
Problem Statement
In this challenge, you are provided with an integer array nums
representing the gaming interface of two players: Player 1 and Player 2. The game begins with both players having a score of 0
. During each turn, a player may choose a number from either end of the array—either the first element or the last. The selected number is then added to the active player's score, and it is removed from the array. The primary objective for each player is to maximize their own total score through strategic selection of numbers, and both players are known to make their choices optimally. The game continues alternating turns until no elements remain in the array.
The task is to determine if Player 1 can either win or tie the game, considering both players' scores when the array is emptied. If Player 1 ends up with a score equal to or greater than Player 2, the output should be true
, indicating a win or tie for Player 1. Conversely, if Player 2 has a superior score, return false
.
Examples
Example 1
Input:
nums = [1,5,2]
Output:
false
Explanation:
Initially, player 1 can choose between 1 and 2. If he chooses 2 (or 1), then player 2 can choose from 1 (or 2) and 5. If player 2 chooses 5, then player 1 will be left with 1 (or 2). So, final score of player 1 is 1 + 2 = 3, and player 2 is 5. Hence, player 1 will never be the winner and you need to return false.
Example 2
Input:
nums = [1,5,233,7]
Output:
true
Explanation:
Player 1 first chooses 1. Then player 2 has to choose between 5 and 7. No matter which number player 2 choose, player 1 can choose 233. Finally, player 1 has more score (234) than player 2 (12), so you need to return True representing player1 can win.
Constraints
1 <= nums.length <= 20
0 <= nums[i] <= 107
Approach and Intuition
Analyzing both the problem constraints and examples provided, we can deduce some initial strategies and intuitions:
Game Dynamics
- When it's a player's turn, choosing either the first or last element impacts not only their own score but also the choice landscape for the opponent. Therefore, each decision is crucial and should be made by forecasting consequent moves, somewhat similar to the game of chess.
Choice Evaluation
- A winning strategy involves evaluating the potential outcome of every decision. For example, if choosing an end element leaves the opponent with a significantly higher-scoring potential, an alternative choice may be optimal.
Example Insights
Example 1 –
[1,5,2]
:- Player 1 choosing either
1
or2
leads to Player 2 eventually selecting5
, leading to Player 1's defeat in score, hence returningfalse
.
- Player 1 choosing either
Example 2 –
[1,5,233,7]
:- Whatever choice Player 2 makes (between
5
and7
), Player 1 can always pick233
on the subsequent turn, ensuring a win or a formidable lead—in this case, leading to a result oftrue
.
- Whatever choice Player 2 makes (between
From these examples, we can understand that the goal isn't just about making the highest immediate score, but about choosing numbers that either limit the opponent's future high scores or setting oneself up for a high score in future rounds.
Solutions
- Java
class Solution {
public boolean canFirstPlayerWin(int[] values) {
int length = values.length;
int[] gameDp = Arrays.copyOf(values, length);
for (int distance = 1; distance < length; ++distance) {
for (int start = 0; start < length - distance; ++start) {
int end = start + distance;
gameDp[start] = Math.max(values[start] - gameDp[start + 1], values[end] - gameDp[start]);
}
}
return gameDp[0] >= 0;
}
}
The given Java method canFirstPlayerWin
determines if the player who starts first in a game can win, provided the sequence of values representing points they can choose from. Here's a succinct breakdown of how the solution works:
- The method receives an array
values
which holds the points available to be selected from during the game. - The key strategy used here involves dynamic programming. An auxiliary array
gameDp
is created that starts with the same values as the input arrayvalues
. - The dynamic programming array is then updated to decide the optimal move at each position by analyzing the possible outcomes for each sub-array of the game.
- This update process uses two nested loops:
- The outer loop, controlled by the variable
distance
, considers increasing lengths of the sub-array starting from each possible point. - The inner loop, controlled by
start
, iterates over all possible starting points of these sub-arrays and updatesgameDp[start]
by choosing the best possible outcome from either end of the sub-array.
- The outer loop, controlled by the variable
- The logic of updating
gameDp[start]
calculates the maximum score difference (favoring the first player) achievable between choosing the currentstart
or the oppositeend
of the sub-array. - The decision-making uses recursion in form of
Math.max(values[start] - gameDp[start + 1], values[end] - gameDp[start])
, which evaluates whether picking the first or last element of the sub-array yields a higher advantage considering the next moves are optimal by the second player. - Finally, the method returns whether the first player can ensure a non-negative score difference (
gameDp[0] >= 0
), implying a win or tie situation.
Effectively, this method evaluates all possible game scenarios and strategically predicts whether starting first is a winning move given both players play optimally.
- Python
class Solution:
def canWin(self, numbers: List[int]) -> bool:
length = len(numbers)
table = numbers[:]
for distance in range(1, length):
for start in range(length - distance):
end = start + distance
table[start] = max(numbers[start] - table[start + 1], numbers[end] - table[start])
return table[0] >= 0
The solution presented uses a dynamic programming approach to solve the "Predict the Winner" problem in Python. This problem involves determining if the first player can secure a win given a set of numbers and optimal moves by both players.
- Start by copying the list of input numbers to a working table called
table
. - Use a nested loop where the outer loop controls the distance between the considered indices and the inner loop iterates through possible starting indices.
- For each pair of indices defined by the
start
andend
variables, update thetable[start]
value. This update chooses the maximum possible score the current player can secure assuming the opposing player also plays optimally. The value is calculated by comparing the scores obtained by either taking the first or the last element from the current subarray. - After populating the table, a check on
table[0]
determines if the first player can win or not. A non-negative value indicates that the first player has a winning strategy.
The overall approach systematically checks all possible game states to ensure the first player's moves are optimal, adjusting dynamically based on the second player's potential responses.
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