Added edmonds_blossom_algorithm.py. For maximum matching in the graph. #12043

[TarunVishwakarma1]

Describe your change:

This PR implements the Edmonds' Blossom Algorithm in the EdmondsBlossomAlgorithm class, which efficiently finds the maximum matching in general graphs, including those with odd-length cycles. The implementation includes methods for handling augmenting paths, updating matchings, and contracting blossoms.

Maximum Matching: Introduces the maximumMatching method, which takes a list of edges and the total number of vertices as input and returns a list of matched pairs of vertices.
Augmenting Path Search: Utilizes a breadth-first search (BFS) approach to locate augmenting paths within the graph, ensuring that it can handle complex graph structures effectively.
Blossom Contraction: Implements mechanisms for contracting blossoms to manage odd-length cycles, which is a key aspect of the algorithm's efficiency.
Detailed Comments: Each method and critical section of the code are thoroughly commented to ensure clarity and understanding of the algorithm's functionality.

• Add an algorithm?
• Fix a bug or typo in an existing algorithm?
• Add or change doctests? -- Note: Please avoid changing both code and tests in a single pull request.
• Documentation change?

Checklist:

• I have read CONTRIBUTING.md.
• This pull request is all my own work -- I have not plagiarized.
• I know that pull requests will not be merged if they fail the automated tests.
• This PR only changes one algorithm file. To ease review, please open separate PRs for separate algorithms.
• All new Python files are placed inside an existing directory.
• All filenames are in all lowercase characters with no spaces or dashes.
• All functions and variable names follow Python naming conventions.
• All function parameters and return values are annotated with Python type hints.
• All functions have doctests that pass the automated testing.
• All new algorithms include at least one URL that points to Wikipedia or another similar explanation.
• If this pull request resolves one or more open issues then the description above includes the issue number(s) with a closing keyword: "Fixes Add Edmonds' Blossom Algorithm for Maximum Matching in Graphs #12043 ".

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[TarunVishwakarma1]

The first 2 PR's were giving errors and had to re-fork the repo, So deleted the pr and created fresh new PR

[algorithms-keeper]

Click here to look at the relevant links ⬇️

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Repository:

• Contributing guidelines
• Project Euler solution guidelines

Python:

• Formatted string literals (f-strings)
• Type hints
• doctest
• unittest
• pytest

Automated review generated by algorithms-keeper. If there's any problem regarding this review, please open an issue about it.

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[yuvashrikarunakaran]

from collections import deque, defaultdict

class EdmondsBlossomAlgorithm:
def init(self, vertices, edges):

    self.vertices = vertices
    self.edges = edges
    self.matching = [-1] * vertices  # Tracks the current matching
    self.parent = [-1] * vertices  # Tracks parent in the BFS tree
    self.base = list(range(vertices))  # Tracks base vertices for blossoms
    self.adjacency_list = defaultdict(list)
    for u, v in edges:
        self.adjacency_list[u].append(v)
        self.adjacency_list[v].append(u)

def find_augmenting_path(self, start):
  
    queue = deque([start])
    visited = [False] * self.vertices
    visited[start] = True

    while queue:
        u = queue.popleft()
        for v in self.adjacency_list[u]:
            if self.base[u] == self.base[v] or self.matching[u] == v:
                continue
            if v == start or (self.matching[v] != -1 and self.parent[self.matching[v]] != -1):
                self.contract_blossom(u, v, start)
            elif self.parent[v] == -1:
                self.parent[v] = u
                if self.matching[v] == -1:
                    return v
                next_vertex = self.matching[v]
                queue.append(next_vertex)
                visited[next_vertex] = True
    return None

def contract_blossom(self, u, v, root):
   
    blossom_base = self.find_common_ancestor(u, v)
    marked = [False] * self.vertices
    while self.base[u] != blossom_base:
        marked[self.base[u]] = True
        self.parent[u] = v
        v = self.matching[u]
        u = self.parent[v]
    while self.base[v] != blossom_base:
        marked[self.base[v]] = True
        self.parent[v] = u
        u = self.matching[v]
        v = self.parent[u]
    for i in range(self.vertices):
        if marked[self.base[i]]:
            self.base[i] = blossom_base
            if not marked[i]:
                self.parent[i] = -1

def find_common_ancestor(self, u, v):
   
    visited = [False] * self.vertices
    while True:
        u = self.base[u]
        visited[u] = True
        if self.matching[u] == -1:
            break
        u = self.parent[self.matching[u]]
    while True:
        v = self.base[v]
        if visited[v]:
            return v
        v = self.parent[self.matching[v]]

def augment_path(self, start, end):
   
    while end != -1:
        prev = self.parent[end]
        next_vertex = self.matching[prev]
        self.matching[end] = prev
        self.matching[prev] = end
        end = next_vertex

def maximum_matching(self):
   
    for vertex in range(self.vertices):
        if self.matching[vertex] == -1:
            self.parent = [-1] * self.vertices
            end = self.find_augmenting_path(vertex)
            if end is not None:
                self.augment_path(vertex, end)
    return [(u, v) for u, v in enumerate(self.matching) if u < v]

if name == "main":
vertices = 6
edges = [(0, 1), (1, 2), (2, 3), (3, 0), (3, 4), (4, 5)]
blossom_algo = EdmondsBlossomAlgorithm(vertices, edges)

max_matching = blossom_algo.maximum_matching()
print("Maximum Matching:", max_matching)

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