      arcs.push_back({rectangles[new_connection].index, r.index});
    }
  }
  return arcs;
}
 
namespace {
struct PostProcessedResult {
  std::vector<Rectangle32> rectangles_sorted_by_x_min;
  std::pair<int32_t, int32_t> bounding_box;  // Always starting at (0,0).
};
 
PostProcessedResult ConvertToRectangle32WithNonZeroSizes(
    absl::Span<const Rectangle> rectangles) {
  // This function is a preprocessing function for algorithms that find overlap
  // between rectangles. It does the following:
  //  - It converts the arbitrary int64_t coordinates into a small integer by
  //    sorting the possible values and assigning them consecutive integers.
  //  - It grows zero size intervals to make them size one. This simplifies
  //    things considerably, since it is hard to reason about degenerated
  //    rectangles in the general algorithm.
  //
  // Note that the last point need to be done with care. Imagine the following
  // example:
  //  +----------+
  //  |          |
  //  |          +--------------+
  //  |          |              |
  //  |          |   p,q    r   |
  //  |          +----*-----*-+-+
  //  |          |            |
  //  |          |            |
  //  |          |            |
  //  |          +------------+
  //  |          |
  //  |          |
  //  +----------+
  // Where p,q and r are points (ie, boxes of size 0x0) and p and q have the
  // same coordinates. We replace them by the following:
  //  +----------+
  //  |          |
  //  |          +----------------------+
  //  |          |                      |
  //  |          |                      |
  //  |          +----+-+---------------+
  //  |          |    |p|
  //  |          |    +-+-+
  //  |          |      |q|
  //  |          |      +-+     +-+
  //  |          |              |r|
  //  |          +--------------+-+---+
  //  |          |                    |
  //  |          |                    |
  //  |          |                    |
  //  |          +--------------------+
  //  |          |
  //  |          |
  //  +----------+
  //
  // That is a pretty radical deformation of the original shape, but it retains
  // the property of whether a pair of rectangles intersect or not.
 
  if (rectangles.empty()) return {};
 
  enum class Event {
    kEnd = 0,
    kPoint = 1,
    kBegin = 2,
  };
  std::vector<std::tuple<IntegerValue, Event, int>> x_events;
  std::vector<std::tuple<IntegerValue, Event, int>> y_events;
  x_events.reserve(rectangles.size() * 2);
  y_events.reserve(rectangles.size() * 2);
  for (int i = 0; i < rectangles.size(); ++i) {
    const Rectangle& r = rectangles[i];
    DCHECK_GE(r.SizeX(), 0);
    DCHECK_GE(r.SizeY(), 0);
    if (r.SizeX() == 0) {
      x_events.push_back({r.x_min, Event::kPoint, i});
    } else {
      x_events.push_back({r.x_min, Event::kBegin, i});
      x_events.push_back({r.x_max, Event::kEnd, i});
    }
    if (r.SizeY() == 0) {
      y_events.push_back({r.y_min, Event::kPoint, i});
    } else {
      y_events.push_back({r.y_min, Event::kBegin, i});
      y_events.push_back({r.y_max, Event::kEnd, i});
    }
  }
  std::sort(y_events.begin(), y_events.end());
 
  std::vector<Rectangle32> rectangles32;
  rectangles32.resize(rectangles.size());
  IntegerValue prev_y = 0;
  Event prev_event = Event::kEnd;
  int cur_index = -1;
  for (int i = 0; i < y_events.size(); ++i) {
    const auto [y, event, index] = y_events[i];
    if ((prev_event != event && prev_event != Event::kEnd) || prev_y != y ||
        event == Event::kPoint || cur_index == -1) {
      ++cur_index;
    }
 
    switch (event) {
      case Event::kBegin:
        rectangles32[index].y_min = cur_index;
        rectangles32[index].index = index;
        break;
      case Event::kEnd:
        rectangles32[index].y_max = cur_index;
        break;
      case Event::kPoint:
        rectangles32[index].y_min = cur_index;
        rectangles32[index].y_max = cur_index + 1;
        rectangles32[index].index = index;
        break;
    }
    prev_event = event;
    prev_y = y;
  }
  const int max_y_index = cur_index + 1;
 
  gtl::STLClearObject(&y_events);
 
  std::sort(x_events.begin(), x_events.end());
  IntegerValue prev_x = 0;
  prev_event = Event::kEnd;
  cur_index = -1;
  for (int i = 0; i < x_events.size(); ++i) {
    const auto [x, event, index] = x_events[i];
    if ((prev_event != event && prev_event != Event::kEnd) || prev_x != x ||
        event == Event::kPoint || cur_index == -1) {
      ++cur_index;
    }
 
    switch (event) {
      case Event::kBegin:
        rectangles32[index].x_min = cur_index;
        break;
      case Event::kEnd:
        rectangles32[index].x_max = cur_index;
        break;
      case Event::kPoint:
        rectangles32[index].x_min = cur_index;
        rectangles32[index].x_max = cur_index + 1;
        break;
    }
    prev_event = event;
    prev_x = x;
  }
  const int max_x_index = cur_index + 1;
 
  std::vector<Rectangle32> sorted_rectangles32;
  sorted_rectangles32.reserve(rectangles.size());
  for (int i = 0; i < x_events.size(); ++i) {
    const auto [x, event, index] = x_events[i];
    if (event == Event::kBegin || event == Event::kPoint) {
      sorted_rectangles32.push_back(rectangles32[index]);
    }
  }
 
  return {sorted_rectangles32, {max_x_index, max_y_index}};
}
template <typename RectangleT>
std::optional<std::pair<int, int>> FindOneIntersectionIfPresentImpl(
    absl::Span<const RectangleT> rectangles) {
  using CoordinateType = std::decay_t<decltype(RectangleT::x_min)>;
  DCHECK(absl::c_is_sorted(rectangles,
                           [](const RectangleT& a, const RectangleT& b) {
                             return a.x_min < b.x_min;
                           }));
 
  // Set of box intersection the sweep line. We only store y_min, other
  // coordinates can be accessed via rectangles[index].coordinate.
  struct Element {
    mutable int index;
    CoordinateType y_min;
    bool operator<(const Element& other) const { return y_min < other.y_min; }
  };
 
  // Note: To use btree_set that has no iterator stability, we have to be
  // a bit careful below.
  absl::btree_set<Element> interval_set;
 
  for (int i = 0; i < rectangles.size(); ++i) {
    const CoordinateType x = rectangles[i].x_min;
    const CoordinateType y_min = rectangles[i].y_min;
    const CoordinateType y_max = rectangles[i].y_max;
 
    // TODO(user): We can handle that, but it require some changes below.
    DCHECK_LE(y_min, y_max);
 
    // Try to add this rectangle to the set, if there is an intersection, lazily
    // remove it if its x_max is already passed, otherwise report the
    // intersection.
    auto [it, inserted] = interval_set.insert({i, y_min});
    if (!inserted) {
      if (rectangles[it->index].x_max <= x) {
        // We just replace if the rectangle at position i is stale.
        it->index = i;
      } else {
        // Intersection.
        return {{it->index, i}};
      }
    } else {
      // If there was no element at position y_min, we need to test if the
      // interval before is stale or if it overlap with the new one.
      if (it != interval_set.begin()) {
        auto it_before = it;
        --it_before;
 
        // Lazy erase stale entry.
        if (rectangles[it_before->index].x_max <= x) {
          // For absl::btree_set we don't have iterator stability, so we do need
          // to re-assign 'it' to the element just after the one we erased.
          it = interval_set.erase(it_before);
        } else {
          DCHECK_LE(it_before->y_min, y_min);
          const CoordinateType y_max_before =
              rectangles[it_before->index].y_max;
          if (y_max_before > y_min) {
            // Intersection.
            return {{it_before->index, i}};
          }
        }
      }
    }
 
    // We handled the part before, now we need to deal with the interval that
    // starts after y_min.
    ++it;
    while (it != interval_set.end()) {
      // Lazy erase stale entry.
      if (rectangles[it->index].x_max <= x) {
        it = interval_set.erase(it);
        continue;
      }
 
      DCHECK_LE(y_min, it->y_min);
      if (y_max > it->y_min) {
        // Intersection.
        return {{it->index, i}};
      }
      break;
    }
  }
 
  return {};
}
 
}  // namespace
 
std::vector<std::pair<int, int>> FindPartialRectangleIntersections(
    absl::Span<const Rectangle> rectangles) {
  auto postprocessed = ConvertToRectangle32WithNonZeroSizes(rectangles);
  return FindPartialRectangleIntersectionsImpl(
      postprocessed.rectangles_sorted_by_x_min,
      postprocessed.bounding_box.second);
}
 
std::optional<std::pair<int, int>> FindOneIntersectionIfPresent(
    absl::Span<const Rectangle> rectangles) {
  return FindOneIntersectionIfPresentImpl(rectangles);
}
 
std::optional<std::pair<int, int>> FindOneIntersectionIfPresentWithZeroArea(
    absl::Span<const Rectangle> rectangles) {
  auto postprocessed = ConvertToRectangle32WithNonZeroSizes(rectangles);
  std::optional<std::pair<int, int>> result = FindOneIntersectionIfPresentImpl(
      absl::MakeConstSpan(postprocessed.rectangles_sorted_by_x_min));
  if (!result.has_value()) return {};
  return {{postprocessed.rectangles_sorted_by_x_min[result->first].index,
           postprocessed.rectangles_sorted_by_x_min[result->second].index}};
}
 
}  // namespace sat
}  // namespace operations_research