Longest Consecutive Sequence — Adjacent Gap ≤ K

LC 128 variant: instead of demanding successive integers, two values are considered consecutive when their absolute difference is at most `k`. Requirements Input: an integer array nums and a positive integer k. Define a "chain" as a sequence of values from nums ordered so that every adjacent pair differs by at most k. Return the length of the longest such chain that can be assembled from distinct elements of nums. Clarify with the interviewer whether duplicates count once or as repeated chain members — the loop report did not pin this down. Notes Reduces to walking the sorted unique values once: extend the current chain while sorted[i] - sorted[i-1] <= k, otherwise restart. O(n log n) time, O(n) space. The hashmap-based LC 128 trick (start a chain only at values with no v - 1 predecessor) does not generalize directly because gaps up to k mean the "start" check would need a sliding window — sort-and-scan is simpler. When k = 1 this collapses back to the original Longest Consecutive Sequence; using that as a sanity test confirms the chain logic before adding the k knob. Preparation Re-solve LC 128 from scratch (hashmap O(n) version), then port to the sort-and-scan variant and benchmark mentally for n = 1e5. Pre-build a 60-second derivation that justifies why sort-and-scan is correct: any optimal chain, once sorted, still satisfies the gap constraint. Practice articulating the duplicate-handling clarifying question — this is exactly the kind of unspecified knob Amazon coding rounds reward you for surfacing.