Borůvka's Algorithm for Minimum Spanning Tree using MapReduce (mrjob)

This project implements Borůvka’s algorithm for computing the Minimum Spanning Tree (MST) of a graph using MapReduce, facilitated via the mrjob Python library. This approach is especially suitable for large-scale distributed environments.

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📌 Overview

Borůvka’s algorithm is an efficient greedy algorithm for MST computation. At each iteration, the algorithm:

1. Finds the lightest outgoing edge from each connected component.
2. Merges the connected components using these lightest edges.
3. Repeats this process until all nodes are part of a single component.

This implementation simulates this iterative process using MapReduce, where each Map step identifies candidate edges and each Reduce step selects the lightest one.

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📁 File Structure

• boruvka_mr.py: The main MapReduce job using mrjob.
• dsu.py: A helper module that implements the Disjoint Set Union (Union-Find) data structure, used to keep track of components.

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⚙️ Requirements

• Python 3.7+
• mrjob (pip install mrjob)

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📦 Installation

pip install mrjob

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🚀 How to Run

The script expects three arguments:

• --adjacency_list: A stringified dictionary of the graph adjacency list.
• --dsu_rank: A stringified list of the DSU ranks.
• --dsu_parent: A stringified list of the DSU parent pointers.

Each round of Borůvka corresponds to a single MapReduce job. Between iterations, the DSU state must be updated manually to reflect new merged components.

🧪 Example Run

python boruvka_mr.py \
    --adjacency_list="{0: [(1, 4), (2, 1)], 1: [(0, 4), (2, 3)], 2: [(0, 1), (1, 3)]}" \
    --dsu_rank="[0, 0, 0]" \
    --dsu_parent="[0, 1, 2]" \
    -r inline

📤 Output

Each output line corresponds to the minimum edge chosen from each component in the current round.

Example output:

0	(0, 2, 1)
1	(1, 2, 3)

This means node 0 selected edge (0, 2, 1) and node 1 selected (1, 2, 3) for MST construction.

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🧠 How It Works

Mapper:

For each vertex and its adjacency list:

• It finds the component representatives of both endpoints using DSU.
• If the endpoints are in different components, it yields a candidate edge.

Reducer:

• For each component, selects the minimum-weight outgoing edge to merge with another component.

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🔄 Iterative Use

Borůvka’s algorithm is inherently iterative. This script runs a single phase of the algorithm. To complete the MST, you must:

1. Run this job.
2. Use the resulting edges to merge DSU components.
3. Update the DSU state (parent, rank) and re-run until all nodes are in one component.

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📚 References

• Borůvka, O. (1926). “O jistém problému minimálním” (On a Certain Minimal Problem), Práce Mor. Prírodoved. Spol. v Brne III (3): 37–58.
• Wikipedia: Borůvka's Algorithm
• mrjob documentation

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🧑‍💻 Author

Rajendraprasad Saravanan
GitHub: @Rajendraprasad7

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📜 License

This project is licensed under the MIT License. See the LICENSE file for details.