Use UnionFind to find top-level blossom of vertex
The run time should now be O(n*m*log(n))
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@ -34,7 +34,7 @@ def maximum_weight_matching(
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no effect on the maximum-weight matching.
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Edges with negative weight are ignored.
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This function takes time O(n**3 + n*m*log(n)),
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This function takes time O(n * (n + m) * log(n)),
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where "n" is the number of vertices and "m" is the number of edges.
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This function uses O(n + m) memory.
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@ -544,15 +544,6 @@ class _MatchingContext:
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# Initially there are no non-trivial blossoms.
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self.nontrivial_blossom: list[_NonTrivialBlossom] = []
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# Every vertex is contained in exactly one top-level blossom
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# (possibly the trivial blossom that contains just that vertex).
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#
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# "vertex_top_blossom[x]" is the top-level blossom that contains
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# vertex "x".
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#
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# Initially all vertices are trivial top-level blossoms.
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self.vertex_top_blossom: list[_Blossom] = self.trivial_blossom.copy()
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# "vertex_set_node[x]" represents the vertex "x" inside the
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# union-find datastructure of its top-level blossom.
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self.vertex_set_node = [b.vertex_set.insert(i, math.inf)
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@ -657,8 +648,8 @@ class _MatchingContext:
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def edge_slack_2x(self, e: int) -> float:
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"""Return 2 times the slack of the edge with index "e"."""
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(x, y, w) = self.graph.edges[e]
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bx = self.vertex_top_blossom[x]
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by = self.vertex_top_blossom[y]
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bx = self.vertex_set_node[x].find()
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by = self.vertex_set_node[y].find()
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return self.edge_slack_2x_info(x, y, bx, by, w)
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#
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@ -930,8 +921,9 @@ class _MatchingContext:
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discovers an augmenting path. In this case it returns an alternating
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path that starts and ends in an unmatched vertex.
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This function takes time O(k) to discover a blossom, where "k" is the
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number of sub-blossoms, or time O(n) to discover an augmenting path.
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This function takes time O(k*log(n)) to discover a blossom,
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where "k" is the number of sub-blossoms,
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or time O(n*log(n)) to discover an augmenting path.
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Returns:
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Alternating path as an ordered list of edges between top-level
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@ -956,7 +948,7 @@ class _MatchingContext:
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while x != -1 or y != -1:
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# Check if we found a common ancestor.
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bx = self.vertex_top_blossom[x]
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bx = self.vertex_set_node[x].find()
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if bx.marker:
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first_common = bx
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break
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@ -984,8 +976,8 @@ class _MatchingContext:
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# If we found a common ancestor, trim the paths so they end there.
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if first_common is not None:
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assert self.vertex_top_blossom[xedges[-1][0]] is first_common
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while self.vertex_top_blossom[yedges[-1][0]] is not first_common:
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assert self.vertex_set_node[xedges[-1][0]].find() is first_common
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while self.vertex_set_node[yedges[-1][0]].find() is not first_common:
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yedges.pop()
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# Fuse the two paths.
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@ -1009,7 +1001,7 @@ class _MatchingContext:
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Assign label S to the new blossom.
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Relabel all T-sub-blossoms as S and add their vertices to the queue.
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This function takes total time O(n**2) per stage.
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This function takes total time O(n*log(n)) per stage.
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"""
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# Check that the path is odd-length.
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@ -1017,12 +1009,12 @@ class _MatchingContext:
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assert len(path.edges) >= 3
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# Construct the list of sub-blossoms (current top-level blossoms).
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subblossoms = [self.vertex_top_blossom[x] for (x, y) in path.edges]
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subblossoms = [self.vertex_set_node[x].find() for (x, y) in path.edges]
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# Check that the path is cyclic.
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# Note the path will not always start and end with the same _vertex_,
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# but it must start and end in the same _blossom_.
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subblossoms_next = [self.vertex_top_blossom[y]
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subblossoms_next = [self.vertex_set_node[y].find()
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for (x, y) in path.edges]
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assert subblossoms[0] == subblossoms_next[-1]
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assert subblossoms[1:] == subblossoms_next[:-1]
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@ -1053,10 +1045,6 @@ class _MatchingContext:
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for sub in subblossoms:
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sub.parent = blossom
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# Update blossom-membership of all vertices in the new blossom.
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for x in blossom.vertices():
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self.vertex_top_blossom[x] = blossom
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# Merge union-find structures.
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blossom.vertex_set.merge([sub.vertex_set for sub in subblossoms])
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@ -1090,7 +1078,7 @@ class _MatchingContext:
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def expand_t_blossom(self, blossom: _NonTrivialBlossom) -> None:
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"""Expand the specified T-blossom.
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This function takes time O(n).
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This function takes total time O(n*log(n)) per stage.
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"""
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assert blossom.parent is None
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@ -1116,12 +1104,6 @@ class _MatchingContext:
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assert sub.vertex_dual_offset == 0
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sub.vertex_dual_offset = vertex_dual_fixup
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if isinstance(sub, _NonTrivialBlossom):
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for x in sub.vertices():
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self.vertex_top_blossom[x] = sub
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else:
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self.vertex_top_blossom[sub.base_vertex] = sub
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# Insert blossom in delta2_queue if necessary.
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prio = sub.vertex_set.min_prio()
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if prio < math.inf:
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@ -1139,7 +1121,7 @@ class _MatchingContext:
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# the alternating tree.
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assert blossom.tree_edge is not None
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(x, y) = blossom.tree_edge
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sub = self.vertex_top_blossom[y]
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sub = self.vertex_set_node[y].find()
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# Assign label T to that sub-blossom.
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self.assign_blossom_label_t(sub)
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@ -1175,7 +1157,7 @@ class _MatchingContext:
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def expand_unlabeled_blossom(self, blossom: _NonTrivialBlossom) -> None:
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"""Expand the specified unlabeled blossom.
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This function takes time O(n).
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This function takes total time O(n*log(n)) per stage.
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"""
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assert blossom.parent is None
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@ -1196,12 +1178,6 @@ class _MatchingContext:
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assert sub.vertex_dual_offset == 0
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sub.vertex_dual_offset = vertex_dual_offset
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if isinstance(sub, _NonTrivialBlossom):
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for x in sub.vertices():
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self.vertex_top_blossom[x] = sub
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else:
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self.vertex_top_blossom[sub.base_vertex] = sub
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# Insert blossom in delta2_queue if necessary.
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prio = sub.vertex_set.min_prio()
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if prio < math.inf:
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@ -1323,7 +1299,7 @@ class _MatchingContext:
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# an unmatched vertex or a blossom with unmatched base.
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assert len(path.edges) % 2 == 1
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for x in (path.edges[0][0], path.edges[-1][1]):
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b = self.vertex_top_blossom[x]
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b = self.vertex_set_node[x].find()
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assert self.vertex_mate[b.base_vertex] == -1
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# The augmenting path looks like this:
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@ -1341,11 +1317,11 @@ class _MatchingContext:
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# Augment the non-trivial blossoms on either side of this edge.
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# No action is necessary for trivial blossoms.
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bx = self.vertex_top_blossom[x]
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bx = self.vertex_set_node[x].find()
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if isinstance(bx, _NonTrivialBlossom):
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self.augment_blossom(bx, self.trivial_blossom[x])
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by = self.vertex_top_blossom[y]
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by = self.vertex_set_node[y].find()
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if isinstance(by, _NonTrivialBlossom):
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self.augment_blossom(by, self.trivial_blossom[y])
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@ -1372,7 +1348,7 @@ class _MatchingContext:
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"""
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# Assign label S to the blossom that contains vertex "x".
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bx = self.vertex_top_blossom[x]
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bx = self.vertex_set_node[x].find()
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self.assign_blossom_label_s(bx)
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self.assign_vertex_label_s(bx)
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@ -1388,7 +1364,7 @@ class _MatchingContext:
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else:
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# Vertex "x" is matched to T-vertex "y".
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by = self.vertex_top_blossom[y]
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by = self.vertex_set_node[y].find()
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assert by.label == _LABEL_T
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# Attach the blossom that contains "x" to the alternating tree.
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@ -1405,14 +1381,14 @@ class _MatchingContext:
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- "y" is an unlabeled, matched vertex.
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- There is a tight edge between vertices "x" and "y".
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"""
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assert self.vertex_top_blossom[x].label == _LABEL_S
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assert self.vertex_set_node[x].find().label == _LABEL_S
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by = self.vertex_top_blossom[y]
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by = self.vertex_set_node[y].find()
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# Expand zero-dual blossoms before assigning label T.
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while isinstance(by, _NonTrivialBlossom) and (by.dual_var == 0):
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self.expand_unlabeled_blossom(by)
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by = self.vertex_top_blossom[y]
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by = self.vertex_set_node[y].find()
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# Assign label T to the unlabeled blossom.
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self.assign_blossom_label_t(by)
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@ -1446,7 +1422,7 @@ class _MatchingContext:
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# but not necessarily in the same vertex within that blossom.
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p = path.edges[0][0]
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q = path.edges[-1][1]
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if self.vertex_top_blossom[p] is self.vertex_top_blossom[q]:
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if self.vertex_set_node[p].find() is self.vertex_set_node[q].find():
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self.make_blossom(path)
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return None
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else:
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@ -1475,7 +1451,7 @@ class _MatchingContext:
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x = self.scan_queue.popleft()
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# Double-check that "x" is an S-vertex.
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bx = self.vertex_top_blossom[x]
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bx = self.vertex_set_node[x].find()
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assert bx.label == _LABEL_S
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# Scan the edges that are incident on "x".
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@ -1489,8 +1465,8 @@ class _MatchingContext:
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# Note: blossom index of vertex "x" may change during
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# this loop, so we need to refresh it here.
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bx = self.vertex_top_blossom[x]
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by = self.vertex_top_blossom[y]
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bx = self.vertex_set_node[x].find()
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by = self.vertex_set_node[y].find()
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# Ignore edges that are internal to a blossom.
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if bx is by:
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@ -1589,8 +1565,8 @@ class _MatchingContext:
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delta3_node = self.delta3_queue.find_min()
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e = delta3_node.data
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(x, y, _w) = self.graph.edges[e]
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bx = self.vertex_top_blossom[x]
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by = self.vertex_top_blossom[y]
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bx = self.vertex_set_node[x].find()
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by = self.vertex_set_node[y].find()
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assert (bx.label == _LABEL_S) and (by.label == _LABEL_S)
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if bx is not by:
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# Found edge between different top-level S-blossoms.
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@ -1634,7 +1610,7 @@ class _MatchingContext:
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thereby increasing the number of matched edges by 1.
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If no such path is found, the matching must already be optimal.
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This function takes time O(n**2 + m * log(n)).
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This function takes time O((n + m) * log(n)).
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Returns:
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True if the matching was successfully augmented.
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@ -1684,7 +1660,7 @@ class _MatchingContext:
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# Use the edge from S-vertex to unlabeled vertex that got
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# unlocked through the delta update.
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(x, y, _w) = self.graph.edges[delta_edge]
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if self.vertex_top_blossom[x].label != _LABEL_S:
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if self.vertex_set_node[x].find().label != _LABEL_S:
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(x, y) = (y, x)
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self.extend_tree_t(x, y)
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