Start writing algorithm description
Implementation details still to be written.
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# Implementation of Maximum Weighted Matching
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## Introduction
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This document describes the implementation of an algorithm that computes a maximum weight matching
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in a general graph in time _O(n<sup>3</sup>)_, where _n_ is the number of vertices in the graph.
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In graph theory, a _matching_ is a subset of edges such that none of them share a common vertex.
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A _maximum cardinality matching_ is a matching that contains the largest possible number of edges
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(or equivalently, the largest possible number of vertices).
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For a graph that has weights attached to its edges, a _maximum weight matching_
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is a matching that achieves the largest possible sum of weights of matched edges.
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An algorithm for maximum weight matching can obviously also be used to calculate a maximum
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cardinality matching by simply assigning weight 1 to all edges.
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Certain computer science problems can be understood as _restrictions_ of maximum weighted matching
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in general graphs.
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Examples are: maximum matching in bipartite graphs, maximum cardinality matching in general graphs,
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and maximum weighted matching in general graphs with edge weights limited to integers
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in a certain range.
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Clearly, an algorithm for maximum weighted matching in general graphs also solves
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all of these restricted problems.
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However, some of the restricted problems can be solved with algorithms that are simpler and/or
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faster than the known algorithms for the general problem.
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The rest of this document does not consider restricted problems.
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My focus is purely on maximum weighted matching in general graphs.
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In this document, _n_ refers to the number of vertices and _m_ refers to the number of edges in the graph.
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## A timeline of matching algorithms
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In 1963, Jack Edmonds published the first polynomial-time algorithm for maximum matching in
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general graphs [[1]](#edmonds1965a) [[2]](#edmonds1965b) .
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Efficient algorithms for bipartite graphs were already known at the time, but generalizations
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to non-bipartite graphs would tend to require an exponential number of steps.
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Edmonds solved this by explicitly detecting _blossoms_ (odd-length alternating cycles) and
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adding special treatment for them.
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He also introduced a linear programming technique to handle weighted matching.
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The resulting maximum weighted matching algorithm runs in time _O(n<sup>4</sup>)_.
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In 1973, Harold N. Gabow published a maximum weighted matching algorithm that runs in
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time _O(n<sup>3</sup>)_ [[3]](#gabow1974) .
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It is based on the ideas of Edmonds, but uses different data structures to reduce the amount
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of work.
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In 1983, Galil, Micali and Gabow published a maximum weighted matching algorithm that runs in
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time _O(n\*m\*log(n))_ [[4]](#galil_micali_gabow1986) .
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It is an implementation of Edmonds' blossom algorithm that uses advanced data structures
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to speed up critical parts of the algorithm.
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This algorithm is asymptotically faster than _O(n<sup>3</sup>)_ for sparse graphs,
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but slower for highly dense graphs.
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In 1983, Zvi Galil published an overview of algorithms for 4 variants of the matching problem
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[[5]](#galil1986) :
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maximum-cardinality resp. maximum-weight matching in bipartite resp. general graphs.
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I like this paper a lot.
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It explains the algorithms from first principles and can be understood without prior knowledge
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of the literature.
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The paper describes a maximum weighted matching algorithm that is similar to Edmonds'
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blossom algorithm, but carefully implemented to run in time _O(n<sup>3</sup>)_.
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It then sketches how advanced data structures can be added to arrive at the Galil-Micali-Gabow
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algorithm that runs in time _O(n\*m\*log(n))_.
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In 1990, Gabow published a maximum weighted matching algorithm that runs in time
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_O(n\*m + n<sup>2</sup>\*log(n))_ [[6]](#gabow1990) .
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It uses several advanced data structures, including Fibonacci heaps.
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Unfortunately I don't understand this algorithm at all.
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## Choosing an algorithm
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I selected the _O(n<sup>3</sup>)_ variant of Edmonds' blossom algorithm as described by
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Galil [[5]](#galil1986) .
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This algorithm is usually credited to Gabow [[3]](#gabow1974), but I find the description
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in [[5]](#galil1986) easier to understand.
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This is generally a fine algorithm.
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One of its strengths is that it is relatively easy to understand, especially compared
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to the more recent algorithms.
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Its run time is asymptotically optimal for complete graphs (graphs that have an edge
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between every pair of vertices).
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On the other hand, the algorithm is suboptimal for sparse graphs.
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It is possible to construct highly sparse graphs, having _m = O(n)_,
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that cause this algorithm to perform _Θ(n<sup>3</sup>)_ steps.
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In such cases the Galil-Micali-Gabow algorithm would be significantly faster.
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Then again, there is a question of how important the worst case is for practical applications.
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I suspect that the simple algorithm typically needs only about _O(n\*m)_ steps when running
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on random sparse graphs with random weights, i.e. much faster than its worst case bound.
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After trading off these properties, I decided that I prefer an algorithm that is understandable
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and has decent performance, over an algorithm that is faster in specific cases but also
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significantly more complicated.
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## Description of the algorithm
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My implementation closely follows the description by Zvi Galil in [[5]](#galil1986) .
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I recommend to read that paper before diving into my description below.
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The paper explains the algorithm in depth and shows how it relates to matching
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in bipartite graphs and non-weighted graphs.
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There are subtle aspects to the algorithm that are tricky to implement correctly but are
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mentioned only briefly in the paper.
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In this section, I describe the algorithm from my own perspective:
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a programmer struggling to implement the algorithm correctly.
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My goal is only to describe the algorithm, not to prove its correctness.
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### Basic concepts
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An edge-weighted, undirected graph _G_ consists of a set _V_ of _n_ vertices and a set _E_
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of _m_ edges.
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Vertices are represented by non-negative integers from _0_ to _n-1_: _V = { 0, 1, ..., n-1 }_
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Edges are represented by tuples: _E = { (x, y, w), ... }_ <br>
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where the edge _(x, y, w)_ is incident on vertices _x_ and _y_ and has weight _w_. <br>
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The order of vertices is irrelevant, i.e. _(x, y, w)_ and _(y, x, w)_ represent the same edge. <br>
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Edge weights may be integers or floating point numbers. <br>
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There can be at most one edge between any pair of vertices. <br>
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A vertex can not have an edge to itself.
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A matching is a subset of edges without any pair of edges sharing a common vertex. <br>
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An edge is matched if it is part of the matching, otherwise it is unmatched. <br>
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A vertex is matched if it is incident to an edge in the matching, otherwise it is unmatched.
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An alternating path is a simple path that alternates between matched and unmatched edges.
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*Figure 1*
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Figure 1 depicts a graph with 6 vertices and 9 edges.
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Wavy lines represent matched edges; straight lines represent edges that are currently unmatched.
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The matching has weight 8 + 7 = 15.
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An example of an alternating path would be 0 - 1 - 4 - 5 - 3 (but there are many others).
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### Augmenting paths
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An augmenting path is an alternating path that begins and ends in two unmatched vertices.
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An augmenting path can be used to extend the matching as follows:
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remove all matched edges on the augmenting path from the matching,
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and add all previously unmatched edges on the augmenting path to the matching.
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The result is again a valid matching, and the number of matched edges has increased by one.
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The matching in figure 1 above has several augmenting paths.
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For example, the edge from vertex 0 to vertex 2 is by itself an augmenting path.
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Augmenting along this path would increase the weight of the matching by 2.
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Another augmenting path is 0 - 1 - 4 - 5 - 3 - 2, which would increase the weight of
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the matching by 3.
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Finally, 0 - 4 - 1 - 2 is also an augmenting path, but it would decrease the weight
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of the matching by 2.
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### Main algorithm
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Our algorithm to compute a maximum weighted matching is as follows:
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- Start with an empty matching (all edges unmatched).
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- Repeat the following steps:
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- Find an augmenting path that provides the largest possible increase of the weight of
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the matching.
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- If there is no augmenting path that increases the weight of the matching,
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end the algorithm.
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The current matching is a maximum-weight matching.
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- Otherwise, use the augmenting path to update the matching.
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- Continue by searching for another augmenting path, etc.
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This algorithm ends when there are no more augmenting paths that increase the weight
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of the matching.
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In some cases, there may still be augmenting paths which do not increase the weight of the matching,
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implying that the maximum-weight matching has fewer edges than the maximum cardinality matching.
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Every iteration of the main loop is called a _stage_.
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Note that the final matching contains at most _n/2_ edges, therefore the algorithm
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performs at most _n/2_ stages.
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The only remaining challenge is finding an augmenting path.
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Specifically, finding an augmenting path that increases the weight of the matching
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as much as possible.
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### Blossoms
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A blossom is an odd-length cycle that alternates between matched and unmatched edges.
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Such cycles complicate the search for augmenting paths.
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To overcome these problems, blossoms must be treated specially.
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The trick is to temporarily replace the vertices and edges that are part of the blossom
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by a single super-vertex.
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This is called _shrinking_ the blossom.
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The search for an augmenting path then continues in the modified graph in which
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the odd-length cycle no longer exists.
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It may later become necessary to _expand_ the blossom (undo the shrinking step).
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For example, the cycle 0 - 1 - 4 - 0 in figure 1 above is an odd-length alternating cycle,
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and therefore a candidate to become a blossom.
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In practice, we do not remove vertices or edges from the graph while shrinking a blossom.
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Instead the graph is left unchanged, but a separate data structure keeps track of blossoms
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and which vertices are contained in which blossoms.
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A graph can contain many blossoms.
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Furthermore, after shrinking a blossom, that blossom can become a sub-blossom
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in a bigger blossom.
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Figure 2 depicts a graph with several nested blossoms.
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 <br>
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*Figure 2: Nested blossoms*
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To describe the algorithm unambiguously, we need precise definitions:
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* A _blossom_ is either a single vertex, or an odd-length alternating cycle of _sub-blossoms_.
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* A _non-trivial blossom_ is a blossom that is not a single vertex.
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* A _top-level blossom_ is a blossom that is not contained inside another blossom.
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It can also be a single vertex which is not part of any blossom.
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* The _base vertex_ of a blossom is the only vertex in the blossom that is not matched
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to another vertex in the same blossom.
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The base vertex is either unmatched, or matched to a vertex outside the blossom.
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In a single-vertex blossom, its only vertex is the base vertex.
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In a non-trivial blossom, the base vertex is equal to the base vertex of the unique sub-blossom
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that begins and ends the alternating cycle.
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* The _parent_ of a non-top-level blossom is the directly enclosing blossom in which
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it occurs as a sub-blossom.
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Top-level blossoms do not have parents.
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Non-trivial blossoms are created by shrinking and destroyed by expanding.
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These are explicit steps of the algorithm.
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This implies that _not_ every odd-length alternating cycle is a blossom.
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The algorithm decides which cycles are blossoms.
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Just before a new blossom is created, its sub-blossoms are initially top-level blossoms.
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After creating the blossom, the sub-blossoms are no longer top-level blossoms because they are contained inside the new blossom, which is now a top-level blossom.
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Every vertex is contained in precisely one top-level blossom.
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This may be either a trivial blossom which contains only that single vertex,
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or a non-trivial blossom into which the vertex has been absorbed through shrinking.
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A vertex may belong to different top-level blossoms over time as blossoms are
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created and destroyed by the algorithm.
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I use the notation _B(x)_ to indicate the top-level blossom that contains vertex _x_.
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### Searching for an augmenting path
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Recall that the matching algorithm repeatedly searches for an augmenting path that
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increases the weight of the matching as much as possible.
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I'm going to postpone the part that deals with "increasing the weight as much as possible".
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For now, all we need to know is this:
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At a given step in the algorithm, some edges in the graph are _tight_ while other edges
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have _slack_.
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An augmenting path that consists only of tight edges is _guaranteed_ to increase the weight
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of the matching as much as possible.
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While searching for an augmenting path, we simply restrict the search to tight edges,
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ignoring all edges that have slack.
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Certain explicit actions of the algorithm cause edges to become tight or slack.
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How this works will be explained later.
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To find an augmenting path, the algorithm searches for alternating paths that start
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in an unmatched vertex.
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The collection of alternating paths forms a forest of trees.
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Each tree is rooted in an unmatched vertex, and all paths from the root to the leaves of a tree
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are alternating paths.
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The nodes in these trees are top-level blossoms.
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To facilitate the search, top-level blossoms are labeled as either _S_ or _T_ or _unlabeled_.
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Label S is assigned to the roots of the alternating trees, and to nodes at even distance
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from the root.
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Label T is assigned to nodes at odd distance from the root of an alternating tree.
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Unlabeled blossoms are not (yet) part of an alternating tree.
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(In some papers, the label S is called _outer_ and T is called _inner_.)
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It is important to understand that labels S and T are assigned only to top-level blossoms,
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not to individual vertices.
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However, it is often relevant to know the label of the top-level blossom that contains
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a given vertex.
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I use the following terms for convenience:
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* an _S-blossom_ is a top-level blossom with label S;
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* a _T-blossom_ is a top-level blossom with label T;
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* an _S-vertex_ is a vertex inside an S-blossom;
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* a _T-vertex_ is a vertex inside a T-blossom.
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Edges that span between two S-blossoms are special.
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If both S-blossoms are part of the same alternating tree, the edge is part of
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an odd-length alternating cycle.
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The lowest common ancestor node in the alternating tree forms the beginning and end
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of the alternating cycle.
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In this case a new blossom must be created by shrinking the cycle.
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If the two S-blossoms are in different alternating trees, the edge that links the blossoms
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is part of an augmenting path between the roots of the two trees.
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 <br>
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*Figure 3: Growing alternating trees*
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The graph in figure 3 contains two unmatched vertices: 0 and 7.
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Both form the root of an alternating tree.
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Blue arrows indicate edges in the alternating trees, pointing away from the root.
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The graph edge between vertices 4 and 6 connects two S-blossoms in the same alternating tree.
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Scanning this edge will cause the creation of a new blossom with base vertex 2.
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The graph edge between vertices 6 and 9 connects two S-blossoms which are part
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of different alternating trees.
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Scanning this edge will discover an augmenting path between vertices 0 and 7.
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Note that all vertices in the graph of figure 3 happen to be top-level blossoms.
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In general, the graph may already contain non-trivial blossoms.
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Alternating trees are constructed over top-level blossoms, not necessarily over
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individual vertices.
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When a vertex becomes an S-vertex, it is added to a queue.
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The search procedure considers these vertices one-by-one and tries to use them to
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either grow the alternating tree (thus adding new vertices to the queue),
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or discover an augmenting path or a new blossom.
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In detail, the search for an augmenting path proceeds as follows:
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- Mark all top-level blossoms as _unlabeled_.
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- Initialize an empty queue _Q_.
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- Assign label S to all top-level blossoms that contain an unmatched vertex.
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Add all vertices inside such blossoms to _Q_.
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- Repeat until _Q_ is empty:
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- Take a vertex _x_ from _Q_.
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- Scan all _tight_ edges _(x, y, w)_ that are incident on vertex _x_.
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- Find the top-level blossom _B(x)_ that contains vertex _x_
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and the top-level blossom _B(y)_ that contains vertex _y_.
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- If _B(x)_ and _B(y)_ are the same blossom, ignore this edge.
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(It is an internal edge in the blossom.)
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- Otherwise, if blossom _B(y)_ is unlabeled, assign label T to it.
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The base vertex of _B(y)_ is matched (otherwise _B(y)_ would have label S).
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Find the vertex _t_ which is matched to the base vertex of _B(y)_ and find
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its top-level blossom _B(t)_ (this blossom is still unlabeled).
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Assign label S to blossom _B(t)_, and add all vertices inside _B(t)_ to _Q_.
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- Otherwise, if blossom _B(y)_ has label T, ignore this edge.
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- Otherwise, if blossom _B(y)_ has label S, there are two scenarios:
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- Either _B(x)_ and _B(y)_ are part of the same alternating tree.
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In that case, we have discovered an odd-length alternating cycle.
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Shrink the cycle into a blossom and assign label S to it.
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Add all vertices inside its T-labeled sub-blossoms to _Q_.
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(Vertices inside S-labeled sub-blossoms have already been added to _Q_
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and must not be added again.)
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Continue the search for an augmenting path.
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- Otherwise, if _B(x)_ and _B(y)_ are part of different alternating trees,
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we have found an augmenting path between the roots of those trees.
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End the search and return the augmenting path.
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- If _Q_ becomes empty before an augmenting path has been found,
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it means that no augmenting path exists (which consists of only tight edges).
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For each top-level blossom, we keep track of its label as well as the edge through
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which it obtained its label (attaching it to the alternating tree).
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These edges are needed to trace back through the alternating tree to construct
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a blossom or an alternating path.
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When an edge between S-blossoms is discovered, it is handled as follows:
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- The two top-level blossoms are _B(x)_ and _B(y)_, both labeled S.
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- Trace up through the alternating trees from _B(x)_ and _B(y)_ towards the root.
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- If the blossoms are part of the same alternating tree, the tracing process
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eventually reaches a lowest common ancestor of _B(x)_ and _B(y)_.
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In that case a new blossom must be created.
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Its alternating cycle starts at the common ancestor, follows the path through
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the alternating tree down to _B(x)_, then via the scanned edge to _B(y)_,
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then through the alternating tree up to the common ancestor.
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(Note it is possible that either _B(x)_ or _B(y)_ is itself the lowest common ancestor.)
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- Otherwise, the blossoms are in different trees.
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The tracing process will eventually reach the roots of both trees.
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At that point we have traced out an augmenting path between the two roots.
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Note that the matching algorithm uses two different types of tree data structures.
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These two types of trees are separate concepts.
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But since they are both trees, there is potential for accidentally confusing them.
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It may be helpful to explicitly distinguish the two types:
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- A forest of _blossom structure trees_ represents the nested structure of blossoms.
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Every node in these trees is a blossom.
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The roots are the top-level blossoms.
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The leaf nodes are the single vertices.
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The child nodes of each non-leaf node represent its sub-blossoms.
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Every vertex is a leaf node in precisely one blossom structure tree.
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A blossom structure tree may consist of just one single node, representing
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a vertex that is not part of any non-trivial blossom.
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- A forest of _alternating trees_ represents an intermediate state during the search
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for an augmenting path.
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Every node in these trees is a top-level blossom.
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The roots are top-level blossoms with an unmatched base vertex.
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Every labeled top-level blossom is part of an alternating tree.
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Unlabeled blossoms are not (yet) part of a tree.
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### Augmenting the matching
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Once an augmenting path has been found, augmenting the matching is relatively easy.
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Simply follow the path, adding previously unmatched edges to the matching and removing
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previously matched edges from the matching.
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A useful data structure to keep track of matched edges is an array `mate[x]` indexed by vertex.
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If vertex _x_ is matched, `mate[x]` contains the index of the vertex to which it is matched.
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If vertex _x_ is unmatched, `mate[x]` contains -1.
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||||
|
||||
A slight complication arises when the augmenting path contains a non-trivial blossom.
|
||||
The search returns an augmenting path over top-level blossoms, without details
|
||||
about the layout of the path within blossoms.
|
||||
Any parts of the path that run through a blossom must be traced in order to update
|
||||
the matched/unmatched status of the edges in the blossom.
|
||||
|
||||
When an augmenting path runs through a blossom, it always runs between the base vertex of
|
||||
the blossom and some sub-blossom (potentially the same sub-blossom that contains the base vertex).
|
||||
If the base vertex is unmatched, it forms the start or end of the augmenting path.
|
||||
Otherwise, the augmenting path enters the blossom though the matched edge of the base vertex.
|
||||
From the opposite direction, the augmenting path enters the blossom through an unmatched edge.
|
||||
It follows that the augmenting path must run through an even number of internal edges
|
||||
of the blossom.
|
||||
Fortunately, every sub-blossom can be reached from the base through an even number of steps
|
||||
by walking around the blossom in the appropriate direction.
|
||||
|
||||
Augmenting along a path through a blossom causes a reorientation of the blossom.
|
||||
Afterwards, it is still a blossom and still consists of an odd-length alternating cycle,
|
||||
but the cycle begins and ends in a different sub-blossom.
|
||||
The blossom also has a different base vertex.
|
||||
(In specific cases where the augmenting path merely "grazes" a blossom,
|
||||
the orientation and base vertex remain unchanged.)
|
||||
|
||||
 <br>
|
||||
*Figure 4: Augmenting path through a blossom*
|
||||
|
||||
Figure 4 shows an augmenting path that runs through a blossom.
|
||||
The augmenting path runs through an even number of internal edges in the blossom,
|
||||
which in this case is not the shortest way around the blossom.
|
||||
After augmenting, the blossom has become reoriented:
|
||||
a different vertex became the base vertex.
|
||||
|
||||
In case of nested blossoms, non-trivial sub-blossoms on the augmenting path must
|
||||
be updated recursively.
|
||||
|
||||
Note that the process of repeatedly augmenting the matching will never cause a matched vertex
|
||||
to become unmatched.
|
||||
Once a vertex is matched, augmenting may cause the vertex to become matched to a _different_
|
||||
vertex, but it can not cause the vertex to become unmatched.
|
||||
|
||||
### Edge slack and dual variables
|
||||
|
||||
We still need a method to determine which edges are _tight_.
|
||||
This is done by means of so-called _dual variables_.
|
||||
|
||||
The purpose of dual variables can be explained by rephrasing the maximum matching problem
|
||||
as an instance of linear programming.
|
||||
I'm not going to do that here.
|
||||
I will describe how the algorithm uses dual variables without explaining why.
|
||||
For the mathematical background, I recommend reading [[5]](#galil1986) .
|
||||
|
||||
Every vertex _x_ has a dual variable _u<sub>x</sub>_.
|
||||
Furthermore, every non-trivial blossom _B_ has a dual variable _z<sub>B</sub>_.
|
||||
These variables contain non-negative numbers which change over time through actions
|
||||
of the algorithm.
|
||||
|
||||
Every edge in the graph imposes a constraint on the dual variables:
|
||||
The weight of the edge between vertices _x_ and _y_ must be less or equal
|
||||
to the sum of the dual variables _u<sub>x</sub>_ plus _u<sub>y</sub>_ plus all
|
||||
_z<sub>B</sub>_ of blossoms _B_ that contain the edge.
|
||||
(A blossom contains an edge if it contains both incident vertices.)
|
||||
This constraint is more clearly expressed in a formula:
|
||||
|
||||
$$ u_x + u_y + \sum_{(x,y) \in B} z_B \ge w_{x,y} $$
|
||||
|
||||
The slack _π<sub>x,y</sub>_ of the edge between vertices _x_ and _y_ is a non-negative number
|
||||
that indicates how much room is left before the edge constraint would be violated:
|
||||
|
||||
$$ \pi_{x,y} = u_x + u_y + \sum_{(x,y) \in B} z_B - w_{x,y} $$
|
||||
|
||||
An edge is _tight_ if and only if its slack is zero.
|
||||
Given the values of the dual variables, it is very easy to calculate the slack of an edge
|
||||
which is not contained in any blossom: simply add the duals of its incident vertices and
|
||||
subtract the weight.
|
||||
To check whether an edge is tight, simply compute its slack and check whether it is zero.
|
||||
|
||||
Calculating the slack of an edge that is contained in one or more blossoms is a little tricky,
|
||||
but fortunately we don't need such calculations.
|
||||
The search for augmenting paths only considers edges that span _between_ top-level blossoms,
|
||||
not edges that are contained inside blossoms.
|
||||
So we never need to check the tightness of internal edges in blossoms.
|
||||
|
||||
A matching has maximum weight if it satisfies all of the following constraints:
|
||||
|
||||
- All dual variables and edge slacks are non-negative:
|
||||
_u<sub>i</sub>_ ≥ 0 , _z<sub>B</sub>_ ≥ 0 , _π<sub>x,y</sub>_ ≥ 0
|
||||
- All matched edges have zero slack: edge _(x, y)_ matched implies _π<sub>x,y</sub>_ = 0
|
||||
- All unmatched vertices have dual variable zero: vertex _x_ unmatched implies _u<sub>x</sub>_ = 0
|
||||
|
||||
The first two constraints are satisfied at all times while the matching algorithm runs.
|
||||
When the algorithm updates dual variables, it ensures that dual variables and edge slacks
|
||||
remain non-negative.
|
||||
It also ensures that matched edges remain tight, and that edges which are part of the odd-length
|
||||
cycle in a blossom remain tight.
|
||||
When the algorithm augments the matching, it uses an augmenting path that consists of
|
||||
only tight edges, thus ensuring that newly matched edges have zero slack.
|
||||
|
||||
The third constraint is initially not satisfied.
|
||||
The algorithm makes progress towards satisfying this constraint in two ways:
|
||||
by augmenting the matching, thus reducing the number of unmatched vertices,
|
||||
and by reducing the value of the dual variables of unmatched vertices.
|
||||
Eventually, either all vertices are matched or all unmatched vertices have zero dual.
|
||||
At that point the maximum weight matching has been found.
|
||||
|
||||
When the matching algorithm is finished, the constraints can be checked to verify
|
||||
that the matching is optimal.
|
||||
This check is simpler and faster than the matching algorithm itself.
|
||||
It can therefore be a useful way to guard against bugs in the matching algorithm.
|
||||
|
||||
### Rules for updating dual variables
|
||||
|
||||
At the start of the matching algorithm, all vertex dual variables _u<sub>i</sub>_
|
||||
are initialized to the same value: half of the greatest edge weight value that
|
||||
occurs on any edge in the graph.
|
||||
|
||||
$$ u_i = {1 \over 2} \cdot \max_{(x,y) \in E} w_{x,y} $$
|
||||
|
||||
Initially, there are no blossoms yet so there are no _z<sub>B</sub>_ to be initialized.
|
||||
When the algorithm creates a new blossom, it initializes its dual variable to
|
||||
_z<sub>B</sub>_ = 0.
|
||||
Note that this does not change the slack of any edge.
|
||||
|
||||
If a search for an augmenting path fails while there are still unmatched vertices
|
||||
with positive dual variables, it may not yet have found the maximum weight matching.
|
||||
In such cases the algorithm updates the dual variables until either
|
||||
an augmenting path gets unlocked or the dual variables of all unmatched vertices reach zero.
|
||||
|
||||
To update the dual variables, the algorithm chooses a value _δ_ that represents how much
|
||||
the duals will change.
|
||||
It then changes dual variables as follows:
|
||||
|
||||
- _u<sub>x</sub> ← u<sub>x</sub> − δ_ for every S-vertex _x_
|
||||
- _u<sub>x</sub> ← u<sub>x</sub> + δ_ for every T-vertex _x_
|
||||
- _z<sub>B</sub> ← z<sub>B</sub> + 2 * δ_ for every non-trivial S-blossom _B_
|
||||
- _z<sub>B</sub> ← z<sub>B</sub> − 2 * δ_ for every non-trivial T-blossom _B_
|
||||
|
||||
Dual variables of unlabeled blossoms and their vertices remain unchanged.
|
||||
Dual variables _z<sub>B</sub>_ of non-trivial sub-blossoms also remain changed;
|
||||
only top-level blossoms have their _z<sub>B</sub>_ updated.
|
||||
|
||||
Note that this update does not change the slack of edges that are either matched,
|
||||
or linked in the alternating tree, or contained in a blossom.
|
||||
However, the update reduces the slack of edges between S blossoms and edges between S-blossoms
|
||||
and unlabeled blossoms.
|
||||
It may cause some of these edges to become tight, allowing them to be used
|
||||
to construct an augmenting path.
|
||||
|
||||
The value of _δ_ is determined as follows:
|
||||
_δ_ = min(_δ<sub>1</sub>_, _δ<sub>2</sub>_, _δ<sub>3</sub>_, _δ<sub>4</sub>_) where
|
||||
|
||||
- _δ<sub>1</sub>_ is the minimum _u<sub>x</sub>_ of any S-vertex _x_.
|
||||
- _δ<sub>2</sub>_ is the minimum slack of any edge between an S-blossom and
|
||||
an unlabeled blossom.
|
||||
- _δ<sub>3</sub>_ is half of the minimum slack of any edge between two different S-blossoms.
|
||||
- _δ<sub>4</sub>_ is half of the minimum _z<sub>B</sub>_ of any T-blossom _B_.
|
||||
|
||||
_δ<sub>1</sub>_ protects against any vertex dual becoming negative.
|
||||
_δ<sub>2</sub>_ and _δ<sub>3</sub>_ together protect against any edge slack
|
||||
becoming negative.
|
||||
_δ<sub>4</sub>_ protects against any blossom dual becoming negative.
|
||||
|
||||
If the dual update is limited by _δ<sub>1</sub>_, it causes the duals of all remaining
|
||||
unmatched vertices to become zero.
|
||||
At that point the maximum matching has been found and the algorithm ends.
|
||||
If the dual update is limited by _δ<sub>2</sub>_ or _δ<sub>3</sub>_, it causes
|
||||
an edge to become tight.
|
||||
The next step is to either add that edge to the alternating tree (_δ<sub>2</sub>_)
|
||||
or use it to construct a blossom or augmenting path (_δ<sub>3</sub>_).
|
||||
If the dual update is limited by _δ<sub>4</sub>_, it causes the dual variable of
|
||||
a T-blossom to become zero.
|
||||
The next step is to expand that blossom.
|
||||
|
||||
A dual update may find that _δ = 0_, implying that the dual variables don't change.
|
||||
This can still be useful since all types of updates have side effects (adding an edge
|
||||
to an alternating tree, or expanding a blossom) that allow the algorithm to make progress.
|
||||
|
||||
During a single _stage_, the algorithm may iterate several times between scanning tight edges and
|
||||
updating dual variables.
|
||||
These iterations are called _substages_.
|
||||
To clarify: A stage is the process of growing alternating trees until an augmenting path is found.
|
||||
A stage ends either by augmenting the matching, or by concluding that no further improvement
|
||||
is possible.
|
||||
Each stage consists of one or more substages.
|
||||
A substage scans tight edges to grow the alternating trees.
|
||||
When a substage runs out of tight edges, it ends by performing a dual variable update.
|
||||
A substage also ends immediately when it finds an augmenting path.
|
||||
At the end of each stage, labels and alternating trees are removed.
|
||||
|
||||
The matching algorithm ends when a substage ends in a dual variable update limited
|
||||
by _δ<sub>1</sub>_.
|
||||
At that point the matching has maximum weight.
|
||||
|
||||
### Expanding a blossom
|
||||
|
||||
There are two scenarios where a blossom must be expanded.
|
||||
One is when the dual variable of a T-blossom becomes zero after a dual update limited
|
||||
by _δ<sub>4</sub>_.
|
||||
In this case the blossom must be expanded, otherwise further dual updates would cause
|
||||
its dual to become negative.
|
||||
|
||||
The other scenario is when the algorithm is about to assign label T to an unlabeled blossom
|
||||
with dual variable zero.
|
||||
A T-blossom with zero dual serves no purpose, potentially blocks an augmenting path,
|
||||
and is likely to be expanded anyway via a _δ<sub>4</sub>=0_ update.
|
||||
It is therefore better to preemptively expand the unlabeled blossom.
|
||||
The step that would have assigned label T to the blossom is then re-evaluated,
|
||||
which will cause it to assign label T to a sub-blossom of the expanded blossom.
|
||||
It may then turn out that this sub-blossom must also be expanded, and this becomes
|
||||
a recursive process until we get to a sub-blossom that is either a single vertex or
|
||||
a blossom with non-zero dual.
|
||||
|
||||
Note that [[5]](#galil1986) specifies that all top-level blossoms with dual variable zero should be
|
||||
expanded after augmenting the matching.
|
||||
This prevents the existence of unlabeled top-level blossoms with zero dual,
|
||||
and therefore prevents the scenario where label T would be assigned to such blossoms.
|
||||
That strategy is definitely correct, but it may lead to unnecessary expanding
|
||||
of blossoms which are then recreated during the search for the next augmenting path.
|
||||
Postponing the expansion of these blossoms until they are about to be labeled T,
|
||||
as described above, may be faster in some cases.
|
||||
|
||||
Expanding an unlabeled top-level blossom _B_ is pretty straightforward.
|
||||
Simply promote all of its sub-blossoms to top-level blossoms, then delete _B_.
|
||||
Note that the slack of all edges remains unchanged, since _z<sub>B</sub> = 0_.
|
||||
|
||||
Expanding a T-blossom is tricky because the labeled blossom is part of an alternating tree.
|
||||
After expanding the blossom, the part of the alternating tree that runs through the blossom
|
||||
must be reconstructed.
|
||||
An alternating path through a blossom always runs through its base vertex.
|
||||
After expanding T-blossom _B_, we can reconstruct the alternating path by following it
|
||||
from the sub-blossom where the path enters _B_ to the sub-blossom that contains the base vertex
|
||||
(choosing the direction around the blossom that takes an even number of steps).
|
||||
We then assign alternating labels T and S to the sub-blossoms along that path
|
||||
and link them into the alternating tree.
|
||||
All vertices of sub-blossoms that got label S are inserted into _Q_.
|
||||
|
||||
 <br>
|
||||
*Figure 5: Expanding a T-blossom*
|
||||
|
||||
### Keeping track of least-slack edges
|
||||
|
||||
To perform a dual variable update, the algorithm needs to compute the values
|
||||
of _δ<sub>1</sub>_, _δ<sub>2</sub>_, _δ<sub>3</sub>_ and _δ<sub>4</sub>_
|
||||
and determine which edge (_δ<sub>2</sub>_, _δ<sub>3</sub>_) or
|
||||
blossom (_δ<sub>4</sub>_) achieves the minimum value.
|
||||
|
||||
The total number of dual updates during a matching may be _Θ(n<sup>2</sup>)_.
|
||||
Since we want to limit the total number of steps of the matching algorithm to _O(n<sup>3</sup>)_,
|
||||
each dual update may use at most _O(n)_ steps.
|
||||
|
||||
We can find _δ<sub>1</sub>_ in _O(n)_ steps by simply looping over all vertices
|
||||
and checking their dual variables.
|
||||
We can find _δ<sub>4</sub>_ in _O(n)_ steps by simply looping over all non-trivial blossoms
|
||||
(since there are fewer than _n_ non-trivial blossoms).
|
||||
We could find _δ<sub>2</sub>_ and _δ<sub>3</sub>_ by simply looping over
|
||||
all edges of the graph in _O(m)_ steps, but that exceeds our budget of _O(n)_ steps.
|
||||
So we need better techniques.
|
||||
|
||||
For _δ<sub>2</sub>_, we determine the least-slack edge between an S-blossom and unlabeled
|
||||
blossom as follows.
|
||||
For every vertex _y_ in any unlabeled blossom, keep track of _e<sub>y</sub>_:
|
||||
the least-slack edge that connects _y_ to any S-vertex.
|
||||
The thing to keep track of is the identity of the edge, not the slack value.
|
||||
This information is kept up-to-date as part of the procedure that considers S-vertices.
|
||||
The scanning procedure eventually considers all edges _(x, y, w)_ where _x_ is an S-vertex.
|
||||
At that moment _e<sub>y</sub>_ is updated if the new edge has lower slack.
|
||||
|
||||
Calculating _δ<sub>2</sub>_ then becomes a matter of looping over all vertices _x_,
|
||||
checking whether _B(x)_ is unlabeled and calculating the slack of _e<sub>x</sub>_.
|
||||
|
||||
One subtle aspect of this technique is that a T-vertex can loose its label when
|
||||
the containing T-blossom gets expanded.
|
||||
At that point, we suddenly need to have kept track of its least-slack edge to any S-vertex.
|
||||
It is therefore necessary to do the same tracking also for T-vertices.
|
||||
So the technique is: For any vertex that is not an S-vertex, track its least-slack edge
|
||||
to any S-vertex.
|
||||
|
||||
Another subtle aspect is that a T-vertex may have a zero-slack edge to an S-vertex.
|
||||
Even though these edges are already tight, they must still be tracked.
|
||||
If the T-vertex loses its label, this edge needs to be reconsidered by the scanning procedure.
|
||||
By including these edges in least-slack edge tracking, they will be rediscovered
|
||||
through a _δ<sub>2</sub>=0_ update after the vertex becomes unlabeled.
|
||||
|
||||
For _δ<sub>3</sub>_, we determine the least-slack edge between any pair of S-blossoms
|
||||
as follows.
|
||||
For every S-blossom _B_, keep track of _e<sub>B</sub>_:
|
||||
the least-slack edge between _B_ and any other S-blossom.
|
||||
Note that this is done per S-blossoms, not per S-vertex.
|
||||
This information is kept up-to-date as part of the procedure that considers S-vertices.
|
||||
Calculating _δ<sub>3</sub>_ then becomes a matter of looping over all S-blossoms _B_
|
||||
and calculating the slack of _e<sub>B</sub>_.
|
||||
|
||||
A complication occurs when S-blossoms are merged.
|
||||
Some of the least-slack edges of the sub-blossoms may be internal edges in the merged blossom,
|
||||
and therefore irrelevant for _δ<sub>3</sub>_.
|
||||
As a result, the proper _e<sub>B</sub>_ of the merged blossom may be different from all
|
||||
least-slack edges of its sub-blossoms.
|
||||
An additional data structure is needed to find _e<sub>B</sub>_ of the merged blossom.
|
||||
|
||||
For every S-blossom _B_, maintain a list _L<sub>B</sub>_ of edges between _B_ and
|
||||
other S-blossoms.
|
||||
The purpose of _L<sub>B</sub>_ is to contain, for every other S-blossom, the least-slack edge
|
||||
between _B_ and that blossom.
|
||||
These lists are kept up-to-date as part of the procedure that considers S-vertices.
|
||||
While considering vertex _x_, if edge _(x, y, w)_ has positive slack,
|
||||
and _B(y)_ is an S-blossom, the edge is inserted in _L<sub>B(x)</sub>_.
|
||||
This may cause _L<sub>B(x)</sub>_ to contain multiple edges to _B(y)_.
|
||||
That's okay as long as it definitely contains the least-slack edge to _B(y)_.
|
||||
|
||||
When a new S-blossom is created, form its list _L<sub>B</sub>_ by merging the lists
|
||||
of its sub-blossoms.
|
||||
Ignore edges that are internal to the merged blossom.
|
||||
If there are multiple edges to the same target blossom, keep only the least-slack of these edges.
|
||||
Then find _e<sub>B</sub>_ of the merged blossom by simply taking the least-slack edge
|
||||
out of _L<sub>B</sub>_.
|
||||
|
||||
|
||||
## Run time of the algorithm
|
||||
|
||||
Every stage of the algorithm either increases the number of matched vertices by 2 or
|
||||
ends the matching.
|
||||
Therefore the number of stages is at most _n/2_.
|
||||
Every stage runs in _O(n<sup>2</sup>)_ steps, therefore the complete algorithm runs in
|
||||
_O(n<sup>3</sup>)_ steps.
|
||||
|
||||
During each stage, edge scanning is driven by the queue _Q_.
|
||||
Every vertex enters _Q_ at most once.
|
||||
Each vertex that enters _Q_ has its incident edges scanned, causing every edge in the graph
|
||||
to be scanned at most twice per stage.
|
||||
Scanning an edge is done in constant time, unless it leads to the discovery of a blossom
|
||||
or an augmenting path, which will be separately accounted for.
|
||||
Therefore edge scanning needs _O(m)_ steps per stage.
|
||||
|
||||
Creating a blossom reduces the number of top-level blossoms by at least 2,
|
||||
thus limiting the number of simultaneously existing blossoms to _O(n)_.
|
||||
Blossoms that are created during a stage become S-blossoms and survive until the end of that stage,
|
||||
therefore _O(n)_ blossoms are created during a stage.
|
||||
Creating a blossom involves tracing the alternating path to the closest common ancestor,
|
||||
and some bookkeeping per sub-blossom,
|
||||
and inserting new S-vertices _Q_,
|
||||
all of which can be done in _O(n)_ steps per blossom creation.
|
||||
The cost of managing least-slack edges between S-blossoms will be separately accounted for.
|
||||
Therefore blossom creation needs _O(n<sup>2</sup>)_ steps per stage
|
||||
(excluding least-slack edge management).
|
||||
|
||||
As part of creating a blossom, a list _L<sub>B</sub>_ of least-slack edges must be formed.
|
||||
This involves processing every element of all least-slack edge lists of its sub-blossoms,
|
||||
and removing redundant edges from the merged list.
|
||||
Merging and removing redundant edges can be done in one sweep via a temporary array indexed
|
||||
by target blossom.
|
||||
Collect the least-slack edges of the sub-blossoms into this array,
|
||||
indexed by the target blossom of the edge,
|
||||
keeping only the edge with lowest slack per target blossom.
|
||||
Then convert the array back into a list by removing unused indices.
|
||||
This takes _O(1)_ steps per candidate edge, plus _O(n)_ steps to manage the temporary array.
|
||||
I choose to shift the cost of collecting the candidate edges from the sub-blossoms to
|
||||
the actions that inserted those edges into the sub-blossom lists.
|
||||
There are two processes which insert edges into _L<sub>B</sub>_: edge scanning and blossom
|
||||
creation.
|
||||
Edge scanning inserts each graph edge at most twice per stage for a total cost of _O(m)_ steps
|
||||
per stage.
|
||||
Blossom creation inserts at most _O(n)_ edges per blossom creation.
|
||||
Therefore the total cost of S-blossom least-slack edge management is
|
||||
_O(m + n<sup>2</sup>) = O(n<sup>2</sup>)_ steps per stage.
|
||||
|
||||
The number of blossom expansions during a stage is _O(n)_.
|
||||
Expanding a blossom involves some bookkeeping per sub-blossom,
|
||||
and reconstructing the alternating path through the blossom,
|
||||
and inserting any new S-vertices into _Q_,
|
||||
all of which can be done in _O(n)_ steps per blossom.
|
||||
Therefore blossom expansion needs _O(n<sup>2</sup>)_ steps per stage.
|
||||
|
||||
The length of an augmenting path is _O(n)_.
|
||||
Tracing the augmenting path and augmenting the matching along the path can be done in _O(n)_ steps.
|
||||
Augmenting through a blossom takes a number of steps that is proportional in the number of
|
||||
its sub-blossoms.
|
||||
Since there are fewer than _n_ non-trivial blossoms, the total cost of augmenting through
|
||||
blossoms is _O(n)_ steps.
|
||||
Therefore the total cost of augmenting is _O(n)_ steps per stage.
|
||||
|
||||
A dual variable update limited by _δ<sub>1</sub>_ ends the algorithm and therefore
|
||||
happens at most once.
|
||||
An update limited by _δ<sub>2</sub>_ labels a previously labeled blossom
|
||||
and therefore happens _O(n)_ times per stage.
|
||||
An update limited by _δ<sub>3</sub>_ either creates a blossom or finds an augmenting path
|
||||
and therefore happens _O(n)_ times per stage.
|
||||
An update limited by _δ<sub>4</sub>_ expands a blossom and therefore happens
|
||||
_O(n)_ times per stage.
|
||||
Therefore the number of dual variable updates is _O(n)_ per stage.
|
||||
The cost of calculating the _δ_ values is _O(n)_ per update as discussed above.
|
||||
Applying changes to the dual variables can be done by looping over all vertices and looping over
|
||||
all top-level blossoms in _O(n)_ steps per update.
|
||||
Therefore the total cost of dual variable updates is _O(n<sup>2</sup>)_ per stage.
|
||||
|
||||
|
||||
## Implementation details
|
||||
|
||||
_TO BE WRITTEN_
|
||||
|
||||
|
||||
## References
|
||||
|
||||
1. <a id="edmonds1965a"></a>
|
||||
Jack Edmonds, "Paths, trees, and flowers." _Canadian Journal of Mathematics vol. 17 no. 3_, 1965.
|
||||
([link](https://doi.org/10.4153/CJM-1965-045-4))
|
||||
([pdf](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/08B492B72322C4130AE800C0610E0E21/S0008414X00039419a.pdf/paths_trees_and_flowers.pdf))
|
||||
|
||||
2. <a id="edmonds1965b"></a>
|
||||
Jack Edmonds, "Maximum matching and a polyhedron with 0,1-vertices." _Journal of research of the National Bureau of Standards vol. 69B_, 1965.
|
||||
([pdf](https://nvlpubs.nist.gov/nistpubs/jres/69B/jresv69Bn1-2p125_A1b.pdf))
|
||||
|
||||
3. <a id="gabow1974"></a>
|
||||
Harold N. Gabow, "Implementation of algorithms for maximum matching on nonbipartite graphs." Ph.D. thesis, Stanford University, 1974.
|
||||
|
||||
4. <a id="galil_micali_gabow1986"></a>
|
||||
Z. Galil, S. Micali, H. Gabow, "An O(EV log V) algorithm for finding a maximal weighted matching in general graphs." _SIAM Journal on Computing vol. 15 no. 1_, 1986.
|
||||
([link](https://epubs.siam.org/doi/abs/10.1137/0215009))
|
||||
([pdf](https://www.researchgate.net/profile/Zvi-Galil/publication/220618201_An_OEVlog_V_Algorithm_for_Finding_a_Maximal_Weighted_Matching_in_General_Graphs/links/56857f5208ae051f9af1e257/An-OEVlog-V-Algorithm-for-Finding-a-Maximal-Weighted-Matching-in-General-Graphs.pdf))
|
||||
|
||||
5. <a id="galil1986"></a>
|
||||
Zvi Galil, "Efficient algorithms for finding maximum matching in graphs." _ACM Computing Surveys vol. 18 no. 1_, 1986.
|
||||
([link](https://dl.acm.org/doi/abs/10.1145/6462.6502))
|
||||
([pdf](https://dl.acm.org/doi/pdf/10.1145/6462.6502))
|
||||
|
||||
6. <a id="gabow1990"></a>
|
||||
Harold N. Gabow, "Data structures for weighted matching and nearest common ancestors with linking." _Proc. 1st ACM-SIAM symposium on discrete algorithms_, 1990.
|
||||
([link](https://dl.acm.org/doi/abs/10.5555/320176.320229))
|
||||
([pdf](https://dl.acm.org/doi/pdf/10.5555/320176.320229))
|
||||
|
||||
|
||||
---
|
||||
Written in 2023 by Joris van Rantwijk.
|
||||
This work is licensed under [CC BY-ND 4.0](https://creativecommons.org/licenses/by-nd/4.0/).
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Reference in New Issue