Check if equal sum components can be obtained from given Graph by removing edges from a Cycle

Given an undirected graph with N vertices and N edges that contain only one cycle, and an array arr[] of size N, where arr[i] denotes the value of the i^th node, the task is to check if the cycle can be divided into two components such that the sum of all the node values in both the components is the same.

Examples:

Input: N = 10, arr[] = {4, 2, 3, 3, 1, 2, 6, 2, 2, 5}, edges[] = {{1, 2}, {1, 5}, {1, 3}, {2, 6}, {2, 7}, {2, 4}, {4, 8}, {4, 3}, {3, 9}, {9, 10} } 

Output: Yes
Explanation: By removing the edges 1-2 and 3-4, the sum of all the node values of both the generated components is equal to 15.

Input: N= 4, arr[] = {1, 2, 3, 3}, edges[] = {{1, 2}, {2, 3}, {3, 4}, {2, 4}} 

Output: No 
Explanation: No possible way exists to obtain two equal sum components by removing an edges from the cycle.

Approach: The idea to solve this problem is to first find the nodes which are part of the cycle. Then, add the value of each node that is not a part of the cycle to its nearest node in the cycle. The final step involves checking whether the cycle can be divided into two equal sum components. Below are the steps:

• The first step is to find all the nodes which are a part of a cycle using detect cycle in an undirected graph using DFS.
• Perform the DFS Traversal on the given graph and do the following:
  • Mark the current node as visited. For each unvisited node connected to the current node recursively perform the DFS Traversal for each node.
  • If the adjacent node of the current node is already visited and is not the same as the previous node of the current node then it means that the current node is part of the cycle. Backtrack till this specific adjacent node is reached to find all nodes that are part of the cycle and store them in a vector inCycle[].
• Find the sum of values for each node in the inCycle[] and add this sum to the current inCycle[] node value.
• Find the total sum of values of all the nodes present in inCycle[] and store it in a variable totalSum. If the value of totalSum is odd then print “No” because the cycle with an odd sum cannot be divided into two equal sum components.
• Otherwise, check if the value of totalSum/2 is present in inCycle[] then print “Yes” Else print “No”.

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Recursive function to find all the
// nodes that are part of the cycle
void findNodesinCycle(int u, bool* vis,
                      int* prev,
                      vector<int> adj[],
                      vector<int>& inCycle)
{
    // Mark current node as visited
    vis[u] = true;

    for (int v : adj[u]) {

        // If node v is not visited
        if (!vis[v]) {

            prev[v] = u;

            // Recursively find cycle
            findNodesinCycle(v, vis, prev,
                             adj, inCycle);

            // If cycle is detected then
            // return the previous node
            if (!inCycle.empty())
                return;
        }

        // If node is already visited
        // and not equal to prev[u]
        else if (v != prev[u]) {
            int curr = u;
            inCycle.push_back(curr);

            // Backtrack all vertices
            // that are part of cycle
            // and store them in inCycle
            while (curr != v) {
                curr = prev[curr];
                inCycle.push_back(curr);
            }

            // As the cycle is detected
            // return the previous node
            return;
        }
    }
}

// Function to add the value of each
// node which is not part of the cycle
// to its nearest node in the cycle
int sumOfnonCycleNodes(
    int u, vector<int> adj[],
    vector<int> inCycle, bool* vis,
    int arr[])
{
    // Mark the current node as visited
    vis[u] = true;

    // Stores the value of required sum
    int sum = 0;

    for (int v : adj[u]) {

        // If v is not already visited
        // and not present in inCycle
        if (!vis[v]

            && find(inCycle.begin(),
                    inCycle.end(), v)
                   == inCycle.end()) {

            // Add to sum and call the
            // function recursively
            sum += (arr[v - 1]
                    + sumOfnonCycleNodes(
                          v, adj, inCycle,
                          vis, arr));
        }
    }

    // Return the value of sum
    return sum;
}

// Function that add the edges
// to the graph
void addEdge(vector<int> adj[],
             int u, int v)
{
    adj[u].push_back(v);
    adj[v].push_back(u);
}

// Utility Function to check if the cycle
// can be divided into two same sum
bool isBreakingPossible(vector<int> adj[],
                        int arr[], int N)
{
    // Stores all the nodes that are
    // part of the cycle
    vector<int> inCycle;

    // Array to check if a node is
    // already visited or not
    bool vis[N + 1];

    // Initialize vis to false
    memset(vis, false, sizeof vis);

    // Array to store the previous node
    // of the current node
    int prev[N + 1];

    // Initialize prev to 0
    memset(prev, 0, sizeof prev);

    // Recursive function call
    findNodesinCycle(1, vis, prev,
                     adj, inCycle);

    memset(vis, false, sizeof vis);

    // Update value of each inCycle
    // node
    for (int u : inCycle) {

        // Add sum of values of all
        // required node to current
        // inCycle node value
        arr[u - 1] += sumOfnonCycleNodes(
            u, adj, inCycle, vis, arr);
    }

    // Stores total sum of values of
    // all nodes present in inCycle
    int tot_sum = 0;

    // Find the total required sum
    for (int node : inCycle) {
        tot_sum += arr[node - 1];
    }

    // If value of tot_sum is odd
    // then return false
    if (tot_sum % 2 != 0)
        return false;

    int req_sum = tot_sum / 2;

    // Create an empty map
    unordered_map<int, int> map;

    // Initialise map[0]
    map[0] = -1;

    // Maintain the sum of values of
    // nodes so far
    int curr_sum = 0;

    for (int i = 0; i < inCycle.size(); i++) {

        // Add the current node value
        // to curr_sum
        curr_sum += arr[inCycle[i] - 1];

        // If curr_sum - req_sum in map
        // then there is a subarray of
        // nodes with sum of their values
        // equal to req_sum
        if (map.find(curr_sum - req_sum)
            != map.end()) {
            return true;
        }

        map[curr_sum] = i;
    }

    // If no such subarray exists
    return false;
}

// Function to check if the cycle can
// be divided into two same sum
void checkCycleDivided(int edges[][2],
                       int arr[],
                       int N)
{
    vector<int> adj[N + 1];

    // Traverse the given edges
    for (int i = 0; i < N; i++) {

        int u = edges[i][0];
        int v = edges[i][1];

        // Add the edges
        addEdge(adj, u, v);
    }

    // Print the result
    cout << (isBreakingPossible(
                 adj, arr, N)
                 ? "Yes"
                 : "No");
}

// Driver Code
int main()
{
    int N = 10;
    int edges[][2] = { { 1, 2 }, { 1, 5 }, { 1, 3 }, { 2, 6 }, { 2, 7 }, { 2, 4 }, { 4, 8 }, { 4, 3 }, { 3, 9 }, { 9, 10 } };
    int arr[] = { 4, 2, 3, 3, 1,
                  2, 6, 2, 2, 5 };

    // Function Call
    checkCycleDivided(edges, arr, N);

    return 0;
}

Java

// Java program for the above approach
import java.util.*;
class GFG
{

  // Recursive function to find all the
  // nodes that are part of the cycle
  static void findNodesinCycle(int u, boolean[] vis,
                               int[] prev,
                               ArrayList<ArrayList<Integer>> adj,
                               ArrayList<Integer> inCycle)
  {

    // Mark current node as visited
    vis[u] = true;
    for (int v : adj.get(u))
    {

      // If node v is not visited
      if (!vis[v])
      {
        prev[v] = u;

        // Recursively find cycle
        findNodesinCycle(v, vis, prev,
                         adj, inCycle);

        // If cycle is detected then
        // return the previous node
        if (inCycle.size() > 0)
          return;
      }

      // If node is already visited
      // and not equal to prev[u]
      else if (v != prev[u])
      {
        int curr = u;
        inCycle.add(curr);

        // Backtrack all vertices
        // that are part of cycle
        // and store them in inCycle
        while (curr != v)
        {
          curr = prev[curr];
          inCycle.add(curr);
        }

        // As the cycle is detected
        // return the previous node
        return;
      }
    }
  }

  // Function to add the value of each
  // node which is not part of the cycle
  // to its nearest node in the cycle
  static int sumOfnonCycleNodes(
    int u, ArrayList<ArrayList<Integer>> adj,
    ArrayList<Integer> inCycle, boolean[] vis,
    int arr[])
  {

    // Mark the current node as visited
    vis[u] = true;

    // Stores the value of required sum
    int sum = 0;

    for (int v : adj.get(u))
    {

      // If v is not already visited
      // and not present in inCycle
      if (!vis[v]
          && !inCycle.contains(v))
      {

        // Add to sum and call the
        // function recursively
        sum += (arr[v - 1] + sumOfnonCycleNodes(
          v, adj, inCycle, vis, arr));
      }
    }

    // Return the value of sum
    return sum;
  }

  // Utility Function to check if the cycle
  // can be divided into two same sum
  static boolean isBreakingPossible(ArrayList<ArrayList<Integer>> adj,
                                    int arr[], int N)
  {

    // Stores all the nodes that are
    // part of the cycle
    ArrayList<Integer> inCycle = new ArrayList<>();

    // Array to check if a node is
    // already visited or not
    boolean[] vis = new boolean[N + 1];

    // Array to store the previous node
    // of the current node
    int[] prev = new int[N + 1];

    // Recursive function call
    findNodesinCycle(1, vis, prev,
                     adj, inCycle);

    Arrays.fill(vis,false);

    // Update value of each inCycle
    // node
    for (Integer u : inCycle)
    {

      // Add sum of values of all
      // required node to current
      // inCycle node value
      arr[u - 1] += sumOfnonCycleNodes(
        u, adj, inCycle, vis, arr);
    }

    // Stores total sum of values of
    // all nodes present in inCycle
    int tot_sum = 0;

    // Find the total required sum
    for (int node : inCycle)
    {
      tot_sum += arr[node - 1];
    }

    // If value of tot_sum is odd
    // then return false
    if (tot_sum % 2 != 0)
      return false;
    int req_sum = tot_sum / 2;

    // Create an empty map
    Map<Integer, Integer> map = new HashMap<>();

    // Initialise map[0]
    map.put(0, -1);

    // Maintain the sum of values of
    // nodes so far
    int curr_sum = 0;
    for (int i = 0; i < inCycle.size(); i++)
    {

      // Add the current node value
      // to curr_sum
      curr_sum += arr[inCycle.get(i) - 1];

      // If curr_sum - req_sum in map
      // then there is a subarray of
      // nodes with sum of their values
      // equal to req_sum
      if (map.containsKey(curr_sum - req_sum))
      {
        return true;
      }
      map.put(curr_sum, i);
    }

    // If no such subarray exists
    return false;
  }

  // Function to check if the cycle can
  // be divided into two same sum
  static void checkCycleDivided(int edges[][],
                                int arr[], int N)
  {
    ArrayList<ArrayList<Integer>> adj = new ArrayList<>();

    for(int i = 0; i <= N; i++)
      adj.add(new ArrayList<>());

    // Traverse the given edges
    for (int i = 0; i < N; i++)
    {

      int u = edges[i][0];
      int v = edges[i][1];

      // Add the edges
      adj.get(u).add(v);
      adj.get(v).add(u);
    }

    // Print the result
    System.out.print(isBreakingPossible(
      adj, arr, N) ? "Yes" : "No");
  }

  // Driver code
  public static void main (String[] args)
  {
    int N = 10;
    int edges[][] = { { 1, 2 }, { 1, 5 },
                     { 1, 3 }, { 2, 6 },
                     { 2, 7 }, { 2, 4 },
                     { 4, 8 }, { 4, 3 },
                     { 3, 9 }, { 9, 10 } };
    int arr[] = { 4, 2, 3, 3, 1,
                 2, 6, 2, 2, 5 };

    // Function Call
    checkCycleDivided(edges, arr, N);
  }
}

// This code is contributed by offbeat

Python3

# Python3 program for the above approach

# Recursive function to find all the
# nodes that are part of the cycle
def findNodesinCycle(u):
    global adj, pre, inCycle, vis
    vis[u] = True
    for v in adj[u]:

        # If node v is not visited
        if (not vis[v]):
            pre[v] = u

            # Recursively find cycle
            findNodesinCycle(v)

            # If cycle is detected then
            # return the previous node
            if (len(inCycle) > 0):
                return

        # If node is already visited
        # and not equal to prev[u]
        elif (v != pre[u]):
            curr = u
            inCycle.append(curr)

            # Backtrack all vertices
            # that are part of cycle
            # and store them in inCycle
            while (curr != v):
                curr = pre[curr]
                inCycle.append(curr)

            # As the cycle is detected
            # return the previous node
            return

# Function to add the value of each
# node which is not part of the cycle
# to its nearest node in the cycle
def sumOfnonCycleNodes(u, arr):
    global adj, pre, inCycle, vis
    vis[u] = True

    # Stores the value of required sum
    sum = 0
    for v in adj[u]:

        # If v is not already visited
        # and not present in inCycle
        if (not vis[v]) and (v not in inCycle):

            # Add to sum and call the
            # function recursively
            sum += (arr[v - 1] + sumOfnonCycleNodes(v, arr))

    # Return the value of sum
    return sum

# Function that add the edges
# to the graph
def addEdge(u, v):
    global adj
    adj[u].append(v)
    adj[v].append(u)

# Utility Function to check if the cycle
# can be divided into two same sum
def isBreakingPossible(arr, N):

    # Stores all the nodes that are
    global adj, vis, pre

    # Recursive function call
    findNodesinCycle(1,)

    for i in range(N + 1):
        vis[i] = False

    # Update value of each inCycle
    # node
    for u in inCycle:

        # Add sum of values of all
        # required node to current
        # inCycle node value
        arr[u - 1] += sumOfnonCycleNodes(u, arr)

    # Stores total sum of values of
    # all nodes present in inCycle
    tot_sum = 0

    # Find the total required sum
    for node in inCycle:
        tot_sum += arr[node - 1]

    # If value of tot_sum is odd
    # then return false
    if (tot_sum % 2 != 0):
        return False

    req_sum = tot_sum // 2

    # Create an empty map
    map = {}

    # Initialise map[0]
    map[0] = -1

    # Maintain the sum of values of
    # nodes so far
    curr_sum = 0
    for i in range(len(inCycle)):

        # Add the current node value
        # to curr_sum
        curr_sum += arr[inCycle[i] - 1]

        # If curr_sum - req_sum in map
        # then there is a subarray of
        # nodes with sum of their values
        # equal to req_sum
        if ((curr_sum - req_sum) in map):
            return True
        map[curr_sum] = i

    # If no such subarray exists
    return False

# Function to check if the cycle can
# be divided into two same sum
def checkCycleDivided(edges, arr, N):
    global adj

    # Traverse the given edges
    for i in range(N):

        u = edges[i][0]
        v = edges[i][1]

        # Add the edges
        addEdge(u, v)

    # Print the result
    print("Yes" if isBreakingPossible(arr, N) else "No")

# Driver Code
if __name__ == '__main__':
    N = 10
    edges= [[1, 2], [1, 5], [1, 3],
            [2, 6], [2, 7], [2, 4],
            [4, 8], [4, 3], [3, 9],
            [9, 10]]
    arr, adj, vis =  [4, 2, 3, 3, 1,
                     2, 6, 2, 2, 5], [[] for i in range(N + 1)], [False for i in range(N+1)]

    inCycle, pre =[], [0 for i in range(N+1)]
    checkCycleDivided(edges, arr, N)

    # This code is contributed by mohit kumar 29

C#

// C# program for the above approach
using System;
using System.Collections.Generic;

public class GFG
{

  // Recursive function to find all the
  // nodes that are part of the cycle
  static void findNodesinCycle(int u, bool[] vis,int[] prev,
                               List<List<int>> adj,List<int> inCycle)
  {

    // Mark current node as visited
    vis[u] = true;

    foreach(int v in adj[u])
    {

      // If node v is not visited
      if (!vis[v])
      {
        prev[v] = u;

        // Recursively find cycle
        findNodesinCycle(v, vis, prev,adj, inCycle);

        // If cycle is detected then
        // return the previous node
        if (inCycle.Count > 0)
          return;
      }

      // If node is already visited
      // and not equal to prev[u]
      else if (v != prev[u])
      {
        int curr = u;
        inCycle.Add(curr);

        // Backtrack all vertices
        // that are part of cycle
        // and store them in inCycle

        while (curr != v)
        {
          curr = prev[curr];
          inCycle.Add(curr);
        }

        // As the cycle is detected
        // return the previous node
        return;
      }
    }
  }

  // Function to add the value of each
  // node which is not part of the cycle
  // to its nearest node in the cycle
  static int sumOfnonCycleNodes(int u, List<List<int>> adj,List<int> inCycle, bool[] vis,int[] arr)
  {
    // Mark the current node as visited
    vis[u] = true;

    // Stores the value of required sum
    int sum = 0;

    foreach(int v in adj[u])
    {
      // If v is not already visited
      // and not present in inCycle
      if (!vis[v] && !inCycle.Contains(v))
      {
        // Add to sum and call the
        // function recursively
        sum += (arr[v - 1] + sumOfnonCycleNodes(v, adj, inCycle, vis, arr));
      }
    }

    // Return the value of sum
    return sum;
  }

  // Utility Function to check if the cycle
  // can be divided into two same sum
  static bool isBreakingPossible(List<List<int>> adj,int[] arr, int N)
  {

    // Stores all the nodes that are
    // part of the cycle
    List<int> inCycle = new List<int>();

    // Array to check if a node is
    // already visited or not
    bool[] vis = new bool[N + 1];

    // Array to store the previous node
    // of the current node
    int[] prev = new int[N + 1];

    // Recursive function call
    findNodesinCycle(1, vis, prev, adj, inCycle);

    // Update value of each inCycle
    // node
    foreach(int u in inCycle)
    {

      // Add sum of values of all
      // required node to current
      // inCycle node value
      arr[u - 1] += sumOfnonCycleNodes(u, adj, inCycle, vis, arr);
    }

    // Stores total sum of values of
    // all nodes present in inCycle
    int tot_sum = 0;

    // Find the total required sum
    foreach(int node in inCycle)
    {
      tot_sum += arr[node - 1];
    }

    // If value of tot_sum is odd
    // then return false
    if (tot_sum % 2 != 0)
      return false;
    int req_sum = tot_sum / 2;

    // Create an empty map
    Dictionary<int, int> map = new Dictionary<int, int>();

    // Initialise map[0]
    map.Add(0, -1);

    // Maintain the sum of values of
    // nodes so far
    int curr_sum = 0;
    for (int i = 0; i < inCycle.Count; i++)
    {

      // Add the current node value
      // to curr_sum
      curr_sum += arr[inCycle[i] - 1];

      // If curr_sum - req_sum in map
      // then there is a subarray of
      // nodes with sum of their values
      // equal to req_sum
      if (map.ContainsKey(curr_sum - req_sum))
      {
        return true;
      }
      map.Add(curr_sum, i);
    }

    // If no such subarray exists
    return false;
  }

  // Function to check if the cycle can
  // be divided into two same sum
  static void checkCycleDivided(int[,] edges,int[] arr, int N)
  {
    List<List<int>> adj = new List<List<int>>();
    for(int i = 0; i <= N; i++)
    {
      adj.Add(new List<int>());

    }

    // Traverse the given edges
    for (int i = 0; i < N; i++)
    {

      int u = edges[i,0];
      int v = edges[i,1];

      // Add the edges
      adj[u].Add(v);
      adj[v].Add(u);
    }

    // Print the result
    Console.Write(isBreakingPossible(adj, arr, N) ? "Yes" : "No");
  }

  // Driver code
  static public void Main (){
    int N = 10;
    int[,] edges = { { 1, 2 }, { 1, 5 },
                    { 1, 3 }, { 2, 6 },
                    { 2, 7 }, { 2, 4 },
                    { 4, 8 }, { 4, 3 },
                    { 3, 9 }, { 9, 10 } };
    int[] arr = { 4, 2, 3, 3, 1,
                 2, 6, 2, 2, 5 };

    // Function Call
    checkCycleDivided(edges, arr, N);
  }
}

// This code is contributed by avanitrachhadiya2155

Javascript

<script>

// JavaScript program for the above approach

// Recursive function to find all the
  // nodes that are part of the cycle
function findNodesinCycle(u,vis,prev,adj,inCycle)
{
    // Mark current node as visited
    vis[u] = true;
    for (let v=0;v< adj[u].length;v++)
    {

      // If node v is not visited
      if (!vis[adj[u][v]])
      {
        prev[adj[u][v]] = u;

        // Recursively find cycle
        findNodesinCycle(adj[u][v], vis, prev,
                         adj, inCycle);

        // If cycle is detected then
        // return the previous node
        if (inCycle.length > 0)
          return;
      }

      // If node is already visited
      // and not equal to prev[u]
      else if (adj[u][v] != prev[u])
      {
        let curr = u;
        inCycle.push(curr);

        // Backtrack all vertices
        // that are part of cycle
        // and store them in inCycle
        while (curr != adj[u][v])
        {
          curr = prev[curr];
          inCycle.push(curr);
        }

        // As the cycle is detected
        // return the previous node
        return;
      }
    }
}

// Function to add the value of each
  // node which is not part of the cycle
  // to its nearest node in the cycle
function sumOfnonCycleNodes(u,adj,inCycle,vis,arr)
{
    // Mark the current node as visited
    vis[u] = true;

    // Stores the value of required sum
    let sum = 0;

    for (let v=0;v< adj[u].length;v++)
    {

      // If v is not already visited
      // and not present in inCycle
      if (!vis[adj[u][v]]
          && !inCycle.includes(adj[u][v]))
      {

        // Add to sum and call the
        // function recursively
        sum += (arr[adj[u][v] - 1] + sumOfnonCycleNodes(
          adj[u][v], adj, inCycle, vis, arr));
      }
    }

    // Return the value of sum
    return sum;
}

// Utility Function to check if the cycle
  // can be divided into two same sum
function isBreakingPossible(adj,arr,N)
{
    // Stores all the nodes that are
    // part of the cycle
    let inCycle = [];

    // Array to check if a node is
    // already visited or not
    let vis = new Array(N + 1);

    // Array to store the previous node
    // of the current node
    let prev = new Array(N + 1);

    // Recursive function call
    findNodesinCycle(1, vis, prev,
                     adj, inCycle);

    for(let i=0;i<vis.length;i++)
    {
        vis[i]=false;
    }

    // Update value of each inCycle
    // node
    for (let u=0;u< inCycle.length;u++)
    {

      // Add sum of values of all
      // required node to current
      // inCycle node value
      arr[inCycle[u] - 1] += sumOfnonCycleNodes(
        inCycle[u], adj, inCycle, vis, arr);
    }

    // Stores total sum of values of
    // all nodes present in inCycle
    let tot_sum = 0;

    // Find the total required sum
    for (let node=0;node< inCycle.length;node++)
    {
      tot_sum += arr[inCycle[node] - 1];
    }

    // If value of tot_sum is odd
    // then return false
    if (tot_sum % 2 != 0)
      return false;
    let req_sum = tot_sum / 2;

    // Create an empty map
    let map = new Map();

    // Initialise map[0]
    map.set(0, -1);

    // Maintain the sum of values of
    // nodes so far
    let curr_sum = 0;
    for (let i = 0; i < inCycle.length; i++)
    {

      // Add the current node value
      // to curr_sum
      curr_sum += arr[inCycle[i] - 1];

      // If curr_sum - req_sum in map
      // then there is a subarray of
      // nodes with sum of their values
      // equal to req_sum
      if (map.has(curr_sum - req_sum))
      {
        return true;
      }
      map.set(curr_sum, i);
    }

    // If no such subarray exists
    return false;
}

// Function to check if the cycle can
  // be divided into two same sum
function checkCycleDivided(edges,arr,N)
{
    let adj = [];

    for(let i = 0; i <= N; i++)
      adj.push([]);

    // Traverse the given edges
    for (let i = 0; i < N; i++)
    {

      let u = edges[i][0];
      let v = edges[i][1];

      // Add the edges
      adj[u].push(v);
      adj[v].push(u);
    }

    // Print the result
    document.write(isBreakingPossible(
      adj, arr, N) ? "Yes" : "No");
}

  // Driver code
let N = 10;
let edges  = [[ 1, 2 ], [ 1, 5 ],
                     [ 1, 3 ], [ 2, 6 ],
                     [ 2, 7 ], [ 2, 4 ],
                     [ 4, 8 ], [ 4, 3 ],
                     [ 3, 9 ], [ 9, 10 ]];
let arr=[4, 2, 3, 3, 1,
                 2, 6, 2, 2, 5];
// Function Call
checkCycleDivided(edges, arr, N);


// This code is contributed by patel2127

</script>

Output:

Yes

Time Complexity: O(N)
Auxiliary Space: O(N)