Distributed Algorithms:

Introduction:

Distributed algorithms are a category of algorithms designed to solve problems in a distributed system, where multiple independent entities, often referred to as nodes or processes, collaborate to achieve a common goal. In distributed systems, these entities may be geographically dispersed, connected by a network, and operate concurrently.

Key Characteristics of Distributed Systems:

Concurrency: Multiple entities execute simultaneously.

Independence: Nodes operate independently and may not have global knowledge of the system.

Communication: Nodes communicate with each other through a network, exchanging messages to coordinate their actions.

Fault Tolerance: Distributed algorithms often address issues such as node failures, network partitions, and other forms of system failures.

Distributed algorithms play a crucial role in enabling the efficient and reliable operation of distributed systems.

They address the challenges posed by the inherently decentralized and concurrent nature of such systems, ensuring that nodes can collaborate effectively to achieve common objectives.

 

Distributed Breadth-First Search (BFS):

Distributed Breadth-First Search (BFS) is an algorithm used to traverse and explore nodes in a distributed graph or network.

BFS is commonly used to find the shortest path between two nodes in an unweighted graph.

In a distributed setting, each node in the graph is a separate computing entity, and the BFS algorithm is distributed across these nodes.

Distributed BFS Algorithm:

1.    Initialization:

·        Each node is assigned a unique identifier.

·        The source node initiates the BFS exploration.

2.    Local Exploration:

·        Each node explores its local neighbors and identifies nodes not yet visited.

·        It creates messages containing information about the discovered nodes and sends them to neighboring nodes.

3.    Message Propagation:

·        Messages propagate through the network, updating the knowledge of each node about the discovered nodes.

4.    Global Queue:

·        A global queue is maintained, storing the nodes to be explored in the next iteration.

5.    Iterative Exploration:

·        Nodes continue to explore and propagate messages until the entire graph is traversed or the desired condition is met.

Example: Consider a distributed graph

A ---- B

|        |

C ---- D

  • Nodes: A, B, C, D
  • Edges: (A, B), (A, C), (B, D), (C, D)

Initialization:

  • Source Node: A

Iteration 1:

  • A explores neighbors B and C.
  • Messages are sent to B and C containing information about discovered nodes (B, C).
  • Global queue: [B, C]

Iteration 2:

  • B explores neighbor D.
  • Message sent to D containing information about discovered node (D).
  • C explores neighbor D.
  • Message sent to D containing information about discovered node (D).
  • Global queue: [C, D]

Iteration 3:

  • C and D have no unexplored neighbors.
  • The algorithm terminates.

Result:

  • Nodes discovered in BFS order: A, B, C, D

 


Distributed Breadth-First Search (BFS) is an algorithm used to traverse and explore nodes in a distributed graph or network. BFS is commonly used to find the shortest path between two nodes in an unweighted graph. In a distributed setting, each node in the graph is a separate computing entity, and the BFS algorithm is distributed across these nodes.

Distributed BFS Algorithm:

1.    Initialization:

·        Each node is assigned a unique identifier.

·        The source node initiates the BFS exploration.

2.    Local Exploration:

·        Each node explores its local neighbors and identifies nodes not yet visited.

·        It creates messages containing information about the discovered nodes and sends them to neighboring nodes.

3.    Message Propagation:

·        Messages propagate through the network, updating the knowledge of each node about the discovered nodes.

4.    Global Queue:

·        A global queue is maintained, storing the nodes to be explored in the next iteration.

5.    Iterative Exploration:

·        Nodes continue to explore and propagate messages until the entire graph is traversed or the desired condition is met.

Example:

Consider a distributed graph:

|A --- B|

|C --- D|

  • Nodes: A, B, C, D
  • Edges: (A, B), (A, C), (B, D), (C, D)

Initialization:

  • Source Node: A

Iteration 1:

  • A explores neighbors B and C.
  • Messages are sent to B and C containing information about discovered nodes (B, C).
  • Global queue: [B, C]

Iteration 2:

  • B explores neighbor D.
  • Message sent to D containing information about discovered node (D).
  • C explores neighbor D.
  • Message sent to D containing information about discovered node (D).
  • Global queue: [C, D]

Iteration 3:

  • C and D have no unexplored neighbors.
  • The algorithm terminates.

Result:

  • Nodes discovered in BFS order: A, B, C, D

In this example, each node performs local exploration and propagates messages to inform other nodes about discovered neighbors. The global queue ensures the order of exploration across the distributed network. This iterative process continues until all reachable nodes are discovered.

 

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:

mathematicaCopy code

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.