Number of cyclic elements in an array where we can jump according to value
Given a array arr[] of n integers. For every value arr[i], we can move to arr[i] + 1 clockwise
considering array elements in cycle. We need to count cyclic elements in the array. An element is cyclic if starting from it and moving to arr[i] + 1 leads to same element.
Examples:
Input : arr[] = {1, 1, 1, 1}
Output : 4
All 4 elements are cyclic elements.
1 -> 3 -> 1
2 -> 4 -> 2
3 -> 1 -> 3
4 -> 2 -> 4
Input : arr[] = {3, 0, 0, 0}
Output : 1
There is one cyclic point 1,
1 -> 1
The path covered starting from 2 is
2 -> 3 -> 4 -> 1 -> 1.
The path covered starting from 3 is
2 -> 3 -> 4 -> 1 -> 1.
The path covered starting from 4 is
4 -> 1 -> 1
One simple solution is to check all elements one by one. We follow simple path starting from every element arr[i], we go to arr[i] + 1. If we come back to a visited element other than arr[i], we do not count arr[i]. Time complexity of this solution is O(n2)
An efficient solution is based on below steps.
1) Create a directed graph using array indexes as nodes. We add an edge from i to node (arr[i] + 1)%n.
2) Once a graph is created we find all strongly connected components using Kosaraju’s Algorithm
3) We finally return sum of counts of nodes in individual strongly connected component.
C++
// C++ program to count cyclic points// in an array using Kosaraju's Algorithm#include <bits/stdc++.h>using namespace std;// Most of the code is taken from below linkclass Graph { int V; list<int>* adj; void fillOrder(int v, bool visited[], stack<int>& Stack); int DFSUtil(int v, bool visited[]);public: Graph(int V); void addEdge(int v, int w); int countSCCNodes(); Graph getTranspose();};Graph::Graph(int V){ this->V = V; adj = new list<int>[V];}// Counts number of nodes reachable// from vint Graph::DFSUtil(int v, bool visited[]){ visited[v] = true; int ans = 1; list<int>::iterator i; for (i = adj[v].begin(); i != adj[v].end(); ++i) if (!visited[*i]) ans += DFSUtil(*i, visited); return ans;}Graph Graph::getTranspose(){ Graph g(V); for (int v = 0; v < V; v++) { list<int>::iterator i; for (i = adj[v].begin(); i != adj[v].end(); ++i) g.adj[*i].push_back(v); } return g;}void Graph::addEdge(int v, int w){ adj[v].push_back(w);}void Graph::fillOrder(int v, bool visited[], stack<int>& Stack){ visited[v] = true; list<int>::iterator i; for (i = adj[v].begin(); i != adj[v].end(); ++i) if (!visited[*i]) fillOrder(*i, visited, Stack); Stack.push(v);}// This function mainly returns total count of // nodes in individual SCCs using Kosaraju's// algorithm.int Graph::countSCCNodes(){ int res = 0; stack<int> Stack; bool* visited = new bool[V]; for (int i = 0; i < V; i++) visited[i] = false; for (int i = 0; i < V; i++) if (visited[i] == false) fillOrder(i, visited, Stack); Graph gr = getTranspose(); for (int i = 0; i < V; i++) visited[i] = false; while (Stack.empty() == false) { int v = Stack.top(); Stack.pop(); if (visited[v] == false) { int ans = gr.DFSUtil(v, visited); if (ans > 1) res += ans; } } return res;}// Returns count of cyclic elements in arr[]int countCyclic(int arr[], int n){ int res = 0; // Create a graph of array elements Graph g(n + 1); for (int i = 1; i <= n; i++) { int x = arr[i-1]; // If i + arr[i-1] jumps beyond last // element, we take mod considering // cyclic array int v = (x + i) % n + 1; // If there is a self loop, we // increment count of cyclic points. if (i == v) res++; g.addEdge(i, v); } // Add nodes of strongly connected components // of size more than 1. res += g.countSCCNodes(); return res;}// Driver codeint main(){ int arr[] = {1, 1, 1, 1}; int n = sizeof(arr)/sizeof(arr[0]); cout << countCyclic(arr, n); return 0;} |
Java
// Python3 program to count cyclic points// in an array using Kosaraju's Algorithm// Counts number of nodes reachable// from vimport java.io.*;import java.util.*;class GFG { static boolean[] visited = new boolean[100]; static Stack<Integer> stack = new Stack<Integer>(); static ArrayList<ArrayList<Integer>> adj = new ArrayList<ArrayList<Integer>>(); static int DFSUtil(int v) { visited[v] = true; int ans = 1; for(int i: adj.get(v)) { if(!visited[i]) { ans += DFSUtil(i); } } return ans; } static void getTranspose() { for(int v = 0; v < 5; v++) { for(int i : adj.get(v)) { adj.get(i).add(v); } } } static void addEdge(int v, int w) { adj.get(v).add(w); } static void fillOrder(int v) { visited[v] = true; for(int i: adj.get(v)) { if(!visited[i]) { fillOrder(i); } } stack.add(v); } // This function mainly returns total count of // nodes in individual SCCs using Kosaraju's // algorithm. static int countSCCNodes() { int res = 0; // stack<int> Stack; // bool* visited = new bool[V]; for(int i = 0; i < 5; i++) { if(visited[i] == false) { fillOrder(i); } } getTranspose(); for(int i = 0; i < 5; i++) { visited[i] = false; } while(stack.size() > 0) { int v = stack.get(stack.size() - 1); stack.remove(stack.size() - 1); if (visited[v] == false) { int ans = DFSUtil(v); if (ans > 1) { res += ans; } } } return res; } // Returns count of cyclic elements in arr[] static int countCyclic(int[] arr, int n) { int res = 0; // Create a graph of array elements for(int i = 1; i < n + 1; i++) { int x = arr[i - 1]; // If i + arr[i-1] jumps beyond last // element, we take mod considering // cyclic array int v = (x + i) % n + 1; // If there is a self loop, we // increment count of cyclic points. if (i == v) res++; addEdge(i, v); } // Add nodes of strongly connected components // of size more than 1. res += countSCCNodes(); return res; } // Driver code public static void main (String[] args) { int arr[] = {1, 1, 1, 1}; int n = arr.length; for(int i = 0; i < 100; i++) { adj.add(new ArrayList<Integer>()); } System.out.println(countCyclic(arr, n)); }}// This code is contributed by avanitrachhadiya2155 |
Python3
# Python3 program to count cyclic points# in an array using Kosaraju's Algorithm# Counts number of nodes reachable# from vdef DFSUtil(v): global visited, adj visited[v] = True ans = 1 for i in adj[v]: if (not visited[i]): ans += DFSUtil(i) return ansdef getTranspose(): global visited, adj for v in range(5): for i in adj[v]: adj[i].append(v)def addEdge(v, w): global visited, adj adj[v].append(w)def fillOrder(v): global Stack, adj, visited visited[v] = True for i in adj[v]: if (not visited[i]): fillOrder(i) Stack.append(v)# This function mainly returns total count of# nodes in individual SCCs using Kosaraju's# algorithm.def countSCCNodes(): global adj, visited, S res = 0 #stack<int> Stack; #bool* visited = new bool[V]; for i in range(5): if (visited[i] == False): fillOrder(i) getTranspose() for i in range(5): visited[i] = False while (len(Stack) > 0): v = Stack[-1] del Stack[-1] if (visited[v] == False): ans = DFSUtil(v) if (ans > 1): res += ans return res# Returns count of cyclic elements in arr[]def countCyclic(arr, n): global adj res = 0 # Create a graph of array elements for i in range(1, n + 1): x = arr[i - 1] # If i + arr[i-1] jumps beyond last # element, we take mod considering # cyclic array v = (x + i) % n + 1 # If there is a self loop, we # increment count of cyclic points. if (i == v): res += 1 addEdge(i, v) # Add nodes of strongly connected components # of size more than 1. res += countSCCNodes() return res# Driver codeif __name__ == '__main__': adj = [[] for i in range(100)] visited = [False for i in range(100)] arr = [ 1, 1, 1, 1 ] Stack = [] n = len(arr) print(countCyclic(arr, n))# This code is contributed by mohit kumar 29 |
C#
// C# program to count cyclic points// in an array using Kosaraju's Algorithm// Counts number of nodes reachable// from vusing System;using System.Collections.Generic;public class GFG{ static bool[] visited = new bool[100]; static List<int> stack = new List<int>(); static List<List<int>> adj = new List<List<int>>(); static int DFSUtil(int v) { visited[v] = true; int ans = 1; foreach(int i in adj[v]) { if(!visited[i]) { ans += DFSUtil(i); } } return ans; } static void getTranspose() { for(int v = 0; v < 5; v++) { foreach(int i in adj[v]) { adj[i].Add(v); } } } static void addEdge(int v, int w) { adj[v].Add(w); } static void fillOrder(int v) { visited[v] = true; foreach(int i in adj[v]) { if(!visited[i]) { fillOrder(i); } } stack.Add(v); } // This function mainly returns total count of // nodes in individual SCCs using Kosaraju's // algorithm. static int countSCCNodes() { int res = 0; // stack<int> Stack; // bool* visited = new bool[V]; for(int i = 0; i < 5; i++) { if(visited[i] == false) { fillOrder(i); } } getTranspose(); for(int i = 0; i < 5; i++) { visited[i] = false; } while(stack.Count > 0) { int v=stack[stack.Count - 1]; stack.Remove(stack.Count - 1); if (visited[v] == false) { int ans = DFSUtil(v); if (ans > 1) { res += ans; } } } return res; } // Returns count of cyclic elements in arr[] static int countCyclic(int[] arr, int n) { int res = 0; // Create a graph of array elements for(int i = 1; i < n + 1; i++) { int x = arr[i - 1]; // If i + arr[i-1] jumps beyond last // element, we take mod considering //cyclic array int v = (x + i) % n + 1; // If there is a self loop, we // increment count of cyclic points. if (i == v) res++; addEdge(i, v); } // Add nodes of strongly connected components // of size more than 1. res += countSCCNodes(); return res; } // Driver code static public void Main () { int[] arr = {1, 1, 1, 1}; int n = arr.Length; for(int i = 0; i < 100; i++) { adj.Add(new List<int>()); } Console.WriteLine(countCyclic(arr, n)); }}// This code is contributed by rag2127 |
Output:
4
Time Complexity : O(n)
Auxiliary space : O(n) Note that there are only O(n) edges.
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