PR: Dinic's Algorithm for Maximum Flow

This PR provides a comprehensive R implementation of Dinic's Algorithm, a highly efficient algorithm to compute the maximum flow in a flow network. The algorithm leverages level graphs and blocking flows to improve performance over classical methods like Ford-Fulkerson.

Overview

Dinic's Algorithm operates in two main phases iteratively:

1. Level Graph Construction (BFS):

   • Perform a Breadth-First Search from the source to label vertices with their distance from the source.
   • Only consider edges with remaining capacity.

2. Blocking Flow Computation (DFS):

   • Send flow along paths in the level graph using Depth-First Search.
   • Continue until no augmenting paths exist in the current level graph.

These steps are repeated until no augmenting path exists from the source to the sink. The approach ensures efficient convergence and allows multiple flows per BFS iteration.

Features

• Computes maximum flow in directed networks.
• Handles networks with multiple paths and varying capacities.
• Uses adjacency lists and capacity/flow matrices for efficiency.
• Finds the minimum cut automatically.
• Suitable for bipartite matching, multi-source/multi-sink networks, and assignment problems.
• Includes robust error handling and validation.

Complexity

• Time Complexity:
  • General graphs: O(V² * E)
  • Unit capacity networks: O(E * √V)
• Space Complexity: O(V + E) for adjacency lists, level arrays, and flow matrices.

Applications

• Network routing and traffic optimization
• Bipartite matching problems
• Image segmentation
• Airline scheduling and logistics
• Project selection and resource allocation

Demonstration

Example 1: Basic 6-Vertex Network

# Create network and add edges
network <- create_flow_network(6)
network <- add_edge(network, 1, 2, 16)
network <- add_edge(network, 1, 3, 13)
network <- add_edge(network, 2, 3, 10)
network <- add_edge(network, 2, 4, 12)
network <- add_edge(network, 3, 2, 4)
network <- add_edge(network, 3, 5, 14)
network <- add_edge(network, 4, 3, 9)
network <- add_edge(network, 4, 6, 20)
network <- add_edge(network, 5, 4, 7)
network <- add_edge(network, 5, 6, 4)

# Compute maximum flow
result <- dinic_max_flow(network, source = 1, sink = 6)
print_max_flow(result, network)
print_flow_edges(network)