CF1779C Least Prefix Sum

Baltic, a famous chess player who is also a mathematician, has an array $ a_1,a_2, \ldots, a_n $ , and he can perform the following operation several (possibly $ 0 $ ) times: - Choose some index $ i $ ( $ 1 \leq i \leq n $ ); - multiply $ a_i $ with $ -1 $ , that is, set $ a_i := -a_i $ . Baltic's favorite number is $ m $ , and he wants $ a_1 + a_2 + \cdots + a_m $ to be the smallest of all non-empty prefix sums. More formally, for each $ k = 1,2,\ldots, n $ it should hold that $ $$$a_1 + a_2 + \cdots + a_k \geq a_1 + a_2 + \cdots + a_m. $ $

Please note that multiple smallest prefix sums may exist and that it is only required that $ a\_1 + a\_2 + \\cdots + a\_m $ is one of them.

Help Baltic find the minimum number of operations required to make $ a\_1 + a\_2 + \\cdots + a\_m$$$ the least of all prefix sums. It can be shown that a valid sequence of operations always exists.

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In the first example, we perform the operation $ a_4 := -a_4 $ . The array becomes $ [-1,-2,-3,4] $ and the prefix sums, $ [a_1, \ a_1+a_2, \ a_1+a_2+a_3, \ a_1+a_2+a_3+a_4] $ , are equal to $ [-1,-3,-6,-2] $ . Thus $ a_1 + a_2 + a_3=-6 $ is the smallest of all prefix sums. In the second example, we perform the operation $ a_3 := -a_3 $ . The array becomes $ [1,2,-3,4] $ with prefix sums equal to $ [1,3,0,4] $ . In the third and fourth examples, $ a_1 + a_2 + \cdots + a_m $ is already the smallest of the prefix sums — no operation needs to be performed. In the fifth example, a valid sequence of operations is: - $ a_3 := -a_3 $ , - $ a_2 := -a_2 $ , - $ a_5 := -a_5 $ . The array becomes $ [-2,-3,5,-5,20] $ and its prefix sums are $ [-2,-5,0,-5,15] $ . Note that $ a_1+a_2=-5 $ and $ a_1+a_2+a_3+a_4=-5 $ are both the smallest of the prefix sums (and this is a valid solution).