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| 1 | +// Copyright 2010-2025 Google LLC |
| 2 | +// Licensed under the Apache License, Version 2.0 (the "License"); |
| 3 | +// you may not use this file except in compliance with the License. |
| 4 | +// You may obtain a copy of the License at |
| 5 | +// |
| 6 | +// http://www.apache.org/licenses/LICENSE-2.0 |
| 7 | +// |
| 8 | +// Unless required by applicable law or agreed to in writing, software |
| 9 | +// distributed under the License is distributed on an "AS IS" BASIS, |
| 10 | +// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. |
| 11 | +// See the License for the specific language governing permissions and |
| 12 | +// limitations under the License. |
| 13 | + |
| 14 | +#include "ortools/graph/dag_connectivity.h" |
| 15 | + |
| 16 | +#include <algorithm> |
| 17 | +#include <cstdint> |
| 18 | +#include <utility> |
| 19 | +#include <vector> |
| 20 | + |
| 21 | +#include "absl/types/span.h" |
| 22 | +#include "ortools/base/container_logging.h" |
| 23 | +#include "ortools/base/logging.h" |
| 24 | +#include "ortools/graph/topologicalsorter.h" |
| 25 | + |
| 26 | +namespace operations_research { |
| 27 | + |
| 28 | +// The algorithm is as follows: |
| 29 | +// 1. Sort the nodes of the graph topologically. If a cycle is detected, |
| 30 | +// terminate |
| 31 | +// 2. Build the adjacency list for the graph, i.e., adj_list[i] is the list |
| 32 | +// of nodes that can be directly reached from i. |
| 33 | +// 3. Create a 2d bool vector x where x[i][j] indicates there is a path from |
| 34 | +// i to j, and for each arc in "arcs", set x[i][j] to true |
| 35 | +// 4. In reverse topological order (leaves first) for each node i, for each |
| 36 | +// child j of i, for each node k reachable for j, set k to be reachable |
| 37 | +// from i as well (x[i][k] = true for all k s.t. x[j][k] is true). |
| 38 | +// |
| 39 | +// The running times of the steps are: |
| 40 | +// 1. O(num_arcs) |
| 41 | +// 2. O(num_arcs) |
| 42 | +// 3. O(num_nodes^2 + num_arcs) |
| 43 | +// 4. O(num_nodes*num_arcs) |
| 44 | +// Thus the total run time is O(num_nodes^2 + num_nodes*num_arcs). |
| 45 | +// |
| 46 | +// Implementation note: typically, step 4 will dominate. To speed up the inner |
| 47 | +// loop, we use Bitset64, allowing use to merge 64 x[k][j] values at a time with |
| 48 | +// the |= operator. |
| 49 | +// |
| 50 | +// For graphs where num_arcs is o(num_nodes), a different data structure could |
| 51 | +// be used in 3, but this isn't really the interesting case (and prevents |=). |
| 52 | +// |
| 53 | +// A further improvement on this algorithm is possible, step four can run in |
| 54 | +// time O(num_nodes*num_arcs_in_transitive_reduction), and as a by product, |
| 55 | +// the transitive reduction can also be produced as output. For details, see |
| 56 | +// "A REDUCT-AND_CLOSURE ALGORITHM FOR GRAPHS" (Alla Goralcikova and |
| 57 | +// Vaclav Koubek 1979). The better typeset paper "AN IMPROVED ALGORITHM FOR |
| 58 | +// TRANSITIVE CLOSURE ON ACYCLIC DIGRAPHS" (Klaus Simon 1988) gives a slight |
| 59 | +// improvement on the result (less memory, same runtime). |
| 60 | +std::vector<Bitset64<int64_t>> ComputeDagConnectivity( |
| 61 | + absl::Span<const std::pair<int, int>> arcs, bool* error_was_cyclic, |
| 62 | + std::vector<int>* error_cycle_out) { |
| 63 | + CHECK(error_was_cyclic != nullptr); |
| 64 | + CHECK(error_cycle_out != nullptr); |
| 65 | + *error_was_cyclic = false; |
| 66 | + error_cycle_out->clear(); |
| 67 | + if (arcs.empty()) return {}; |
| 68 | + int num_nodes = 0; |
| 69 | + for (const std::pair<int, int>& arc : arcs) { |
| 70 | + CHECK_GE(arc.first, 0); |
| 71 | + CHECK_GE(arc.second, 0); |
| 72 | + num_nodes = std::max(num_nodes, arc.first + 1); |
| 73 | + num_nodes = std::max(num_nodes, arc.second + 1); |
| 74 | + } |
| 75 | + DenseIntStableTopologicalSorter sorter(num_nodes); |
| 76 | + for (const auto& arc : arcs) { |
| 77 | + sorter.AddEdge(arc.first, arc.second); |
| 78 | + } |
| 79 | + std::vector<int> topological_order; |
| 80 | + int next; |
| 81 | + while (sorter.GetNext(&next, error_was_cyclic, error_cycle_out)) { |
| 82 | + topological_order.push_back(next); |
| 83 | + } |
| 84 | + if (*error_was_cyclic) return {}; |
| 85 | + std::vector<std::vector<int>> adjacency_list(num_nodes); |
| 86 | + for (const auto& arc : arcs) { |
| 87 | + adjacency_list[arc.first].push_back(arc.second); |
| 88 | + } |
| 89 | + |
| 90 | + std::vector<Bitset64<int64_t>> connectivity(num_nodes); |
| 91 | + for (Bitset64<int64_t>& bitset : connectivity) { |
| 92 | + bitset.Resize(num_nodes); |
| 93 | + } |
| 94 | + for (const auto& arc : arcs) { |
| 95 | + connectivity[arc.first].Set(arc.second); |
| 96 | + } |
| 97 | + |
| 98 | + // Iterate over the nodes in reverse topological order. |
| 99 | + std::reverse(topological_order.begin(), topological_order.end()); |
| 100 | + // NOTE(user): these two loops visit every arc in the graph, and each |
| 101 | + // union is over a set of size given by the number of nodes. This gives the |
| 102 | + // runtime in step 4 of O(num_nodes*num_arcs) |
| 103 | + for (const int node : topological_order) { |
| 104 | + for (const int child : adjacency_list[node]) { |
| 105 | + connectivity[node].Union(connectivity[child]); |
| 106 | + } |
| 107 | + } |
| 108 | + return connectivity; |
| 109 | +} |
| 110 | + |
| 111 | +std::vector<Bitset64<int64_t>> ComputeDagConnectivityOrDie( |
| 112 | + absl::Span<const std::pair<int, int>> arcs) { |
| 113 | + bool error_was_cyclic = false; |
| 114 | + std::vector<int> error_cycle; |
| 115 | + std::vector<Bitset64<int64_t>> result = |
| 116 | + ComputeDagConnectivity(arcs, &error_was_cyclic, &error_cycle); |
| 117 | + CHECK(!error_was_cyclic) << "Graph should have been acyclic but has cycle: " |
| 118 | + << gtl::LogContainer(error_cycle); |
| 119 | + return result; |
| 120 | +} |
| 121 | + |
| 122 | +} // namespace operations_research |
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