#3096

Medium

LeetCode Problem Workspace

Minimum Levels to Gain More Points

Determine the minimum number of levels Alice must play in a binary game array to secure more points than Bob using prefix sums.

array prefix sum

Problem Statement

You are given a binary array possible of length n representing levels in a game. Each value indicates whether a level can be cleared: 1 means clearable and 0 means impossible. Alice and Bob play sequentially, with Alice starting from level 0 and Bob playing the remaining levels. Each cleared level gives 1 point, and each failed level subtracts 1 point.

Determine the minimum number of levels Alice must play at the start so that she ends with more points than Bob. If it is impossible for Alice to gain more points regardless of her moves, return -1. Convert all 0 values in the array to -1 to track net points and use prefix sums to evaluate potential splits efficiently.

Examples

Example 1

Input: possible = [1,0,1,0]

Output: 1

Let's look at all the levels that Alice can play up to: Alice must play a minimum of 1 level to gain more points.

Example 2

Input: possible = [1,1,1,1,1]

Output: 3

Let's look at all the levels that Alice can play up to: Alice must play a minimum of 3 levels to gain more points.

Example 3

Input: possible = [0,0]

Output: -1

The only possible way is for both players to play 1 level each. Alice plays level 0 and loses 1 point. Bob plays level 1 and loses 1 point. As both players have equal points, Alice can't gain more points than Bob.

Constraints

2 <= n == possible.length <= 105
possible[i] is either 0 or 1.

Solution Approach

Transform the Array

Change all 0 values in possible to -1 to represent failed levels. This allows calculation of net points by simple addition, linking directly to the Array plus Prefix Sum pattern.

Compute Prefix Sums

Calculate prefix sums from the start of the array. Each prefix sum represents Alice's total if she plays that many levels. This step efficiently captures Alice's potential point advantage over Bob.

Identify Minimum Levels

Iterate over prefix sums to find the smallest number of levels where Alice's score exceeds Bob's remaining points. If no such prefix sum exists, return -1, highlighting the key failure mode of insufficient playable levels.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity is O(n) because each element is processed once during the transformation and prefix sum calculation. Space complexity is O(n) for storing prefix sums, though it can be optimized to O(1) using a running total instead of an array.

What Interviewers Usually Probe

Look for efficient conversion of 0s to -1s to simplify net gain calculation.
Expect prefix sum usage to determine Alice's advantage quickly without nested loops.
Watch out for edge cases where all levels are impossible or Alice cannot surpass Bob.

Common Pitfalls or Variants

Common pitfalls

Forgetting to convert 0s to -1s, which leads to incorrect net point computation.
Not considering the case where Alice cannot gain more points and returning an invalid index.
Calculating Bob's points incorrectly by not accounting for sequential play split.

Follow-up variants

Find minimum levels Alice must play when some levels have variable points instead of 1 or -1.
Determine maximum points Alice can achieve using the same prefix sum approach.
Apply the array plus prefix sum pattern to multiple players instead of just Alice and Bob.

FAQ

What is the key idea behind Minimum Levels to Gain More Points?

The main idea is to convert impossible levels to -1 and use prefix sums to determine the minimum levels Alice must play to outscore Bob.

How does changing 0s to -1s affect the solution?

It simplifies net score calculation, allowing direct addition in prefix sums to track Alice's advantage efficiently.

Can this solution handle very large arrays?

Yes, the O(n) time complexity ensures even arrays up to 10^5 elements are processed efficiently.

What if all levels are impossible for Alice?

In that case, Alice cannot gain more points than Bob, and the function should return -1.

Why is this considered an Array plus Prefix Sum pattern problem?

Because the solution relies on transforming the array and computing prefix sums to evaluate cumulative point advantage at each split point.

terminal

Solution

Solution 1: Enumeration

First, we calculate the sum of the scores that both players can get, denoted as $s$.

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class Solution:
    def minimumLevels(self, possible: List[int]) -> int:
        s = sum(-1 if x == 0 else 1 for x in possible)
        t = 0
        for i, x in enumerate(possible[:-1], 1):
            t += -1 if x == 0 else 1
            if t > s - t:
                return i
        return -1

Continue Practicing

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