The Celebrity Problem

In a party of N people, only one person is known to everyone. Such a person may be present in the party, if yes, (s)he doesn’t know anyone in the party. We can only ask questions like “does A know B? “. Find the stranger (celebrity) in the minimum number of questions.

We can describe the problem input as an array of numbers/characters representing persons in the party. We also have a hypothetical function HaveAcquaintance(A, B) which returns true if A knows B, false otherwise. How can we solve the problem.
Examples:

Input:
MATRIX = { {0, 0, 1, 0},
           {0, 0, 1, 0},
           {0, 0, 0, 0},
           {0, 0, 1, 0} }
Output:id = 2
Explanation: The person with ID 2 does not 
know anyone but everyone knows him

Input:
MATRIX = { {0, 0, 1, 0},
           {0, 0, 1, 0},
           {0, 1, 0, 0},
           {0, 0, 1, 0} }
Output: No celebrity
Explanation: There is no celebrity.

We measure the complexity in terms of calls made to HaveAcquaintance().

Method 1: This uses Graph to arrive at the particular solution.

  • Approach:
    Model the solution using graphs. Initialize indegree and outdegree of every vertex as 0. If A knows B, draw a directed edge from A to B, increase indegree of B and outdegree of A by 1. Construct all possible edges of the graph for every possible pair [i, j]. There are NC2 pairs. If a celebrity is present in the party, there will be one sink node in the graph with outdegree of zero and indegree of N-1.
  • Algorithm:
    1. Create two arrays indegree and outdegree, to store the indegree and outdegree
    2. Run a nested loop, the outer loop from 0 to n and inner loop from 0 to n.
    3. For every pair i, j check if i knows j then increase the outdegree of i and indegree of j
    4. For every pair i, j check if j knows i then increase the outdegree of j and indegree of i
    5. Run a loop from 0 to n and find the id where the indegree is n-1 and outdegree is 0
  • Implementation:



    filter_none

    edit
    close

    play_arrow

    link
    brightness_4
    code

    // C++ program to find celebrity
    #include <bits/stdc++.h>
    #include <list>
    using namespace std;
       
    // Max # of persons in the party
    #define N 8
       
    // Person with 2 is celebrity
    bool MATRIX[N][N] = {{0, 0, 1, 0},
                        {0, 0, 1, 0},
                        {0, 0, 0, 0},
                        {0, 0, 1, 0}};
       
    bool knows(int a, int b)
    {
        return MATRIX[a][b];
    }
       
    // Returns -1 if celebrity
    // is not present. If present,
    // returns id (value from 0 to n-1).
    int findCelebrity(int n)
    {
        //the graph needs not be constructed
        //as the edges can be found by
        //using knows function
          
        //degree array;
        int indegree[n]={0},outdegree[n]={0};
          
        //query for all edges
        for(int i=0; i<n; i++)
        {
            for(int j=0; j<n; j++)
            {
                int x = knows(i,j);
                  
                //set the degrees
                outdegree[i]+=x;
                indegree[j]+=x;
            }
        }
          
        //find a person with indegree n-1
        //and out degree 0
        for(int i=0; i<n; i++)
        if(indegree[i] == n-1 && outdegree[i] == 0)
            return i;
          
        return -1;
    }
       
    // Driver code
    int main()
    {
        int n = 4;
        int id = findCelebrity(n);
        id == -1 ? cout << "No celebrity" :
                   cout << "Celebrity ID " << id;
        return 0;
    }

    chevron_right

    
    

    Output :

    Celebrity ID 2
  • Complexity Analysis:

    • Time Complexity: O(n2).
      A nested loop is run traversing the array, SO the time complexity is O(n2)
    • Space Complexity: O(n).
      Since extra space of size n is required.
  • Method 2: This method uses recursion to solve the above problem.

    • Approach :
      The problem can be decomposed into a combination of smaller instances. Say, if the celebrity of N-1 persons is known, can the solution to N? There are two possibilities, Celebrity(N-1) may know N, or N already knew Celebrity(N-1). In the former case, N will be a celebrity if N. In the latter case, check that Celebrity(N-1) doesn’t know N.
      The above-mentioned approach use Recursion to find the celebrity among n persons. It is necessary to find a celebrity among n-1 persons. So while calculating the celebrity of i persons the function calls itself to find the celebrity of i-1 persons and continues doing so until the base case is found.
    • Algorithm :

      1. Create a recursive function that takes an integer n.
      2. Check the base case, if the value of n is 0 then return 0.
      3. Call the recursive function and get the ID of celebrity from the first n-1 elements.
      4. If the id is -1 then assign n as celebrity and return the value.
      5. If the celebrity of first n-1 elements knows n-1 and n-1 does not know the celebrity then return n-1, (0 based indexing)
      6. If the celebrity of first n-1 elements does not know n-1 and n-1 knows the celebrity then return id of celebrity of n-1 elements, (0 based indexing)
      7. Else return -1
      8. Create a wrapper function and check whether the id returned by the function is really the celebrity or not.
    • Implementation:

      filter_none

      edit
      close

      play_arrow

      link
      brightness_4
      code

      // C++ program to find celebrity
      #include <bits/stdc++.h>
      #include <list>
      using namespace std;
        
      // Max # of persons in the party
      #define N 8
        
      // Person with 2 is celebrity
      bool MATRIX[N][N] = {{0, 0, 1, 0},
                          {0, 0, 1, 0},
                          {0, 1, 0, 0},
                          {0, 0, 1, 0}};
        
      bool knows(int a, int b)
      {
          return MATRIX[a][b];
      }
        
      // Returns -1 if celebrity
      // is not present. If present,
      // returns id (value from 0 to n-1).
      int findCelebrity(int n)
      {
          //base case
          if(n == 1)
          return n - 1;
            
          //find the celebrity with n-1
          //persons
          int id = findCelebrity(n-1);
            
          //if there are no celebrities
          if(id == -1)
              return n-1;
            
          // if the celebrity knows the
          //nth person
          else if(knows(id, n-1) && !knows(n-1, id))
          {
              return n-1; 
          }
          //if the nth person knows the
          //celebrity then return the id
          else if(knows(n-1, id) && !knows(id, n-1))
          {
              return id;
          }
        
          //if there is no celebrity 
          return -1;
      }
        
      // Returns -1 if celebrity
      // is not present. If present,
      // returns id (value from 0 to n-1).
      // a wrapper over findCelebrity
      int Celebrity(int n)
      {
          //find the celebrity
          int id = findCelebrity(n);
            
          //check if the celebrity found
          //is really the celebrity
          if(id == -1)
              return id;
          else
          {
              int c1=0, c2=0;
        
              //check the id is really the
              //celebrity
              for(int i=0; i<n; i++)
              if(i != id)
              {
                  c1+=knows(id, i);
                  c2+=knows(i, id);
              }
                
              //if the person is known to
              //everyone.
              if(c1==0 && c2==n-1)
                  return id;
                
              return -1;
          }
      }
        
      // Driver code
      int main()
      {
          int n = 4;
          int id = Celebrity(n);
          id == -1 ? cout << "No celebrity" :
                  cout << "Celebrity ID " << id;
          return 0;
      }

      chevron_right

      
      

      Output :

      Celebrity ID 2
    • Complexity Analysis:

      • Time Complexity: O(n).
        The recursive function is called n times, so the time complexity is O(n).
      • Space Complexity: O(1).
        As no extra space is required.

    Method 3: This method uses Stack to arrive at the following solution.



    • Approach: There are some observations based on elimination technique (Refer Polya’s How to Solve It book).

      • If A knows B, then A can’t be a celebrity. Discard A, and B may be celebrity.
      • If A doesn’t know B, then B can’t be a celebrity. Discard B, and A may be celebrity.
      • Repeat above two steps till there is only one person.
      • Ensure the remained person is a celebrity. (What is the need of this step?)

      The stack can be used to verify celebrity. Insert all the elements in the stack. If the size of the stack is greater than 1 then pop top 2 elements of the stack (represent them as A and B). Check if A knows B, then A can’t be a celebrity. If A doesn’t know B, then B can’t be a celebrity. So one of the element gets discarded and push the other element back in the stack. In this way, only one element will be left. Check if the element left in the stack is celebrity or not. If yes print the id else print -1.

    • Algorithm:
      1. Create a stack and push all the id’s in the stack.
      2. Run a loop while there are more than 1 element in the stack.
      3. Pop top two element from the stack (represent them as A and B)
      4. Check if A knows B, then A can’t be a celebrity and push B in stack. Check if A doesn’t know B, then B can’t be a celebrity push A in stack
      5. Assign the remaining element in the stack as the celebrity
      6. Run a loop from 0 to n-1 and find the count of persons who knows the celebrity and the number of people whom the celebrity knows. if the count of persons who knows the celebrity is n-1 and the count of people whom the celebrity knows is 0 then return the id of celebrity else return -1.
    • Implementation:

      C++

      filter_none

      edit
      close

      play_arrow

      link
      brightness_4
      code

      // C++ program to find celebrity
      #include <bits/stdc++.h>
      #include <list>
      using namespace std;
        
      // Max # of persons in the party
      #define N 8
        
      // Person with 2 is celebrity
      bool MATRIX[N][N] = {{0, 0, 1, 0},
                          {0, 0, 1, 0},
                          {0, 0, 0, 0},
                          {0, 0, 1, 0}};
        
      bool knows(int a, int b)
      {
          return MATRIX[a][b];
      }
        
      // Returns -1 if celebrity
      // is not present. If present,
      // returns id (value from 0 to n-1).
      int findCelebrity(int n)
      {
          // Handle trivial 
          // case of size = 2
        
          stack<int> s;
        
          int C; // Celebrity
        
          // Push everybody to stack
          for (int i = 0; i < n; i++)
              s.push(i);
        
          // Extract top 2
          int A = s.top();
          s.pop();
          int B = s.top();
          s.pop();
        
          // Find a potential celevrity
          while (s.size() > 1)
          {
              if (knows(A, B))
              {
                  A = s.top();
                  s.pop();
              }
              else
              {
                  B = s.top();
                  s.pop();
              }
          }
        
          // Potential candidate?
          C = s.top();
          s.pop();
        
          // Last candidate was not 
          // examined, it leads one 
          // excess comparison (optimize)
          if (knows(C, B))
              C = B;
        
          if (knows(C, A))
              C = A;
        
          // Check if C is actually
          // a celebrity or not
          for (int i = 0; i < n; i++)
          {
              // If any person doesn't 
              // know 'a' or 'a' doesn't 
              // know any person, return -1
              if ( (i != C) &&
                      (knows(C, i) || 
                       !knows(i, C)) )
                  return -1;
          }
        
          return C;
      }
        
      // Driver code
      int main()
      {
          int n = 4;
          int id = findCelebrity(n);
          id == -1 ? cout << "No celebrity" :
                     cout << "Celebrity ID " << id;
          return 0;
      }

      chevron_right

      
      

      Java

      filter_none

      edit
      close

      play_arrow

      link
      brightness_4
      code

      // Java program to find celebrity using
      // stack data structure
        
      import java.util.Stack;
        
      class GFG 
      {
          // Person with 2 is celebrity
          static int MATRIX[][] = { { 0, 0, 1, 0 },
                                  { 0, 0, 1, 0 },
                                  { 0, 0, 0, 0 }, 
                                  { 0, 0, 1, 0 } };
        
          // Returns true if a knows 
          // b, false otherwise
          static boolean knows(int a, int b) 
          {
              boolean res = (MATRIX[a][b] == 1) ? 
                                           true
                                           false;
              return res;
          }
        
          // Returns -1 if celebrity 
          // is not present. If present,
          // returns id (value from 0 to n-1).
          static int findCelebrity(int n) 
          {
              Stack<Integer> st = new Stack<>();
              int c;
        
              // Step 1 :Push everybody
              // onto stack
              for (int i = 0; i < n; i++) 
              {
                  st.push(i);
              }
        
              while (st.size() > 1
              {
                  // Step 2 :Pop off top
                  // two persons from the 
                  // stack, discard one 
                  // person based on return
                  // status of knows(A, B).
                  int a = st.pop();
                  int b = st.pop();
        
                  // Step 3 : Push the 
                  // remained person onto stack.
                  if (knows(a, b)) 
                  {
                      st.push(b);
                  }
        
                  else
                      st.push(a);
              }
        
              c = st.pop();
        
              // Step 5 : Check if the last 
              // person is celebrity or not
              for (int i = 0; i < n; i++) 
              {
                  // If any person doesn't
                  //  know 'c' or 'a' doesn't 
                  // know any person, return -1
                  if (i != c && (knows(c, i) || 
                                !knows(i, c)))
                      return -1;
              }
              return c;
          }
        
          // Driver Code
          public static void main(String[] args) 
          {
              int n = 4;
              int result = findCelebrity(n);
              if (result == -1
              {
                  System.out.println("No Celebrity");
              
              else
                  System.out.println("Celebrity ID "
                                              result);
          }
      }
        
      // This code is contributed 
      // by Rishabh Mahrsee

      chevron_right

      
      


      Output :

      Celebrity ID 2
    • Complexity Analysis:

      • Time Complexity: O(n).
        Total number of comparisons 3(N-1), so the time complexity is O(n).
      • Space Complexity: O(n).
        n extra space is needed to store the stack.

    Method 4: This method uses Two Pointers technique to solve the above problem.

    • Approach: The idea is to use two pointers, one from start and one from the end. Assume the start person is A, and the end person is B. If A knows B, then A must not be the celebrity. Else, B must not be the celebrity. At the end of the loop, only one index will be left as a celebrity. Go through each person again and check whether this is the celebrity.
      The Two Pointer approach can be used where two pointers can be assigned, one at the start and other at the end and the elements can be compared and the search space can be reduced.
    • Algorithm :
      1. Create two indices a and b, where a = 0 and b = n-1
      2. Run a loop until a is less than b.
      3. Check if a knows b, then a can’t be celebrity. so increment a, i.e. a++
      4. Else b cannot be celebrity, so decrement b, i.e. b–
      5. Assign a as the celebrity
      6. Run a loop from 0 to n-1 and find the count of persons who knows the celebrity and the number of people whom the celebrity knows. if the count of persons who knows the celebrity is n-1 and the count of people whom the celebrity knows is 0 then return the id of celebrity else return -1.
    • Implementation.

      C++

      filter_none

      edit
      close

      play_arrow

      link
      brightness_4
      code

      // C++ program to find 
      // celebrity in O(n) time
      // and O(1) extra space
      #include <bits/stdc++.h>
      using namespace std;
        
      // Max # of persons in the party
      #define N 8
        
      // Person with 2 is celebrity
      bool MATRIX[N][N] = {{0, 0, 1, 0},
                           {0, 0, 1, 0},
                           {0, 0, 0, 0},
                           {0, 0, 1, 0}
      };
        
      bool knows(int a, int b)
      {
          return MATRIX[a][b];
      }
        
      // Returns id of celebrity
      int findCelebrity(int n)
      {
          // Initialize two pointers 
          // as two corners
          int a = 0;
          int b = n - 1;
        
          // Keep moving while 
          // the two pointers
          // don't become same. 
          while (a < b)
          {
              if (knows(a, b))
                  a++;
              else
                  b--;
          }
        
          // Check if a is actually
          // a celebrity or not
          for (int i = 0; i < n; i++)
          {
              // If any person doesn't 
              // know 'a' or 'a' doesn't
              // know any person, return -1
              if ( (i != a) &&
                      (knows(a, i) || 
                      !knows(i, a)) )
                  return -1;
          }
        
          return a;
      }
        
      // Driver code
      int main()
      {
          int n = 4;
          int id = findCelebrity(n);
          id == -1 ? cout << "No celebrity" :
                     cout << "Celebrity ID " 
                          << id;
          return 0;
      }

      chevron_right

      
      

      Java

      filter_none

      edit
      close

      play_arrow

      link
      brightness_4
      code

      // Java program to find 
      // celebrity using two 
      // pointers
        
      class GFG 
      {
          // Person with 2 is celebrity
          static int MATRIX[][] = { { 0, 0, 1, 0 },
                                     { 0, 0, 1, 0 }, 
                                    { 0, 0, 0, 0 },
                                    { 0, 0, 1, 0 } };
        
          // Returns true if a knows
          // b, false otherwise
          static boolean knows(int a, int b) 
          {
              boolean res = (MATRIX[a][b] == 1) ? 
                                           true
                                           false;
              return res;
          }
        
          // Returns -1 if celebrity 
          // is not present. If present, 
          // returns id (value from 0 to n-1).
          static int findCelebrity(int n) 
          {
              // Initialize two pointers 
              // as two corners
              int a = 0;
              int b = n - 1;
                
              // Keep moving while 
              // the two pointers
              // don't become same.
              while (a < b) 
              {
                  if (knows(a, b))
                      a++;
                  else
                      b--;
              }
        
              // Check if a is actually 
              // a celebrity or not
              for (int i = 0; i < n; i++) 
              {
                  // If any person doesn't 
                  // know 'a' or 'a' doesn't
                  // know any person, return -1
                  if (i != a && (knows(a, i) || 
                                 !knows(i, a)))
                      return -1;
              }
              return a;
          }
        
          // Driver Code
          public static void main(String[] args) 
          {
              int n = 4;
              int result = findCelebrity(n);
              if (result == -1
              {
                  System.out.println("No Celebrity");
              
              else
                  System.out.println("Celebrity ID "
                                              result);
          }
      }
        
      // This code is contributed by Rishabh Mahrsee

      chevron_right

      
      

      C#

      filter_none

      edit
      close

      play_arrow

      link
      brightness_4
      code

      // C# program to find 
      // celebrity using two 
      // pointers
      using System;
        
      class GFG 
      {
          // Person with 2 is celebrity
          static int [,]MATRIX = {{ 0, 0, 1, 0 },
                                  { 0, 0, 1, 0 }, 
                                  { 0, 0, 0, 0 },
                                  { 0, 0, 1, 0 }};
        
          // Returns true if a knows
          // b, false otherwise
          static bool knows(int a, int b) 
          {
              bool res = (MATRIX[a, b] == 1) ? 
                                        true
                                        false;
              return res;
          }
        
          // Returns -1 if celebrity 
          // is not present. If present, 
          // returns id (value from 0 to n-1).
          static int findCelebrity(int n) 
          {
              // Initialize two pointers 
              // as two corners
              int a = 0;
              int b = n - 1;
                
              // Keep moving while 
              // the two pointers
              // don't become same.
              while (a < b) 
              {
                  if (knows(a, b))
                      a++;
                  else
                      b--;
              }
        
              // Check if a is actually 
              // a celebrity or not
              for (int i = 0; i < n; i++) 
              {
                  // If any person doesn't 
                  // know 'a' or 'a' doesn't
                  // know any person, return -1
                  if (i != a && (knows(a, i) || 
                                !knows(i, a)))
                      return -1;
              }
              return a;
          }
        
          // Driver Code
          public static void Main() 
          {
              int n = 4;
              int result = findCelebrity(n);
              if (result == -1) 
              {
                  Console.WriteLine("No Celebrity");
              
              else
                  Console.WriteLine("Celebrity ID "
                                             result);
          }
      }
        
      // This code is contributed by anuj_67.

      chevron_right

      
      

      PHP

      filter_none

      edit
      close

      play_arrow

      link
      brightness_4
      code

      <?php
      // PHP program to find 
      // celebrity in O(n) time
      // and O(1) extra space
        
        
      // Max # of persons 
      // in the party $N = 8;
        
      // Person with 2 is celebrity
      $MATRIX = array(array(0, 0, 1, 0),
                      array(0, 0, 1, 0),
                      array(0, 0, 0, 0),
                      array(0, 0, 1, 0));
        
      function knows( $a, $b)
      {
          global $MATRIX;
          return $MATRIX[$a][$b];
      }
        
      // Returns id of celebrity
      function findCelebrity( $n)
      {
          // Initialize two 
          // pointers as two corners
          $a = 0;
          $b = $n - 1;
        
          // Keep moving while 
          // the two pointers
          // don't become same. 
          while ($a < $b)
          {
              if (knows($a, $b))
                  $a++;
              else
                  $b--;
          }
        
          // Check if a is actually
          // a celebrity or not
          for ( $i = 0; $i < $n; $i++)
          {
              // If any person doesn't 
              // know 'a' or 'a' doesn't 
              // know any person, return -1
              if ( ($i != $a) and
                      (knows($a, $i) || 
                      !knows($i, $a)) )
                  return -1;
          }
        
          return $a;
      }
        
      // Driver code
      $n = 4;
      $id = findCelebrity($n);
      if($id == -1)
      echo "No celebrity" ;
      else
      echo "Celebrity ID " , $id;
        
      // This code is contributed by anuj_67.
      ?>

      chevron_right

      
      


      Output :

      Celebrity ID 2
    • Complexity Analysis:

      • Time Complexity: O(n).
        Total number of comparisons 2(N-1), so the time complexity is O(n).
      • Space Complexity : O(1).
        No extra space is required.

    Thanks to Sissi Peng for suggesting this method.

    Related Article:
    Number of sink nodes in a graph

    Exercises:

    1. Write code to find celebrity. Don’t use any data structures like graphs, stack, etc… you have access to N and HaveAcquaintance(int, int) only.
    2. Implement the algorithm using Queues. What is your observation? Compare your solution with Finding Maximum and Minimum in an array and Tournament Tree. What are minimum number of comparisons do we need (optimal number of calls to HaveAcquaintance())?

    Venki. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

    Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.




    My Personal Notes arrow_drop_up

    Improved By : vt_m, andrew1234, rohitkumar52