Check if the array can be sorted using swaps between given indices only

Given an array arr[] of size N consisting of distinct integers from range [0, N – 1] arranged in a random order. Also given a few pairs where each pair denotes the indices where the elements of the array can be swapped. There is no limit on the number of swaps allowed. The task is to find if it is possible to arrange the array in ascending order using these swaps. If possible then print Yes else print No.

Examples:

Input: arr[] = {0, 4, 3, 2, 1, 5}, pairs[][] = {{1, 4}, {2, 3}}
Output: Yes
swap(arr[1], arr[4]) -> arr[] = {0, 1, 3, 2, 4, 5}
swap(arr[2], arr[3]) -> arr[] = {0, 1, 2, 3, 4, 5}

Input: arr[] = {1, 2, 3, 0, 4}, pairs[][] = {{2, 3}}
Output: No

Approach: The given problem can be considered as a graph problem where N denotes the total number of nodes in the graph and each swapping pair denotes an undirected edge in the graph. We have to find out if it is possible to convert the input array in the form of {0, 1, 2, 3, …, N – 1}.
Let us call the above array as B. Now find out all the connected components of both the arrays and if the elements differ for at least one component then the answer is No else the answer is Yes.

Below is the implementation of the above approach:

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function that returns true if the array elements
// can be sorted with the given operation
bool canBeSorted(int N, vector<int> a, int P, 
vector<pair<int, int> > vp)
{
  
    // To create the adjacency list of the graph
    vector<int> v[N];
  
    // Boolean array to mark the visited nodes
    bool vis[N] = { false };
  
    // Creating adjacency list for undirected graph
    for (int i = 0; i < P; i++) {
        v[vp[i].first].push_back(vp[i].second);
        v[vp[i].second].push_back(vp[i].first);
    }
  
    for (int i = 0; i < N; i++) {
  
        // If not already visited
        // then apply BFS
        if (!vis[i]) {
            queue<int> q;
            vector<int> v1;
            vector<int> v2;
  
            // Set visited to true
            vis[i] = true;
  
            // Push the node to the queue
            q.push(i);
  
            // While queue is not empty
            while (!q.empty()) {
                int u = q.front();
                v1.push_back(u);
                v2.push_back(a[u]);
                q.pop();
  
                // Check all the adjacent nodes
                for (auto s : v[u]) {
  
                    // If not visited
                    if (!vis[s]) {
  
                        // Set visited to true
                        vis[s] = true;
                        q.push(s);
                    }
                }
            }
            sort(v1.begin(), v1.end());
            sort(v2.begin(), v2.end());
  
            // If the connected component does not
            // contain same elements then return false
            if (v1 != v2)
                return false;
        }
    }
    return true;
}
  
// Driver code
int main()
{
    vector<int> a = { 0, 4, 3, 2, 1, 5 };
    int n = a.size();
    vector<pair<int, int> > vp = { { 1, 4 }, { 2, 3 } };
    int p = vp.size();
  
    if (canBeSorted(n, a, p, vp))
        cout << "Yes";
    else
        cout << "No";
  
    return 0;
}

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Output:

Yes


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