1809D · Binary String Sorting

Problem: 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.

Input Format: 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 $3 \cdot 10^5$.

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

Note: 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.