kaustubh-nair / Blossom-Algorithm

Blossom Algorithm

Implementation and animation of the Blossom algorithm to construct a maximum matching in any graph. The blossom algorithm is a polynomial time maximum graph matching algorithm that has a complexity O(|V²|x|E|). We have used NetworkX along with Matplotlib to maintain and visualize the graph. The animation is interactive and the steps of the algorithm can be viewed one-by-one by pressing the enter key while the program is executing. A video of the algorithm can be viewed in the video/ folder.

Team members

1. Bishal Pandia IMT2017010
2. Kaustubh Nair IMT2017025
3. Sarthak Khoche IMT2017038

Prerequisites and Setup

To install dependencies, run

$ pip install -r requirements.txt

Usage

To run the algorithm, in your terminal, enter

$ python main.py

Press the enter key to update graph and view the next step in the algorithm.

Animations

Here are some animations of the algorithm:

Algorithm

A step-by-step run through of the whole algorithm

Initialization

A sample graph is first initialized with the help of src/sample_graphs.py

Finding free vertices

All the free vertices with matched attribute as False are shown (yellow)

Finding random vertex

A random vertex is picked from all the free vertices (red)

Breadth First Search

With the random vertex as root, we start the BFS and highlight the next vertex (red)

If the next vertex is unmatched, we construct an augmenting path between them (yellow)

And then invert the augmenting path to increase its length by 1 (green)

In the case that next vertex is matched

We add the matched edge to the augmenting path initially (yellow)

We check the next vertex from root, if it is matched as well, that suggests presence of an odd cycle. We construct a blossom and shrink it to a supernode (visualization of blossom can be seen in Animations above)

If the next vertex from the root is unmatched, weconstruct an augmenting path from that edge instead (yellow)

And then invert it (green)

We keep repeating these step until we have no free vertex left. That is when we obtain the maximum matching (red) of the graph according to Edmonds' Blossom Algorithm

Brief about the code

Here is an overview about the files related to the implementation:

src/sample_graphs.py contains some example graphs which can be used for the algorithm.

main.py contains code regarding initialization of the graph, running the algorithm and animation of the graph.

src/blossom.py contains the implementation of the Blossom algorithm.

src/helpers.py contains helpers used for the animation.