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Problem: Minimum Sum of Absolute Differences in Subsequences
Description:
Given an array of length N (where 1≤N≤1e5) and an integer k (where 1≤k≤50), find the minimum possible value of the sum of absolute differences of adjacent elements for all subsequences of length k.
Mathematically, for a subsequence {ai1,ai2,…,aik} where i1<i2<…<iki1<i2<…<ik, we need to minimize the value of:
This is a relatively standard dynamic programming problem. Let dp[i][j] represent the minimum total cost of a subsequence of length j ending on the ith element. Our transitions are simply
dp[i][j]= min k<i (dp[k][j-1]+ abs(a[k]-a[i]))
Naively this can be computed in O(N^2*K). To optimize this we can use a segment tree similar to its use in CSES pizzeria queries. This solution will run in O(NKlogN) time.