CF1809D Binary String Sorting

You are given a binary string $ s $ consisting of only characters 0 and/or 1. You can perform several operations on this string (possibly zero). There are two types of operations: - choose two consecutive elements and swap them. In order to perform this operation, you pay $ 10^{12} $ coins; - choose any element from the string and remove it. In order to perform this operation, you pay $ 10^{12}+1 $ coins. Your task is to calculate the minimum number of coins required to sort the string $ s $ in non-decreasing order (i. e. transform $ s $ so that $ s_1 \le s_2 \le \dots \le s_m $ , where $ m $ is the length of the string after applying all operations). An empty string is also considered sorted in non-decreasing order.

The first line contains a single integer $ t $ ( $ 1 \le t \le 10^4 $ ) — the number of test cases. The only line of each test case contains the string $ s $ ( $ 1 \le |s| \le 3 \cdot 10^5 $ ), consisting of only characters 0 and/or 1. The sum of lengths of all given strings doesn't exceed $ 3 \cdot 10^5 $ .

For each test case, print a single integer — the minimum number of coins required to sort the string $ s $ in non-decreasing order.

In the first example, you have to remove the $ 1 $ -st element, so the string becomes equal to 00. In the second example, the string is already sorted. In the third example, you have to swap the $ 2 $ -nd and the $ 3 $ -rd elements, so the string becomes equal to 0011. In the fourth example, you have to swap the $ 3 $ -rd and the $ 4 $ -th elements, so the string becomes equal to 00011101, and then remove the $ 7 $ -th element, so the string becomes equal to 0001111. In the fifth example, you have to remove the $ 1 $ -st element, so the string becomes equal to 001101, and then remove the $ 5 $ -th element, so the string becomes equal to 00111. In the sixth example, the string is already sorted.