Given a Binary Tree, find size of the Largest Independent Set(LIS) in it. A subset of all tree nodes is an independent set if there is no edge between any two nodes of the subset.
For example, consider the following binary tree. The largest independent set(LIS) is {10, 40, 60, 70, 80} and size of the LIS is 5.
A Dynamic Programming solution solves a given problem using solutions of subproblems in bottom up manner. Can the given problem be solved using solutions to subproblems? If yes, then what are the subproblems? Can we find largest independent set size (LISS) for a node X if we know LISS for all descendants of X? If a node is considered as part of LIS, then its children cannot be part of LIS, but its grandchildren can be. Following is optimal substructure property.
1) Optimal Substructure:
Let LISS(X) indicates size of largest independent set of a tree with root X.
LISS(X) = MAX { (1 + sum of LISS for all grandchildren of X),
(sum of LISS for all children of X) }
The idea is simple, there are two possibilities for every node X, either X is a member of the set or not a member. If X is a member, then the value of LISS(X) is 1 plus LISS of all grandchildren. If X is not a member, then the value is sum of LISS of all children.
2) Overlapping Subproblems
Following is recursive implementation that simply follows the recursive structure mentioned above.
C++
// A naive recursive implementation of// Largest Independent Set problem #include <bits/stdc++.h>using namespace std; // A utility function to find // max of two integers int max(int x, int y) { return (x > y) ? x : y; } /* A binary tree node has data, pointer to left child and a pointer to right child */class node { public: int data; node *left, *right; }; // The function returns size of the // largest independent set in a given // binary tree int LISS(node *root) { if (root == NULL) return 0; // Calculate size excluding the current node int size_excl = LISS(root->left) + LISS(root->right); // Calculate size including the current node int size_incl = 1; if (root->left) size_incl += LISS(root->left->left) + LISS(root->left->right); if (root->right) size_incl += LISS(root->right->left) + LISS(root->right->right); // Return the maximum of two sizes return max(size_incl, size_excl); } // A utility function to create a node node* newNode( int data ) { node* temp = new node(); temp->data = data; temp->left = temp->right = NULL; return temp; } // Driver Codeint main() { // Let us construct the tree // given in the above diagram node *root = newNode(20); root->left = newNode(8); root->left->left = newNode(4); root->left->right = newNode(12); root->left->right->left = newNode(10); root->left->right->right = newNode(14); root->right = newNode(22); root->right->right = newNode(25); cout << "Size of the Largest" << " Independent Set is " << LISS(root); return 0; } // This is code is contributed // by rathbhupendra |
C
// A naive recursive implementation of Largest Independent Set problem#include <stdio.h>#include <stdlib.h>// A utility function to find max of two integersint max(int x, int y) { return (x > y)? x: y; }/* A binary tree node has data, pointer to left child and a pointer to right child */struct node{ int data; struct node *left, *right;};// The function returns size of the largest independent set in a given // binary treeint LISS(struct node *root){ if (root == NULL) return 0; // Calculate size excluding the current node int size_excl = LISS(root->left) + LISS(root->right); // Calculate size including the current node int size_incl = 1; if (root->left) size_incl += LISS(root->left->left) + LISS(root->left->right); if (root->right) size_incl += LISS(root->right->left) + LISS(root->right->right); // Return the maximum of two sizes return max(size_incl, size_excl);}// A utility function to create a nodestruct node* newNode( int data ){ struct node* temp = (struct node *) malloc( sizeof(struct node) ); temp->data = data; temp->left = temp->right = NULL; return temp;}// Driver program to test above functionsint main(){ // Let us construct the tree given in the above diagram struct node *root = newNode(20); root->left = newNode(8); root->left->left = newNode(4); root->left->right = newNode(12); root->left->right->left = newNode(10); root->left->right->right = newNode(14); root->right = newNode(22); root->right->right = newNode(25); printf ("Size of the Largest Independent Set is %d ", LISS(root)); return 0;} |
Java
// A naive recursive implementation of// Largest Independent Set problemimport java.io.*;class GFG { // A utility function to find // max of two integers static int max(int x, int y) { return (x > y) ? x : y; } /* A binary tree node has data, pointer to left child and a pointer to right child */ static class Node { int data; Node left, right; }; // The function returns size of the // largest independent set in a given // binary tree static int LISS(Node root) { if (root == null) return 0; // Calculate size excluding the current node int size_excl = LISS(root.left) + LISS(root.right); // Calculate size including the current node int size_incl = 1; if (root.left != null) size_incl += LISS(root.left.left) + LISS(root.left.right); if (root.right != null) size_incl += LISS(root.right.left) + LISS(root.right.right); // Return the maximum of two sizes return max(size_incl, size_excl); } // A utility function to create a node static Node newNode(int data) { Node temp = new Node(); temp.data = data; temp.left = temp.right = null; return temp; } // Driver Code public static void main(String args[]) { // Let us construct the tree // given in the above diagram Node root = newNode(20); root.left = newNode(8); root.left.left = newNode(4); root.left.right = newNode(12); root.left.right.left = newNode(10); root.left.right.right = newNode(14); root.right = newNode(22); root.right.right = newNode(25); System.out.println("Size of the Largest" + " Independent Set is " + LISS(root)); }}// This code has been contributed by 29AjayKumar |
Python3
# A naive recursive implementation of# Largest Independent Set problem # A utility function to find # max of two integers def max(x, y): if(x > y): return x else: return y# A binary tree node has data, #pointer to left child and a #pointer to right childclass node : def __init__(self): self.data = 0 self.left = self.right = None# The function returns size of the # largest independent set in a given # binary tree def LISS(root): if (root == None) : return 0 # Calculate size excluding the current node size_excl = LISS(root.left) + LISS(root.right) # Calculate size including the current node size_incl = 1 if (root.left != None): size_incl += LISS(root.left.left) + \ LISS(root.left.right) if (root.right != None): size_incl += LISS(root.right.left) + \ LISS(root.right.right) # Return the maximum of two sizes return max(size_incl, size_excl) # A utility function to create a node def newNode( data ) : temp = node() temp.data = data temp.left = temp.right = None return temp # Driver Code# Let us construct the tree # given in the above diagram root = newNode(20) root.left = newNode(8) root.left.left = newNode(4) root.left.right = newNode(12) root.left.right.left = newNode(10) root.left.right.right = newNode(14) root.right = newNode(22) root.right.right = newNode(25) print( "Size of the Largest" , " Independent Set is " , LISS(root) )# This code is contributed by Arnab Kundu |
C#
// C# program for calculating LISS // using dynamic programming using System;class LisTree { /* A binary tree node has data, pointer to left child and a pointer to right child */ public class node { public int data, liss; public node left, right; public node(int data) { this.data = data; this.liss = 0; } } // A memoization function returns size // of the largest independent set in // a given binary tree static int liss(node root) { if (root == null) return 0; if (root.liss != 0) return root.liss; if (root.left == null && root.right == null) return root.liss = 1; // Calculate size excluding the // current node int liss_excl = liss(root.left) + liss(root.right); // Calculate size including the // current node int liss_incl = 1; if (root.left != null) { liss_incl += (liss(root.left.left) + liss(root.left.right)); } if (root.right != null) { liss_incl += (liss(root.right.left) + liss(root.right.right)); } // Maximum of two sizes is LISS, // store it for future uses. return root.liss = Math.Max(liss_excl, liss_incl); } // Driver code public static void Main(String[] args) { // Let us construct the tree given // in the above diagram node root = new node(20); root.left = new node(8); root.left.left = new node(4); root.left.right = new node(12); root.left.right.left = new node(10); root.left.right.right = new node(14); root.right = new node(22); root.right.right = new node(25); Console.WriteLine("Size of the Largest Independent Set is " + liss(root)); } } // This code is contributed by Princi Singh |
Javascript
<script>// A naive recursive implementation of// Largest Independent Set problem// A utility function to find// max of two integersfunction max(x, y){ return(x > y) ? x : y;}// 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; }}// The function returns size of the// largest independent set in a given// binary treefunction LISS(root){ if (root == null) return 0; // Calculate size excluding the current node let size_excl = LISS(root.left) + LISS(root.right); // Calculate size including the current node let size_incl = 1; if (root.left != null) size_incl += LISS(root.left.left) + LISS(root.left.right); if (root.right != null) size_incl += LISS(root.right.left) + LISS(root.right.right); // Return the maximum of two sizes return max(size_incl, size_excl);}// Driver Code// Let us construct the tree// given in the above diagramlet root = new Node(20);root.left = new Node(8);root.left.left = new Node(4);root.left.right = new Node(12);root.left.right.left = new Node(10);root.left.right.right = new Node(14);root.right = new Node(22);root.right.right = new Node(25);document.write("Size of the Largest" + " Independent Set is " + LISS(root));// This code is contributed by avanitrachhadiya2155</script> |
Output
Size of the Largest Independent Set is 5
Time Complexity of the above naive recursive approach is exponential. It should be noted that the above function computes the same subproblems again and again. For example, LISS of node with value 50 is evaluated for node with values 10 and 20 as 50 is grandchild of 10 and child of 20.
Auxiliary Space: O(n) ,here n is the number of nodes in given Binary tree.
Since same subproblems are called again, this problem has Overlapping Subproblems property. So LISS problem has both properties (see this and this) of a dynamic programming problem. Like other typical Dynamic Programming(DP) problems, recomputations of same subproblems can be avoided by storing the solutions to subproblems and solving problems in bottom up manner.
Following are implementation of Dynamic Programming based solution. In the following solution, an additional field ‘liss’ is added to tree nodes. The initial value of ‘liss’ is set as 0 for all nodes. The recursive function LISS() calculates ‘liss’ for a node only if it is not already set.
C++
/* Dynamic programming based program for Largest Independent Set problem */#include <bits/stdc++.h>using namespace std; // A utility function to find max of two integers int max(int x, int y) { return (x > y)? x: y; } /* A binary tree node has data, pointerto left child and a pointer to right child */class node { public: int data; int liss; node *left, *right; }; // A memoization function returns size // of the largest independent set in // a given binary tree int LISS(node *root) { if (root == NULL) return 0; if (root->liss) return root->liss; if (root->left == NULL && root->right == NULL) return (root->liss = 1); // Calculate size excluding the current node int liss_excl = LISS(root->left) + LISS(root->right); // Calculate size including the current node int liss_incl = 1; if (root->left) liss_incl += LISS(root->left->left) + LISS(root->left->right); if (root->right) liss_incl += LISS(root->right->left) + LISS(root->right->right); // Maximum of two sizes is LISS, store it for future uses. root->liss = max(liss_incl, liss_excl); return root->liss; } // A utility function to create a node node* newNode(int data) { node* temp = new node(); temp->data = data; temp->left = temp->right = NULL; temp->liss = 0; return temp; } // Driver codeint main() { // Let us construct the tree // given in the above diagram node *root = newNode(20); root->left = newNode(8); root->left->left = newNode(4); root->left->right = newNode(12); root->left->right->left = newNode(10); root->left->right->right = newNode(14); root->right = newNode(22); root->right->right = newNode(25); cout << "Size of the Largest Independent Set is " << LISS(root); return 0; } // This code is contributed by rathbhupendra |
C
/* Dynamic programming based program for Largest Independent Set problem */#include <stdio.h>#include <stdlib.h>// A utility function to find max of two integersint max(int x, int y) { return (x > y)? x: y; }/* A binary tree node has data, pointer to left child and a pointer to right child */struct node{ int data; int liss; struct node *left, *right;};// A memoization function returns size of the largest independent set in// a given binary treeint LISS(struct node *root){ if (root == NULL) return 0; if (root->liss) return root->liss; if (root->left == NULL && root->right == NULL) return (root->liss = 1); // Calculate size excluding the current node int liss_excl = LISS(root->left) + LISS(root->right); // Calculate size including the current node int liss_incl = 1; if (root->left) liss_incl += LISS(root->left->left) + LISS(root->left->right); if (root->right) liss_incl += LISS(root->right->left) + LISS(root->right->right); // Maximum of two sizes is LISS, store it for future uses. root->liss = max(liss_incl, liss_excl); return root->liss;}// A utility function to create a nodestruct node* newNode(int data){ struct node* temp = (struct node *) malloc( sizeof(struct node) ); temp->data = data; temp->left = temp->right = NULL; temp->liss = 0; return temp;}// Driver program to test above functionsint main(){ // Let us construct the tree given in the above diagram struct node *root = newNode(20); root->left = newNode(8); root->left->left = newNode(4); root->left->right = newNode(12); root->left->right->left = newNode(10); root->left->right->right = newNode(14); root->right = newNode(22); root->right->right = newNode(25); printf ("Size of the Largest Independent Set is %d ", LISS(root)); return 0;} |
Java
// Java program for calculating LISS // using dynamic programmingimport java.io.*;public class LisTree { /* A binary tree node has data, pointer to left child and a pointer to right child */ static class node { int data, liss; node left, right; public node(int data) { this.data = data; this.liss = 0; } } // A memoization function returns size // of the largest independent set in // a given binary tree static int liss(node root) { if (root == null) return 0; if (root.liss != 0) return root.liss; if (root.left == null && root.right == null) return root.liss = 1; // Calculate size excluding the // current node int liss_excl = liss(root.left) + liss(root.right); // Calculate size including the // current node int liss_incl = 1; if (root.left != null) { liss_incl += (liss(root.left.left) + liss(root.left.right)); } if (root.right != null) { liss_incl += (liss(root.right.left) + liss(root.right.right)); } // Maximum of two sizes is LISS, // store it for future uses. return root.liss = Math.max(liss_excl, liss_incl); } public static void main(String[] args) { // Let us construct the tree given // in the above diagram node root = new node(20); root.left = new node(8); root.left.left = new node(4); root.left.right = new node(12); root.left.right.left = new node(10); root.left.right.right = new node(14); root.right = new node(22); root.right.right = new node(25); System.out.println("Size of the Largest Independent Set is " + liss(root)); }}// This code is contributed by Rishabh Mahrsee |
Python3
# Python3 program for calculating LISS# using dynamic programming# A binary tree node has data,# pointer to left child and a# pointer to right childclass node: def __init__(self, data): self.data = data self.left = self.right = None self.liss = 0# A memoization function returns size# of the largest independent set in# a given binary treedef liss(root): if root == None: return 0 if (root.liss): return root.liss if (root.left == None and root.right == None): root.liss = 1 return root.liss # Calculate size excluding the # current node liss_excl = (liss(root.left) + liss(root.right)) # Calculate size including the # current node liss_incl = 1 if root.left != None: liss_incl += (liss(root.left.left) + liss(root.left.right)) if root.right != None: liss_incl += (liss(root.right.left) + liss(root.right.right)) # Maximum of two sizes is LISS, # store it for future uses. root.liss = max(liss_excl, liss_incl) return root.liss # Driver Code# Let us construct the tree given# in the above diagramroot = node(20)root.left = node(8)root.left.left = node(4)root.left.right = node(12)root.left.right.left = node(10)root.left.right.right = node(14)root.right = node(22)root.right.right = node(25)print("Size of the Largest Independent "\ "Set is ", liss(root))# This code is contributed by nishthagoel712 |
C#
// C# program for calculating LISS // using dynamic programmingusing System; public class LisTree { /* A binary tree node has data, pointer to left child and a pointer to right child */ public class node { public int data, liss; public node left, right; public node(int data) { this.data = data; this.liss = 0; } } // A memoization function returns size // of the largest independent set in // a given binary tree static int liss(node root) { if (root == null) return 0; if (root.liss != 0) return root.liss; if (root.left == null && root.right == null) return root.liss = 1; // Calculate size excluding the // current node int liss_excl = liss(root.left) + liss(root.right); // Calculate size including the // current node int liss_incl = 1; if (root.left != null) { liss_incl += (liss(root.left.left) + liss(root.left.right)); } if (root.right != null) { liss_incl += (liss(root.right.left) + liss(root.right.right)); } // Maximum of two sizes is LISS, // store it for future uses. return root.liss = Math.Max(liss_excl, liss_incl); } // Driver code public static void Main(String[] args) { // Let us construct the tree given // in the above diagram node root = new node(20); root.left = new node(8); root.left.left = new node(4); root.left.right = new node(12); root.left.right.left = new node(10); root.left.right.right = new node(14); root.right = new node(22); root.right.right = new node(25); Console.WriteLine("Size of the Largest Independent Set is " + liss(root)); }}/* This code is contributed by PrinciRaj1992 */ |
Javascript
<script> // JavaScript program for calculating LISS // using dynamic programming /* A binary tree node has data, pointer to left child and a pointer to right child */ class node { constructor(data) { this.data = data; this.liss = 0; this.left = null; this.right = null; } } // A memoization function returns size // of the largest independent set in // a given binary tree function liss(root) { if (root == null) return 0; if (root.liss != 0) return root.liss; if (root.left == null && root.right == null) return (root.liss = 1); // Calculate size excluding the // current node var liss_excl = liss(root.left) + liss(root.right); // Calculate size including the // current node var liss_incl = 1; if (root.left != null) { liss_incl += liss(root.left.left) + liss(root.left.right); } if (root.right != null) { liss_incl += liss(root.right.left) + liss(root.right.right); } // Maximum of two sizes is LISS, // store it for future uses. return (root.liss = Math.max(liss_excl, liss_incl)); } // Driver code // Let us construct the tree given // in the above diagram var root = new node(20); root.left = new node(8); root.left.left = new node(4); root.left.right = new node(12); root.left.right.left = new node(10); root.left.right.right = new node(14); root.right = new node(22); root.right.right = new node(25); document.write( "Size of the Largest Independent Set is " + liss(root) ); </script> |
Output
Size of the Largest Independent Set is 5
Time Complexity: O(n) where n is the number of nodes in given Binary tree.
Auxiliary Space: O(n)
Following extensions to above solution can be tried as an exercise.
1) Extend the above solution for n-ary tree.
2) The above solution modifies the given tree structure by adding an additional field ‘liss’ to tree nodes. Extend the solution so that it doesn’t modify the tree structure.
3) The above solution only returns size of LIS, it doesn’t print elements of LIS. Extend the solution to print all nodes that are part of LIS.
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