Add Push-Relabel Algorithm in R

[nachiket2005]

Overview

The push_relabel function computes the maximum flow from a given source to a sink vertex in a directed graph.
Unlike the Ford–Fulkerson or Edmonds–Karp methods that rely on augmenting paths, the Push–Relabel algorithm maintains a preflow and adjusts vertex heights to push excess flow efficiently through the network.

This implementation:

• Initializes a preflow from the source
• Repeatedly pushes excess flow to neighboring vertices
• Relabels vertices when no more pushes are possible
• Uses a discharge loop to efficiently manage active vertices

It is designed for clarity and educational use, closely matching the behavior of class-based implementations in other languages while being idiomatic to R.

---

Features

• Implements Push–Relabel (Preflow–Push) method for maximum flow
• Supports directed graphs represented as square capacity matrices
• Includes Push, Relabel, and Discharge operations
• Uses 1-based indexing for compatibility with R conventions
• Tracks vertex heights, excess flows, and residual capacities
• Returns detailed results including:
  • max_flow — total maximum flow value
  • flow — final flow matrix
  • excess — excess flow per vertex
  • height — vertex heights after termination
• Includes ready-to-run example usage

---

Complexity

• Time Complexity: O(V³) in the generic implementation
  • (Performance can be improved with heuristics like gap relabeling or global relabeling, not included here.)
• Space Complexity: O(V²) for flow and capacity matrices

---

Demonstration

Run the following example in an R session to test the implementation:

source('R/graph_algorithms/push_relabel.r')

graph <- matrix(c(
  0, 16, 13, 0, 0, 0,
  0, 0, 10, 12, 0, 0,
  0, 4, 0, 0, 14, 0,
  0, 0, 9, 0, 0, 20,
  0, 0, 0, 7, 0, 4,
  0, 0, 0, 0, 0, 0
), nrow = 6, byrow = TRUE)

source <- 1L
sink <- 6L

res <- push_relabel(graph, source, sink)
cat("Maximum Flow:", res$max_flow, "\n")

[Copilot]

Pull Request Overview

This PR adds multiple new algorithm implementations to TheAlgorithms/R repository, introducing both graph algorithms and mathematical functions. While the PR title mentions only the Push-Relabel algorithm, it actually contains several new implementations.

• Adds four new graph algorithms: Push-Relabel maximum flow, Edmonds-Karp maximum flow, Hamiltonian path finder, and Boruvka's MST
• Implements Fast Fourier Transform in the mathematics section
• Provides comprehensive documentation for all new algorithms

Reviewed Changes

Copilot reviewed 9 out of 9 changed files in this pull request and generated 8 comments.

Show a summary per file

File	Description
graph_algorithms/push_relabel.r	Implements Push-Relabel algorithm for maximum flow computation
graph_algorithms/edmonds_karp.r	Implements Edmonds-Karp algorithm for maximum flow computation
graph_algorithms/hamiltonian_path.r	Implements backtracking algorithm to find Hamiltonian paths
graph_algorithms/boruvka_mst.r	Implements Boruvka's algorithm for minimum spanning tree
mathematics/fast_fourier_transform.r	Implements recursive Cooley-Tukey FFT algorithm
documentation/push_relabel.md	Documentation for Push-Relabel algorithm
documentation/hamiltonian_path.md	Documentation for Hamiltonian path algorithm
documentation/fast_fourier_transform.md	Documentation for FFT algorithm
documentation/boruvka_mst.md	Documentation for Boruvka's MST algorithm

[siriak]

Please remove separate docs