G. Underpalindromity

Codeforces

IDCF10175G

Time2000ms

Memory256MB

Difficulty—

English · Original

Formal · Original

Let us call _underpalindromity_ of array b of length k the minimal number of times one need to increment some elements bj by 1 so that the array b would become a palindrome, that is, b1 = bk, b2 = bk - 1, and so on. The array of length n, consisting of integers, is given. Consider all its subarrays of length k, and for each of these subarrays its _underpalindromity_ pi. It's needed to calculate sum of all pi (1 ≤ i ≤ n - k + 1). The first line contains two integers n and k (1 ≤ k ≤ n ≤ 200000) — the length of the array and the length of subarrays. The second line contains n integers ai ( - 108 ≤ ai ≤ 108) — the elements of the array. Output a single integer — sum of _underpalindromities_ of all subarrays of length k. ## Input The first line contains two integers n and k (1 ≤ k ≤ n ≤ 200000) — the length of the array and the length of subarrays.The second line contains n integers ai ( - 108 ≤ ai ≤ 108) — the elements of the array. ## Output Output a single integer — sum of _underpalindromities_ of all subarrays of length k. [samples]

**Definitions** Let $ n, k \in \mathbb{Z} $ with $ 1 \leq k \leq n \leq 200000 $. Let $ A = (a_1, a_2, \dots, a_n) $ be a sequence of integers, where $ a_i \in [-10^8, 10^8] $. For a subarray $ B_i = (a_i, a_{i+1}, \dots, a_{i+k-1}) $ of length $ k $, define its **underpalindromity** $ p_i $ as: $$ p_i = \sum_{j=1}^{\lfloor k/2 \rfloor} \left| a_{i+j-1} - a_{i+k-j} \right| $$ **Constraints** 1. $ 1 \leq k \leq n \leq 200000 $ 2. $ -10^8 \leq a_i \leq 10^8 $ for all $ i \in \{1, \dots, n\} $ **Objective** Compute: $$ \sum_{i=1}^{n - k + 1} p_i = \sum_{i=1}^{n - k + 1} \sum_{j=1}^{\lfloor k/2 \rfloor} \left| a_{i+j-1} - a_{i+k-j} \right| $$

API Response (JSON)

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