Distributed Minimum Spanning Tree (MST) Algorithm:

Distributed Minimum Spanning Tree algorithms aim to construct a minimum spanning tree across a distributed network. The goal is to connect all nodes with the minimum total edge weight, ensuring that there are no cycles. One such algorithm is the Distributed Minimum Spanning Tree (DMST) algorithm.

Algorithm Overview:

1.    Initialization:

·        Each node is assigned a unique identifier.

·        Initially, each node considers itself as a separate tree.

2.    Local Computation:

·        Each node locally computes the minimum-weight edge connecting it to its neighbors.

3.    Message Passing:

·        Nodes exchange messages containing information about their local minimum-weight edges with their neighbors.

4.    Message Processing:

·        Upon receiving messages, nodes compare the received minimum-weight edges with their locally computed edges.

·        Nodes update their information based on the minimum-weight edges received.

5.    Tree Formation:

·        Nodes gradually form a minimum spanning tree by incorporating edges with the minimum weight.

6.    Termination:

·        The algorithm terminates when all nodes have agreed upon the minimum spanning tree.

Example:

Consider a distributed network with nodes A, B, C, D, and the following weighted edges:

  • (A, B): 2
  • (A, C): 1
  • (A, D): 4
  • (B, C): 2
  • (B, D): 3
  • (C, D): 5

Algorithm Execution:

1.    Initialization:

·        Each node starts as a separate tree.

2.    Local Computation:

·        Nodes compute their local minimum-weight edges.

·        A: (A, C) with weight 1

·        B: (A, B) with weight 2

·        C: (A, C) with weight 1

·        D: (A, D) with weight 4

3.    Message Passing:

·        Nodes exchange messages with their neighbors, containing information about their locally computed edges.

4.    Message Processing:

·        Nodes process received messages and update their information based on the minimum-weight edges.

5.    Tree Formation:

·        Nodes gradually incorporate minimum-weight edges, forming a spanning tree.

·        A: (A, C) with weight 1

·        B: (A, B) with weight 2

·        C: (A, C) with weight 1

·        D: (A, C) with weight 1

6.    Termination:

·        The algorithm terminates when all nodes agree upon the minimum spanning tree.

Resulting Minimum Spanning Tree:

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A / \ C D

In this example, the DMST algorithm constructs a minimum spanning tree connecting all nodes with the minimum total edge weight. Each node contributes its locally computed minimum-weight edge, and through message passing and processing, the nodes collectively form a distributed minimum spanning tree.