Added the implementation of the Edmond Karp algorithm along with test…

… cases (#252)

* Added the implementation for bijection method binary to decimal and euler method

* Added the implementation of the edmondkarp along with tests

* Update graph/edmondkarp.ts

Co-authored-by: Lars Müller <[email protected]>

* Update edmondkarp.ts

* Update graph/edmondkarp.ts

Co-authored-by: Lars Müller <[email protected]>

* Implement the furhter changes to make the algorithm more refined

* Updated the test cases 

Updated the test cases

* Updated the code and rewrite some functions 

Updated the code and rewrite some functions

* Implement the optimal stack queue implementation in the edmond karp algorithm

* removed the bisection_method.ts, decimal_convert.ts, euler_method.ts

* Revert changes to package.json and package-lock.json

---------

Co-authored-by: Lars Müller <[email protected]>

graph/edmonds_karp.ts

@@ -0,0 +1,97 @@
+import { StackQueue } from '../data_structures/queue/stack_queue'
+
+/**
+ * @function edmondsKarp
+ * @description Compute the maximum flow from a source node to a sink node using the Edmonds-Karp algorithm.
+ * @Complexity_Analysis
+ * Time complexity: O(V * E^2) where V is the number of vertices and E is the number of edges.
+ * Space Complexity: O(E) due to residual graph representation.
+ * @param {[number, number][][]} graph - The graph in adjacency list form.
+ * @param {number} source - The source node.
+ * @param {number} sink - The sink node.
+ * @return {number} - The maximum flow from the source node to the sink node.
+ * @see https://en.wikipedia.org/wiki/Edmonds%E2%80%93Karp_algorithm
+ */
+export default function edmondsKarp(
+  graph: [number, number][][],
+  source: number,
+  sink: number
+): number {
+  const n = graph.length
+
+  // Initialize residual graph
+  const residualGraph: [number, number][][] = Array.from(
+    { length: n },
+    () => []
+  )
+
+  // Build residual graph from the original graph
+  for (let u = 0; u < n; u++) {
+    for (const [v, cap] of graph[u]) {
+      if (cap > 0) {
+        residualGraph[u].push([v, cap]) // Forward edge
+        residualGraph[v].push([u, 0]) // Reverse edge with 0 capacity
+      }
+    }
+  }
+
+  const findAugmentingPath = (parent: (number | null)[]): number => {
+    const visited = Array(n).fill(false)
+    const queue = new StackQueue<number>()
+    queue.enqueue(source)
+    visited[source] = true
+    parent[source] = null
+
+    while (queue.length() > 0) {
+      const u = queue.dequeue()
+      for (const [v, cap] of residualGraph[u]) {
+        if (!visited[v] && cap > 0) {
+          parent[v] = u
+          visited[v] = true
+          if (v === sink) {
+            // Return the bottleneck capacity along the path
+            let pathFlow = Infinity
+            let current = v
+            while (parent[current] !== null) {
+              const prev = parent[current]!
+              const edgeCap = residualGraph[prev].find(
+                ([node]) => node === current
+              )![1]
+              pathFlow = Math.min(pathFlow, edgeCap)
+              current = prev
+            }
+            return pathFlow
+          }
+          queue.enqueue(v)
+        }
+      }
+    }
+    return 0
+  }
+
+  let maxFlow = 0
+  const parent = Array(n).fill(null)
+
+  while (true) {
+    const pathFlow = findAugmentingPath(parent)
+    if (pathFlow === 0) break // No augmenting path found
+
+    // Update the capacities and reverse capacities in the residual graph
+    let v = sink
+    while (parent[v] !== null) {
+      const u = parent[v]!
+      // Update capacity of the forward edge
+      const forwardEdge = residualGraph[u].find(([node]) => node === v)!
+      forwardEdge[1] -= pathFlow
+      // Update capacity of the reverse edge
+      const reverseEdge = residualGraph[v].find(([node]) => node === u)!
+      reverseEdge[1] += pathFlow
+
+      v = u
+    }
+
+    maxFlow += pathFlow
+  }
+
+  return maxFlow
+}

graph/test/edmonds_karp.test.ts

@@ -0,0 +1,82 @@
+import edmondsKarp from '../edmonds_karp'
+
+describe('Edmonds-Karp Algorithm', () => {
+  it('should find the maximum flow in a simple graph', () => {
+    const graph: [number, number][][] = [
+      [
+        [1, 3],
+        [2, 2]
+      ], // Node 0: Edges to node 1 (capacity 3), and node 2 (capacity 2)
+      [[3, 2]], // Node 1: Edge to node 3 (capacity 2)
+      [[3, 3]], // Node 2: Edge to node 3 (capacity 3)
+      [] // Node 3: No outgoing edges
+    ]
+    const source = 0
+    const sink = 3
+    const maxFlow = edmondsKarp(graph, source, sink)
+    expect(maxFlow).toBe(4)
+  })
+
+  it('should find the maximum flow in a more complex graph', () => {
+    const graph: [number, number][][] = [
+      [
+        [1, 10],
+        [2, 10]
+      ], // Node 0: Edges to node 1 and node 2 (both capacity 10)
+      [
+        [3, 4],
+        [4, 8]
+      ], // Node 1: Edges to node 3 (capacity 4), and node 4 (capacity 8)
+      [[4, 9]], // Node 2: Edge to node 4 (capacity 9)
+      [[5, 10]], // Node 3: Edge to node 5 (capacity 10)
+      [[5, 10]], // Node 4: Edge to node 5 (capacity 10)
+      [] // Node 5: No outgoing edges (sink)
+    ]
+    const source = 0
+    const sink = 5
+    const maxFlow = edmondsKarp(graph, source, sink)
+    expect(maxFlow).toBe(14)
+  })
+
+  it('should return 0 when there is no path from source to sink', () => {
+    const graph: [number, number][][] = [
+      [], // Node 0: No outgoing edges
+      [], // Node 1: No outgoing edges
+      [] // Node 2: No outgoing edges (sink)
+    ]
+    const source = 0
+    const sink = 2
+    const maxFlow = edmondsKarp(graph, source, sink)
+    expect(maxFlow).toBe(0)
+  })
+
+  it('should handle graphs with no edges', () => {
+    const graph: [number, number][][] = [
+      [], // Node 0: No outgoing edges
+      [], // Node 1: No outgoing edges
+      [] // Node 2: No outgoing edges
+    ]
+    const source = 0
+    const sink = 2
+    const maxFlow = edmondsKarp(graph, source, sink)
+    expect(maxFlow).toBe(0)
+  })
+
+  it('should handle graphs with self-loops', () => {
+    const graph: [number, number][][] = [
+      [
+        [0, 10],
+        [1, 10]
+      ], // Node 0: Self-loop with capacity 10, and edge to node 1 (capacity 10)
+      [
+        [1, 10],
+        [2, 10]
+      ], // Node 1: Self-loop and edge to node 2
+      [] // Node 2: No outgoing edges (sink)
+    ]
+    const source = 0
+    const sink = 2
+    const maxFlow = edmondsKarp(graph, source, sink)
+    expect(maxFlow).toBe(10)
+  })
+})