LeetCode 837 New 21 Game - Implementation

Naive: for each $dp[i]$, sum last $maxPts$ values. $O(n \cdot maxPts)$ total. improvement: maintain a running sum $S$ of the last $maxPts$ probabilities. $dp[i] = S / maxPts$. Update $S$: add $dp[i]$ (if $i < k$, since we keep drawing), subtract $dp[i - maxPts]$ (if valid). Final complexity: $O(n)$ time, $O(n)$ space. The sliding window avoids recomputing the sum. The sliding window sum avoids recomputing the last maxPts probabilities. This is a common DP improvement.

Time complexity: $O(n)$.

Space complexity: $O(n)$.