diff --git a/lemon/cost_scaling.h b/lemon/cost_scaling.h --- a/lemon/cost_scaling.h +++ b/lemon/cost_scaling.h @@ -201,10 +201,11 @@ TEMPLATE_DIGRAPH_TYPEDEFS(GR); typedef std::vector IntVector; - typedef std::vector BoolVector; typedef std::vector ValueVector; typedef std::vector CostVector; typedef std::vector LargeCostVector; + typedef std::vector BoolVector; + // Note: vector is used instead of vector for efficiency reasons private: @@ -248,6 +249,7 @@ // Parameters of the problem bool _have_lower; Value _sum_supply; + int _sup_node_num; // Data structures for storing the digraph IntNodeMap _node_id; @@ -276,6 +278,12 @@ LargeCost _epsilon; int _alpha; + IntVector _buckets; + IntVector _bucket_next; + IntVector _bucket_prev; + IntVector _rank; + int _max_rank; + // Data for a StaticDigraph structure typedef std::pair IntPair; StaticDigraph _sgr; @@ -828,6 +836,11 @@ } } + _sup_node_num = 0; + for (NodeIt n(_graph); n != INVALID; ++n) { + if (sup[n] > 0) ++_sup_node_num; + } + // Find a feasible flow using Circulation Circulation, ValueArcMap, ValueNodeMap> circ(_graph, low, cap, sup); @@ -862,7 +875,7 @@ } for (int a = _first_out[_root]; a != _res_arc_num; ++a) { int ra = _reverse[a]; - _res_cap[a] = 1; + _res_cap[a] = 0; _res_cap[ra] = 0; _cost[a] = 0; _cost[ra] = 0; @@ -876,7 +889,14 @@ void start(Method method) { // Maximum path length for partial augment const int MAX_PATH_LENGTH = 4; - + + // Initialize data structures for buckets + _max_rank = _alpha * _res_node_num; + _buckets.resize(_max_rank); + _bucket_next.resize(_res_node_num + 1); + _bucket_prev.resize(_res_node_num + 1); + _rank.resize(_res_node_num + 1); + // Execute the algorithm switch (method) { case PUSH: @@ -915,63 +935,175 @@ } } } + + // Initialize a cost scaling phase + void initPhase() { + // Saturate arcs not satisfying the optimality condition + for (int u = 0; u != _res_node_num; ++u) { + int last_out = _first_out[u+1]; + LargeCost pi_u = _pi[u]; + for (int a = _first_out[u]; a != last_out; ++a) { + int v = _target[a]; + if (_res_cap[a] > 0 && _cost[a] + pi_u - _pi[v] < 0) { + Value delta = _res_cap[a]; + _excess[u] -= delta; + _excess[v] += delta; + _res_cap[a] = 0; + _res_cap[_reverse[a]] += delta; + } + } + } + + // Find active nodes (i.e. nodes with positive excess) + for (int u = 0; u != _res_node_num; ++u) { + if (_excess[u] > 0) _active_nodes.push_back(u); + } + + // Initialize the next arcs + for (int u = 0; u != _res_node_num; ++u) { + _next_out[u] = _first_out[u]; + } + } + + // Early termination heuristic + bool earlyTermination() { + const double EARLY_TERM_FACTOR = 3.0; + + // Build a static residual graph + _arc_vec.clear(); + _cost_vec.clear(); + for (int j = 0; j != _res_arc_num; ++j) { + if (_res_cap[j] > 0) { + _arc_vec.push_back(IntPair(_source[j], _target[j])); + _cost_vec.push_back(_cost[j] + 1); + } + } + _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end()); + + // Run Bellman-Ford algorithm to check if the current flow is optimal + BellmanFord bf(_sgr, _cost_map); + bf.init(0); + bool done = false; + int K = int(EARLY_TERM_FACTOR * std::sqrt(double(_res_node_num))); + for (int i = 0; i < K && !done; ++i) { + done = bf.processNextWeakRound(); + } + return done; + } + + // Global potential update heuristic + void globalUpdate() { + int bucket_end = _root + 1; + + // Initialize buckets + for (int r = 0; r != _max_rank; ++r) { + _buckets[r] = bucket_end; + } + Value total_excess = 0; + for (int i = 0; i != _res_node_num; ++i) { + if (_excess[i] < 0) { + _rank[i] = 0; + _bucket_next[i] = _buckets[0]; + _bucket_prev[_buckets[0]] = i; + _buckets[0] = i; + } else { + total_excess += _excess[i]; + _rank[i] = _max_rank; + } + } + if (total_excess == 0) return; + + // Search the buckets + int r = 0; + for ( ; r != _max_rank; ++r) { + while (_buckets[r] != bucket_end) { + // Remove the first node from the current bucket + int u = _buckets[r]; + _buckets[r] = _bucket_next[u]; + + // Search the incomming arcs of u + LargeCost pi_u = _pi[u]; + int last_out = _first_out[u+1]; + for (int a = _first_out[u]; a != last_out; ++a) { + int ra = _reverse[a]; + if (_res_cap[ra] > 0) { + int v = _source[ra]; + int old_rank_v = _rank[v]; + if (r < old_rank_v) { + // Compute the new rank of v + LargeCost nrc = (_cost[ra] + _pi[v] - pi_u) / _epsilon; + int new_rank_v = old_rank_v; + if (nrc < LargeCost(_max_rank)) + new_rank_v = r + 1 + int(nrc); + + // Change the rank of v + if (new_rank_v < old_rank_v) { + _rank[v] = new_rank_v; + _next_out[v] = _first_out[v]; + + // Remove v from its old bucket + if (old_rank_v < _max_rank) { + if (_buckets[old_rank_v] == v) { + _buckets[old_rank_v] = _bucket_next[v]; + } else { + _bucket_next[_bucket_prev[v]] = _bucket_next[v]; + _bucket_prev[_bucket_next[v]] = _bucket_prev[v]; + } + } + + // Insert v to its new bucket + _bucket_next[v] = _buckets[new_rank_v]; + _bucket_prev[_buckets[new_rank_v]] = v; + _buckets[new_rank_v] = v; + } + } + } + } + + // Finish search if there are no more active nodes + if (_excess[u] > 0) { + total_excess -= _excess[u]; + if (total_excess <= 0) break; + } + } + if (total_excess <= 0) break; + } + + // Relabel nodes + for (int u = 0; u != _res_node_num; ++u) { + int k = std::min(_rank[u], r); + if (k > 0) { + _pi[u] -= _epsilon * k; + _next_out[u] = _first_out[u]; + } + } + } /// Execute the algorithm performing augment and relabel operations void startAugment(int max_length = std::numeric_limits::max()) { // Paramters for heuristics - const int BF_HEURISTIC_EPSILON_BOUND = 1000; - const int BF_HEURISTIC_BOUND_FACTOR = 3; + const int EARLY_TERM_EPSILON_LIMIT = 1000; + const double GLOBAL_UPDATE_FACTOR = 3.0; + const int global_update_freq = int(GLOBAL_UPDATE_FACTOR * + (_res_node_num + _sup_node_num * _sup_node_num)); + int next_update_limit = global_update_freq; + + int relabel_cnt = 0; + // Perform cost scaling phases - IntVector pred_arc(_res_node_num); - std::vector path_nodes; + std::vector path; for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ? 1 : _epsilon / _alpha ) { - // "Early Termination" heuristic: use Bellman-Ford algorithm - // to check if the current flow is optimal - if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) { - _arc_vec.clear(); - _cost_vec.clear(); - for (int j = 0; j != _res_arc_num; ++j) { - if (_res_cap[j] > 0) { - _arc_vec.push_back(IntPair(_source[j], _target[j])); - _cost_vec.push_back(_cost[j] + 1); - } - } - _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end()); - - BellmanFord bf(_sgr, _cost_map); - bf.init(0); - bool done = false; - int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(_res_node_num)); - for (int i = 0; i < K && !done; ++i) - done = bf.processNextWeakRound(); - if (done) break; - } - - // Saturate arcs not satisfying the optimality condition - for (int a = 0; a != _res_arc_num; ++a) { - if (_res_cap[a] > 0 && - _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) { - Value delta = _res_cap[a]; - _excess[_source[a]] -= delta; - _excess[_target[a]] += delta; - _res_cap[a] = 0; - _res_cap[_reverse[a]] += delta; - } + // Early termination heuristic + if (_epsilon <= EARLY_TERM_EPSILON_LIMIT) { + if (earlyTermination()) break; } - // Find active nodes (i.e. nodes with positive excess) - for (int u = 0; u != _res_node_num; ++u) { - if (_excess[u] > 0) _active_nodes.push_back(u); - } - - // Initialize the next arcs - for (int u = 0; u != _res_node_num; ++u) { - _next_out[u] = _first_out[u]; - } - + // Initialize current phase + initPhase(); + // Perform partial augment and relabel operations while (true) { // Select an active node (FIFO selection) @@ -981,46 +1113,44 @@ } if (_active_nodes.size() == 0) break; int start = _active_nodes.front(); - path_nodes.clear(); - path_nodes.push_back(start); // Find an augmenting path from the start node + path.clear(); int tip = start; - while (_excess[tip] >= 0 && - int(path_nodes.size()) <= max_length) { + while (_excess[tip] >= 0 && int(path.size()) < max_length) { int u; - LargeCost min_red_cost, rc; - int last_out = _sum_supply < 0 ? - _first_out[tip+1] : _first_out[tip+1] - 1; + LargeCost min_red_cost, rc, pi_tip = _pi[tip]; + int last_out = _first_out[tip+1]; for (int a = _next_out[tip]; a != last_out; ++a) { - if (_res_cap[a] > 0 && - _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) { - u = _target[a]; - pred_arc[u] = a; + u = _target[a]; + if (_res_cap[a] > 0 && _cost[a] + pi_tip - _pi[u] < 0) { + path.push_back(a); _next_out[tip] = a; tip = u; - path_nodes.push_back(tip); goto next_step; } } // Relabel tip node - min_red_cost = std::numeric_limits::max() / 2; + min_red_cost = std::numeric_limits::max(); + if (tip != start) { + int ra = _reverse[path.back()]; + min_red_cost = _cost[ra] + pi_tip - _pi[_target[ra]]; + } for (int a = _first_out[tip]; a != last_out; ++a) { - rc = _cost[a] + _pi[_source[a]] - _pi[_target[a]]; + rc = _cost[a] + pi_tip - _pi[_target[a]]; if (_res_cap[a] > 0 && rc < min_red_cost) { min_red_cost = rc; } } _pi[tip] -= min_red_cost + _epsilon; - - // Reset the next arc of tip _next_out[tip] = _first_out[tip]; + ++relabel_cnt; // Step back if (tip != start) { - path_nodes.pop_back(); - tip = path_nodes.back(); + tip = _source[path.back()]; + path.pop_back(); } next_step: ; @@ -1028,11 +1158,11 @@ // Augment along the found path (as much flow as possible) Value delta; - int u, v = path_nodes.front(), pa; - for (int i = 1; i < int(path_nodes.size()); ++i) { + int pa, u, v = start; + for (int i = 0; i != int(path.size()); ++i) { + pa = path[i]; u = v; - v = path_nodes[i]; - pa = pred_arc[v]; + v = _target[pa]; delta = std::min(_res_cap[pa], _excess[u]); _res_cap[pa] -= delta; _res_cap[_reverse[pa]] += delta; @@ -1041,6 +1171,12 @@ if (_excess[v] > 0 && _excess[v] <= delta) _active_nodes.push_back(v); } + + // Global update heuristic + if (relabel_cnt >= next_update_limit) { + globalUpdate(); + next_update_limit += global_update_freq; + } } } } @@ -1048,98 +1184,70 @@ /// Execute the algorithm performing push and relabel operations void startPush() { // Paramters for heuristics - const int BF_HEURISTIC_EPSILON_BOUND = 1000; - const int BF_HEURISTIC_BOUND_FACTOR = 3; + const int EARLY_TERM_EPSILON_LIMIT = 1000; + const double GLOBAL_UPDATE_FACTOR = 2.0; + const int global_update_freq = int(GLOBAL_UPDATE_FACTOR * + (_res_node_num + _sup_node_num * _sup_node_num)); + int next_update_limit = global_update_freq; + + int relabel_cnt = 0; + // Perform cost scaling phases BoolVector hyper(_res_node_num, false); + LargeCostVector hyper_cost(_res_node_num); for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ? 1 : _epsilon / _alpha ) { - // "Early Termination" heuristic: use Bellman-Ford algorithm - // to check if the current flow is optimal - if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) { - _arc_vec.clear(); - _cost_vec.clear(); - for (int j = 0; j != _res_arc_num; ++j) { - if (_res_cap[j] > 0) { - _arc_vec.push_back(IntPair(_source[j], _target[j])); - _cost_vec.push_back(_cost[j] + 1); - } - } - _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end()); - - BellmanFord bf(_sgr, _cost_map); - bf.init(0); - bool done = false; - int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(_res_node_num)); - for (int i = 0; i < K && !done; ++i) - done = bf.processNextWeakRound(); - if (done) break; + // Early termination heuristic + if (_epsilon <= EARLY_TERM_EPSILON_LIMIT) { + if (earlyTermination()) break; } - - // Saturate arcs not satisfying the optimality condition - for (int a = 0; a != _res_arc_num; ++a) { - if (_res_cap[a] > 0 && - _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) { - Value delta = _res_cap[a]; - _excess[_source[a]] -= delta; - _excess[_target[a]] += delta; - _res_cap[a] = 0; - _res_cap[_reverse[a]] += delta; - } - } - - // Find active nodes (i.e. nodes with positive excess) - for (int u = 0; u != _res_node_num; ++u) { - if (_excess[u] > 0) _active_nodes.push_back(u); - } - - // Initialize the next arcs - for (int u = 0; u != _res_node_num; ++u) { - _next_out[u] = _first_out[u]; - } + + // Initialize current phase + initPhase(); // Perform push and relabel operations while (_active_nodes.size() > 0) { - LargeCost min_red_cost, rc; + LargeCost min_red_cost, rc, pi_n; Value delta; int n, t, a, last_out = _res_arc_num; + next_node: // Select an active node (FIFO selection) - next_node: n = _active_nodes.front(); - last_out = _sum_supply < 0 ? - _first_out[n+1] : _first_out[n+1] - 1; - + last_out = _first_out[n+1]; + pi_n = _pi[n]; + // Perform push operations if there are admissible arcs if (_excess[n] > 0) { for (a = _next_out[n]; a != last_out; ++a) { if (_res_cap[a] > 0 && - _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) { + _cost[a] + pi_n - _pi[_target[a]] < 0) { delta = std::min(_res_cap[a], _excess[n]); t = _target[a]; // Push-look-ahead heuristic Value ahead = -_excess[t]; - int last_out_t = _sum_supply < 0 ? - _first_out[t+1] : _first_out[t+1] - 1; + int last_out_t = _first_out[t+1]; + LargeCost pi_t = _pi[t]; for (int ta = _next_out[t]; ta != last_out_t; ++ta) { if (_res_cap[ta] > 0 && - _cost[ta] + _pi[_source[ta]] - _pi[_target[ta]] < 0) + _cost[ta] + pi_t - _pi[_target[ta]] < 0) ahead += _res_cap[ta]; if (ahead >= delta) break; } if (ahead < 0) ahead = 0; // Push flow along the arc - if (ahead < delta) { + if (ahead < delta && !hyper[t]) { _res_cap[a] -= ahead; _res_cap[_reverse[a]] += ahead; _excess[n] -= ahead; _excess[t] += ahead; _active_nodes.push_front(t); hyper[t] = true; + hyper_cost[t] = _cost[a] + pi_n - pi_t; _next_out[n] = a; goto next_node; } else { @@ -1162,18 +1270,18 @@ // Relabel the node if it is still active (or hyper) if (_excess[n] > 0 || hyper[n]) { - min_red_cost = std::numeric_limits::max() / 2; + min_red_cost = hyper[n] ? -hyper_cost[n] : + std::numeric_limits::max(); for (int a = _first_out[n]; a != last_out; ++a) { - rc = _cost[a] + _pi[_source[a]] - _pi[_target[a]]; + rc = _cost[a] + pi_n - _pi[_target[a]]; if (_res_cap[a] > 0 && rc < min_red_cost) { min_red_cost = rc; } } _pi[n] -= min_red_cost + _epsilon; + _next_out[n] = _first_out[n]; hyper[n] = false; - - // Reset the next arc - _next_out[n] = _first_out[n]; + ++relabel_cnt; } // Remove nodes that are not active nor hyper @@ -1183,6 +1291,14 @@ !hyper[_active_nodes.front()] ) { _active_nodes.pop_front(); } + + // Global update heuristic + if (relabel_cnt >= next_update_limit) { + globalUpdate(); + for (int u = 0; u != _res_node_num; ++u) + hyper[u] = false; + next_update_limit += global_update_freq; + } } } }