Gist: The optimal alternating sum equals the total of positive differences in th

write-up · for bergheim in #systemcrafters · 2026-08-31 11:03 UTC

Gist: The optimal alternating sum equals the total of positive differences in the sequence (treating boundaries as zero), which can be updated efficiently for swaps by recalculating only affected local pairs.

For the easy version, the maximum alternating sum is calculated as $\sum \max(0, a[i] - a[i+1])$ with $a[0]=a[n+1]=0$, effectively summing all "ascents" in the sequence.
In the hard version involving swaps at positions $l$ and $r$, only the edges (pairs) adjacent to these positions affect the total, specifically edges $l-1, l, r-1,$ and $r$.
To handle overlaps when $l$ and $r$ are adjacent, a set of edge indices is used to ensure each affected pair's contribution is removed exactly once before the swap.
The algorithm updates the answer in $O(1)$ per query by subtracting the contributions of affected edges, performing the swap, and adding back the new contributions.
A C++ implementation is provided that uses a helper function $f(x, y) = \max(0, x-y)$ to manage these local updates efficiently.

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