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Implement heap-based edge tracking for delta3

This commit is contained in:
Joris van Rantwijk 2024-05-25 17:51:23 +02:00
parent 91d4afb271
commit 13b6b76d47
1 changed files with 60 additions and 198 deletions
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python/mwmatching.py
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@ -10,6 +10,8 @@ import collections
from collections.abc import Sequence
from typing import NamedTuple, Optional
from datastruct import UnionFindQueue, PriorityQueue
def maximum_weight_matching(
edges: Sequence[tuple[int, int, float]]
@ -32,7 +34,7 @@ def maximum_weight_matching(
no effect on the maximum-weight matching.
Edges with negative weight are ignored.
This function takes time O(n**3), where "n" is the number of vertices.
This function takes time O(n**3 + n*m*log(n)), where "n" is the number of vertices.
This function uses O(n + m) memory, where "m" is the number of edges.
Parameters:
@ -380,11 +382,6 @@ class _Blossom:
# "tree_edge = None" if the blossom is the root of an alternating tree.
self.tree_edge: Optional[tuple[int, int]] = None
# For a top-level S-blossom,
# "best_edge" is the edge index of the least-slack edge to
# a different S-blossom, or -1 if no such edge has been found.
self.best_edge: int = -1
# "marker" is a temporary variable used to discover common
# ancestors in the blossom tree. It is normally False, except
# when used by "trace_alternating_paths()".
@ -450,11 +447,6 @@ class _NonTrivialBlossom(_Blossom):
# New blossoms start with dual variable 0.
self.dual_var: float = 0
# For a non-trivial, top-level S-blossom,
# "best_edge_set" is a list of least-slack edges between this blossom
# and other S-blossoms.
self.best_edge_set: Optional[list[int]] = None
def vertices(self) -> list[int]:
"""Return a list of vertex indices contained in the blossom."""
@ -530,6 +522,15 @@ class _MatchingContext:
# Initially all vertices are trivial top-level blossoms.
self.vertex_top_blossom: list[_Blossom] = self.trivial_blossom.copy()
# Running sum of applied delta steps times 2.
self.delta_sum_2x: float = 0
# Queue containing edges between S-vertices in different top-level
# blossoms. The priority of each edge is equal to its slack
# plus two times the running sum of delta updates.
self.delta3_queue = PriorityQueue()
self.delta3_set: set[int] = set()
# Every vertex has a variable in the dual LPP.
#
# "vertex_dual_2x[x]" is 2 times the dual variable of vertex "x".
@ -563,6 +564,23 @@ class _MatchingContext:
assert self.vertex_top_blossom[x] is not self.vertex_top_blossom[y]
return self.vertex_dual_2x[x] + self.vertex_dual_2x[y] - 2 * w
def edge_slack(self, e: int) -> float:
"""Return the slack of the edge with index "e".
This function must only be used for edges between two vertices
in different top-level S-blossoms. If all edge weights are
integers, the result is also an integer.
"""
(x, y, w) = self.graph.edges[e]
assert self.vertex_top_blossom[x] is not self.vertex_top_blossom[y]
dual_2x = self.vertex_dual_2x[x] + self.vertex_dual_2x[y]
if self.graph.integer_weights:
assert dual_2x % 2 == 0
dual = dual_2x // 2
else:
dual = dual_2x / 2
return dual - w
#
# Least-slack edge tracking:
#
@ -576,16 +594,6 @@ class _MatchingContext:
# 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.
#
# For each top-level S-blossom, we keep track of the least-slack edge
# to any S-vertex not in the same blossom.
#
# Furthermore, for each top-level S-blossom, we keep a list of least-slack
# edges to other top-level S-blossoms. For any pair of top-level
# S-blossoms, the least-slack edge between them is contained in the edge
# list of at least one of the blossoms. An edge list may contain multiple
# edges to the same S-blossom. Such redundant edges are pruned during
# blossom merging to limit the number of tracked edges.
#
# 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
@ -604,12 +612,8 @@ class _MatchingContext:
for x in range(num_vertex):
self.vertex_best_edge[x] = -1
for blossom in self.trivial_blossom:
blossom.best_edge = -1
for blossom in self.nontrivial_blossom:
blossom.best_edge = -1
blossom.best_edge_set = None
self.delta3_queue.clear()
self.delta3_set.clear()
def lset_add_vertex_edge(self, y: int, e: int, slack: float) -> None:
"""Add edge "e" from an S-vertex to unlabeled vertex or T-vertex "y".
@ -650,155 +654,6 @@ class _MatchingContext:
return (best_index, best_slack)
@staticmethod
def lset_new_blossom(blossom: _Blossom) -> None:
"""Start tracking edges for a new S-blossom."""
assert blossom.best_edge == -1
if isinstance(blossom, _NonTrivialBlossom):
assert blossom.best_edge_set is None
blossom.best_edge_set = []
def lset_add_blossom_edge(
self,
blossom: _Blossom,
e: int,
slack: float
) -> None:
"""Add edge "e" between the specified S-blossom and another S-blossom.
This function takes time O(1) per call.
This function is called O(m) times per stage.
"""
# Track least-slack edge between this blossom and any other S-blossom.
if blossom.best_edge == -1:
blossom.best_edge = e
else:
best_slack = self.edge_slack_2x(blossom.best_edge)
if slack < best_slack:
blossom.best_edge = e
# Regardless of whether this edge is currently the least-slack edge,
# this edge may later become the least-slack edge if other edges
# become unavailable when S-blossoms are merged.
#
# We therefore add the edge to a list of potential future least-slack
# edges for this blossom. We do this only for non-trivial blossoms.
if isinstance(blossom, _NonTrivialBlossom):
assert blossom.best_edge_set is not None
blossom.best_edge_set.append(e)
def lset_merge_blossoms(self, blossom: _NonTrivialBlossom) -> None:
"""Update least-slack edge tracking after merging sub-blossoms
into a new S-blossom.
This function takes total time O(n**2) per stage.
"""
num_vertex = self.graph.num_vertex
# Calculate the set of least-slack edges to other S-blossoms.
# We basically merge the edge lists from all sub-blossoms, but reject
# edges that are internal to this blossom, and trim the set such that
# there is at most one edge to each external S-blossom.
#
# Sub-blossoms that were formerly labeled T can be ignored; their
# vertices are in the queue and will discover neighbouring S-blossoms
# via the edge scan process.
#
# Build a temporary array holding the least-slack edge index to
# each top-level S-blossom. This array is indexed by the base vertex
# of the blossoms.
best_edge_to_blossom: list[int] = num_vertex * [-1]
zero_slack: float = 0
best_slack_to_blossom: list[float] = num_vertex * [zero_slack]
# And find the overall least-slack edge to any other S-blossom.
best_edge = -1
best_slack: float = 0
# Add the least-slack edges of every S-sub-blossom.
for sub in blossom.subblossoms:
if sub.label != _LABEL_S:
continue
if isinstance(sub, _NonTrivialBlossom):
# Pull the edge list from the sub-blossom.
assert sub.best_edge_set is not None
sub_edge_set = sub.best_edge_set
# Delete the edge list from the sub-blossom.
sub.best_edge_set = None
else:
# Trivial blossoms don't have a list of least-slack edges,
# so we just look at all adjacent edges. This happens at most
# once per vertex per stage.
# It adds up to O(m) time per stage.
sub_edge_set = self.graph.adjacent_edges[sub.base_vertex]
# Add edges to the temporary array.
for e in sub_edge_set:
(x, y, _w) = self.graph.edges[e]
bx = self.vertex_top_blossom[x]
by = self.vertex_top_blossom[y]
assert (bx is blossom) or (by is blossom)
# Reject internal edges in this blossom.
if bx is by:
continue
# Set bx = blossom at the other end of this edge.
bx = by if (bx is blossom) else bx
# Reject edges that don't link to an S-blossom.
if bx.label != _LABEL_S:
continue
# Keep only the least-slack edge to "bx".
slack = self.edge_slack_2x(e)
bx_base = bx.base_vertex
if ((best_edge_to_blossom[bx_base] == -1)
or (slack < best_slack_to_blossom[bx_base])):
best_edge_to_blossom[bx_base] = e
best_slack_to_blossom[bx_base] = slack
# Update the overall least-slack edge to any S-blossom.
if (best_edge == -1) or (slack < best_slack):
best_edge = e
best_slack = slack
# Extract a compact list of least-slack edge indices.
# We can not keep the temporary array because that would blow up
# memory use to O(n**2).
best_edge_set = [e for e in best_edge_to_blossom if e != -1]
blossom.best_edge_set = best_edge_set
# Keep the overall least-slack edge.
blossom.best_edge = best_edge
def lset_get_best_blossom_edge(self) -> tuple[int, float]:
"""Return the index and slack of the least-slack edge between
any pair of top-level S-blossoms.
This function takes time O(n) per call.
This function takes total time O(n**2) per stage.
Returns:
Tuple (edge_index, slack_2x) if there is a least-slack edge,
or (-1, 0) if there is no suitable edge.
"""
best_index = -1
best_slack: float = 0
for blossom in self.trivial_blossom + self.nontrivial_blossom:
if (blossom.label == _LABEL_S) and (blossom.parent is None):
e = blossom.best_edge
if e != -1:
slack = self.edge_slack_2x(e)
if (best_index == -1) or (slack < best_slack):
best_index = e
best_slack = slack
return (best_index, best_slack)
#
# General support routines:
#
@ -954,9 +809,6 @@ class _MatchingContext:
if sub.label == _LABEL_T:
self.queue.extend(sub.vertices())
# Merge least-slack edges for the S-sub-blossoms.
self.lset_merge_blossoms(blossom)
@staticmethod
def find_path_through_blossom(
blossom: _NonTrivialBlossom,
@ -1251,9 +1103,6 @@ class _MatchingContext:
# Attach the blossom that contains "x" to the alternating tree.
bx.tree_edge = (y, x)
# Start least-slack edge tracking for the S-blossom.
self.lset_new_blossom(bx)
# Add all vertices inside the newly labeled S-blossom to the queue.
self.queue.extend(bx.vertices())
@ -1378,7 +1227,10 @@ class _MatchingContext:
elif ylabel == _LABEL_S:
# Update tracking of least-slack edges between S-blossoms.
self.lset_add_blossom_edge(bx, e, slack)
if e not in self.delta3_set:
prio = self.edge_slack(e) + self.delta_sum_2x
self.delta3_set.add(e)
self.delta3_queue.insert(prio, e)
if ylabel != _LABEL_S:
# Update tracking of least-slack edges from vertex "y" to
@ -1435,20 +1287,28 @@ class _MatchingContext:
# Compute delta3: half minimum slack of any edge between two top-level
# S-blossoms.
(e, slack) = self.lset_get_best_blossom_edge()
if e != -1:
if self.graph.integer_weights:
# If all edge weights are even integers, the slack
# of any edge between two S blossoms is also an even
# integer. Therefore the delta is an integer.
assert slack % 2 == 0
slack = slack // 2
else:
slack = slack / 2
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_top_blossom[x]
by = self.vertex_top_blossom[y]
assert (bx.label == _LABEL_S) and (by.label == _LABEL_S)
if bx is not by:
# Found edge between different top-level S-blossoms.
slack = self.edge_slack(e)
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_set.remove(e)
# Compute delta4: half minimum dual variable of a top-level T-blossom.
for blossom in self.nontrivial_blossom:
@ -1465,6 +1325,8 @@ class _MatchingContext:
num_vertex = self.graph.num_vertex
self.delta_sum_2x += delta_2x
# Apply delta to dual variables of all vertices.
for x in range(num_vertex):
xlabel = self.vertex_top_blossom[x].label
@ -1498,7 +1360,7 @@ class _MatchingContext:
thereby increasing the number of matched edges by 1.
If no such path is found, the matching must already be optimal.
This function takes time O(n**2).
This function takes time O(n**2 + m * log(n)).
Returns:
True if the matching was successfully augmented.