Suppose you are at a party with n people (labeled from 0 to n - 1) and among them, there may exist one celebrity. The definition of a celebrity is that all the other n - 1 people know him/her, but he/she does not know any of them.

Now you want to find out who the celebrity is or verify that there is not one. The only thing you are allowed to do is to ask questions like: “Hi, A. Do you know B?” to get information about whether A knows B. You need to find out the celebrity (or verify there is not one) by asking as few questions as possible (in the asymptotic sense).

You are given a helper function bool knows(a, b) which tells you whether person a knows person b. Implement a function int findCelebrity(n). There will be exactly one celebrity if he/she is in the party. Return the celebrity’s label if there is a celebrity in the party. If there is no celebrity, return -1.

Examples

Example 1:

Input: graph = [[1,1,0],[0,1,0],[1,1,1]]
Output: 1
Explanation: There are three persons labeled with 0, 1 and 2. 
graph[i][j] = 1 means person i knows person j, otherwise graph[i][j] = 0 means person i does not know person j.
The celebrity is the person labeled as 1 because both 0 and 2 know him but 1 does not know anybody.

Example 2:

Input: graph = [[1,0,1],[1,1,0],[0,1,1]]
Output: -1
Explanation: There is no celebrity.

Constraints

  • n == graph.length == graph[i].length
  • 2 <= n <= 100
  • graph[i][j] is 0 or 1.
  • graph[i][i] == 1 (everyone knows themselves)

Thinking Process

  1. Elimination Property: If candidate knows someone, they can’t be celebrity
  • Model entities as nodes and relationships as edges.
  • Pick traversal (BFS/DFS) or shortest-path (Dijkstra) based on weights.
  • Union-Find helps when connectivity updates are frequent.
Two pointers 1 3 5 7 9 L R move L/R based on comparison

Common Approaches

Typical techniques for this pattern:

Approach Time Space Notes
BFS / DFS traversal O(V+E) O(V) Connectivity, flood fill
Dijkstra O((V+E)log V) O(V) Non-negative edge weights
Union-Find (DSU) (this problem) O(α(n)) O(n) Dynamic connectivity
Topological sort O(V+E) O(V) DAG ordering, cycle detection

Solution

Time Complexity: O(n) - makes at most 3n calls to knows()
Space Complexity: O(1)

The key insight is to use a two-pass approach:

  1. First pass: Find a candidate celebrity by eliminating people who cannot be the celebrity
  2. Second pass: Verify that the candidate is indeed a celebrity

Solution: Two-Pass with Candidate Elimination

# Forward declaration of the knows API
# def knows(a, b): return bool

class Solution:
    def findCelebrity(self, n):
        # First pass: find candidate
        candidate = 0

        for i in range(1, n):
            if knows(candidate, i):
                candidate = i

        # Second pass: verify candidate
        for i in range(n):
            if i != candidate:
                # Celebrity should not know anyone
                if knows(candidate, i):
                    return -1

                # Everyone should know the celebrity
                if not knows(i, candidate):
                    return -1

        return candidate

Solution Explanation

Approach: Union-Find (DSU) (this problem)

Key idea: 1. Elimination Property: If candidate knows someone, they can’t be celebrity

How the code works:

  1. Elimination Property: If candidate knows someone, they can’t be celebrity
    • Model entities as nodes and relationships as edges.
    • Pick traversal (BFS/DFS) or shortest-path (Dijkstra) based on weights.
    • Union-Find helps when connectivity updates are frequent.

Walkthrough — input graph = [[1,1,0],[0,1,0],[1,1,1]], expected output 1:

There are three persons labeled with 0, 1 and 2. graph[i][j] = 1 means person i knows person j, otherwise graph[i][j] = 0 means person i does not know person j. The celebrity is the person labeled as 1 because both 0 and 2 know him but 1 does not know anybody.

| Approach | Time | Space | Calls to knows() | Pros | Cons | |———-|——|——-|——————|——|——| | Two-Pass | O(n) | O(1) | ~3n | Optimal | Requires two passes | | Brute Force | O(n²) | O(1) | ~n² | Simple | Inefficient |

Algorithm Breakdown

First Pass: Find Candidate

candidate = 0
for (i = 1 i < n i += 1) :
if knows(candidate, i):
    candidate = i

Why:

  • Start with person 0 as candidate
  • If candidate knows person i, candidate cannot be celebrity
  • Update candidate to i (who might be celebrity)
  • After loop, candidate is the only possible celebrity

Invariant: After processing person i, candidate doesn’t know anyone from i+1 to n-1.

Second Pass: Verify Candidate

for (i = 0 i < n i += 1) :
if i != candidate:
    if (knows(candidate, i)) return -1  # Celebrity knows someone
    if (not knows(i, candidate)) return -1   # Someone doesn't know celebrity

Why:

  • Check celebrity doesn’t know anyone (except themselves)
  • Check everyone knows the celebrity
  • If either fails, return -1 (no celebrity)

Complexity

| Approach | Time | Space | Calls to knows() | Pros | Cons | |———-|——|——-|——————|——|——| | Two-Pass | O(n) | O(1) | ~3n | Optimal | Requires two passes | | Brute Force | O(n²) | O(1) | ~n² | Simple | Inefficient |

Implementation Details

Why First Pass Works

Key Observation:

  • If knows(candidate, i) is true, candidate cannot be celebrity
  • But i might be celebrity (we don’t know if i knows others yet)
  • We can safely eliminate candidate and try i

Invariant Maintenance: After processing person i:

  • Candidate doesn’t know anyone from i+1 to n-1
  • All people before candidate have been eliminated

Why Verification is Necessary

Example where first pass finds wrong candidate:

Graph:
    0  1  2
0 [ 1  1  0 ]
1 [ 0  1  0 ]
2 [ 0  0  1 ]

First Pass:
- candidate = 0
- knows(0, 1) = true → candidate = 1
- knows(1, 2) = false → candidate = 1

But person 1 knows person 0! So verification fails.
Actually, there's no celebrity in this graph.

Optimization: Early Termination

The current solution can be slightly optimized:

class Solution:
    def findCelebrity(self, n):
        for i in range(n):
            isCelebrity = True

            # Check if i knows anyone
            for j in range(n):
                if i != j and knows(i, j):
                    isCelebrity = False
                    break

            if not isCelebrity:
                continue

            # Check if everyone knows i
            for j in range(n):
                if i != j and not knows(j, i):
                    isCelebrity = False
                    break

            if isCelebrity:
                return i

        return -1

Why: Separating verification into two loops allows early termination.

Common Mistakes

  1. No celebrity: Return -1 after verification fails
  2. Celebrity is person 0: First pass keeps candidate = 0
  3. Celebrity is last person: First pass updates to last person
  4. Everyone knows everyone: No celebrity (candidate knows someone)

  5. Nobody knows anyone: No celebrity (nobody knows candidate)

  6. Skipping verification: Must verify candidate meets both conditions
  7. Wrong elimination logic: Must check knows(candidate, i), not knows(i, candidate)
  8. Not handling self-loops: knows(i, i) is always true (everyone knows themselves)
  9. Assuming candidate exists: Must return -1 if verification fails
  10. Incorrect loop bounds: Must check all n people in verification

Optimization Tips

  1. Two-pass approach: Optimal O(n) solution
  2. Early termination: Can return -1 as soon as verification fails
  3. Single candidate: Only need to verify one person after first pass

Real-World Applications

  1. Social Networks: Finding influential people
  2. Recommendation Systems: Identifying key nodes
  3. Graph Analysis: Finding nodes with specific properties
  4. Network Topology: Identifying central nodes

Pattern Recognition

This problem demonstrates the “Candidate Elimination” pattern:

1. Start with a candidate
2. Use elimination property to narrow down candidates
3. Verify final candidate meets all conditions
4. Return result or indicate no solution

Similar problems:

  • Finding majority element
  • Finding unique elements
  • Graph problems with special node properties

Why Two-Pass Works

First Pass Guarantee:

  • After first pass, candidate is the only person who could be celebrity
  • All others have been eliminated (they know someone after them)

Why Only One Candidate:

  • If two people could be celebrities:
    • Person A doesn’t know person B → A could be celebrity
    • Person B doesn’t know person A → B could be celebrity
    • But then A doesn’t know B AND B doesn’t know A
    • This violates the condition that everyone knows the celebrity
  • Therefore, at most one candidate remains

Verification Necessity:

  • First pass only checks one direction (candidate doesn’t know others)
  • Must verify other direction (everyone knows candidate)
  • Must verify candidate doesn’t know anyone before them

Step-by-Step Trace: n = 4, Celebrity is person 2

Graph:
    0  1  2  3
0 [ 1  1  1  0 ]
1 [ 0  1  1  0 ]
2 [ 0  0  1  0 ]
3 [ 1  1  1  1 ]

First Pass:
- candidate = 0
- i = 1: knows(0, 1) = true → candidate = 1
- i = 2: knows(1, 2) = true → candidate = 2
- i = 3: knows(2, 3) = false → candidate = 2
- Candidate: 2

Second Pass:
- i = 0: knows(2, 0) = false ✓, knows(0, 2) = true ✓
- i = 1: knows(2, 1) = false ✓, knows(1, 2) = true ✓
- i = 3: knows(2, 3) = false ✓, knows(3, 2) = true ✓
- Verification passed: return 2

Mathematical Insight

Graph Theory Perspective:

  • Celebrity has in-degree = n-1 (everyone knows them)
  • Celebrity has out-degree = 0 (knows nobody, except self)
  • In a directed graph, at most one such node can exist

Why:

  • If two nodes have out-degree 0, they don’t know each other
  • But celebrity must be known by everyone
  • Contradiction → at most one celebrity

Key Takeaways

  1. Elimination Property: If candidate knows someone, they can’t be celebrity
  2. Single Candidate: After first pass, at most one candidate remains
  3. Verification Needed: Must verify candidate meets both celebrity conditions
  4. Efficient: Only O(n) calls to knows() instead of O(n²)

References

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