⚡ Match Highlights: dsa/check if chords of a circle are symmetric after some rotation

Given two Integers N and M, N indicating equidistant points on circumference of a circle and M indicating number of chords formed with those points. Also given is a vector of pairs C containing position of chords. The task is to rotate the circle by any degree, say X, where 0 < X < 360, and check if the chords of are still symmetric to the original circle.

Example:

Input: N = 12, M = 6, C = {{1, 3}, {3, 7}, {5, 7}, {7, 11}, {9, 11}, {11, 3}};
Output: YES

Original
After Rotation

Input: N = 10, M = 3, C = {{1, 2}, {3, 2}, {7, 2}}
Output: NO

No rotational symmetry possible

Naive Approach: Rotate for every distance K in the range[1, N] and check for each point [a, b] if the rotated point [a + K, b + K] exits.
If there exists any k then print YES else print NO
Time Complexity: O(N*M)

Efficient Approach: It is enough to check for the divisors of N.
Let us suppose if we rotate the image by K units then the whole image will be divided into N/K blocks. Then if K is not a divisor of N, there will be an asymmetric block of length less than K and the image will never be symmetric to the original figure.
So calculate all the divisors of N and check for each chord the rotated chord exists or not.

Below is the implementation of the above approach:

C++

// C++ Program to for the above approach

#include <bits/stdc++.h>
using namespace std;

// Utility function to calculate
// divisors of a number in O(sqrt(N))
vector<int> calculateDivisors(int N)
{
    vector<int> div;
    for (int i = 1; i * i <= N; i++) {
        if (N % i == 0) {
            div.push_back(i);
            if (N / i != i && i != 1) {
                div.push_back(N / i);
            }
        }
    }
    return div;
}
int checkRotationallySymmetric(
    vector<pair<int, int> > A,
    int N, int M)
{
    // Maintain a set to check quickly
    // the presence of a chord
    set<pair<int, int> > st;
    for (int i = 0; i < M; i++) {
        --A[i].first, --A[i].second;
        if (A[i].first > A[i].second) {
            swap(A[i].first, A[i].second);
        }
        st.insert(A[i]);
    }

    // Calculate the divisors of N.
    vector<int> div = calculateDivisors(N);

    // Iterate through the divisors
    for (auto x : div) {
        bool exist = 1;
        for (int i = 0; i < M; i++) {
            int dx = (A[i].first + x) % N;
            int dy = (A[i].second + x) % N;
            if (dx > dy) {
                swap(dx, dy);
            }
            if (st.find({ dx, dy }) != st.end()) {

                // There exists a valid
                // chord after rotation
            }
            else {

                // There is no valid chord after rotation
                exist = false;
                break;
            }
        }

        // if there exist another chord after
        // rotation for every other chord print
        // YES and exit the function
        if (exist) {
            cout << "YES";
            return 0;
        }
    }
    cout << "NO";
    return 0;
}

// Driver Code
int main()
{
    int N = 12, M = 6;
    vector<pair<int, int> > C
        = { { 1, 3 }, { 3, 7 }, { 5, 7 },
             { 7, 11 }, { 9, 11 }, { 11, 3 } };
    checkRotationallySymmetric(C, N, M);
    return 0;
}

Java

import java.util.ArrayList;
import java.util.HashSet;
import java.util.List;
import java.util.Set;

class GFG {

  // function to calculate divisors
  static List<Integer> calculateDivisors(int N)
  {
    List<Integer> div = new ArrayList<>();

    // checking all possible divisors
    for (int i = 1; i * i <= N; i++) {
      if (N % i == 0) {
        div.add(i);
        if (N / i != i && i != 1) {
          div.add(N / i);
        }
      }
    }
    return div;
  }

  static int checkRotationallySymmetric(List<int[]> A,
                                        int N, int M)
  {
    // Maintain a set to check quickly
    // the presence of a chord
    Set<String> st = new HashSet<>();
    for (int i = 0; i < M; i++) {
      A.get(i)[0] = A.get(i)[0] - 1;
      A.get(i)[1] = A.get(i)[1] - 1;
      if (A.get(i)[0] > A.get(i)[1]) {
        int temp = A.get(i)[0];
        A.get(i)[0] = A.get(i)[1];
        A.get(i)[1] = temp;
      }
      st.add(A.get(i)[0] + " " + A.get(i)[1]);
    }

    // Calculate the divisors of N.
    List<Integer> div = calculateDivisors(N);

    // Iterate through the divisors
    for (int x : div) {
      boolean exist = true;
      for (int i = 0; i < M; i++) {
        int dx = (A.get(i)[0] + x) % N;
        int dy = (A.get(i)[1] + x) % N;
        if (dx > dy) {
          int temp = dx;
          dx = dy;
          dy = temp;
        }
        if (st.contains(dx + " " + dy)) {

          // There exists a valid chord after
          // rotation
        }
        else {

          // There is no valid chord after
          // rotation
          exist = false;
          break;
        }
      }

      // if there exist another chord after rotation
      // for every other chord print YES and exit the
      // function
      if (exist) {
        System.out.println("YES");
        return 0;
      }
    }
    System.out.println("NO");
    return 0;
  }

  // Driver Code
  public static void main(String[] args)
  {
    int N = 12, M = 6;
    List<int[]> C = new ArrayList<>();
    C.add(new int[] { 1, 3 });
    C.add(new int[] { 3, 7 });
    C.add(new int[] { 5, 7 });
    C.add(new int[] { 7, 11 });
    C.add(new int[] { 9, 11 });
    C.add(new int[] { 11, 3 });

    // function call
    checkRotationallySymmetric(C, N, M);
  }
}

// This code is contributed by phasing17

Python3

# Python3 program to implement the approach
from typing import List, Tuple

def calculateDivisors(N: int) -> List[int]:
    """
    Utility function to calculate
    divisors of a number in O(sqrt(N))
    """
    div = []
    for i in range(1, int((N ** 0.5) + 1)):
        if N % i == 0:
            div.append(i)
            if N // i != i and i != 1:
                div.append(N // i)
    return div

def checkRotationallySymmetric(A: List[Tuple[int, int]], N: int, M: int) -> None:
    """
    Main function to check rotationally symmetric
    """
    # Maintain a set to check quickly
    # the presence of a chord
    st = set()
    for i in range(M):
        A[i] = (A[i][0] - 1, A[i][1] - 1)
        if A[i][0] > A[i][1]:
            A[i] = (A[i][1], A[i][0])
        st.add(tuple(A[i]))

    # Calculate the divisors of N.
    div = calculateDivisors(N)

    # Iterate through the divisors
    for j in range(len(div)):
        exist = True
        for i in range(M):
            dx = (A[i][0] + div[j]) % N
            dy = (A[i][1] + div[j]) % N
            if dx > dy:
                # swapping dx and dy. 
                temp = dx
                dx = dy
                dy = temp
            if tuple((dx, dy)) in st:
                # There exists a valid chord after rotation
                pass
            else:
                # There is no valid chord after rotation
                exist = False
                break
        # if there exist another chord after rotation for every other chord print YES and exit the function
        if exist:
            print("YES")
            return
    print("NO")
    return

# Driver code
N = 12
M = 6
C = [(1, 3), (3, 7), (5, 7), (7, 11), (9, 11), (11, 3)]
checkRotationallySymmetric(C, N, M)


# This code is contributed by phasing17

// C# code to implement the above approach
using System;
using System.Collections.Generic;

class GFG {

  // function to calculate divisors
  static List<int> calculateDivisors(int N)
  {
    List<int> div = new List<int>();

    // checking all possible divisors
    for (int i = 1; i * i <= N; i++) {
      if (N % i == 0) {
        div.Add(i);
        if (N / i != i && i != 1) {
          div.Add(N / i);
        }
      }
    }
    return div;
  }

  static int
    checkRotationallySymmetric(List<Tuple<int, int> > A,
                               int N, int M)
  {
    // Maintain a set to check quickly
    // the presence of a chord
    HashSet<Tuple<int, int> > st
      = new HashSet<Tuple<int, int> >();
    for (int i = 0; i < M; i++) {
      A[i] = new Tuple<int, int>(A[i].Item1 - 1,
                                 A[i].Item2 - 1);
      if (A[i].Item1 > A[i].Item2) {
        A[i] = new Tuple<int, int>(A[i].Item2,
                                   A[i].Item1);
      }
      st.Add(A[i]);
    }

    // Calculate the divisors of N.
    List<int> div = calculateDivisors(N);

    // Iterate through the divisors
    foreach(var x in div)
    {
      bool exist = true;
      for (int i = 0; i < M; i++) {
        int dx = (A[i].Item1 + x) % N;
        int dy = (A[i].Item2 + x) % N;
        if (dx > dy) {
          Tuple<int, int> temp
            = new Tuple<int, int>(dy, dx);
          dx = temp.Item1;
          dy = temp.Item2;
        }
        if (st.Contains(
          new Tuple<int, int>(dx, dy))) {

          // There exists a valid
          // chord after rotation
        }
        else {

          // There is no valid chord after
          // rotation
          exist = false;
          break;
        }
      }

      // if there exist another chord after
      // rotation for every other chord print
      // YES and exit the function
      if (exist) {
        Console.WriteLine("YES");
        return 0;
      }
    }
    Console.WriteLine("NO");
    return 0;
  }

  // Driver Code
  static void Main(string[] args)
  {
    int N = 12, M = 6;
    List<Tuple<int, int> > C
      = new List<Tuple<int, int> >{
      new Tuple<int, int>(1, 3),
      new Tuple<int, int>(3, 7),
      new Tuple<int, int>(5, 7),
      new Tuple<int, int>(7, 11),
      new Tuple<int, int>(9, 11),
      new Tuple<int, int>(11, 3)
      };

    // function call
    checkRotationallySymmetric(C, N, M);
  }
}

// This code is contributed by phasing17

JavaScript

// JavaScript Program to for the above approach

// Utility function to calculate
// divisors of a number in O(sqrt(N))
function calculateDivisors(N)
{
    let div = new Array();
    for (let i = 1; i * i <= N; i++) {
        if (N % i == 0) {
            div.push(i);
            if (Math.floor(N / i) != i && i != 1) {
                div.push(Math.floor(N / i));
            }
        }
    }

    return div;
}
function checkRotationallySymmetric( A, N, M)
{
    // Maintain a set to check quickly
    // the presence of a chord
    let st = new Set();
    for (let i = 0; i < M; i++) {
        A[i][0] = A[i][0] - 1;
        A[i][1] = A[i][1] - 1;
        if (A[i][0] > A[i][1]) {
            let temp = A[i][0];
            A[i][0] = A[i][1];
            A[i][1] = temp;
        }
        st.add(A[i].join(''));
    }

    // Calculate the di   ors of N.
    let div = calculateDivisors(N);
    
    // Iterate through the divisors
    for (let j = 0; j < div.length; j++){
        
        let exist = true;
        for (let i = 0; i < M; i++) {
            let dx = (A[i][0] + div[j]) % N;
            let dy = (A[i][1] + div[j]) % N;
            if (dx > dy) {
                
                // swapping dx and dy. 
                let temp = dx;
                dx = dy;
                dy = temp;
            }
            
            if (st.has([dx, dy].join(''))) {
                // There exists a valid
                // chord after rotation
            }
            else {
                // console.log([dx, dy]);
                // There is no valid chord after rotation
                exist = false;
                break;
            }
        }

        // if there exist another chord after
        // rotation for every other chord print
        // YES and exit the function
        if (exist) {
            console.log("YES");
            return 0;
        }
    }
    console.log("NO");
    return 0;
}

// Driver Code
let N = 12, M = 6;
let C
    = [[1, 3 ], [3, 7 ], [5, 7 ],
         [7, 11 ], [9, 11], [11, 3 ]];
checkRotationallySymmetric(C, N, M);

// The code is contributed by Gautam goel (gautamgoel962)

Output

YES

Time Complexity: O(M*sqrt(N)*log M)
Space Complexity: O(M)

Check if chords of a Circle are symmetric after some rotation

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