Clean up least-slack edge tracking

python/mwmatching.py

@@ -78,7 +78,7 @@ def maximum_weight_matching(
     # of matched edges by 1.
     #
     # This loop runs through at most (n/2 + 1) iterations.
-    # Each iteration takes time O(n**2).
+    # Each iteration takes time O((n + m) * log(n)).
     while ctx.run_stage():
         pass
 
@@ -554,6 +554,8 @@ class _MatchingContext:
 
         # "vertex_set_node[x]" represents the vertex "x" inside the
         # union-find datastructure of its top-level blossom.
+        #
+        # Initially, each vertex belongs to its owwn trivial top-level blossom.
         self.vertex_set_node = [b.vertex_set.insert(i, math.inf)
                                 for (i, b) in enumerate(self.trivial_blossom)]
 
@@ -587,7 +589,7 @@ class _MatchingContext:
         self.delta_sum_2x: float = 0
 
         # Queue containing unlabeled top-level blossoms that have an edge to
-        # an S-blossom. The priority of a blossom is 2 times the least slack
+        # an S-blossom. The priority of a blossom is 2 times its least slack
         # to an S blossom, plus 2 times the running sum of delta steps.
         self.delta2_queue: PriorityQueue[_Blossom] = PriorityQueue()
 
@@ -623,11 +625,15 @@ class _MatchingContext:
             del blossom.vertex_set
             blossom.tree_blossoms = None
 
+    #
+    # Least-slack edge tracking:
+    #
+
     def edge_pseudo_slack_2x(self, e: int) -> float:
         """Return 2 times the pseudo-slack of the specified edge.
 
         The pseudo-slack of an edge is related to its true slack, but
-        distorted in a way that makes it invariant under delta steps.
+        adjusted in a way that makes it invariant under delta steps.
 
         If the edge connects two S-vertices in different top-level blossoms,
         the true slack is the pseudo-slack minus 2 times the running sum
@@ -641,47 +647,33 @@ class _MatchingContext:
         (x, y, w) = self.graph.edges[e]
         return self.vertex_dual_2x[x] + self.vertex_dual_2x[y] - 2 * w
 
-    #
+    def delta2_add_edge(self, e: int, y: int, by: _Blossom) -> None:
-    # Least-slack edge tracking:
+        """Add edge "e" for delta2 tracking.
-    #
-    # To calculate delta steps, the matching algorithm needs to find
-    #  - the least-slack edge between any S-vertex and an unlabeled vertex;
-    #  - the least-slack edge between any pair of top-level S-blossoms.
-    #
-    # For each unlabeled vertex and each T-vertex, we keep track of the
-    # least-slack edge to any S-vertex. Tracking for unlabeled vertices
-    # serves to provide the least-slack edge for the delta step.
-    # Tracking for T-vertices is done because such vertices can turn into
-    # unlabeled vertices if they are part of a T-blossom that gets expanded.
-    #
-    # Note: For a given vertex or blossom, the identity of the least-slack
-    # edge to any S-blossom remains unchanged during a delta step.
-    # Although the delta step changes edge slacks, it changes the slack
-    # of every edge to an S-vertex by the same amount. Therefore the edge
-    # that had least slack before the delta step, will still have least slack
-    # after the delta step.
-    #
 
-    # TODO -- rename function, maybe refactor
+        Edge "e" connects an S-vertex to a T-vertex or unlabeled vertex "y".
-    def lset_add_vertex_edge(self, y: int, by: _Blossom, e: int) -> None:
-        """Add edge "e" from an S-vertex to unlabeled vertex or T-vertex "y".
 
         This function takes time O(log(n)).
         """
+
         prio = self.edge_pseudo_slack_2x(e)
 
         improved = (self.vertex_sedge_queue[y].empty()
                     or (self.vertex_sedge_queue[y].find_min().prio > prio))
 
+        # Insert edge in the S-edge queue of vertex "y".
         assert self.vertex_sedge_node[e] is None
         self.vertex_sedge_node[e] = self.vertex_sedge_queue[y].insert(prio, e)
 
+        # Continue if the new edge becomes the least-slack S-edge for "y".
         if not improved:
             return
 
+        # Update the priority of "y" in its UnionFindQueue.
         prev_min = by.vertex_set.min_prio()
         self.vertex_set_node[y].set_prio(prio)
 
+        # If the blossom is unlabeled and the new edge becomes its least-slack
+        # S-edge, insert or update the blossom in the global delta2 queue.
         if (by.label == _LABEL_NONE) and (prio < prev_min):
             prio += by.vertex_dual_offset
             if by.delta2_node is None:
@@ -689,20 +681,99 @@ class _MatchingContext:
             elif prio < by.delta2_node.prio:
                 self.delta2_queue.decrease_prio(by.delta2_node, prio)
 
-    # TODO -- rename function, maybe refactor
+    def delta2_remove_edge(self, e: int, y: int, by: _Blossom) -> None:
-    def lset_get_best_vertex_edge(self) -> tuple[int, float]:
+        """Remove edge "e" from delta2 tracking.
-        """Return the index and slack of the least-slack edge between
+
-        any S-vertex and unlabeled vertex.
+        This function is called if an S-vertex becomes unlabeled,
+        and edge "e" connects that vertex to vertex "y" which is a T-vertex
+        or unlabeled vertex.
+
+        This function takes time O(log(n)).
+        """
+        vertex_sedge_node = self.vertex_sedge_node[e]
+        if vertex_sedge_node is not None:
+            # Delete edge from the S-edge queue of vertex "y".
+            vertex_sedge_queue = self.vertex_sedge_queue[y]
+            vertex_sedge_queue.delete(vertex_sedge_node)
+            self.vertex_sedge_node[e] = None
+
+            if vertex_sedge_queue.empty():
+                prio = math.inf
+            else:
+                prio = vertex_sedge_queue.find_min().prio
+
+            # If necessary, update the priority of "y" in its UnionFindQueue.
+            if prio > self.vertex_set_node[y].prio:
+                self.vertex_set_node[y].set_prio(prio)
+                if by.label == _LABEL_NONE:
+                    # Update or delete the blossom in the global delta2 queue.
+                    assert by.delta2_node is not None
+                    prio = by.vertex_set.min_prio()
+                    if prio < math.inf:
+                        prio += by.vertex_dual_offset
+                        if prio > by.delta2_node.prio:
+                            self.delta2_queue.increase_prio(
+                                by.delta2_node, prio)
+                    else:
+                        self.delta2_queue.delete(by.delta2_node)
+                        by.delta2_node = None
+
+    def delta2_enable_blossom(self, blossom: _Blossom) -> None:
+        """Enable delta2 tracking for "blossom".
+
+        This function is called when a blossom becomes an unlabeled top-level
+        blossom. If the blossom has at least one edge to an S-vertex,
+        the blossom will be inserted in the global delta2 queue.
+
+        This function takes time O(log(n)).
+        """
+        assert blossom.delta2_node is None
+        prio = blossom.vertex_set.min_prio()
+        if prio < math.inf:
+            prio += blossom.vertex_dual_offset
+            blossom.delta2_node = self.delta2_queue.insert(prio, blossom)
+
+    def delta2_disable_blossom(self, blossom: _Blossom) -> None:
+        """Disable delta2 tracking for "blossom".
+
+        The blossom will be removed from the global delta2 queue.
+        This function is called when a blossom stops being an unlabeled
+        top-level blossom.
+
+        This function takes time O(log(n)).
+        """
+        if blossom.delta2_node is not None:
+            self.delta2_queue.delete(blossom.delta2_node)
+            blossom.delta2_node = None
+
+    def delta2_clear_vertex(self, x: int) -> None:
+        """Clear delta2 tracking for vertex "x".
+
+        This function is called when "x" becomes an S-vertex.
+        It is assumed that the blossom containing "x" has already been
+        disabled for delta2 tracking.
+
+        This function takes time O(k * log(n)),
+        where "k" is the number of edges incident on "x".
+        """
+        self.vertex_sedge_queue[x].clear()
+        for e in self.graph.adjacent_edges[x]:
+            self.vertex_sedge_node[e] = None
+        self.vertex_set_node[x].set_prio(math.inf)
+
+    def delta2_get_min_edge(self) -> tuple[int, float]:
+        """Find the least-slack edge between any S-vertex and any unlabeled
+        vertex.
 
         This function takes time O(log(n)).
 
         Returns:
-            Tuple (edge_index, slack_2x) if there is a least-slack edge,
+            Tuple (edge_index, slack_2x) if there is an S-to-unlabeled edge,
-            or (-1, 0) if there is no suitable edge.
+            or (-1, Inf) if there is no such edge.
         """
 
         if self.delta2_queue.empty():
-            return (-1, 0)
+            return (-1, math.inf)
 
         delta2_node = self.delta2_queue.find_min()
         blossom = delta2_node.data
@@ -716,6 +787,74 @@ class _MatchingContext:
 
         return (e, slack_2x)
 
+    def delta3_add_edge(self, e: int) -> None:
+        """Add edge "e" for delta3 tracking.
+
+        This function is called if a vertex becomes an S-vertex and edge "e"
+        connects it to an S-vertex in a different top-level blossom.
+
+        This function takes time O(log(n)).
+        """
+        # The edge may already be in the delta3 queue, if it was previously
+        # discovered in the opposite direction.
+        if self.delta3_node[e] is None:
+            # Priority is edge slack plus 2 times the running sum of
+            # delta steps.
+            prio_2x = self.edge_pseudo_slack_2x(e)
+            if self.graph.integer_weights:
+                # If all edge weights are integers, the slack of
+                # any edge between S-vertices is also an integer.
+                assert prio_2x % 2 == 0
+                prio = prio_2x // 2
+            else:
+                prio = prio_2x / 2
+            self.delta3_node[e] = self.delta3_queue.insert(prio, e)
+
+    def delta3_remove_edge(self, e: int) -> None:
+        """Remove edge "e" from delta3 tracking.
+
+        This function is called if a former S-vertex becomes unlabeled,
+        and edge "e" connects it to another S-vertex.
+
+        This function takes time O(log(n)).
+        """
+        delta3_node = self.delta3_node[e]
+        if delta3_node is not None:
+            self.delta3_queue.delete(delta3_node)
+            self.delta3_node[e] = None
+
+    def delta3_get_min_edge(self) -> tuple[int, float]:
+        """Find the least-slack edge between any pair of S-vertices in
+        different top-level blossoms.
+
+        This function takes time O(1 + k * log(n)),
+        where "k" is the number of intra-blossom edges removed from the queue.
+
+        Returns:
+            Tuple (edge_index, slack) if there is an S-to-S edge,
+            or (-1, Inf) if there is no suitable edge.
+        """
+        while not self.delta3_queue.empty():
+            delta3_node = self.delta3_queue.find_min()
+            e = delta3_node.data
+            (x, y, _w) = self.graph.edges[e]
+            bx = self.vertex_set_node[x].find()
+            by = self.vertex_set_node[y].find()
+            assert (bx.label == _LABEL_S) and (by.label == _LABEL_S)
+            if bx is not by:
+                slack = delta3_node.prio - self.delta_sum_2x
+                return (e, slack)
+
+            # Reject edges between vertices within the same top-level blossom.
+            # Although intra-blossom edges are never inserted into the queue,
+            # existing edges in the queue may become intra-blossom when
+            # a new blossom is formed.
+            self.delta3_queue.delete(delta3_node)
+            self.delta3_node[e] = None
+
+        # If the queue is empty, no suitable edge exists.
+        return (-1, math.inf)
+
     #
     # General support routines:
     #
@@ -734,10 +873,7 @@ class _MatchingContext:
         assert blossom.label == _LABEL_NONE
         blossom.label = _LABEL_S
 
-        # Delete blossom from delta2 queue.
+        self.delta2_disable_blossom(blossom)
-        if blossom.delta2_node is not None:
-            self.delta2_queue.delete(blossom.delta2_node)
-            blossom.delta2_node = None
 
         # Prepare for lazy updating of S-blossom dual variable.
         if isinstance(blossom, _NonTrivialBlossom):
@@ -765,14 +901,7 @@ class _MatchingContext:
         blossom.vertex_dual_offset = 0
         for x in vertices:
             self.vertex_dual_2x[x] += vertex_dual_fixup
-
+            self.delta2_clear_vertex(x)
-            # Clean up tracking of edges from vertex "x" to S-vertices.
-            # We maintain that tracking only for unlabeled vertices and
-            # T-vertices.
-            self.vertex_sedge_queue[x].clear()
-            for e in self.graph.adjacent_edges[x]:
-                self.vertex_sedge_node[e] = None
-            self.vertex_set_node[x].set_prio(math.inf)
 
     def assign_blossom_label_t(self, blossom: _Blossom) -> None:
         """Assign label T to an unlabeled top-level blossom."""
@@ -781,10 +910,7 @@ class _MatchingContext:
         assert blossom.label == _LABEL_NONE
         blossom.label = _LABEL_T
 
-        # Delete blossom from delta2 queue.
+        self.delta2_disable_blossom(blossom)
-        if blossom.delta2_node is not None:
-            self.delta2_queue.delete(blossom.delta2_node)
-            blossom.delta2_node = None
 
         if isinstance(blossom, _NonTrivialBlossom):
 
@@ -838,47 +964,17 @@ class _MatchingContext:
             (p, q, _w) = edges[e]
             y = p if p != x else q
 
-            # If this edge was in the delta3_queue, remove it since
+            self.delta3_remove_edge(e)
-            # this is no longer an edge between S-vertices.
-            delta3_node = self.delta3_node[e]
-            if delta3_node is not None:
-                self.delta3_queue.delete(delta3_node)
-                self.delta3_node[e] = None
 
             by = self.vertex_set_node[y].find()
             if by.label == _LABEL_S:
                 # This is an edge between "x" and an S-vertex.
                 # Add this edge to "vertex_sedge_queue[x]".
                 # Update delta2 tracking accordingly.
-                self.lset_add_vertex_edge(x, bx, e)
+                self.delta2_add_edge(e, x, bx)
 
             else:
-                # This is no longer an edge between "y" and an S-vertex.
+                self.delta2_remove_edge(e, y, by)
-                # Remove this edge from "vertex_sedge_queue[y]".
-                # Update delta2 tracking accordingly.
-                # TODO -- untangle this mess
-                vertex_sedge_node = self.vertex_sedge_node[e]
-                if vertex_sedge_node is not None:
-                    vertex_sedge_queue = self.vertex_sedge_queue[y]
-                    vertex_sedge_queue.delete(vertex_sedge_node)
-                    self.vertex_sedge_node[e] = None
-                    if vertex_sedge_queue.empty():
-                        prio = math.inf
-                    else:
-                        prio = vertex_sedge_queue.find_min().prio
-                    if prio > self.vertex_set_node[y].prio:
-                        self.vertex_set_node[y].set_prio(prio)
-                        if by.label == _LABEL_NONE:
-                            assert by.delta2_node is not None
-                            prio = by.vertex_set.min_prio()
-                            if prio < math.inf:
-                                prio += by.vertex_dual_offset
-                                if prio > by.delta2_node.prio:
-                                    self.delta2_queue.increase_prio(
-                                        by.delta2_node, prio)
-                            else:
-                                self.delta2_queue.delete(by.delta2_node)
-                                by.delta2_node = None
 
     def reset_blossom_label(self, blossom: _Blossom) -> None:
         """Remove blossom label."""
@@ -907,14 +1003,7 @@ class _MatchingContext:
         elif blossom.label == _LABEL_T:
 
             self.remove_blossom_label_t(blossom)
-
+            self.delta2_enable_blossom(blossom)
-            # Since the blossom is now unlabeled, insert it in delta2_queue
-            # if it has at least one edge to an S-vertex.
-            assert blossom.delta2_node is None
-            prio = blossom.vertex_set.min_prio()
-            if prio < math.inf:
-                prio += blossom.vertex_dual_offset
-                blossom.delta2_node = self.delta2_queue.insert(prio, blossom)
 
     def _check_alternating_tree_consistency(self) -> None:
         """TODO -- remove this function, only for debugging"""
@@ -1161,12 +1250,7 @@ class _MatchingContext:
             assert sub.vertex_dual_offset == 0
             sub.vertex_dual_offset = vertex_dual_fixup
 
-            # Insert blossom in delta2_queue if necessary.
+            self.delta2_enable_blossom(sub)
-            prio = sub.vertex_set.min_prio()
-            if prio < math.inf:
-                assert sub.delta2_node is None
-                prio += sub.vertex_dual_offset
-                sub.delta2_node = self.delta2_queue.insert(prio, sub)
 
         # The expanding blossom was part of an alternating tree, linked to
         # a parent node in the tree via one of its subblossoms, and linked to
@@ -1224,10 +1308,7 @@ class _MatchingContext:
         assert blossom.parent is None
         assert blossom.label == _LABEL_NONE
 
-        # Remove blossom from delta2 heap.
+        self.delta2_disable_blossom(blossom)
-        assert blossom.delta2_node is not None
-        self.delta2_queue.delete(blossom.delta2_node)
-        blossom.delta2_node = None
 
         # Split union-find structure.
         blossom.vertex_set.split()
@@ -1244,12 +1325,7 @@ class _MatchingContext:
             assert sub.vertex_dual_offset == 0
             sub.vertex_dual_offset = vertex_dual_offset
 
-            # Insert blossom in delta2_queue if necessary.
+            self.delta2_enable_blossom(sub)
-            prio = sub.vertex_set.min_prio()
-            if prio < math.inf:
-                assert sub.delta2_node is None
-                prio += sub.vertex_dual_offset
-                sub.delta2_node = self.delta2_queue.insert(prio, sub)
 
         # Delete the expanded blossom.
         self.nontrivial_blossom.remove(blossom)
@@ -1546,25 +1622,9 @@ class _MatchingContext:
                     continue
 
                 if by.label == _LABEL_S:
-                    # Update tracking of least-slack edges between S-blossoms.
+                    self.delta3_add_edge(e)
-                    # Priority is edge slack plus 2 times the running sum of
-                    # delta steps.
-                    if self.delta3_node[e] is None:
-                        prio_2x = self.edge_pseudo_slack_2x(e)
-                        if self.graph.integer_weights:
-                            # If all edge weights are integers, the slack of
-                            # any edge between S-vertices is also an integer.
-                            assert prio_2x % 2 == 0
-                            prio = prio_2x // 2
                 else:
-                            prio = prio_2x / 2
+                    self.delta2_add_edge(e, y, by)
-                        self.delta3_node[e] = self.delta3_queue.insert(prio, e)
-                else:
-                    # Update tracking of least-slack edges from vertex "y" to
-                    # any S-vertex. We do this for T-vertices and unlabeled
-                    # vertices. Edges which already have zero slack are still
-                    # tracked.
-                    self.lset_add_vertex_edge(y, by, e)
 
         self.scan_queue.clear()
 
@@ -1604,7 +1664,7 @@ class _MatchingContext:
         # Compute delta2: minimum slack of any edge between an S-vertex and
         # an unlabeled vertex.
         # This takes time O(log(n)).
-        (e, slack) = self.lset_get_best_vertex_edge()
+        (e, slack) = self.delta2_get_min_edge()
         if (e != -1) and (slack <= delta_2x):
             delta_type = 2
             delta_2x = slack
@@ -1612,31 +1672,12 @@ class _MatchingContext:
 
         # Compute delta3: half minimum slack of any edge between two top-level
         # S-blossoms.
-        #
+        # This takes total time O(m * log(n)) per stage.
-        # This loop iterates O(m) times per stage.
+        (e, slack) = self.delta3_get_min_edge()
-        # Each iteration takes time O(log(n)).
+        if (e != -1) and (slack <= delta_2x):
-        while not self.delta3_queue.empty():
-            delta3_node = self.delta3_queue.find_min()
-            e = delta3_node.data
-            (x, y, _w) = self.graph.edges[e]
-            bx = self.vertex_set_node[x].find()
-            by = self.vertex_set_node[y].find()
-            assert (bx.label == _LABEL_S) and (by.label == _LABEL_S)
-            if bx is not by:
-                # Found edge between different top-level S-blossoms.
-                slack = delta3_node.prio - self.delta_sum_2x
-                if slack <= delta_2x:
             delta_type = 3
             delta_2x = slack
             delta_edge = e
-                break
-
-            # Reject edges between vertices within the same top-level blossom.
-            # Although intra-blossom edges are never inserted into the queue,
-            # existing edges in the queue may become intra-blossom when
-            # a new blossom is formed.
-            self.delta3_queue.delete(delta3_node)
-            self.delta3_node[e] = None
 
         # Compute delta4: half minimum dual variable of a top-level T-blossom.
         # This takes time O(log(n)).