Maximum matching by Edmonds: Difference between revisions
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== Recursion anchor == | == Recursion anchor == | ||
The graph search does not run into a [[Matchings in graphs#Blossoms|blossom]]. | The graph search does not find an augmenting path, nor does it run into a [[Matchings in graphs#Blossoms|blossom]]. | ||
== | == Recursive step == | ||
'''Abstract view:''' | '''Abstract view:''' | ||
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# The repeated graph search is terminated. | # The repeated graph search is terminated. | ||
# A blossom <math>B</math> is identified. | # A blossom <math>B</math> is identified. | ||
# <math> | # <math>B</math> is [[Matchings in graphs#Blossoms|shrunken]], giving <math>G'=(V',E')</math> and the restriction <math>M'</math> of <math>M</math> to <math>G'</math>. | ||
# The procedure is called recursively with <math>G'</math> and <math>M'</math>. | # The procedure is called recursively with <math>G'</math> and <math>M'</math>. | ||
# Exactly one node of <math> | # Exactly one node of <math>B</math> is matched by <math>M'</math>, so extend <math>M'</math> in the obvious, unique way by <math>\lfloor|B|/2\rfloor</math> edges of <math>B</math>. | ||
# This extension of <math>M'</math> is returned. | # This extension of <math>M'</math> is returned. | ||
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'''Implementation of step 2:''' | '''Implementation of step 2:''' | ||
Let <math>w</math> be the node on which the two different parities occur, and let <math>v</math> be the node from which <math>w</math> is seen at that moment. In the arborescence constructed so far, let <math>u</math> denote the node such that the paths from the start node to <math>v</math> and <math>w</math> coincide up to <math>u</math> and part at <math>u</math>. Note that the incoming edge of <math>u</math> in the arborescence is matched because, otherwise, the arborescence could | Let <math>w</math> be the node on which the two different parities occur, and let <math>v</math> be the node from which <math>w</math> is seen at that moment. In the arborescence constructed so far, let <math>u</math> denote the node such that the paths from the start node to <math>v</math> and <math>w</math> coincide up to <math>u</math> and part at <math>u</math>. Note that the incoming edge of <math>u</math> in the arborescence is matched because, otherwise, the arborescence could not branch at <math>u</math>. Therefore, the concatenation of the branch from <math>u</math> to <math>v</math>, the branch from <math>u</math> to <math>w</math>, and <math>\{v,w\}</math> is a blossom, and the search entered this blossom via its stem. | ||
'''Proof:''' | '''Proof:''' | ||
First we show: If <math>M</math> is not maximum, the graph search will either find an augmenting path or run into a blossom via its stem. Suppose the search does not find an augmenting path. Due to the proof in the [[Classical bipartite cardinality matching#Induction step|induction step]] of the [[Classical bipartite cardinality matching|classical bipartite cardinality matching algorithm]], the graph search runs into an odd cycle. Since we chose a graph search strategy that never returns to the start node, the remark on that proof implies that this odd cycle is a blossom, and that the search entered this blossom via its stem. | |||
It remains to show that the returned extension of <math>M'</math> is cardinality-maximal for <math>G</math> provided <math>M'</math> is cardinality-maximal fpr <math>G'</math>. However, this follows immediately from the fundamental [[Matchings in graphs#Blossoms|statement about blossoms]]. | |||
== Correctness == | == Correctness == | ||
Follows immediately from the invariant. | |||
== Complexity == | == Complexity == | ||
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'''Proof:''' | '''Proof:''' | ||
Due to the variant, the recursion depth is in <math>\mathcal{O}(n)</math>. Each recursive step is dominated by the graph traversal. | Due to the variant, the recursion depth is in <math>\mathcal{O}(n)</math>. Each recursive step is dominated by the graph traversal. The [[Basic graph definitions#Adjacency, incidence, and degree|statement and remark on isolated nodes]] apply. | ||
Latest revision as of 09:25, 22 February 2015
Abstract view
Algorithmic problem: Cardinality-maximal matching
Type of algorithm: recursion.
Invariant: Each recursive step returns a cardinality-maximal matching of its input graph.
Variant: The number of nodes decreases.
Recursion anchor
The graph search does not find an augmenting path, nor does it run into a blossom.
Recursive step
Abstract view: Apply the modified repeated graph search from the classical algorithm for the restriction to bipartite graphs. More specifically, choose a BFS or another graph traversal strategy that will never return to the start node (this requirement excludes DFS).
The search maintains a boolean flag even/odd, which indicates the parity of the distance of the current node from the start node of the current search. Also, each node has a boolean label even/odd. When this node is seen for the first time, its label is set according to the flag. If a labeled node is seen again, but with another distance parity than before:
- The repeated graph search is terminated.
- A blossom [math]\displaystyle{ B }[/math] is identified.
- [math]\displaystyle{ B }[/math] is shrunken, giving [math]\displaystyle{ G'=(V',E') }[/math] and the restriction [math]\displaystyle{ M' }[/math] of [math]\displaystyle{ M }[/math] to [math]\displaystyle{ G' }[/math].
- The procedure is called recursively with [math]\displaystyle{ G' }[/math] and [math]\displaystyle{ M' }[/math].
- Exactly one node of [math]\displaystyle{ B }[/math] is matched by [math]\displaystyle{ M' }[/math], so extend [math]\displaystyle{ M' }[/math] in the obvious, unique way by [math]\displaystyle{ \lfloor|B|/2\rfloor }[/math] edges of [math]\displaystyle{ B }[/math].
- This extension of [math]\displaystyle{ M' }[/math] is returned.
Remark: In typical formulations of this algorithm, the graph search is not left open but the search strategy applied in Hopcroft-Karp is hard-coded.
Implementation of step 2: Let [math]\displaystyle{ w }[/math] be the node on which the two different parities occur, and let [math]\displaystyle{ v }[/math] be the node from which [math]\displaystyle{ w }[/math] is seen at that moment. In the arborescence constructed so far, let [math]\displaystyle{ u }[/math] denote the node such that the paths from the start node to [math]\displaystyle{ v }[/math] and [math]\displaystyle{ w }[/math] coincide up to [math]\displaystyle{ u }[/math] and part at [math]\displaystyle{ u }[/math]. Note that the incoming edge of [math]\displaystyle{ u }[/math] in the arborescence is matched because, otherwise, the arborescence could not branch at [math]\displaystyle{ u }[/math]. Therefore, the concatenation of the branch from [math]\displaystyle{ u }[/math] to [math]\displaystyle{ v }[/math], the branch from [math]\displaystyle{ u }[/math] to [math]\displaystyle{ w }[/math], and [math]\displaystyle{ \{v,w\} }[/math] is a blossom, and the search entered this blossom via its stem.
Proof: First we show: If [math]\displaystyle{ M }[/math] is not maximum, the graph search will either find an augmenting path or run into a blossom via its stem. Suppose the search does not find an augmenting path. Due to the proof in the induction step of the classical bipartite cardinality matching algorithm, the graph search runs into an odd cycle. Since we chose a graph search strategy that never returns to the start node, the remark on that proof implies that this odd cycle is a blossom, and that the search entered this blossom via its stem.
It remains to show that the returned extension of [math]\displaystyle{ M' }[/math] is cardinality-maximal for [math]\displaystyle{ G }[/math] provided [math]\displaystyle{ M' }[/math] is cardinality-maximal fpr [math]\displaystyle{ G' }[/math]. However, this follows immediately from the fundamental statement about blossoms.
Correctness
Follows immediately from the invariant.
Complexity
Statement: The asymptotic worst-case complexity is in [math]\displaystyle{ \mathcal{O}(n\!\cdot\!m) }[/math], where [math]\displaystyle{ n=|V| }[/math] and [math]\displaystyle{ m=|E| }[/math].
Proof: Due to the variant, the recursion depth is in [math]\displaystyle{ \mathcal{O}(n) }[/math]. Each recursive step is dominated by the graph traversal. The statement and remark on isolated nodes apply.