Pretty Good Proportion

#### World Finals, 2015

Submit Solution(Code Jam Page)

### Problem

I have a sequence of N binary digits. I am looking for a substring with just the right proportion of 0s and 1s, but it may not exist, so I will settle for something that's just pretty good.

Can you find a substring where the fraction of 1s is as close as possible to the given fraction F? Output the earliest possible index at which such a substring starts.

### Input

The first line of the input gives the number of test cases, T. T test cases follow. Each one starts with a line containing N and F. F will be a decimal fraction between 0 and 1 inclusive, with exactly 6 digits after the decimal point. The next line contains N digits, each being either 0 or 1.

### Output

For each test case, output one line containing "Case #x: y", where x is the test case number (starting from 1) and y is the 0-based index of the start of the substring with the fraction of 1s that is as close as possible to F. If there are multiple possible answers, output the smallest correct value.

### Limits

1 ≤ T ≤ 100.
0 ≤ F ≤ 1
F will have exactly 6 digits after the decimal point.

1 ≤ N ≤ 1000.

1 ≤ N ≤ 500,000.

### Sample

 Input Output ```5 12 0.666667 001001010111 11 0.400000 10000100011 9 0.000000 111110111 5 1.000000 00000 15 0.333333 000000000011000 ``` ```Case #1: 5 Case #2: 5 Case #3: 5 Case #4: 0 Case #5: 6 ```

In Case #1, there is no substring that has exactly a 1-proportion of exactly 666667/1000000. The closest we can get is 2/3. The input string has 5 substrings that achieve it -- 3 substrings of length 3 that start at indices 5, 7, and 8 (101, 101, and 011); as well as two substrings of length 6 that start at indices 5 and 6 (101011 and 010111). The smallest of these indices is 5.

Points Correct Attempted
5pt 26 26
22pt 10 18