Watch 10 video solutions for The Earliest and Latest Rounds Where Players Compete, a hard level problem involving Dynamic Programming, Memoization. This walkthrough by codestorywithMIK has 7,825 views views. Want to try solving it yourself? Practice on FleetCode or read the detailed text solution.
There is a tournament where n players are participating. The players are standing in a single row and are numbered from 1 to n based on their initial standing position (player 1 is the first player in the row, player 2 is the second player in the row, etc.).
The tournament consists of multiple rounds (starting from round number 1). In each round, the ith player from the front of the row competes against the ith player from the end of the row, and the winner advances to the next round. When the number of players is odd for the current round, the player in the middle automatically advances to the next round.
1, 2, 4, 6, 7
1 competes against player 7.2 competes against player 6.4 automatically advances to the next round.After each round is over, the winners are lined back up in the row based on the original ordering assigned to them initially (ascending order).
The players numbered firstPlayer and secondPlayer are the best in the tournament. They can win against any other player before they compete against each other. If any two other players compete against each other, either of them might win, and thus you may choose the outcome of this round.
Given the integers n, firstPlayer, and secondPlayer, return an integer array containing two values, the earliest possible round number and the latest possible round number in which these two players will compete against each other, respectively.
Example 1:
Input: n = 11, firstPlayer = 2, secondPlayer = 4 Output: [3,4] Explanation: One possible scenario which leads to the earliest round number: First round: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 Second round: 2, 3, 4, 5, 6, 11 Third round: 2, 3, 4 One possible scenario which leads to the latest round number: First round: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 Second round: 1, 2, 3, 4, 5, 6 Third round: 1, 2, 4 Fourth round: 2, 4
Example 2:
Input: n = 5, firstPlayer = 1, secondPlayer = 5 Output: [1,1] Explanation: The players numbered 1 and 5 compete in the first round. There is no way to make them compete in any other round.
Constraints:
2 <= n <= 281 <= firstPlayer < secondPlayer <= nProblem Overview: You are given n players in a knockout tournament where the first player faces the last, the second faces the second-last, and so on. Two specific players always win until they face each other. The task is to compute the earliest and latest possible round where these two players can compete, considering all valid outcomes of other matches.
Approach 1: Simulation Approach (Exponential Time, Moderate Space)
This approach directly simulates tournament rounds while exploring every possible outcome for matches that do not involve the two target players. In each round, players are paired from both ends of the lineup and winners move to the next round. When neither of the tracked players participates in a match, both outcomes are explored, generating multiple possible player configurations for the next round. The process continues until the two players meet, tracking the minimum and maximum round numbers. Time complexity grows exponentially because many bracket configurations are explored, while space complexity is proportional to the number of simulated states.
Approach 2: Recursive DP with Memoization (O(n^3) time, O(n^3) space)
The optimized solution models the tournament using recursion and caches repeated states with memoization. Instead of simulating full player lists, the state is defined by (n, firstPlayer, secondPlayer). In each round, players are paired symmetrically and only the relative positions of the tracked players matter. You iterate through all possible numbers of winners that could appear before each tracked player in the next round. These combinations represent different outcomes of unrelated matches. Each resulting configuration recursively computes the earliest and latest meeting round for the reduced bracket. Storing results in a DP cache avoids recomputing identical states, reducing the complexity to roughly O(n^3) time and O(n^3) space.
This approach relies heavily on dynamic programming and recursive state transitions similar to tournament bracket DP problems. The key insight is that player identities do not matter except for the two tracked positions; only their indices relative to the bracket affect future rounds.
Recommended for interviews: The recursive DP with memoization solution is the expected approach. Brute-force simulation demonstrates understanding of the tournament mechanics, but interviewers look for the state-compression insight that reduces the problem to positions and uses DP caching to avoid exponential exploration.
| Approach | Time | Space | When to Use |
|---|---|---|---|
| Simulation Approach | Exponential | O(states) | Useful for understanding tournament mechanics or small n where exploring all match outcomes is feasible. |
| Recursive DP with Memoization | O(n^3) | O(n^3) | General optimal solution. Efficiently computes earliest and latest meeting rounds by caching repeated bracket states. |