CF1550C Manhattan Subarrays

Suppose you have two points $ p = (x_p, y_p) $ and $ q = (x_q, y_q) $ . Let's denote the Manhattan distance between them as $ d(p, q) = |x_p - x_q| + |y_p - y_q| $ . Let's say that three points $ p $ , $ q $ , $ r $ form a bad triple if $ d(p, r) = d(p, q) + d(q, r) $ . Let's say that an array $ b_1, b_2, \dots, b_m $ is good if it is impossible to choose three distinct indices $ i $ , $ j $ , $ k $ such that the points $ (b_i, i) $ , $ (b_j, j) $ and $ (b_k, k) $ form a bad triple. You are given an array $ a_1, a_2, \dots, a_n $ . Calculate the number of good subarrays of $ a $ . A subarray of the array $ a $ is the array $ a_l, a_{l + 1}, \dots, a_r $ for some $ 1 \le l \le r \le n $ . Note that, according to the definition, subarrays of length $ 1 $ and $ 2 $ are good.

N/A

N/A

In the first test case, it can be proven that any subarray of $ a $ is good. For example, subarray $ [a_2, a_3, a_4] $ is good since it contains only three elements and: - $ d((a_2, 2), (a_4, 4)) = |4 - 3| + |2 - 4| = 3 $ $ < $ $ d((a_2, 2), (a_3, 3)) + d((a_3, 3), (a_4, 4)) = 3 + 1 + 2 + 1 = 7 $ ; - $ d((a_2, 2), (a_3, 3)) $ $ < $ $ d((a_2, 2), (a_4, 4)) + d((a_4, 4), (a_3, 3)) $ ; - $ d((a_3, 3), (a_4, 4)) $ $ < $ $ d((a_3, 3), (a_2, 2)) + d((a_2, 2), (a_4, 4)) $ ; In the second test case, for example, subarray $ [a_1, a_2, a_3, a_4] $ is not good, since it contains a bad triple $ (a_1, 1) $ , $ (a_2, 2) $ , $ (a_4, 4) $ : - $ d((a_1, 1), (a_4, 4)) = |6 - 9| + |1 - 4| = 6 $ ; - $ d((a_1, 1), (a_2, 2)) = |6 - 9| + |1 - 2| = 4 $ ; - $ d((a_2, 2), (a_4, 4)) = |9 - 9| + |2 - 4| = 2 $ ; So, $ d((a_1, 1), (a_4, 4)) = d((a_1, 1), (a_2, 2)) + d((a_2, 2), (a_4, 4)) $ .