2924. Find Champion II

Difficulty: Medium

Topics: Graph

There are n teams numbered from 0 to n - 1 in a tournament; each team is also a node in a DAG.

You are given the integer n and a 0-indexed 2D integer array edges of length m representing the DAG, where >dges[i] = [ui, vi] indicates that there is a directed edge from team ui to team vi in the graph.

A directed edge from a to b in the graph means that team a is stronger than team b and team b is weaker than team a.

Team a will be the champion of the tournament if there is no team b that is stronger than team a.

Return the team that will be the champion of the tournament if there is a unique champion, otherwise, return -1.

Notes

• A cycle is a series of nodes a1, a2, ..., an, an+1 such that node a1 is the same node as node an+1, the nodes a1, a2, ..., an are distinct, and there is a directed edge from the node ai to node ai+1 for every i in the range [1, n].
• A DAG is a directed graph that does not have any cycle.

Example 1:

• Input: n = 3, edges = [[0,1],[1,2]]
• Output: 0
• Explanation: Team 1 is weaker than team 0. Team 2 is weaker than team 1. So the champion is team 0.

Example 2:

• Input: n = 4, edges = [[0,2],[1,3],[1,2]]
• Output: -1
• Explanation: Team 2 is weaker than team 0 and team 1. Team 3 is weaker than team 1. But team 1 and team 0 are not weaker than any other teams. So the answer is -1.

Constraints:

• 1 <= n <= 100
• m == edges.length
• 0 <= m <= n * (n - 1) / 2
• edges[i].length == 2
• 0 <= edge[i][j] <= n - 1
• edges[i][0] != edges[i][1]
• The input is generated such that if team a is stronger than team b, team b is not stronger than team a.
• The input is generated such that if team a is stronger than team b and team b is stronger than team c, then team a is stronger than team c.

Hint:

1. The champion(s) should have in-degree 0 in the DAG.

Solution:

We need to identify the team(s) with an in-degree of 0 in the given Directed Acyclic Graph (DAG). Teams with no incoming edges represent teams that no other team is stronger than, making them candidates for being the champion. If there is exactly one team with an in-degree of 0, it is the unique champion. If there are multiple or no such teams, the result is -1.

Let's implement this solution in PHP: 2924. Find Champion II

<?php
/**
 * @param Integer $n
 * @param Integer[][] $edges
 * @return Integer
 */
function findChampion($n, $edges) {
    // Initialize in-degrees for all teams
    $inDegree = array_fill(0, $n, 0);

    // Calculate the in-degree for each team
    foreach ($edges as $edge) {
        $inDegree[$edge[1]]++;
    }

    // Find teams with in-degree 0
    $candidates = [];
    for ($i = 0; $i < $n; $i++) {
        if ($inDegree[$i] == 0) {
            $candidates[] = $i;
        }
    }

    // If exactly one team has in-degree 0, return it; otherwise, return -1
    return count($candidates) === 1 ? $candidates[0] : -1;
}

// Example 1
$n1 = 3;
$edges1 = [[0, 1], [1, 2]];
echo "Example 1 Output: " . findChampion($n1, $edges1) . PHP_EOL; // Output: 0

// Example 2
$n2 = 4;
$edges2 = [[0, 2], [1, 3], [1, 2]];
echo "Example 2 Output: " . findChampion($n2, $edges2) . PHP_EOL; // Output: -1
?>

Explanation:

1. Input Parsing:

   • n is the number of teams.
   • edges is the list of directed edges in the graph.

2. Initialize In-degree:

   • Create an array inDegree of size n initialized to 0.

3. Calculate In-degree:

   • For each edge [u, v], increment the in-degree of v (team v has one more incoming edge).

4. Find Candidates:

   • Iterate through the inDegree array and collect indices where the in-degree is 0. These indices represent teams with no other stronger teams.

5. Determine Champion:

   • If exactly one team has in-degree 0, it is the unique champion.
   • If multiple teams or no teams have in-degree 0, return -1.

Example Walkthrough

Example 1:

• Input: n = 3, edges = [[0, 1], [1, 2]]
• In-degree: [0, 1, 1]
• Teams with in-degree 0: [0]
• Output: 0 (Team 0 is the unique champion).

Example 2:

• Input: n = 4, edges = [[0, 2], [1, 3], [1, 2]]
• In-degree: [0, 0, 2, 1]
• Teams with in-degree 0: [0, 1]
• Output: -1 (There are multiple potential champions).

Complexity Analysis

1. Time Complexity:

   • Computing in-degrees: O(m), where m is the number of edges.
   • Checking teams: O(n), where n is the number of teams.
   • Total: O(n + m).

2. Space Complexity:

   • inDegree array: O(n).

This solution is efficient and works within the given constraints.

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