Stone Game VI

Stone Game VI - Problem

Array Math Greedy Sorting Heap (Priority Queue) Game Theory Medium

Alice and Bob take turns playing a game, with Alice starting first.

There are n stones in a pile. On each player's turn, they can remove a stone from the pile and receive points based on the stone's value. Alice and Bob may value the stones differently.

You are given two integer arrays of length n, aliceValues and bobValues. Each aliceValues[i] and bobValues[i] represents how Alice and Bob, respectively, value the i-th stone.

The winner is the person with the most points after all the stones are chosen. If both players have the same amount of points, the game results in a draw. Both players will play optimally. Both players know the other's values.

Determine the result of the game, and:

If Alice wins, return 1.
If Bob wins, return -1.
If the game results in a draw, return 0.

Input & Output

Example 1 — Alice Wins

$ Input: aliceValues = [1,3], bobValues = [3,1]

› Output: 1

💡 Note: Combined values: [4,4]. Alice picks first stone (gains 1), Bob picks second (gains 1). Alice: 1, Bob: 1. Tie, so result is 0. Wait - let me recalculate: If Alice picks stone 0 (A:1,B:3), then Bob picks stone 1 (A:3,B:1) and gets 1 point. Alice:1, Bob:1 = tie. But if Alice picks stone 1 first (A:3,B:3), Bob gets stone 0 (A:1,B:3) and gets 3. Alice:3, Bob:3 = tie. Actually both lead to tie, so result is 0.

Example 2 — Different Values

$ Input: aliceValues = [1,2], bobValues = [3,1]

› Output: 0

💡 Note: Combined values: stone 0 has sum 4, stone 1 has sum 3. Alice picks stone 0 (gains 1), Bob picks stone 1 (gains 1). Alice: 1, Bob: 1. Draw.

Example 3 — Clear Winner

$ Input: aliceValues = [2,4,3], bobValues = [1,1,2]

› Output: 1

💡 Note: Combined values: [3,5,5]. Sorted order: stone 1 (sum 5), stone 2 (sum 5), stone 0 (sum 3). Alice picks stone 1 (gains 4), Bob picks stone 2 (gains 2), Alice picks stone 0 (gains 2). Alice: 6, Bob: 2. Alice wins.

Constraints

n == aliceValues.length == bobValues.length
1 ≤ n ≤ 10⁵
1 ≤ aliceValues[i], bobValues[i] ≤ 100

Visualization

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Asked in

G Google 12 a Amazon 8 f Facebook 6

The key insight is that picking a stone gives you points AND denies your opponent their points for that stone. The optimal strategy is to sort stones by combined value (aliceValues[i] + bobValues[i]) and have players pick greedily from highest impact stones first. Best approach uses greedy sorting in O(n log n) time, O(n) space.

Common Approaches

✓ Brute Force - Game Tree Exploration

⏱️ Time: O(n!) Space: O(n)

Use recursion to simulate the game where each player alternately picks stones optimally. At each turn, try all remaining stones and choose the one that leads to the best outcome for the current player.

Greedy - Combined Value Priority

⏱️ Time: O(n log n) Space: O(n)

The key insight is that when a player picks a stone, they gain their value for it AND prevent the opponent from getting their value. So the total impact of picking stone i is aliceValues[i] + bobValues[i]. Sort by this combined value and simulate optimal play.

Brute Force - Game Tree Exploration — Algorithm Steps

Recursively simulate the game for each player's turn
For each turn, try picking every remaining stone
Choose the stone that maximizes current player's advantage

Visualization

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Step-by-Step Walkthrough

Initial State

All stones available, Alice's turn

Recursive Search

Try each stone, simulate opponent's optimal response

Backtrack

Return best result for current player

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <stdbool.h>

int gameResult(bool* used, int* aliceValues, int* bobValues, int n,
              int aliceScore, int bobScore, bool isAliceTurn) {
    bool allUsed = true;
    for (int i = 0; i < n; i++) {
        if (!used[i]) {
            allUsed = false;
            break;
        }
    }
    
    if (allUsed) {
        if (aliceScore > bobScore) return 1;
        else if (bobScore > aliceScore) return -1;
        else return 0;
    }
    
    if (isAliceTurn) {
        int bestResult = -2;
        for (int i = 0; i < n; i++) {
            if (!used[i]) {
                used[i] = true;
                int result = gameResult(used, aliceValues, bobValues, n,
                                      aliceScore + aliceValues[i], bobScore, false);
                used[i] = false;
                if (result > bestResult) {
                    bestResult = result;
                }
            }
        }
        return bestResult;
    } else {
        int bestResult = 2;
        for (int i = 0; i < n; i++) {
            if (!used[i]) {
                used[i] = true;
                int result = gameResult(used, aliceValues, bobValues, n,
                                      aliceScore, bobScore + bobValues[i], true);
                used[i] = false;
                if (result < bestResult) {
                    bestResult = result;
                }
            }
        }
        return bestResult;
    }
}

int solution(int* aliceValues, int* bobValues, int n) {
    bool* used = (bool*)calloc(n, sizeof(bool));
    int result = gameResult(used, aliceValues, bobValues, n, 0, 0, true);
    free(used);
    return result;
}

int main() {
    char line1[1000], line2[1000];
    fgets(line1, sizeof(line1), stdin);
    fgets(line2, sizeof(line2), stdin);
    
    // Simple parsing for arrays
    int aliceValues[100], bobValues[100];
    int n = 0;
    
    char* token = strtok(line1 + 1, ",]");
    while (token != NULL) {
        aliceValues[n] = atoi(token);
        token = strtok(NULL, ",]");
        n++;
    }
    
    token = strtok(line2 + 1, ",]");
    int idx = 0;
    while (token != NULL) {
        bobValues[idx] = atoi(token);
        token = strtok(NULL, ",]");
        idx++;
    }
    
    printf("%d\n", solution(aliceValues, bobValues, n));
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n!)

Each player can choose from n, n-1, n-2... stones, leading to n! possible game sequences

⚠ Quadratic Growth

Space Complexity

O(n)

Recursion depth equals number of stones, plus array to track used stones

⚡ Linearithmic Space

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Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Brute Force - Game Tree Exploration — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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