Given an integer Q representing the number of queries and an array where each query has an integer N. Our task is to iterate through each query and find the number of paths such that bitwise AND of all nodes on that path is odd.
A binary tree is constructed with N vertices numbered 1 through N. For each i, from 2 to N, there is an edge between vertex i and vertex i / 2 (rounded off).
Examples:
Input: Q = 2, [5, 2]
Output: 1 0
Explanation:
For first query the binary tree will be
1
/ \
2 3
/ \
4 5
The path which satisfies
the condition is 1 -> 3.
Hence only 1 path.
For the second query,
the binary tree will be
1
/
2
There no such path that
satisfies the condition.
Input: Q = 3, [3, 7, 13]
Output: 1 3 4
Approach: The idea is to use Dynamic Programming and precomputation of answers for all values from 1 to the maximum value of N in the query.
- Firstly, observe that if a bitwise AND of a path is odd, then none elements of that path can be even. Hence, the required path should have odd elements.
- We know that for an ith node (except for node 1) the parent node will be i/2 (rounded off). Maintain a dp array for storing the answer for the ith node. Another array will be used to store the number of odd elements from the current node until the parents are odd.
- While computing the dp array, the first condition is if the ith node value is even, then dp[i] = dp[i – 1] because the ith node will not contribute to the answer, so dp[i] will be the answer to (i-1)th node. Secondly, if ith node is odd, then dp[i] = dp[i-1] + nC2 -(n-1)C2. On simplification dp[i] = dp[i-1] + (number of odd elements till now) – 1.
Below is the implementation of the above approach:
C++
// C++ implementation to count// paths in Binary Tree// with odd bitwise AND#include <bits/stdc++.h> using namespace std;// Function to count number of paths// in binary tree such that bitwise// AND of all nodes is Oddvoid compute(vector<int> query) { // vector v for storing // the count of odd numbers // vector dp to store the // count of bitwise odd paths // till that vertex vector<int> v(100001), dp(100001); v[1] = 1, v[2] = 0; dp[1] = 0, dp[2] = 0; // Precomputing for each value for (int i = 3; i < 100001; i++) { // check for odd value if (i % 2 != 0) { if ((i / 2) % 2 == 0) { v[i] = 1; dp[i] = dp[i - 1]; } else { // Number of odd elements will // be +1 till the parent node v[i] = v[i / 2] + 1; dp[i] = dp[i - 1] + v[i] - 1; } } // For even case else { // Since node is even // Number of odd elements // will be 0 v[i] = 0; // Even value node will // not contribute in answer // hence dp[i] = previous answer dp[i] = dp[i - 1]; } } // Printing the answer // for each query for (auto x : query) cout << dp[x] << endl;}// Driver codeint main(){ // vector to store queries vector<int> query = { 5, 2 }; compute(query); return 0;} |
Java
// Java implementation to count// paths in Binary Tree// with odd bitwise ANDclass GFG{// Function to count number of paths// in binary tree such that bitwise// AND of all nodes is Oddstatic void compute(int[] query){ // v for storing the count // of odd numbers // dp to store the count of // bitwise odd paths // till that vertex int []v = new int[100001]; int []dp = new int[100001]; v[1] = 1; v[2] = 0; dp[1] = 0; dp[2] = 0; // Precomputing for each value for(int i = 3; i < 100001; i++) { // Check for odd value if (i % 2 != 0) { if ((i / 2) % 2 == 0) { v[i] = 1; dp[i] = dp[i - 1]; } else { // Number of odd elements will // be +1 till the parent node v[i] = v[i / 2] + 1; dp[i] = dp[i - 1] + v[i] - 1; } } // For even case else { // Since node is even // Number of odd elements // will be 0 v[i] = 0; // Even value node will // not contribute in answer // hence dp[i] = previous answer dp[i] = dp[i - 1]; } } // Printing the answer // for each query for(int x : query) System.out.print(dp[x] + "\n");}// Driver codepublic static void main(String[] args){ // To store queries int []query = { 5, 2 }; compute(query);}}// This code is contributed by Princi Singh |
Python3
# Python3 implementation to count# paths in Binary Tree with odd # bitwise AND# Function to count number of paths# in binary tree such that bitwise# AND of all nodes is Odddef compute(query): # vector v for storing # the count of odd numbers # vector dp to store the # count of bitwise odd paths # till that vertex v = [None] * 100001 dp = [None] * 100001 v[1] = 1 v[2] = 0 dp[1] = 0 dp[2] = 0 # Precomputing for each value for i in range(3, 100001): # Check for odd value if (i % 2 != 0): if ((i // 2) % 2 == 0): v[i] = 1 dp[i] = dp[i - 1] else: # Number of odd elements will # be +1 till the parent node v[i] = v[i // 2] + 1 dp[i] = dp[i - 1] + v[i] - 1 # For even case else: # Since node is even # Number of odd elements # will be 0 v[i] = 0 # Even value node will # not contribute in answer # hence dp[i] = previous answer dp[i] = dp[i - 1] # Printing the answer # for each query for x in query: print(dp[x])# Driver code# Vector to store queriesquery = [ 5, 2 ]compute(query)# This code is contributed by sanjoy_62 |
C#
// C# implementation to count// paths in Binary Tree// with odd bitwise ANDusing System;class GFG{// Function to count number of paths// in binary tree such that bitwise// AND of all nodes is Oddstatic void compute(int[] query){ // v for storing the count // of odd numbers // dp to store the count of // bitwise odd paths // till that vertex int []v = new int[100001]; int []dp = new int[100001]; v[1] = 1; v[2] = 0; dp[1] = 0; dp[2] = 0; // Precomputing for each value for(int i = 3; i < 100001; i++) { // Check for odd value if (i % 2 != 0) { if ((i / 2) % 2 == 0) { v[i] = 1; dp[i] = dp[i - 1]; } else { // Number of odd elements will // be +1 till the parent node v[i] = v[i / 2] + 1; dp[i] = dp[i - 1] + v[i] - 1; } } // For even case else { // Since node is even // Number of odd elements // will be 0 v[i] = 0; // Even value node will // not contribute in answer // hence dp[i] = previous answer dp[i] = dp[i - 1]; } } // Printing the answer // for each query foreach(int x in query) Console.Write(dp[x] + "\n");}// Driver codepublic static void Main(String[] args){ // To store queries int []query = { 5, 2 }; compute(query);}}// This code is contributed by Amit Katiyar |
Javascript
<script>// Javascript implementation to count// paths in Binary Tree// with odd bitwise AND// Function to count number of paths// in binary tree such that bitwise// AND of all nodes is Oddfunction compute(query) { // vector v for storing // the count of odd numbers // vector dp to store the // count of bitwise odd paths // till that vertex var v = Array(100001).fill(0); var dp = Array(100001).fill(0); v[1] = 1, v[2] = 0; dp[1] = 0, dp[2] = 0; // Precomputing for each value for (var i = 3; i < 100001; i++) { // check for odd value if (i % 2 != 0) { if (parseInt(i / 2) % 2 == 0) { v[i] = 1; dp[i] = dp[i - 1]; } else { // Number of odd elements will // be +1 till the parent node v[i] = v[parseInt(i / 2)] + 1; dp[i] = dp[i - 1] + v[i] - 1; } } // For even case else { // Since node is even // Number of odd elements // will be 0 v[i] = 0; // Even value node will // not contribute in answer // hence dp[i] = previous answer dp[i] = dp[i - 1]; } } // Printing the answer // for each query query.forEach(x => { document.write( dp[x] + "<br>"); });}// Driver code// vector to store queriesvar query = [5, 2];compute(query);</script> |
Output:
1 0
Time Complexity: O( Nmax + Q*(1) ), where Nmax is the maximum value of N. Q*(1) since we are precomputing each query.
Auxiliary Space: O(Nmax)
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