CF1536D Omkar and Medians

Uh oh! Ray lost his array yet again! However, Omkar might be able to help because he thinks he has found the OmkArray of Ray's array. The OmkArray of an array $ a $ with elements $ a_1, a_2, \ldots, a_{2k-1} $ , is the array $ b $ with elements $ b_1, b_2, \ldots, b_{k} $ such that $ b_i $ is equal to the median of $ a_1, a_2, \ldots, a_{2i-1} $ for all $ i $ . Omkar has found an array $ b $ of size $ n $ ( $ 1 \leq n \leq 2 \cdot 10^5 $ , $ -10^9 \leq b_i \leq 10^9 $ ). Given this array $ b $ , Ray wants to test Omkar's claim and see if $ b $ actually is an OmkArray of some array $ a $ . Can you help Ray? The median of a set of numbers $ a_1, a_2, \ldots, a_{2i-1} $ is the number $ c_{i} $ where $ c_{1}, c_{2}, \ldots, c_{2i-1} $ represents $ a_1, a_2, \ldots, a_{2i-1} $ sorted in nondecreasing order.

N/A

N/A

In the second case of the first sample, the array $ [4] $ will generate an OmkArray of $ [4] $ , as the median of the first element is $ 4 $ . In the fourth case of the first sample, the array $ [3, 2, 5] $ will generate an OmkArray of $ [3, 3] $ , as the median of $ 3 $ is $ 3 $ and the median of $ 2, 3, 5 $ is $ 3 $ . In the fifth case of the first sample, the array $ [2, 1, 0, 3, 4, 4, 3] $ will generate an OmkArray of $ [2, 1, 2, 3] $ as - the median of $ 2 $ is $ 2 $ - the median of $ 0, 1, 2 $ is $ 1 $ - the median of $ 0, 1, 2, 3, 4 $ is $ 2 $ - and the median of $ 0, 1, 2, 3, 3, 4, 4 $ is $ 3 $ . In the second case of the second sample, the array $ [1, 0, 4, 3, 5, -2, -2, -2, -4, -3, -4, -1, 5] $ will generate an OmkArray of $ [1, 1, 3, 1, 0, -2, -1] $ , as - the median of $ 1 $ is $ 1 $ - the median of $ 0, 1, 4 $ is $ 1 $ - the median of $ 0, 1, 3, 4, 5 $ is $ 3 $ - the median of $ -2, -2, 0, 1, 3, 4, 5 $ is $ 1 $ - the median of $ -4, -2, -2, -2, 0, 1, 3, 4, 5 $ is $ 0 $ - the median of $ -4, -4, -3, -2, -2, -2, 0, 1, 3, 4, 5 $ is $ -2 $ - and the median of $ -4, -4, -3, -2, -2, -2, -1, 0, 1, 3, 4, 5, 5 $ is $ -1 $ For all cases where the answer is NO, it can be proven that it is impossible to find an array $ a $ such that $ b $ is the OmkArray of $ a $ .