Given a binary tree with N nodes numbered [1, N], the task is to find the size of the smallest Dominating set of that tree.
A set of nodes is said to be a dominating node if every node in the binary tree not present in the set is an immediate child/parent to any node in that set.
Examples:
Input: 1 / 2 / \ 4 3 / 5 / \ 6 7 / \ \ 8 9 10 Output: 3 Explanation: Smallest dominating set is {2, 6, 7} Input: 1 / \ 2 3 / \ / \ 4 5 6 7 / \ / 8 9 10 Output: 4 Explanation: One of the smallest dominating set = {2, 3, 6, 4}
Approach:
In order to solve this problem we are using a dynamic programming approach by defining the following two states for every node:
- The first state compulsory tells us whether it is compulsory to choose the node in the set or not.
- The second state covered, tells us whether the node’s parent/child is in the set or not.
If it is compulsory to choose the node, we choose it and mark its children as covered. Otherwise, we have an option to choose it or reject it and then update its children as covered or not accordingly. Check the states for every node and find the required size of the set accordingly.
Below code is the implementation of the above approach:
/* C++ program to find the size of the minimum dominating set of the tree */ #include <bits/stdc++.h> using namespace std;
#define N 1005 // Definition of a tree node struct Node {
int data;
Node *left, *right;
}; /* Helper function that allocates a new node */ Node* newNode( int data)
{ Node* node = new Node();
node->data = data;
node->left = node->right = NULL;
return node;
} // DP array to precompute // and store the results int dp[N][5][5];
// minDominatingSettion to return the size of // the minimum dominating set of the array int minDominatingSet(Node* root, int covered,
int compulsory)
{ // Base case
if (!root)
return 0;
// Setting the compulsory value if needed
if (!root->left and !root->right and !covered)
compulsory = true ;
// Check if the answer is already computed
if (dp[root->data][covered][compulsory] != -1)
return dp[root->data][covered][compulsory];
// If it is compulsory to select
// the node
if (compulsory) {
// Choose the node and set its children as covered
return dp[root->data]
[covered]
[compulsory]
= 1
+ minDominatingSet(
root->left, 1, 0)
+ minDominatingSet(
root->right, 1, 0);
}
// If it is covered
if (covered) {
return dp[root->data]
[covered]
[compulsory]
= min(
1
+ minDominatingSet(
root->left, 1, 0)
+ minDominatingSet(
root->right, 1, 0),
minDominatingSet(
root->left, 0, 0)
+ minDominatingSet(
root->right, 0, 0));
}
// If the current node is neither covered nor
// needs to be selected compulsorily
int ans = 1
+ minDominatingSet(
root->left, 1, 0)
+ minDominatingSet(
root->right, 1, 0);
if (root->left) {
ans = min(ans,
minDominatingSet(
root->left, 0, 1)
+ minDominatingSet(
root->right, 0, 0));
}
if (root->right) {
ans = min(ans,
minDominatingSet(
root->left, 0, 0)
+ minDominatingSet(
root->right, 0, 1));
}
// Store the result
return dp[root->data]
[covered]
[compulsory]
= ans;
} // Driver code signed main()
{ // initialising the DP array
memset (dp, -1, sizeof (dp));
// Constructing the tree
Node* root = newNode(1);
root->left = newNode(2);
root->left->left = newNode(3);
root->left->right = newNode(4);
root->left->left->left = newNode(5);
root->left->left->left->left = newNode(6);
root->left->left->left->right = newNode(7);
root->left->left->left->right->right = newNode(10);
root->left->left->left->left->left = newNode(8);
root->left->left->left->left->right = newNode(9);
cout << minDominatingSet(root, 0, 0) << endl;
return 0;
} |
// Java program to find the size of the //minimum dominating set of the tree import java.util.*;
class GFG{
static final int N = 1005 ;
// Definition of a tree node static class Node
{ int data;
Node left, right;
}; // Helper function that allocates a // new node static Node newNode( int data)
{ Node node = new Node();
node.data = data;
node.left = node.right = null ;
return node;
} // DP array to precompute // and store the results static int [][][]dp = new int [N][ 5 ][ 5 ];
// minDominatingSettion to return the size of // the minimum dominating set of the array static int minDominatingSet(Node root,
int covered,
int compulsory)
{ // Base case
if (root == null )
return 0 ;
// Setting the compulsory value if needed
if (root.left != null &&
root.right != null &&
covered > 0 )
compulsory = 1 ;
// Check if the answer is already computed
if (dp[root.data][covered][compulsory] != - 1 )
return dp[root.data][covered][compulsory];
// If it is compulsory to select
// the node
if (compulsory > 0 )
{
// Choose the node and set its
// children as covered
return dp[root.data][covered][compulsory] = 1 +
minDominatingSet(root.left, 1 , 0 ) +
minDominatingSet(root.right, 1 , 0 );
}
// If it is covered
if (covered > 0 )
{
return dp[root.data][covered]
[compulsory] = Math.min( 1 +
minDominatingSet(root.left, 1 , 0 ) +
minDominatingSet(root.right, 1 , 0 ),
minDominatingSet(root.left, 0 , 0 )+
minDominatingSet(root.right, 0 , 0 ));
}
// If the current node is neither covered nor
// needs to be selected compulsorily
int ans = 1 + minDominatingSet(root.left, 1 , 0 ) +
minDominatingSet(root.right, 1 , 0 );
if (root.left != null )
{
ans = Math.min(ans,
minDominatingSet(root.left, 0 , 1 ) +
minDominatingSet(root.right, 0 , 0 ));
}
if (root.right != null )
{
ans = Math.min(ans,
minDominatingSet(root.left, 0 , 0 ) +
minDominatingSet(root.right, 0 , 1 ));
}
// Store the result
return dp[root.data][covered][compulsory] = ans;
} // Driver code public static void main(String[] args)
{ // Initialising the DP array
for ( int i = 0 ; i < N; i++)
{
for ( int j = 0 ; j < 5 ; j++)
{
for ( int l = 0 ; l < 5 ; l++)
dp[i][j][l] = - 1 ;
}
}
// Constructing the tree
Node root = newNode( 1 );
root.left = newNode( 2 );
root.left.left = newNode( 3 );
root.left.right = newNode( 4 );
root.left.left.left = newNode( 5 );
root.left.left.left.left = newNode( 6 );
root.left.left.left.right = newNode( 7 );
root.left.left.left.right.right = newNode( 10 );
root.left.left.left.left.left = newNode( 8 );
root.left.left.left.left.right = newNode( 9 );
System.out.print(minDominatingSet(
root, 0 , 0 ) + "\n" );
} } // This code is contributed by amal kumar choubey |
# Python3 program to find the size of the # minimum dominating set of the tree */ N = 1005
# Definition of a tree node class Node:
def __init__( self , data):
self .data = data
self .left = None
self .right = None
# Helper function that allocates a # new node def newNode(data):
node = Node(data)
return node
# DP array to precompute # and store the results dp = [[[ - 1 for i in range ( 5 )] for j in range ( 5 )] for k in range (N)];
# minDominatingSettion to return the size of # the minimum dominating set of the array def minDominatingSet(root, covered, compulsory):
# Base case
if ( not root):
return 0 ;
# Setting the compulsory value if needed
if ( not root.left and not root.right and not covered):
compulsory = True ;
# Check if the answer is already computed
if (dp[root.data][covered][compulsory] ! = - 1 ):
return dp[root.data][covered][compulsory];
# If it is compulsory to select
# the node
if (compulsory):
dp[root.data][covered][compulsory] = 1 + minDominatingSet(root.left, 1 , 0 ) + minDominatingSet(root.right, 1 , 0 );
# Choose the node and set its children as covered
return dp[root.data][covered][compulsory]
# If it is covered
if (covered):
dp[root.data][covered][compulsory] = min ( 1 + minDominatingSet(root.left, 1 , 0 ) + minDominatingSet(root.right, 1 , 0 ),minDominatingSet(root.left, 0 , 0 ) + minDominatingSet(root.right, 0 , 0 ));
return dp[root.data][covered][compulsory]
# If the current node is neither covered nor
# needs to be selected compulsorily
ans = 1 + minDominatingSet(root.left, 1 , 0 ) + minDominatingSet(root.right, 1 , 0 );
if (root.left):
ans = min (ans, minDominatingSet(root.left, 0 , 1 ) + minDominatingSet(root.right, 0 , 0 ));
if (root.right):
ans = min (ans, minDominatingSet( root.left, 0 , 0 ) + minDominatingSet(root.right, 0 , 1 ));
# Store the result
dp[root.data][covered][compulsory] = ans;
return ans
# Driver code if __name__ = = '__main__' :
# Constructing the tree
root = newNode( 1 );
root.left = newNode( 2 );
root.left.left = newNode( 3 );
root.left.right = newNode( 4 );
root.left.left.left = newNode( 5 );
root.left.left.left.left = newNode( 6 );
root.left.left.left.right = newNode( 7 );
root.left.left.left.right.right = newNode( 10 );
root.left.left.left.left.left = newNode( 8 );
root.left.left.left.left.right = newNode( 9 );
print (minDominatingSet(root, 0 , 0 ))
# This code is contributed by rutvik_56
|
// C# program to find the size of the //minimum dominating set of the tree using System;
class GFG{
static readonly int N = 1005;
// Definition of a tree node public class Node
{ public
int data;
public
Node left, right;
}; // Helper function that allocates a // new node public static Node newNode( int data)
{ Node node = new Node();
node.data = data;
node.left = node.right = null ;
return node;
} // DP array to precompute // and store the results static int [,,]dp = new int [N, 5, 5];
// minDominatingSettion to return the size of // the minimum dominating set of the array static int minDominatingSet(Node root,
int covered,
int compulsory)
{ // Base case
if (root == null )
return 0;
// Setting the compulsory value if needed
if (root.left != null &&
root.right != null &&
covered > 0)
compulsory = 1;
// Check if the answer is already computed
if (dp[root.data, covered, compulsory] != -1)
return dp[root.data, covered, compulsory];
// If it is compulsory to select
// the node
if (compulsory > 0)
{
// Choose the node and set its
// children as covered
return dp[root.data, covered, compulsory] = 1 +
minDominatingSet(root.left, 1, 0) +
minDominatingSet(root.right, 1, 0);
}
// If it is covered
if (covered > 0)
{
return dp[root.data, covered, compulsory] = Math.Min(1 +
minDominatingSet(root.left, 1, 0) +
minDominatingSet(root.right, 1, 0),
minDominatingSet(root.left, 0, 0)+
minDominatingSet(root.right, 0, 0));
}
// If the current node is neither covered nor
// needs to be selected compulsorily
int ans = 1 + minDominatingSet(root.left, 1, 0) +
minDominatingSet(root.right, 1, 0);
if (root.left != null )
{
ans = Math.Min(ans,
minDominatingSet(root.left, 0, 1) +
minDominatingSet(root.right, 0, 0));
}
if (root.right != null )
{
ans = Math.Min(ans,
minDominatingSet(root.left, 0, 0) +
minDominatingSet(root.right, 0, 1));
}
// Store the result
return dp[root.data, covered, compulsory] = ans;
} // Driver code public static void Main(String[] args)
{ // Initialising the DP array
for ( int i = 0; i < N; i++)
{
for ( int j = 0; j < 5; j++)
{
for ( int l = 0; l < 5; l++)
dp[i, j, l] = -1;
}
}
// Constructing the tree
Node root = newNode(1);
root.left = newNode(2);
root.left.left = newNode(3);
root.left.right = newNode(4);
root.left.left.left = newNode(5);
root.left.left.left.left = newNode(6);
root.left.left.left.right = newNode(7);
root.left.left.left.right.right = newNode(10);
root.left.left.left.left.left = newNode(8);
root.left.left.left.left.right = newNode(9);
Console.Write(minDominatingSet
root, 0, 0) + "\n" );
} } // This code is contributed by Rohit_ranjan |
<script> // JavaScript program to find the size of the
//minimum dominating set of the tree
let N = 1005;
// Definition of a tree node
class Node
{
constructor(data) {
this .left = null ;
this .right = null ;
this .data = data;
}
}
// Helper function that allocates a
// new node
function newNode(data)
{
let node = new Node(data);
return node;
}
// DP array to precompute
// and store the results
let dp = new Array(N);
// minDominatingSettion to return the size of
// the minimum dominating set of the array
function minDominatingSet(root, covered, compulsory)
{
// Base case
if (root == null )
return 0;
// Setting the compulsory value if needed
if (root.left != null &&
root.right != null &&
covered > 0)
compulsory = 1;
// Check if the answer is already computed
if (dp[root.data][covered][compulsory] != -1)
return dp[root.data][covered][compulsory];
// If it is compulsory to select
// the node
if (compulsory > 0)
{
// Choose the node and set its
// children as covered
return dp[root.data][covered][compulsory] = 1 +
minDominatingSet(root.left, 1, 0) +
minDominatingSet(root.right, 1, 0);
}
// If it is covered
if (covered > 0)
{
return dp[root.data][covered]
[compulsory] = Math.min(1 +
minDominatingSet(root.left, 1, 0) +
minDominatingSet(root.right, 1, 0),
minDominatingSet(root.left, 0, 0)+
minDominatingSet(root.right, 0, 0));
}
// If the current node is neither covered nor
// needs to be selected compulsorily
let ans = 1 + minDominatingSet(root.left, 1, 0) +
minDominatingSet(root.right, 1, 0);
if (root.left != null )
{
ans = Math.min(ans,
minDominatingSet(root.left, 0, 1) +
minDominatingSet(root.right, 0, 0));
}
if (root.right != null )
{
ans = Math.min(ans,
minDominatingSet(root.left, 0, 0) +
minDominatingSet(root.right, 0, 1));
}
// Store the result
dp[root.data][covered][compulsory] = ans;
return dp[root.data][covered][compulsory];
}
// Initialising the DP array
for (let i = 0; i < N; i++)
{
dp[i] = new Array(5);
for (let j = 0; j < 5; j++)
{
dp[i][j] = new Array(5);
for (let l = 0; l < 5; l++)
dp[i][j][l] = -1;
}
}
// Constructing the tree
let root = newNode(1);
root.left = newNode(2);
root.left.left = newNode(3);
root.left.right = newNode(4);
root.left.left.left = newNode(5);
root.left.left.left.left = newNode(6);
root.left.left.left.right = newNode(7);
root.left.left.left.right.right = newNode(10);
root.left.left.left.left.left = newNode(8);
root.left.left.left.left.right = newNode(9);
document.write(minDominatingSet(root, 0, 0));
</script> |
3
Time Complexity: O(N)
Auxiliary Space: O(N)