Tarjan’s off-line lowest common ancestors algorithm
Prerequisite : LCA basics, Disjoint Set Union by Rank and Path Compression
We are given a tree(can be extended to a DAG) and we have many queries of form LCA(u, v), i.e., find LCA of nodes ‘u’ and ‘v’.
We can perform those queries in O(N + QlogN) time using RMQ, where O(N) time for pre-processing and O(log N) for answering the queries, where
N = number of nodes and
Q = number of queries to be answered.
Can we do better than this? Can we do in linear(almost) time? Yes.
The article presents an offline algorithm which performs those queries in approximately O(N + Q) time. Although, this is not exactly linear, as there is an Inverse Ackermann function involved in the time complexity analysis. For more details on Inverse Ackermann function see this. Just as a summary, we can say that the Inverse Ackermann Function remains less than 4, for any value of input size that can be written in physical inverse. Thus, we consider this as almost linear.
We consider the input tree as shown below. We will Pre-Process the tree and fill two arrays- child[] and sibling[] according to the below explanation-
Let we want to process these queries- LCA(5,4), LCA(1,3), LCA(2,3)
Now, after pre-processing, we perform a LCA walk starting from the root of the tree(here- node ‘1’). But prior to the LCA walk, we colour all the nodes with WHITE. During the whole LCA walk, we use three disjoint set union functions- makeSet(), findSet(), unionSet().
These functions use the technique of union by rank and path compression to improve the running time. During the LCA walk, our queries gets processed and outputted (in a random order). After the LCA walk of the whole tree, all the nodes gets coloured BLACK.
Tarjan Offline LCA Algorithm steps from CLRS, Section-21-3, Pg 584, 2nd /3rd edition.
Note- The queries may not be processed in the original order. We can easily modify the process and sort them according to the input order.
The below pictures clearly depict all the steps happening. The red arrow shows the direction of travel of our recursive function LCA().
As, we can clearly see from the above pictures, the queries are processed in the following order, LCA(5,4), LCA(2,3), LCA(1,3) which is not in the same order as the input(LCA(5,4), LCA(1,3), LCA(2,3)).
Below is the implementation of the above approach:
C++
// A C++ Program to implement Tarjan Offline LCA Algorithm #include <bits/stdc++.h> #define V 5 // number of nodes in input tree #define WHITE 1 // COLOUR 'WHITE' is assigned value 1 #define BLACK 2 // COLOUR 'BLACK' is assigned value 2 /* A binary tree node has data, pointer to left child and a pointer to right child */ struct Node { int data; Node* left, *right; }; /* subset[i].parent-->Holds the parent of node-'i' subset[i].rank-->Holds the rank of node-'i' subset[i].ancestor-->Holds the LCA queries answers subset[i].child-->Holds one of the child of node-'i' if present, else -'0' subset[i].sibling-->Holds the right-sibling of node-'i' if present, else -'0' subset[i].color-->Holds the colour of node-'i' */ struct subset { int parent, rank, ancestor, child, sibling, color; }; // Structure to represent a query // A query consists of (L,R) and we will process the // queries offline a/c to Tarjan's offline LCA algorithm struct Query { int L, R; }; /* Helper function that allocates a new node with the given data and NULL left and right pointers. */ Node* newNode( int data) { Node* node = new Node; node->data = data; node->left = node->right = NULL; return (node); } //A utility function to make set void makeSet( struct subset subsets[], int i) { if (i < 1 || i > V) return ; subsets[i].color = WHITE; subsets[i].parent = i; subsets[i].rank = 0; return ; } // A utility function to find set of an element i // (uses path compression technique) int findSet( struct subset subsets[], int i) { // find root and make root as parent of i (path compression) if (subsets[i].parent != i) subsets[i].parent = findSet (subsets, subsets[i].parent); return subsets[i].parent; } // A function that does union of two sets of x and y // (uses union by rank) void unionSet( struct subset subsets[], int x, int y) { int xroot = findSet (subsets, x); int yroot = findSet (subsets, y); // Attach smaller rank tree under root of high rank tree // (Union by Rank) if (subsets[xroot].rank < subsets[yroot].rank) subsets[xroot].parent = yroot; else if (subsets[xroot].rank > subsets[yroot].rank) subsets[yroot].parent = xroot; // If ranks are same, then make one as root and increment // its rank by one else { subsets[yroot].parent = xroot; (subsets[xroot].rank)++; } } // The main function that prints LCAs. u is root's data. // m is size of q[] void lcaWalk( int u, struct Query q[], int m, struct subset subsets[]) { // Make Sets makeSet(subsets, u); // Initially, each node's ancestor is the node // itself. subsets[findSet(subsets, u)].ancestor = u; int child = subsets[u].child; // This while loop doesn't run for more than 2 times // as there can be at max. two children of a node while (child != 0) { lcaWalk(child, q, m, subsets); unionSet (subsets, u, child); subsets[findSet(subsets, u)].ancestor = u; child = subsets[child].sibling; } subsets[u].color = BLACK; for ( int i = 0; i < m; i++) { if (q[i].L == u) { if (subsets[q[i].R].color == BLACK) { printf ( "LCA(%d %d) -> %d\n" , q[i].L, q[i].R, subsets[findSet(subsets,q[i].R)].ancestor); } } else if (q[i].R == u) { if (subsets[q[i].L].color == BLACK) { printf ( "LCA(%d %d) -> %d\n" , q[i].L, q[i].R, subsets[findSet(subsets,q[i].L)].ancestor); } } } return ; } // This is basically an inorder traversal and // we preprocess the arrays-> child[] // and sibling[] in "struct subset" with // the tree structure using this function. void preprocess(Node * node, struct subset subsets[]) { if (node == NULL) return ; // Recur on left child preprocess(node->left, subsets); if (node->left != NULL&&node->right != NULL) { /* Note that the below two lines can also be this- subsets[node->data].child = node->right->data; subsets[node->right->data].sibling = node->left->data; This is because if both left and right children of node-'i' are present then we can store any of them in subsets[i].child and correspondingly its sibling*/ subsets[node->data].child = node->left->data; subsets[node->left->data].sibling = node->right->data; } else if ((node->left != NULL && node->right == NULL) || (node->left == NULL && node->right != NULL)) { if (node->left != NULL && node->right == NULL) subsets[node->data].child = node->left->data; else subsets[node->data].child = node->right->data; } //Recur on right child preprocess (node->right, subsets); } // A function to initialise prior to pre-processing and // LCA walk void initialise( struct subset subsets[]) { // Initialising the structure with 0's memset (subsets, 0, (V+1) * sizeof ( struct subset)); // We colour all nodes WHITE before LCA Walk. for ( int i=1; i<=V; i++) subsets[i].color=WHITE; return ; } // Prints LCAs for given queries q[0..m-1] in a tree // with given root void printLCAs(Node *root, Query q[], int m) { // Allocate memory for V subsets and nodes struct subset * subsets = new subset[V+1]; // Creates subsets and colors them WHITE initialise(subsets); // Preprocess the tree preprocess(root, subsets); // Perform a tree walk to process the LCA queries // offline lcaWalk(root->data , q, m, subsets); } // Driver program to test above functions int main() { /* We construct a binary tree :- 1 / \ 2 3 / \ 4 5 */ Node *root = newNode(1); root->left = newNode(2); root->right = newNode(3); root->left->left = newNode(4); root->left->right = newNode(5); // LCA Queries to answer Query q[] = {{5, 4}, {1, 3}, {2, 3}}; int m = sizeof (q)/ sizeof (q[0]); printLCAs(root, q, m); return 0; } |
Java
// A Java Program to implement Tarjan Offline LCA Algorithm import java.util.Arrays; class GFG { static final int V = 5 ; // number of nodes in input tree static final int WHITE = 1 ; // COLOUR 'WHITE' is assigned value 1 static final int BLACK = 2 ; // COLOUR 'BLACK' is assigned value 2 /* A binary tree node has data, pointer to left child and a pointer to right child */ static class Node { int data; Node left, right; }; /* subset[i].parent-.Holds the parent of node-'i' subset[i].rank-.Holds the rank of node-'i' subset[i].ancestor-.Holds the LCA queries answers subset[i].child-.Holds one of the child of node-'i' if present, else -'0' subset[i].sibling-.Holds the right-sibling of node-'i' if present, else -'0' subset[i].color-.Holds the colour of node-'i' */ static class subset { int parent; int rank; int ancestor; int child; int sibling; int color; }; // Structure to represent a query // A query consists of (L,R) and we will process the // queries offline a/c to Tarjan's offline LCA algorithm static class Query { int L, R; Query( int L, int R) { this .L = L; this .R = R; } }; /* Helper function that allocates a new node with the given data and null left and right pointers. */ static Node newNode( int data) { Node node = new Node(); node.data = data; node.left = node.right = null ; return (node); } // A utility function to make set static void makeSet(subset subsets[], int i) { if (i < 1 || i > V) return ; subsets[i].color = WHITE; subsets[i].parent = i; subsets[i].rank = 0 ; return ; } // A utility function to find set of an element i // (uses path compression technique) static int findSet(subset subsets[], int i) { // find root and make root as parent of i (path compression) if (subsets[i].parent != i) subsets[i].parent = findSet (subsets, subsets[i].parent); return subsets[i].parent; } // A function that does union of two sets of x and y // (uses union by rank) static void unionSet(subset subsets[], int x, int y) { int xroot = findSet (subsets, x); int yroot = findSet (subsets, y); // Attach smaller rank tree under root of high rank tree // (Union by Rank) if (subsets[xroot].rank < subsets[yroot].rank) subsets[xroot].parent = yroot; else if (subsets[xroot].rank > subsets[yroot].rank) subsets[yroot].parent = xroot; // If ranks are same, then make one as root and increment // its rank by one else { subsets[yroot].parent = xroot; (subsets[xroot].rank)++; } } // The main function that prints LCAs. u is root's data. // m is size of q[] static void lcaWalk( int u, Query q[], int m, subset subsets[]) { // Make Sets makeSet(subsets, u); // Initially, each node's ancestor is the node // itself. subsets[findSet(subsets, u)].ancestor = u; int child = subsets[u].child; // This while loop doesn't run for more than 2 times // as there can be at max. two children of a node while (child != 0 ) { lcaWalk(child, q, m, subsets); unionSet (subsets, u, child); subsets[findSet(subsets, u)].ancestor = u; child = subsets[child].sibling; } subsets[u].color = BLACK; for ( int i = 0 ; i < m; i++) { if (q[i].L == u) { if (subsets[q[i].R].color == BLACK) { System.out.printf( "LCA(%d %d)->%d\n" , q[i].L, q[i].R, subsets[findSet(subsets,q[i].R)].ancestor); } } else if (q[i].R == u) { if (subsets[q[i].L].color == BLACK) { System.out.printf( "LCA(%d %d)->%d\n" , q[i].L, q[i].R, subsets[findSet(subsets,q[i].L)].ancestor); } } } return ; } // This is basically an inorder traversal and // we preprocess the arrays. child[] // and sibling[] in "subset" with // the tree structure using this function. static void preprocess(Node node, subset subsets[]) { if (node == null ) return ; // Recur on left child preprocess(node.left, subsets); if (node.left != null && node.right != null ) { /* Note that the below two lines can also be this- subsets[node.data].child = node.right.data; subsets[node.right.data].sibling = node.left.data; This is because if both left and right children of node-'i' are present then we can store any of them in subsets[i].child and correspondingly its sibling*/ subsets[node.data].child = node.left.data; subsets[node.left.data].sibling = node.right.data; } else if ((node.left != null && node.right == null ) || (node.left == null && node.right != null )) { if (node.left != null && node.right == null ) subsets[node.data].child = node.left.data; else subsets[node.data].child = node.right.data; } // Recur on right child preprocess (node.right, subsets); } // A function to initialise prior to pre-processing and // LCA walk static void initialise(subset subsets[]) { // We colour all nodes WHITE before LCA Walk. for ( int i = 1 ; i < subsets.length; i++) { subsets[i] = new subset(); subsets[i].color = WHITE; } return ; } // Prints LCAs for given queries q[0..m-1] in a tree // with given root static void printLCAs(Node root, Query q[], int m) { // Allocate memory for V subsets and nodes subset []subsets = new subset[V + 1 ]; // Creates subsets and colors them WHITE initialise(subsets); // Preprocess the tree preprocess(root, subsets); // Perform a tree walk to process the LCA queries // offline lcaWalk(root.data , q, m, subsets); } // Driver code public static void main(String[] args) { /* We construct a binary tree :- 1 / \ 2 3 / \ 4 5 */ Node root = newNode( 1 ); root.left = newNode( 2 ); root.right = newNode( 3 ); root.left.left = newNode( 4 ); root.left.right = newNode( 5 ); // LCA Queries to answer Query q[] = new Query[ 3 ]; q[ 0 ] = new Query( 5 , 4 ); q[ 1 ] = new Query( 1 , 3 ); q[ 2 ] = new Query( 2 , 3 ); int m = q.length; printLCAs(root, q, m); } } // This code is contributed by gauravrajput1 |
Python3
# A Python3 program to implement Tarjan # Offline LCA Algorithm # Number of nodes in input tree V = 5 # COLOUR 'WHITE' is assigned value 1 WHITE = 1 # COLOUR 'BLACK' is assigned value 2 BLACK = 2 # A binary tree node has data, pointer # to left child and a pointer to right child class Node: def __init__( self ): self .data = 0 self .left = None self .right = None ''' subset[i].parent-.Holds the parent of node-'i' subset[i].rank-.Holds the rank of node-'i' subset[i].ancestor-.Holds the LCA queries answers subset[i].child-.Holds one of the child of node-'i' if present, else -'0' subset[i].sibling-.Holds the right-sibling of node-'i' if present, else -'0' subset[i].color-.Holds the colour of node-'i' ''' class subset: def __init__( self ): self .parent = 0 self .rank = 0 self .ancestor = 0 self .child = 0 self .sibling = 0 self .color = 0 # Structure to represent a query # A query consists of (L,R) and we # will process the queries offline # a/c to Tarjan's offline LCA algorithm class Query: def __init__( self , L, R): self .L = L self .R = R # Helper function that allocates a new node # with the given data and None left and # right pointers. def newNode(data): node = Node() node.data = data node.left = node.right = None return (node) # A utility function to make set def makeSet(subsets, i): if (i < 1 or i > V): return subsets[i].color = WHITE subsets[i].parent = i subsets[i].rank = 0 return # A utility function to find set of an element i # (uses path compression technique) def findSet(subsets, i): # Find root and make root as parent # of i (path compression) if (subsets[i].parent ! = i): subsets[i].parent = findSet(subsets, subsets[i].parent) return subsets[i].parent # A function that does union of two sets # of x and y (uses union by rank) def unionSet(subsets, x, y): xroot = findSet(subsets, x) yroot = findSet(subsets, y) # Attach smaller rank tree under root of # high rank tree (Union by Rank) if (subsets[xroot].rank < subsets[yroot].rank): subsets[xroot].parent = yroot else if (subsets[xroot].rank > subsets[yroot].rank): subsets[yroot].parent = xroot # If ranks are same, then make one as root # and increment its rank by one else : subsets[yroot].parent = xroot (subsets[xroot].rank) + = 1 # The main function that prints LCAs. u is # root's data. m is size of q[] def lcaWalk(u, q, m, subsets): # Make Sets makeSet(subsets, u) # Initially, each node's ancestor is the node # itself. subsets[findSet(subsets, u)].ancestor = u child = subsets[u].child # This while loop doesn't run for more than 2 times # as there can be at max. two children of a node while (child ! = 0 ): lcaWalk(child, q, m, subsets) unionSet(subsets, u, child) subsets[findSet(subsets, u)].ancestor = u child = subsets[child].sibling subsets[u].color = BLACK for i in range (m): if (q[i].L = = u): if (subsets[q[i].R].color = = BLACK): print ( "LCA(%d %d) -> %d" % (q[i].L, q[i].R, subsets[findSet(subsets, q[i].R)].ancestor)) else if (q[i].R = = u): if (subsets[q[i].L].color = = BLACK): print ( "LCA(%d %d) -> %d" % (q[i].L, q[i].R, subsets[findSet(subsets, q[i].L)].ancestor)) return # This is basically an inorder traversal and # we preprocess the arrays. child[] # and sibling[] in "struct subset" with # the tree structure using this function. def preprocess(node, subsets): if (node = = None ): return # Recur on left child preprocess(node.left, subsets) if (node.left ! = None and node.right ! = None ): ''' Note that the below two lines can also be this- subsets[node.data].child = node.right.data; subsets[node.right.data].sibling = node.left.data; This is because if both left and right children of node-'i' are present then we can store any of them in subsets[i].child and correspondingly its sibling''' subsets[node.data].child = node.left.data subsets[node.left.data].sibling = node.right.data else if ((node.left ! = None and node.right = = None ) or (node.left = = None and node.right ! = None )): if (node.left ! = None and node.right = = None ): subsets[node.data].child = node.left.data else : subsets[node.data].child = node.right.data # Recur on right child preprocess(node.right, subsets) # A function to initialise prior to pre-processing and # LCA walk def initialise(subsets): # Initialising the structure with 0's # memset(subsets, 0, (V+1) * sizeof(struct subset)); # We colour all nodes WHITE before LCA Walk. for i in range ( 1 , V + 1 ): subsets[i].color = WHITE return # Prints LCAs for given queries q[0..m-1] in a tree # with given root def printLCAs(root, q, m): # Allocate memory for V subsets and nodes subsets = [subset() for _ in range (V + 1 )] # Creates subsets and colors them WHITE initialise(subsets) # Preprocess the tree preprocess(root, subsets) # Perform a tree walk to process the LCA queries # offline lcaWalk(root.data, q, m, subsets) # Driver code if __name__ = = "__main__" : ''' We construct a binary tree :- 1 / \ 2 3 / \ 4 5 ''' root = newNode( 1 ) root.left = newNode( 2 ) root.right = newNode( 3 ) root.left.left = newNode( 4 ) root.left.right = newNode( 5 ) # LCA Queries to answer q = [Query( 5 , 4 ), Query( 1 , 3 ), Query( 2 , 3 )] m = len (q) printLCAs(root, q, m) # This code is contributed by sanjeev2552 |
C#
// A C# Program to implement Tarjan Offline LCA Algorithm using System; public class GFG { static readonly int V = 5; // number of nodes in input tree static readonly int WHITE = 1; // COLOUR 'WHITE' is assigned value 1 static readonly int BLACK = 2; // COLOUR 'BLACK' is assigned value 2 /* A binary tree node has data, pointer to left child and a pointer to right child */ public class Node { public int data; public Node left, right; }; /* subset[i].parent-.Holds the parent of node-'i' subset[i].rank-.Holds the rank of node-'i' subset[i].ancestor-.Holds the LCA queries answers subset[i].child-.Holds one of the child of node-'i' if present, else -'0' subset[i].sibling-.Holds the right-sibling of node-'i' if present, else -'0' subset[i].color-.Holds the colour of node-'i' */ public class subset { public int parent; public int rank; public int ancestor; public int child; public int sibling; public int color; }; // Structure to represent a query // A query consists of (L,R) and we will process the // queries offline a/c to Tarjan's offline LCA algorithm public class Query { public int L, R; public Query( int L, int R) { this .L = L; this .R = R; } }; /* Helper function that allocates a new node with the given data and null left and right pointers. */ static Node newNode( int data) { Node node = new Node(); node.data = data; node.left = node.right = null ; return (node); } // A utility function to make set static void makeSet(subset []subsets, int i) { if (i < 1 || i > V) return ; subsets[i].color = WHITE; subsets[i].parent = i; subsets[i].rank = 0; return ; } // A utility function to find set of an element i // (uses path compression technique) static int findSet(subset []subsets, int i) { // find root and make root as parent of i (path compression) if (subsets[i].parent != i) subsets[i].parent = findSet (subsets, subsets[i].parent); return subsets[i].parent; } // A function that does union of two sets of x and y // (uses union by rank) static void unionSet(subset []subsets, int x, int y) { int xroot = findSet (subsets, x); int yroot = findSet (subsets, y); // Attach smaller rank tree under root of high rank tree // (Union by Rank) if (subsets[xroot].rank < subsets[yroot].rank) subsets[xroot].parent = yroot; else if (subsets[xroot].rank > subsets[yroot].rank) subsets[yroot].parent = xroot; // If ranks are same, then make one as root and increment // its rank by one else { subsets[yroot].parent = xroot; (subsets[xroot].rank)++; } } // The main function that prints LCAs. u is root's data. // m is size of q[] static void lcaWalk( int u, Query []q, int m, subset []subsets) { // Make Sets makeSet(subsets, u); // Initially, each node's ancestor is the node // itself. subsets[findSet(subsets, u)].ancestor = u; int child = subsets[u].child; // This while loop doesn't run for more than 2 times // as there can be at max. two children of a node while (child != 0) { lcaWalk(child, q, m, subsets); unionSet (subsets, u, child); subsets[findSet(subsets, u)].ancestor = u; child = subsets[child].sibling; } subsets[u].color = BLACK; for ( int i = 0; i < m; i++) { if (q[i].L == u) { if (subsets[q[i].R].color == BLACK) { Console.WriteLine( "LCA(" + q[i].L + " " + q[i].R+ ") -> " + subsets[findSet(subsets, q[i].R)].ancestor); } } else if (q[i].R == u) { if (subsets[q[i].L].color == BLACK) { Console.WriteLine( "LCA(" + q[i].L + " " + q[i].R + ") -> " + subsets[findSet(subsets, q[i].L)].ancestor); } } } return ; } // This is basically an inorder traversal and // we preprocess the arrays. child[] // and sibling[] in "subset" with // the tree structure using this function. static void preprocess(Node node, subset []subsets) { if (node == null ) return ; // Recur on left child preprocess(node.left, subsets); if (node.left != null && node.right != null ) { /* Note that the below two lines can also be this- subsets[node.data].child = node.right.data; subsets[node.right.data].sibling = node.left.data; This is because if both left and right children of node-'i' are present then we can store any of them in subsets[i].child and correspondingly its sibling*/ subsets[node.data].child = node.left.data; subsets[node.left.data].sibling = node.right.data; } else if ((node.left != null && node.right == null ) || (node.left == null && node.right != null )) { if (node.left != null && node.right == null ) subsets[node.data].child = node.left.data; else subsets[node.data].child = node.right.data; } // Recur on right child preprocess (node.right, subsets); } // A function to initialise prior to pre-processing and // LCA walk static void initialise(subset []subsets) { // We colour all nodes WHITE before LCA Walk. for ( int i = 1; i < subsets.Length; i++) { subsets[i] = new subset(); subsets[i].color = WHITE; } return ; } // Prints LCAs for given queries q[0..m-1] in a tree // with given root static void printLCAs(Node root, Query []q, int m) { // Allocate memory for V subsets and nodes subset []subsets = new subset[V + 1]; // Creates subsets and colors them WHITE initialise(subsets); // Preprocess the tree preprocess(root, subsets); // Perform a tree walk to process the LCA queries // offline lcaWalk(root.data, q, m, subsets); } // Driver code public static void Main(String[] args) { /* We construct a binary tree :- 1 / \ 2 3 / \ 4 5 */ Node root = newNode(1); root.left = newNode(2); root.right = newNode(3); root.left.left = newNode(4); root.left.right = newNode(5); // LCA Queries to answer Query []q = new Query[3]; q[0] = new Query(5, 4); q[1] = new Query(1, 3); q[2] = new Query(2, 3); int m = q.Length; printLCAs(root, q, m); } } // This code is contributed by Rajput-Ji |
Javascript
<script> // A Javascript Program to implement Tarjan Offline LCA Algorithm let V = 5; // number of nodes in input tree let WHITE = 1; // COLOUR 'WHITE' is assigned value 1 let BLACK = 2; // COLOUR 'BLACK' is assigned value 2 /* A binary tree node has data, pointer to left child and a pointer to right child */ class Node { constructor(data) { this .data = data; this .left = this .right = null ; } } /* subset[i].parent-.Holds the parent of node-'i' subset[i].rank-.Holds the rank of node-'i' subset[i].ancestor-.Holds the LCA queries answers subset[i].child-.Holds one of the child of node-'i' if present, else -'0' subset[i].sibling-.Holds the right-sibling of node-'i' if present, else -'0' subset[i].color-.Holds the colour of node-'i' */ class subset { constructor() { this .parent = 0; this .rank = 0; this .ancestor = 0; this .child = 0; this .sibling = 0; this .color = 0; } } // Structure to represent a query // A query consists of (L,R) and we will process the // queries offline a/c to Tarjan's offline LCA algorithm class Query { constructor(L, R) { this .L = L; this .R = R; } } /* Helper function that allocates a new node with the given data and null left and right pointers. */ function newNode(data) { let node = new Node(); node.data = data; node.left = node.right = null ; return (node); } // A utility function to make set function makeSet(subsets,i) { if (i < 1 || i > V) return ; subsets[i].color = WHITE; subsets[i].parent = i; subsets[i].rank = 0; return ; } // A utility function to find set of an element i // (uses path compression technique) function findSet(subsets, i) { // find root and make root as parent of i (path compression) if (subsets[i].parent != i) subsets[i].parent = findSet (subsets, subsets[i].parent); return subsets[i].parent; } // A function that does union of two sets of x and y // (uses union by rank) function unionSet(subsets, x, y) { let xroot = findSet (subsets, x); let yroot = findSet (subsets, y); // Attach smaller rank tree under root of high rank tree // (Union by Rank) if (subsets[xroot].rank < subsets[yroot].rank) subsets[xroot].parent = yroot; else if (subsets[xroot].rank > subsets[yroot].rank) subsets[yroot].parent = xroot; // If ranks are same, then make one as root and increment // its rank by one else { subsets[yroot].parent = xroot; (subsets[xroot].rank)++; } } // The main function that prints LCAs. u is root's data. // m is size of q[] function lcaWalk(u,q,m,subsets) { // Make Sets makeSet(subsets, u); // Initially, each node's ancestor is the node // itself. subsets[findSet(subsets, u)].ancestor = u; let child = subsets[u].child; // This while loop doesn't run for more than 2 times // as there can be at max. two children of a node while (child != 0) { lcaWalk(child, q, m, subsets); unionSet (subsets, u, child); subsets[findSet(subsets, u)].ancestor = u; child = subsets[child].sibling; } subsets[u].color = BLACK; for (let i = 0; i < m; i++) { if (q[i].L == u) { if (subsets[q[i].R].color == BLACK) { document.write( "LCA(" + q[i].L+ " " +q[i].R+ ") -> " ,subsets[findSet(subsets,q[i].R)].ancestor+ "<br>" ); } } else if (q[i].R == u) { if (subsets[q[i].L].color == BLACK) { document.write( "LCA(" + q[i].L+ " " +q[i].R+ ") -> " ,subsets[findSet(subsets,q[i].L)].ancestor+ "<br>" ); } } } return ; } // This is basically an inorder traversal and // we preprocess the arrays. child[] // and sibling[] in "subset" with // the tree structure using this function. function preprocess(node,subsets) { if (node == null ) return ; // Recur on left child preprocess(node.left, subsets); if (node.left != null && node.right != null ) { /* Note that the below two lines can also be this- subsets[node.data].child = node.right.data; subsets[node.right.data].sibling = node.left.data; This is because if both left and right children of node-'i' are present then we can store any of them in subsets[i].child and correspondingly its sibling*/ subsets[node.data].child = node.left.data; subsets[node.left.data].sibling = node.right.data; } else if ((node.left != null && node.right == null ) || (node.left == null && node.right != null )) { if (node.left != null && node.right == null ) subsets[node.data].child = node.left.data; else subsets[node.data].child = node.right.data; } // Recur on right child preprocess (node.right, subsets); } // A function to initialise prior to pre-processing and // LCA walk function initialise(subsets) { // We colour all nodes WHITE before LCA Walk. for (let i = 1; i < subsets.length; i++) { subsets[i] = new subset(); subsets[i].color = WHITE; } return ; } // Prints LCAs for given queries q[0..m-1] in a tree // with given root function printLCAs(root, q, m) { // Allocate memory for V subsets and nodes let subsets = new Array(V + 1); // Creates subsets and colors them WHITE initialise(subsets); // Preprocess the tree preprocess(root, subsets); // Perform a tree walk to process the LCA queries // offline lcaWalk(root.data , q, m, subsets); } // Driver code /* We construct a binary tree :- 1 / \ 2 3 / \ 4 5 */ let root = newNode(1); root.left = newNode(2); root.right = newNode(3); root.left.left = newNode(4); root.left.right = newNode(5); // LCA Queries to answer let q = new Array(3); q[0] = new Query(5, 4); q[1] = new Query(1, 3); q[2] = new Query(2, 3); let m = q.length; printLCAs(root, q, m); // This code is contributed by patel2127 </script> |
Output :
LCA(5 4) -> 2 LCA(2 3) -> 1 LCA(1 3) -> 1
Time Complexity : Super-linear, i.e- barely slower than linear. O(N + Q) time, where O(N) time for pre-processing and almost O(1) time for answering the queries.
Auxiliary Space : We use a many arrays- parent[], rank[], ancestor[] which are used in Disjoint Set Union Operations each with the size equal to the number of nodes. We also use the arrays- child[], sibling[], color[] which are useful in this offline algorithm. Hence, we use O(N).
For convenience, all these arrays are put up in a structure- struct subset to hold these arrays.
References
https://en.wikipedia.org/wiki/Tarjan%27s_off-line_lowest_common_ancestors_algorithm
CLRS, Section-21-3, Pg 584, 2nd /3rd edition
http://wcipeg.com/wiki/Lowest_common_ancestor#Offline