Related Articles

# Number of cyclic elements in an array where we can jump according to value

• Difficulty Level : Hard
• Last Updated : 20 May, 2021

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 ``using` `namespace` `std;` `// Most of the code is taken from below link``// https://www.geeksforgeeks.org/strongly-connected-components/``class` `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 v``int` `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 code``int` `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 v``import` `java.io.*;``import` `java.util.*;``class` `GFG``{``  ``static` `boolean``[] visited = ``new` `boolean``[``100``];``  ``static` `Stack stack = ``new` `Stack();``  ``static` `ArrayList> adj =``    ``new` `ArrayList>();` `  ``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 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());` `    ``}``    ``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 v``def` `DFSUtil(v):``    ` `    ``global` `visited, adj` `    ``visited[v] ``=` `True``    ``ans ``=` `1` `    ``for` `i ``in` `adj[v]:``        ``if` `(``not` `visited[i]):``           ``ans ``+``=` `DFSUtil(i)``           ` `    ``return` `ans` `def` `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 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 code``if` `__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 v``using` `System;``using` `System.Collections.Generic;``public` `class` `GFG``{` `  ``static` `bool``[] visited = ``new` `bool``[100];``  ``static` `List<``int``> stack = ``new` `List<``int``>();``  ``static` `List> adj = ``new` `List>();``  ``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 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`

## Javascript

 ``

Output:

`4`

Time Complexity : O(n)
Auxiliary space : O(n) Note that there are only O(n) edges.

This article is contributed by Mohak Agrawal. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.