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https://github.com/Z3Prover/z3
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Implement three pivot rules
This commit is contained in:
Anh-Dung Phan
parent
e412d6175d
commit
0d6ffe6b31
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@ -23,8 +23,6 @@ Notes:
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It remains unclear how to convert DL assignment to a basic feasible solution of Network Simplex.
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A naive approach is to run an algorithm on max flow in order to get a spanning tree.
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The network_simplex class hasn't had multiple pivoting strategies yet.
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--*/
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#ifndef _NETWORK_FLOW_H_
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@ -36,20 +34,207 @@ Notes:
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namespace smt {
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enum pivot_rule {
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// First eligible edge pivot rule
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// Edges are traversed in a wraparound fashion
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FIRST_ELIGIBLE,
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// Best eligible edge pivot rule
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// The best edge is selected in every iteration
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BEST_ELIGIBLE,
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// Candidate list pivot rule
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// Major iterations: candidate list is built from eligible edges (in a wraparound way)
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// Minor iterations: the best edge is selected from the list
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CANDIDATE_LIST
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};
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// Solve minimum cost flow problem using Network Simplex algorithm
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template<typename Ext>
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class network_flow : private Ext {
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private:
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enum edge_state {
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LOWER = 1,
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BASIS = 0,
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UPPER = -1
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};
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typedef dl_var node;
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typedef dl_edge<Ext> edge;
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typedef dl_graph<Ext> graph;
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typedef typename Ext::numeral numeral;
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typedef typename Ext::fin_numeral fin_numeral;
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class pivot_rule_impl {
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protected:
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graph & m_graph;
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svector<edge_state> & m_states;
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vector<numeral> & m_potentials;
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edge_id & m_enter_id;
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public:
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pivot_rule_impl() {}
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pivot_rule_impl(graph & g, vector<numeral> & potentials,
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svector<edge_state> & states, edge_id & enter_id)
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: m_graph(g),
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m_potentials(potentials),
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m_states(states),
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m_enter_id(enter_id) {
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}
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bool choose_entering_edge() {return false;};
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};
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class first_eligible_pivot : pivot_rule_impl {
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private:
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edge_id m_next_edge;
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public:
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first_eligible_pivot(graph & g, vector<numeral> & potentials,
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svector<edge_state> & states, edge_id & enter_id) :
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pivot_rule_impl(g, potentials, states, enter_id),
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m_next_edge(0) {
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}
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bool choose_entering_edge() {
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TRACE("network_flow", tout << "choose_entering_edge...\n";);
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unsigned num_edges = m_graph.get_num_edges();
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for (unsigned i = m_next_edge; i < m_next_edge + num_edges; ++i) {
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edge_id id = (i >= num_edges) ? (i - num_edges) : i;
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node src = m_graph.get_source(id);
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node tgt = m_graph.get_target(id);
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if (m_states[id] != BASIS) {
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numeral cost = m_potentials[src] - m_potentials[tgt] - m_graph.get_weight(id);
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if (cost.is_pos()) {
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m_enter_id = id;
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TRACE("network_flow", {
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tout << "Found entering edge " << id << " between node ";
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tout << src << " and node " << tgt << " with reduced cost = " << cost << "...\n";
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});
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return true;
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}
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}
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}
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TRACE("network_flow", tout << "Found no entering edge...\n";);
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return false;
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};
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};
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class best_eligible_pivot : pivot_rule_impl {
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public:
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best_eligible_pivot(graph & g, vector<numeral> & potentials,
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svector<edge_state> & states, edge_id & enter_id) :
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pivot_rule_impl(g, potentials, states, enter_id) {
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}
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bool choose_entering_edge() {
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TRACE("network_flow", tout << "choose_entering_edge...\n";);
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unsigned num_edges = m_graph.get_num_edges();
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numeral max = numeral::zero();
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for (unsigned i = 0; i < num_edges; ++i) {
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node src = m_graph.get_source(i);
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node tgt = m_graph.get_target(i);
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if (m_states[i] != BASIS) {
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numeral cost = m_potentials[src] - m_potentials[tgt] - m_graph.get_weight(i);
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if (cost > max) {
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max = cost;
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m_enter_id = i;
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}
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}
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}
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if (max.is_pos()) {
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TRACE("network_flow", {
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tout << "Found entering edge " << m_enter_id << " between node ";
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tout << m_graph.get_source(m_enter_id) << " and node " << m_graph.get_target(m_enter_id);
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tout << " with reduced cost = " << max << "...\n";
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});
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return true;
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}
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TRACE("network_flow", tout << "Found no entering edge...\n";);
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return false;
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};
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};
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class candidate_list_pivot : pivot_rule_impl {
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private:
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edge_id m_next_edge;
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svector<edge_id> m_candidates;
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unsigned num_candidates;
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static const unsigned NUM_CANDIDATES = 10;
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public:
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candidate_list_pivot(graph & g, vector<numeral> & potentials,
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svector<edge_state> & states, edge_id & enter_id) :
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pivot_rule_impl(g, potentials, states, enter_id),
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m_next_edge(0),
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num_candidates(NUM_CANDIDATES),
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m_candidates(num_candidates) {
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}
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bool choose_entering_edge() {
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if (m_candidates.empty()) {
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// Build the candidate list
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unsigned num_edges = m_graph.get_num_edges();
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numeral max = numeral::zero();
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unsigned count = 0;
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for (unsigned i = m_next_edge; i < m_next_edge + num_edges; ++i) {
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edge_id id = (i >= num_edges) ? i - num_edges : i;
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node src = m_graph.get_source(id);
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node tgt = m_graph.get_target(id);
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if (m_states[id] != BASIS) {
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numeral cost = m_potentials[src] - m_potentials[tgt] - m_graph.get_weight(id);
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if (cost.is_pos()) {
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m_candidates[count++] = id;
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if (cost > max) {
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max = cost;
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m_enter_id = id;
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}
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}
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if (count >= num_candidates) break;
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}
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}
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m_next_edge = m_enter_id;
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if (max.is_pos()) {
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TRACE("network_flow", {
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tout << "Found entering edge " << m_enter_id << " between node ";
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tout << m_graph.get_source(m_enter_id) << " and node " << m_graph.get_target(m_enter_id);
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tout << " with reduced cost = " << max << "...\n";
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});
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return true;
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}
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TRACE("network_flow", tout << "Found no entering edge...\n";);
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return false;
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}
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else {
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numeral max = numeral::zero();
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unsigned last = m_candidates.size();
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for (unsigned i = 0; i < last; ++i) {
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edge_id id = m_candidates[i];
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node src = m_graph.get_source(id);
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node tgt = m_graph.get_target(id);
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if (m_states[id] != BASIS) {
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numeral cost = m_potentials[src] - m_potentials[tgt] - m_graph.get_weight(id);
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if (cost > max) {
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max = cost;
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m_enter_id = id;
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}
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// Remove stale candidates
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if (!cost.is_pos()) {
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m_candidates[i] = m_candidates[--last];
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}
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}
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}
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if (max.is_pos()) {
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TRACE("network_flow", {
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tout << "Found entering edge " << m_enter_id << " between node ";
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tout << m_graph.get_source(m_enter_id) << " and node " << m_graph.get_target(m_enter_id);
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tout << " with reduced cost = " << max << "...\n";
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});
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return true;
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}
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TRACE("network_flow", tout << "Found no entering edge...\n";);
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return false;
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}
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};
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};
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graph m_graph;
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thread_spanning_tree<Ext> m_tree;
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@ -76,9 +261,7 @@ namespace smt {
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void update_flows();
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// If all reduced costs are non-negative, return false since the current spanning tree is optimal
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// Otherwise return true and update m_entering_edge
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bool choose_entering_edge();
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bool choose_entering_edge(pivot_rule pr);
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// Send as much flow as possible around the cycle, the first basic edge with flow 0 will leave
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// Return false if the problem is unbounded
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@ -99,7 +282,7 @@ namespace smt {
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// Minimize cost flows
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// Return true if found an optimal solution, and return false if unbounded
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bool min_cost();
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bool min_cost(pivot_rule pr = FIRST_ELIGIBLE);
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// Compute the optimal solution
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numeral get_optimal_solution(vector<numeral> & result, bool is_dual);
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@ -129,30 +129,6 @@ namespace smt {
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TRACE("network_flow", tout << pp_vector("Flows", m_flows, true););
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}
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template<typename Ext>
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bool network_flow<Ext>::choose_entering_edge() {
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TRACE("network_flow", tout << "choose_entering_edge...\n";);
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unsigned num_edges = m_graph.get_num_edges();
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for (unsigned i = 0; i < num_edges; ++i) {
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node src = m_graph.get_source(i);
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node tgt = m_graph.get_target(i);
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if (m_states[i] != BASIS) {
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numeral cost = m_potentials[src] - m_potentials[tgt] - m_graph.get_weight(i);
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// TODO: add multiple pivoting strategies
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if (cost.is_pos()) {
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m_enter_id = i;
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TRACE("network_flow", {
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tout << "Found entering edge " << i << " between node ";
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tout << src << " and node " << tgt << " with reduced cost = " << cost << "...\n";
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});
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return true;
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}
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}
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}
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TRACE("network_flow", tout << "Found no entering edge...\n";);
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return false;
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}
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template<typename Ext>
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bool network_flow<Ext>::choose_leaving_edge() {
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TRACE("network_flow", tout << "choose_leaving_edge...\n";);
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@ -191,12 +167,29 @@ namespace smt {
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m_tree.update(m_enter_id, m_leave_id);
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}
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// FIXME: should declare pivot as a pivot_rule_impl and refactor
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template<typename Ext>
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bool network_flow<Ext>::choose_entering_edge(pivot_rule pr) {
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if (pr == FIRST_ELIGIBLE) {
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first_eligible_pivot pivot(m_graph, m_potentials, m_states, m_enter_id);
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return pivot.choose_entering_edge();
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}
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else if (pr == BEST_ELIGIBLE) {
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best_eligible_pivot pivot(m_graph, m_potentials, m_states, m_enter_id);
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return pivot.choose_entering_edge();
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}
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else {
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candidate_list_pivot pivot(m_graph, m_potentials, m_states, m_enter_id);
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return pivot.choose_entering_edge();
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}
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}
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// Minimize cost flows
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// Return true if found an optimal solution, and return false if unbounded
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template<typename Ext>
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bool network_flow<Ext>::min_cost() {
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bool network_flow<Ext>::min_cost(pivot_rule pr) {
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initialize();
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while (choose_entering_edge()) {
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while (choose_entering_edge(pr)) {
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bool bounded = choose_leaving_edge();
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if (!bounded) return false;
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update_flows();
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@ -162,7 +162,7 @@ namespace smt {
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tout << u << ", " << v << ") leaves\n";
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});
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node old_pred = m_pred[q];
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node old_pred = m_pred[q];
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// Update stem nodes from q to v
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if (q != v) {
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for (node n = q; n != u; ) {
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@ -175,18 +175,23 @@ namespace smt {
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old_pred = next_old_pred;
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}
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}
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else {
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node x = get_final(p);
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node y = m_thread[x];
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node z = get_final(q);
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node t = m_thread[get_final(v)];
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node r = find_rev_thread(v);
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m_thread[z] = y;
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m_thread[x] = q;
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m_thread[r] = t;
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}
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m_pred[q] = p;
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// Old threads: alpha -> q -*-> f(q) -> beta | p -*-> f(p) -> gamma
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// New threads: alpha -> beta | p -*-> f(p) -> q -*-> f(q) -> gamma
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node f_p = get_final(p);
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node f_q = get_final(q);
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node alpha = find_rev_thread(q);
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node beta = m_thread[f_q];
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node gamma = m_thread[f_p];
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if (q != gamma) {
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m_thread[alpha] = beta;
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m_thread[f_p] = q;
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m_thread[f_q] = gamma;
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}
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m_pred[q] = p;
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m_tree[q] = enter_id;
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m_root_t2 = q;
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@ -211,7 +216,6 @@ namespace smt {
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Spanning tree of m_graph + root is represented using:
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svector<edge_state> m_states; edge_id |-> edge_state
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svector<bool> m_upwards; node |-> bool
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svector<node> m_pred; node |-> node
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svector<int> m_depth; node |-> int
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svector<node> m_thread; node |-> node
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@ -224,9 +228,7 @@ namespace smt {
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m_thread is a linked list traversing all nodes.
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Furthermore, the nodes linked in m_thread follows a
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depth-first traversal order.
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m_upwards direction of edge from i to m_pred[i] m_graph
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*/
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template<typename Ext>
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bool thread_spanning_tree<Ext>::check_well_formed() {
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