algorithm: binary lifting (#1218)

* Algorithm: BinaryLifting

* Update BinaryLifting.js

* made the requested changes

* added more comments

Graphs/BinaryLifting.js

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+/**
+ * Author: Adrito Mukherjee
+ * Binary Lifting implementation in Javascript
+ * Binary Lifting is a technique that is used to find the kth ancestor of a node in a rooted tree with N nodes
+ * The technique requires preprocessing the tree in O(N log N) using dynamic programming
+ * The techniqe can answer Q queries about kth ancestor of any node in O(Q log N)
+ * It is faster than the naive algorithm that answers Q queries with complexity O(Q K)
+ * It can be used to find Lowest Common Ancestor of two nodes in O(log N)
+ * Tutorial on Binary Lifting: https://codeforces.com/blog/entry/100826
+ */
+
+class BinaryLifting {
+  constructor (root, tree) {
+    this.root = root
+    this.connections = new Map()
+    this.up = new Map() // up[node][i] stores the 2^i-th  parent of node
+    for (const [i, j] of tree) {
+      this.addEdge(i, j)
+    }
+    this.log = Math.ceil(Math.log2(this.connections.size))
+    this.dfs(root, root)
+  }
+
+  addNode (node) {
+    // Function to add a node to the tree (connection represented by set)
+    this.connections.set(node, new Set())
+  }
+
+  addEdge (node1, node2) {
+    // Function to add an edge (adds the node too if they are not present in the tree)
+    if (!this.connections.has(node1)) {
+      this.addNode(node1)
+    }
+    if (!this.connections.has(node2)) {
+      this.addNode(node2)
+    }
+    this.connections.get(node1).add(node2)
+    this.connections.get(node2).add(node1)
+  }
+
+  dfs (node, parent) {
+    // The dfs function calculates 2^i-th ancestor of all nodes for i ranging from 0 to this.log
+    // We make use of the fact the two consecutive jumps of length 2^(i-1) make the total jump length 2^i
+    this.up.set(node, new Map())
+    this.up.get(node).set(0, parent)
+    for (let i = 1; i < this.log; i++) {
+      this.up
+        .get(node)
+        .set(i, this.up.get(this.up.get(node).get(i - 1)).get(i - 1))
+    }
+    for (const child of this.connections.get(node)) {
+      if (child !== parent) this.dfs(child, node)
+    }
+  }
+
+  kthAncestor (node, k) {
+    // if value of k is more than or equal to the number of total nodes, we return the root of the graph
+    if (k >= this.connections.size) {
+      return this.root
+    }
+    // if i-th bit is set in the binary representation of k, we jump from a node to its 2^i-th ancestor
+    // so after checking all bits of k, we will have made jumps of total length k, in just log k steps
+    for (let i = 0; i < this.log; i++) {
+      if (k & (1 << i)) {
+        node = this.up.get(node).get(i)
+      }
+    }
+    return node
+  }
+}
+
+function binaryLifting (root, tree, queries) {
+  const graphObject = new BinaryLifting(root, tree)
+  const ancestors = []
+  for (const [node, k] of queries) {
+    const ancestor = graphObject.kthAncestor(node, k)
+    ancestors.push(ancestor)
+  }
+  return ancestors
+}
+
+export default binaryLifting

Graphs/test/BinaryLifting.test.js

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+import binaryLifting from '../BinaryLifting'
+
+// The graph for Test Case 1 looks like this:
+//
+//         0
+//        /|\
+//       / | \
+//      1  3  5
+//     / \     \
+//    2   4     6
+//         \
+//          7
+//         / \
+//        11  8
+//             \
+//              9
+//               \
+//                10
+
+test('Test case 1', () => {
+  const root = 0
+  const graph = [
+    [0, 1],
+    [0, 3],
+    [0, 5],
+    [5, 6],
+    [1, 2],
+    [1, 4],
+    [4, 7],
+    [7, 11],
+    [7, 8],
+    [8, 9],
+    [9, 10]
+  ]
+  const queries = [
+    [2, 1],
+    [6, 1],
+    [7, 2],
+    [8, 2],
+    [10, 2],
+    [10, 3],
+    [10, 5],
+    [11, 3]
+  ]
+  const kthAncestors = binaryLifting(root, graph, queries)
+  expect(kthAncestors).toEqual([1, 5, 1, 4, 8, 7, 1, 1])
+})
+
+// The graph for Test Case 2 looks like this:
+//
+//         0
+//        / \
+//       1   2
+//      / \   \
+//     3   4   5
+//    /       / \
+//   6       7   8
+
+test('Test case 2', () => {
+  const root = 0
+  const graph = [
+    [0, 1],
+    [0, 2],
+    [1, 3],
+    [1, 4],
+    [2, 5],
+    [3, 6],
+    [5, 7],
+    [5, 8]
+  ]
+  const queries = [
+    [2, 1],
+    [3, 1],
+    [3, 2],
+    [6, 2],
+    [7, 3],
+    [8, 2],
+    [8, 3]
+  ]
+  const kthAncestors = binaryLifting(root, graph, queries)
+  expect(kthAncestors).toEqual([0, 1, 0, 1, 0, 2, 0])
+})