feat: add tarjan algorithm for strongly connected components

graph/tarjan.ts

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+/**
+ * @function tarjan
+ * @description Given a graph, find the strongly connected components(SCC) in reverse topological order. A set of nodes form a SCC if there is a path between all pairs of points within that set.
+ * @Complexity_Analysis
+ * Time complexity: O(V + E). We perform a DFS of (V + E)
+ * Space Complexity: O(V). We hold numerous structures all of which at worst holds O(V) nodes.
+ * @param {[number, number][][]} graph - The graph in adjacency list form
+ * @return {number[][]} - An array of SCCs, where an SCC is an array with the indices of each node within that SCC. The order of SCCs in the array are in reverse topological order.
+ * @see https://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm
+ */
+export const tarjan = (graph: number[][]): number[][] => {
+  if (graph.length === 0) {
+    return [];
+  }
+
+  let index = 0;
+  // The order in which we discover nodes
+  let discovery: number[] = Array(graph.length);
+  // For each node, holds the furthest ancestor it can reach
+  let low: number[] = Array(graph.length).fill(undefined);
+  // Holds the nodes we have visited in a DFS traversal and are considering to group into a SCC
+  let stack: number[] = [];
+  // Holds the elements in the stack.
+  let stackContains = Array(graph.length).fill(false);
+  let sccs: number[][] = [];
+
+  const dfs = (node: number) => {
+    discovery[node] = index;
+    low[node] = index;
+    ++index;
+    stack.push(node);
+    stackContains[node] = true;
+
+    for (const child of graph[node]) {
+      if (low[child] === undefined) {
+        dfs(child);
+        if (low[child] < low[node]) {
+          // Child node loops back to this node's ancestor. Update the low node.
+          low[node] = low[child];
+        }
+      } else if (stackContains[child] && low[node] > discovery[child]) {
+        // Found a backedge. Update the low for this node if needed.
+        low[node] = discovery[child];
+      }
+    }
+
+    if (discovery[node] == low[node]) {
+      // node is the root of a SCC. Gather the SCC's nodes from the stack.
+      let scc: number[] = [];
+      let i;
+      for (i = stack.length - 1; stack[i] != node; --i) {
+        scc.push(stack[i]);
+        stackContains[stack[i]] = false;
+        stack.pop();
+      }
+      scc.push(stack[i]);
+      stackContains[stack[i]] = false;
+      stack.pop();
+      sccs.push(scc);
+    }
+  }
+
+  for (let i = 0; i < graph.length; ++i) {
+    if (low[i] === undefined) {
+      dfs(i);
+    }
+  }
+  return sccs;
+}
+

graph/test/tarjan.test.ts

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+import { tarjan } from "../tarjan";
+
+describe("tarjan", () => {
+
+  it("it should return no sccs for empty graph", () => {
+    expect(tarjan([])).toStrictEqual([]);
+  });
+
+  it("it should return one scc for graph with one element", () => {
+    expect(tarjan([[]])).toStrictEqual([[0]]);
+  });
+
+  it("it should return one scc for graph with element that points to itself", () => {
+    expect(tarjan([[0]])).toStrictEqual([[0]]);
+  });
+
+  it("it should return one scc for two element graph with cycle", () => {
+    expect(tarjan([[1], [0]])).toStrictEqual([[1, 0]]);
+  });
+
+  it("should return one scc for each element for straight line", () => {
+    expect(tarjan([[1], [2], [3], []])).toStrictEqual([[3], [2], [1], [0]]);
+  });
+
+  it("should return sccs for straight line with backedge in middle", () => {
+    expect(tarjan([[1], [2], [3, 0], []])).toStrictEqual([[3], [2, 1, 0]]);
+  });
+
+  it("should return sccs for straight line with backedge from end to middle", () => {
+    expect(tarjan([[1], [2], [3], [1]])).toStrictEqual([[3, 2, 1], [0]]);
+  });
+
+  it("should return scc for each element for graph with no edges", () => {
+    expect(tarjan([[], [], [], []])).toStrictEqual([[0], [1], [2], [3]]);
+  });
+
+  it("should return sccs disconnected graph", () => {
+    expect(tarjan([[1, 2], [0, 2], [0, 1], []])).toStrictEqual([[2, 1, 0], [3]]);
+  });
+
+  it("should return sccs disconnected graph", () => {
+    expect(tarjan([[1, 2], [0, 2], [0, 1], [4], [5], [3]])).toStrictEqual([[2, 1, 0], [5, 4, 3]]);
+  });
+
+  it("should return single scc", () => {
+    expect(tarjan([[1], [2], [3], [0, 4], [3]])).toStrictEqual([[4, 3, 2, 1, 0]]);
+  });
+
+  it("should return one scc for complete connected graph", () => {
+    const input = [[1, 2, 3, 4], [0, 2, 3, 4], [0, 1, 3, 4], [0, 1, 2, 4], [0, 1, 2, 3]];
+    expect(tarjan(input)).toStrictEqual([[4, 3, 2, 1, 0]]);
+  });
+
+  it("should return sccs", () => {
+    const input = [[1], [2], [0, 3], [4], []];
+    expect(tarjan(input)).toStrictEqual([[4], [3], [2, 1, 0]]);
+  });
+
+  it("should return sccs", () => {
+    const input = [[1], [2], [0, 3, 4], [0], [5], [6, 7], [2, 4], [8], [5, 9], [5]];
+    const expected = [[9, 8, 7, 6, 5, 4, 3, 2, 1, 0]];
+    expect(tarjan(input)).toStrictEqual(expected);
+  });
+
+  it("should return sccs", () => {
+    const input = [[1], [0, 2], [0, 3], [4], [5, 7], [6], [4, 7], []];
+    const expected = [[7], [6, 5, 4], [3], [2, 1, 0]];
+    expect(tarjan(input)).toStrictEqual(expected);
+  });
+
+  it("should return sccs where first scc cannot reach second scc", () => {
+    const input = [[1], [2], [0], [4], [5], [2, 3]];
+    const expected = [[2, 1, 0], [5, 4, 3]];
+    expect(tarjan(input)).toStrictEqual(expected);
+  });
+
+})
+