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.

Implementation:

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 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<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 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<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 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<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

Javascript

<script>
// Javascript program to count cyclic points
// in an array using Kosaraju's Algorithm

// Counts number of nodes reachable
// from v

    let visited = new Array(100);
    for(let i=0;i<100;i++)
    {
        visited[i]=false;
    }
    let stack = [];
    let adj=[];

function DFSUtil(v)
{
    visited[v] = true;
    let ans = 1;
    for(let i=0;i<adj[v].length;i++)
    {
      if(!visited[adj[v][i]])
      {
        ans += DFSUtil(adj[v][i]);
      }
    }
    return ans;
}

function getTranspose()
{
    for(let v = 0; v < 5; v++)
    {
      for(let i=0;i<adj[v].length;i++)
      {
        adj[adj[v][i]].push(v);
      }
    }
}

function addEdge(v,w)
{
    adj[v].push(w);
}

function fillOrder(v)
{
    visited[v] = true;
    for(let i=0;i<adj[v].length;i++)
    {
      if(!visited[adj[v][i]])
      {
        fillOrder(adj[v][i]);
      }
    }
    stack.push(v);
}

// This function mainly returns total count of
  // nodes in individual SCCs using Kosaraju's
  // algorithm.
function countSCCNodes()
{
let res = 0;

    // stack<int> Stack;
    // bool* visited = new bool[V];
    for(let i = 0; i < 5; i++)
    {
      if(visited[i] == false)
      {
        fillOrder(i);
      }
    }
    getTranspose();
    for(let i = 0; i < 5; i++)
    {
      visited[i] = false;
    }
    while(stack.length > 0)
    {
      let v = stack[stack.length - 1];
      stack.pop();
      if (visited[v] == false)
      {
        let ans = DFSUtil(v);
        if (ans > 1)
        {
          res += ans;
        }
      }
    }
    return res;
}

// Returns count of cyclic elements in arr[]
function countCyclic(arr,n)
{
    let res = 0;

    // Create a graph of array elements
    for(let i = 1; i < n + 1; i++)
    {
      let x = arr[i - 1];

      // If i + arr[i-1] jumps beyond last
      // element, we take mod considering
      // cyclic array
      let 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
let arr=[1, 1, 1, 1];
let n = arr.length;
for(let i = 0; i < 100; i++)
{
    adj.push([]);

}
document.write(countCyclic(arr, n));



// This code is contributed by unknown2108
</script>

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 review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.