Given a tree of N nodes and an integer K, the task is to find the total number of paths having length K.
Examples:
Input: N = 5, K = 2
tree = 1
/ \
2 5
/ \
3 4
Output: 4
Explanation: The paths having length 2 are
1 – 2 – 3, 1 – 2 – 4, 2 – 1 – 5, 3 – 2 – 4Input: N = 2, K = 2
tree = 1
/
2
Output: 0
Explanation: There is no path in the tree having length 2.
Intuition: The main idea is to find the K-length paths from each node and add them.
- Find the number of K-length paths ‘originating’ from a given node ‘node’. ‘Originating’ here means, ‘node’ will have the smallest depth among all the nodes in the path. For example, 2-length paths originating from 1 are shown in the below diagram.
- Sum above value for all the nodes and it will be the required answer.
Naive Approach: To compute the K-length paths originating from ‘node’ two DFS are used. Say this entire process is: paths_originating_from(node)
- Suppose ‘node’ has multiple children and currently child ‘u’ is being processed.
- For all the previous children, the frequency of nodes at a particular depth has been calculated. More formally, freq[d] gives the number of nodes at depth ‘d’ when only children of ‘node’ before ‘u’ have been processed.
- If there is a node ‘x’ at depth ‘d’, number of K length paths originating from ‘node’ and passing through ‘x’ will be freq[K – d].
- The first DFSF would contribute to the final answer, and the second DFS would update the freq[] array for future use.
- Sum-up ‘paths_originating_from(node)’ for all nodes of the tree, this will be the required answer.
See the image below to understand the 2nd point better.
Below is the implementation of the above approach.
C++
// C++ code to implement above approach#include <bits/stdc++.h>using namespace std;int mx_depth = 0, ans = 0;int N, K;vector<int> freq;vector<vector<int> > g;// This dfs is responsible for calculating ans // and updating freq vectorvoid dfs(int node, int par, int depth, bool contri){ if (depth > K) return; mx_depth = max(mx_depth, depth); if (contri) { ans += freq[K - depth]; } else { freq[depth]++; } for (auto nebr : g[node]) { if (nebr != par) { dfs(nebr, node, depth + 1, contri); } }}// Function to calculate K length paths // originating from nodevoid paths_originating_from(int node, int par){ mx_depth = 0; freq[0] = 1; // For every not-removed nebr, // calculate its contribution, // then update freq vector for it for (auto nebr : g[node]) { if (nebr != par) { dfs(nebr, node, 1, true); dfs(nebr, node, 1, false); } } // Re-initialize freq vector for (int i = 0; i <= mx_depth; ++i) { freq[i] = 0; } // Repeat the same for children for (auto nebr : g[node]) { if (nebr != par) { paths_originating_from(nebr, node); } }}// Utility method to add edges to treevoid edge(int a, int b){ a--; b--; g[a].push_back(b); g[b].push_back(a);}// Driver codeint main(){ N = 5, K = 2; freq = vector<int>(N); g = vector<vector<int> >(N); edge(1, 2); edge(1, 5); edge(2, 3); edge(2, 4); paths_originating_from(0, -1); cout << ans << endl;} |
Java
import java.io.*;import java.util.*;public class Main { static int N, K, mx_depth = 0, ans = 0; ; static void edge(int a, int b, ArrayList<ArrayList<Integer> > g) { a--; b--; g.get(a).add(b); g.get(b).add(a); } // This dfs is responsible for calculating ans // and updating freq vector static void dfs(int node, int par, int depth, boolean contri, ArrayList<Integer> freq, ArrayList<ArrayList<Integer> > g) { if (depth > K) return; mx_depth = Math.max(mx_depth, depth); if (contri) { ans += freq.get(K - depth); } else { freq.set(depth, freq.get(depth) + 1); } for (int i = 0; i < g.get(node).size(); i++) { int nebr = g.get(node).get(i); if (nebr != par) { dfs(nebr, node, depth + 1, contri, freq, g); } } } // Function to calculate K length paths // originating from node static void paths_originating_from(int node, int par, ArrayList<Integer> freq, ArrayList<ArrayList<Integer> > g) { mx_depth = 0; freq.set(0, 1); // For every not-removed nebr, // calculate its contribution, // then update freq vector for it for (int i = 0; i < g.get(node).size(); i++) { int nebr = g.get(node).get(i); if (nebr != par) { dfs(nebr, node, 1, true, freq, g); dfs(nebr, node, 1, false, freq, g); } } // Re-initialize freq vector for (int i = 0; i <= mx_depth; ++i) { freq.set(i, 0); } // Repeat the same for children for (int i = 0; i < g.get(node).size(); i++) { int nebr = g.get(node).get(i); if (nebr != par) { paths_originating_from(nebr, node, freq, g); } } } public static void main(String[] args) { N = 5; K = 2; ArrayList<Integer> freq = new ArrayList<Integer>(); for (int i = 0; i < N; i++) freq.add(0); ArrayList<ArrayList<Integer> > g = new ArrayList<ArrayList<Integer> >(); for (int i = 0; i < N; i++) g.add(new ArrayList<Integer>()); edge(1, 2, g); edge(1, 5, g); edge(2, 3, g); edge(2, 4, g); paths_originating_from(0, -1, freq, g); System.out.println(ans); }}// This code is contributed by garg28harsh. |
Python3
# Python code to implement above approachmx_depth = 0ans = 0N = 5K = 2freq = [0] * Ng = [[] for _ in range(N)]# This dfs is responsible for calculating ans# and updating freq vectordef dfs(node, par, depth, contri): global mx_depth, ans, freq if depth > K: return mx_depth = max(mx_depth, depth) if contri: ans += freq[K - depth] else: freq[depth] += 1 for nebr in g[node]: if nebr != par: dfs(nebr, node, depth + 1, contri)# Function to calculate K length paths# originating from nodedef paths_originating_from(node, par): global mx_depth, freq mx_depth = 0 freq[0] = 1 # For every not-removed nebr, # calculate its contribution, # then update freq vector for it for nebr in g[node]: if nebr != par: dfs(nebr, node, 1, True) dfs(nebr, node, 1, False) # Re-initialize freq vector freq = [0] * (mx_depth + 1) # Repeat the same for children for nebr in g[node]: if nebr != par: paths_originating_from(nebr, node)# Utility method to add edges to treedef edge(a, b): a -= 1 b -= 1 g[a].append(b) g[b].append(a)# Driver codeedge(1, 2)edge(1, 5)edge(2, 3)edge(2, 4)paths_originating_from(0, -1)print(ans) |
Javascript
// Javascript code to implement above approachlet max_Depth = 0, ans = 0, N, K, freq = [];let g = new Array(1000).fill(0).map(() => new Array(0))// This dfs is responsible for calculating ans // and updating freq vectorfunction dfs(node, par, depth, contri) { if (depth > K) return; mx_depth = Math.max(mx_depth, depth); if (contri) { ans += freq[K - depth]; } else { freq[depth]++; } for (let nebr of g[node]) { if (nebr != par) { dfs(nebr, node, depth + 1, contri); } }}// Function to calculate K length paths // originating from nodefunction paths_originating_from(node, par) { mx_depth = 0; freq[0] = 1; // For every not-removed nebr, // calculate its contribution, // then update freq vector for it for (let nebr of g[node]) { if (nebr != par) { dfs(nebr, node, 1, true); dfs(nebr, node, 1, false); } } // Re-initialize freq vector for (let i = 0; i <= mx_depth; ++i) { freq[i] = 0; } // Repeat the same for children for (let nebr of g[node]) { if (nebr != par) { paths_originating_from(nebr, node); } }}// Utility method to add edges to treefunction edge(a, b) { a--; b--; g[a].push(b); g[b].push(a);}// Driver codeN = 5;K = 2;freq = new Array(N).fill(0);g = new Array(1000).fill(0).map(() => new Array());edge(1, 2);edge(1, 5);edge(2, 3);edge(2, 4);paths_originating_from(0, -1);console.log(ans) |
C#
// C# code to implement above approachusing System;using System.Collections.Generic;class Program { static int mx_depth = 0, ans = 0; static int N, K; static List<int> freq; static List<List<int> > g; // This dfs is responsible for calculating ans // and updating freq vector static void dfs(int node, int par, int depth, bool contri) { if (depth > K) return; mx_depth = Math.Max(mx_depth, depth); if (contri) { ans += freq[K - depth]; } else { freq[depth]++; } foreach(int nebr in g[node]) { if (nebr != par) { dfs(nebr, node, depth + 1, contri); } } } // Function to calculate K length paths // originating from node static void paths_originating_from(int node, int par) { mx_depth = 0; freq[0] = 1; // For every not-removed nebr, // calculate its contribution, // then update freq vector for it foreach(int nebr in g[node]) { if (nebr != par) { dfs(nebr, node, 1, true); dfs(nebr, node, 1, false); } } // Re-initialize freq vector for (int i = 0; i <= mx_depth; ++i) { freq[i] = 0; } // Repeat the same for children foreach(int nebr in g[node]) { if (nebr != par) { paths_originating_from(nebr, node); } } } // Utility method to add edges to tree static void edge(int a, int b) { a--; b--; g[a].Add(b); g[b].Add(a); } // Driver code static void Main() { N = 5; K = 2; freq = new List<int>(N); g = new List<List<int> >(N); for (int i = 0; i < N; i++) { freq.Add(0); g.Add(new List<int>()); } edge(1, 2); edge(1, 5); edge(2, 3); edge(2, 4); paths_originating_from(0, -1); Console.WriteLine(ans); }}// This code is contributed by rutikbhosale. |
Output
4
Time Complexity: O(N * H) where H is the height of the tree which can be N at max
Auxiliary Space: O(N)
Efficient Approach: This approach is based on the concept of Centroid Decomposition. The steps are as follows:
- Find the centroid of current tree T.
- All ‘not-removed’ nodes reachable from T belong to its sub-tree. Call paths_originating_from(T), then mark T as ‘removed’.
- Repeat the above process for all ‘not-removed’ neighbors of T.
The following figure shows a tree with current centroid and its sub-tree. Note that nodes with thick borders have already been selected as centroids previously and do not belong to the sub-tree of the current centroid.
Below is the implementation of the above approach.
C++
// C++ code to implement above approach#include <bits/stdc++.h>using namespace std;// Struct for centroid decompositionstruct CD { // 1. mx_depth will be used to store // the height of a node // 2. g[] is adjacency list for tree // 3. freq[] stores frequency of nodes // at particular height, it is maintained // for children of a node int n, k, mx_depth, ans; vector<bool> removed; vector<int> size, freq; vector<vector<int> > g; // Constructor for struct CD(int n1, int k1) { n = n1; k = k1; ans = mx_depth = 0; g.resize(n); size.resize(n); freq.resize(n); removed.assign(n, false); } // Utility method to add edges to tree void edge(int u, int v) { u--; v--; g[u].push_back(v); g[v].push_back(u); } // Finds size of a subtree, // ignoring removed nodes in the way int get_size(int node, int par) { if (removed[node]) return 0; size[node] = 1; for (auto nebr : g[node]) { if (nebr != par) { size[node] += get_size(nebr, node); } } return size[node]; } // Calculates centroid of a subtree // of 'node' of size 'sz' int get_centroid(int node, int par, int sz) { for (auto nebr : g[node]) { if (nebr != par && !removed[nebr] && size[nebr] > sz / 2) { return get_centroid(nebr, node, sz); } } return node; } // Decompose the tree // into various centroids void decompose(int node, int par) { get_size(node, -1); // c is centroid of subtree 'node' int c = get_centroid(node, par, size[node]); // Find paths_originating_from 'c' paths_originating_from(c); // Mark this centroid as removed removed = true; // Find other centroids for (auto nebr : g) { if (!removed[nebr]) { decompose(nebr, c); } } } // This dfs is responsible for // calculating ans and // updating freq vector void dfs(int node, int par, int depth, bool contri) { if (depth > k) return; mx_depth = max(mx_depth, depth); if (contri) { ans += freq[k - depth]; } else { freq[depth]++; } for (auto nebr : g[node]) { if (nebr != par && !removed[nebr]) { dfs(nebr, node, depth + 1, contri); } } } // Function to find K-length paths // originating from node void paths_originating_from(int node) { mx_depth = 0; freq[0] = 1; // For every not-removed nebr, // calculate its contribution, // then update freq vector for it for (auto nebr : g[node]) { if (!removed[nebr]) { dfs(nebr, node, 1, true); dfs(nebr, node, 1, false); } } // Re-initialize freq vector for (int i = 0; i <= mx_depth; ++i) { freq[i] = 0; } }};// Driver codeint main(){ int N = 5, K = 2; CD cd_s(N, K); cd_s.edge(1, 2); cd_s.edge(1, 5); cd_s.edge(2, 3); cd_s.edge(2, 4); cd_s.decompose(0, -1); cout << cd_s.ans; return 0;} |
Java
// Java Code to implement above approachimport java.util.*;public class Solution { // Struct for centroid decomposition static class CD { // 1. mx_depth will be used to store // the height of a node // 2. g[] is adjacency list for tree // 3. freq[] stores frequency of nodes // at particular height, it is maintained // for children of a node int n, k, mx_depth, ans; boolean[] removed; int[] size, freq; ArrayList<ArrayList<Integer> > g; // Constructor for struct CD(int n1, int k1) { n = n1; k = k1; ans = mx_depth = 0; g = new ArrayList<>(); for (int i = 0; i < n; i++) { g.add(new ArrayList<>()); } size = new int[n]; freq = new int[n]; removed = new boolean[n]; } // Utility method to add edges to tree public void edge(int u, int v) { u--; v--; g.get(u).add(v); g.get(v).add(u); } // Finds size of a subtree, // ignoring removed nodes in the way public int get_size(int node, int par) { if (removed[node]) return 0; size[node] = 1; for (Integer nebr : g.get(node)) { if (nebr != par) { size[node] += get_size(nebr, node); } } return size[node]; } // Calculates centroid of a subtree // of 'node' of size 'sz' int get_centroid(int node, int par, int sz) { for (Integer nebr : g.get(node)) { if (nebr != par && !removed[nebr] && size[nebr] > sz / 2) { return get_centroid(nebr, node, sz); } } return node; } // Decompose the tree // into various centroids public void decompose(int node, int par) { get_size(node, -1); // c is centroid of subtree 'node' int c = get_centroid(node, par, size[node]); // Find paths_originating_from 'c' paths_originating_from(c); // Mark this centroid as removed removed = true; // Find other centroids for (Integer nebr : g.get(c)) { if (!removed[nebr]) { decompose(nebr, c); } } } // This dfs is responsible for // calculating ans and // updating freq vector void dfs(int node, int par, int depth, boolean contri) { if (depth > k) return; mx_depth = Math.max(mx_depth, depth); if (contri) { ans += freq[k - depth]; } else { freq[depth]++; } for (Integer nebr : g.get(node)) { if (nebr != par && !removed[nebr]) { dfs(nebr, node, depth + 1, contri); } } } // Function to find K-length paths // originating from node void paths_originating_from(int node) { mx_depth = 0; freq[0] = 1; // For every not-removed nebr, // calculate its contribution, // then update freq vector for it for (Integer nebr : g.get(node)) { if (!removed[nebr]) { dfs(nebr, node, 1, true); dfs(nebr, node, 1, false); } } // Re-initialize freq vector for (int i = 0; i <= mx_depth; ++i) { freq[i] = 0; } } }; // Driver code public static void main(String[] args) { int N = 5, K = 2; CD cd_s = new CD(N, K); cd_s.edge(1, 2); cd_s.edge(1, 5); cd_s.edge(2, 3); cd_s.edge(2, 4); cd_s.decompose(0, -1); System.out.println(cd_s.ans); }}// This code is contributed by karandeep1234 |
Python3
# Python code to implement above approach# Struct for centroid decompositionclass CD: # Constructor for struct def __init__(self, n1, k1): self.n = n1 self.k = k1 self.ans = self.mx_depth = 0 self.removed = [False]*n1 self.size = [0]*n1 self.freq = [0]*n1 self.g = [[] for i in range(n1)] # Utility method to add edges to tree def edge(self, u, v): u -= 1 v -= 1 self.g[u].append(v) self.g[v].append(u) # Finds size of a subtree, # ignoring removed nodes in the way def get_size(self, node, par): if self.removed[node]: return 0 self.size[node] = 1 for nebr in self.g[node]: if nebr != par: self.size[node] += self.get_size(nebr, node) return self.size[node] # Calculates centroid of a subtree # of 'node' of size 'sz' def get_centroid(self, node, par, sz): for nebr in self.g[node]: if nebr != par and not self.removed[nebr] and self.size[nebr] > sz / 2: return self.get_centroid(nebr, node, sz) return node # Decompose the tree # into various centroids def decompose(self, node, par): self.get_size(node, -1) # c is centroid of subtree 'node' c = self.get_centroid(node, par, self.size[node]) # Find paths_originating_from 'c' self.paths_originating_from(c) # Mark this centroid as removed self.removed = True # Find other centroids for nebr in self.g: if not self.removed[nebr]: self.decompose(nebr, c) # This dfs is responsible for # calculating ans and # updating freq vector def dfs(self, node, par, depth, contri): if depth > self.k: return self.mx_depth = max(self.mx_depth, depth) if contri: self.ans += self.freq[self.k - depth] else: self.freq[depth] += 1 for nebr in self.g[node]: if nebr != par and not self.removed[nebr]: self.dfs(nebr, node, depth + 1, contri) # Function to find K-length paths # originating from node def paths_originating_from(self, node): self.mx_depth = 0 self.freq[0] = 1 # For every not-removed nebr, # calculate its contribution, # then update freq vector for it for nebr in self.g[node]: if not self.removed[nebr]: self.dfs(nebr, node, 1, True) self.dfs(nebr, node, 1, False) # Re-initialize freq vector for i in range(self.mx_depth+1): self.freq[i] = 0# Driver codeif __name__ == "__main__": N = 5 K = 2 cd_s = CD(N, K) cd_s.edge(1, 2) cd_s.edge(1, 5) cd_s.edge(2, 3) cd_s.edge(2, 4) cd_s.decompose(0, -1) print(cd_s.ans) |
Javascript
class CD { constructor(n1, k1) { this.n = n1; this.k = k1; this.ans = this.mx_depth = 0; this.removed = Array(n1).fill(false); this.size = Array(n1).fill(0); this.freq = Array(n1).fill(0); this.g = Array(n1).fill().map(() => []); } edge(u, v) { u -= 1; v -= 1; this.g[u].push(v); this.g[v].push(u); } get_size(node, par) { if (this.removed[node]) { return 0; } this.size[node] = 1; for (let nebr of this.g[node]) { if (nebr !== par) { this.size[node] += this.get_size(nebr, node); } } return this.size[node]; } get_centroid(node, par, sz) { for (let nebr of this.g[node]) { if (!this.removed[nebr] && nebr !== par && this.size[nebr] > sz / 2) { return this.get_centroid(nebr, node, sz); } } return node; } decompose(node, par) { this.get_size(node, -1); let c = this.get_centroid(node, par, this.size[node]); this.paths_originating_from(c); this.removed = true; for (let nebr of this.g) { if (!this.removed[nebr]) { this.decompose(nebr, c); } } } dfs(node, par, depth, contri) { if (depth > this.k) { return; } this.mx_depth = Math.max(this.mx_depth, depth); if (contri) { this.ans += this.freq[this.k - depth]; } else { this.freq[depth] += 1; } for (let nebr of this.g[node]) { if (nebr !== par && !this.removed[nebr]) { this.dfs(nebr, node, depth + 1, contri); } } } paths_originating_from(node) { this.mx_depth = 0; this.freq[0] = 1; for (let nebr of this.g[node]) { if (!this.removed[nebr]) { this.dfs(nebr, node, 1, true); this.dfs(nebr, node, 1, false); } } for (let i = 0; i < this.mx_depth + 1; i++) { this.freq[i] = 0; } }}const N = 5;const K = 2;const cd_s = new CD(N, K);cd_s.edge(1, 2);cd_s.edge(1, 5);cd_s.edge(2, 3);cd_s.edge(2, 4);cd_s.decompose(0, -1);console.log(cd_s.ans); |
C#
// C++ code to implement above approachusing System;using System.Collections.Generic;class CD { // 1. mx_depth will be used to store // the height of a node // 2. g[] is adjacency list for tree // 3. freq[] stores frequency of nodes // at particular height, it is maintained // for children of a node private int n, k, mx_depth, ans; private List<bool> removed; private List<int> size, freq; private List<List<int> > g; // Constructor for struct public CD(int n1, int k1) { n = n1; k = k1; ans = mx_depth = 0; g = new List<List<int> >(); for (int i = 0; i < n; i++) g.Add(new List<int>()); size = new List<int>(); size.AddRange(new int[n]); freq = new List<int>(); freq.AddRange(new int[n]); removed = new List<bool>(); removed.AddRange(new bool[n]); } // Utility method to add edges to tree public void Edge(int u, int v) { u--; v--; g[u].Add(v); g[v].Add(u); } // Finds size of a subtree, // ignoring removed nodes in the way private int GetSize(int node, int par) { if (removed[node]) return 0; size[node] = 1; foreach(int nebr in g[node]) { if (nebr != par) size[node] += GetSize(nebr, node); } return size[node]; } // Calculates centroid of a subtree // of 'node' of size 'sz' private int GetCentroid(int node, int par, int sz) { foreach(int nebr in g[node]) { if (!removed[nebr] && nebr != par && size[nebr] > sz / 2) return GetCentroid(nebr, node, sz); } return node; } // Decompose the tree // into various centroids private void Decompose(int node, int par) { GetSize(node, -1); // c is centroid of subtree 'node' int c = GetCentroid(node, par, size[node]); // Find paths_originating_from 'c' PathsOriginatingFrom(c); // Mark this centroid as removed removed = true; // Find other centroids foreach(int nebr in g) { if (!removed[nebr]) Decompose(nebr, c); } } // This dfs is responsible for // calculating ans and // updating freq vector private void Dfs(int node, int par, int depth, bool contri) { if (depth > k) return; mx_depth = Math.Max(mx_depth, depth); if (contri) ans += freq[k - depth]; else freq[depth]++; foreach(int nebr in g[node]) { if (!removed[nebr] && nebr != par) Dfs(nebr, node, depth + 1, contri); } } // Function to find K-length paths // originating from node private void PathsOriginatingFrom(int node) { mx_depth = 0; freq[0] = 1; // For every not-removed nebr, // calculate its contribution, // then update freq vector for it foreach(int nebr in g[node]) { if (!removed[nebr]) { Dfs(nebr, node, 1, true); Dfs(nebr, node, 1, false); } } for (int i = 0; i <= mx_depth; i++) freq[i] = 0; } public int Solve() { Decompose(0, -1); return ans; }}// Driver codeclass Program { static void Main(string[] args) { int N = 5, K = 2; CD cd_s = new CD(N, K); cd_s.Edge(1, 2); cd_s.Edge(1, 5); cd_s.Edge(2, 3); cd_s.Edge(2, 4); Console.WriteLine(cd_s.Solve()); }}// This code is generated by Chetan Bargal |
Output
4
Time Complexity: O(N * log(N)) where log N is the height of the tree
Auxiliary Space: O(N)
Feeling lost in the world of random DSA topics, wasting time without progress? It’s time for a change! Join our DSA course, where we’ll guide you on an exciting journey to master DSA efficiently and on schedule.
Ready to dive in? Explore our Free Demo Content and join our DSA course, trusted by over 100,000 neveropen!
Ready to dive in? Explore our Free Demo Content and join our DSA course, trusted by over 100,000 neveropen!
