Solve two DS&A optimization problems
Overview
This question evaluates algorithm design and combinatorial optimization skills by combining array-partition sum maximization with constrained subset selection, focusing on reasoning about interval sums, feasibility under per-item constraints, and trade-offs in selecting indices or team members.
Problem 1 — Maximize alternating-sum over four array partitions: Given an integer array arr[1..n] (1-based). Choose indices a, b, c with 1 ≤ a ≤ b ≤ c ≤ n+1. For any half-open interval [l, r), define sum[l, r) as the sum of arr[l..r−1], with sum[l, l) = 0. Cutting at a, b, c partitions the array into four contiguous segments in order: S4 = sum[1, a), S1 = sum[a, b), S2 = sum[b, c), S3 = sum[c, n+ 1). Define grossValue(a, b, c) = S1 − S2 + S3 − S4. Compute the maximum possible grossValue over all valid (a, b, c). State your algorithm and its time/space complexity; handle edge cases where cuts coincide or lie at the ends.
Problem 2 — Select largest team under lower/higher‑skill constraints: There are n developers with distinct skill levels 1..n (developer i has skill i). Arrays lowerSkill[1..n] and higherSkill[1..n] are given. A developer i will join the team only if at most lowerSkill[i] chosen teammates have skill lower than i, and at most higherSkill[i] chosen teammates have skill higher than i. Find the maximum possible team size (and optionally one valid team) such that every selected developer’s constraints are satisfied. Describe the algorithm and analyze its complexity.