All Tracks Problem
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Bob has arranged N boxes in a straight line. Each box has a Latin Character \(('a' - 'z')\) written on it. He wants to chain \(26\) boxes in such a way that the chain starts from box with alphabet a and contains all Latin Characters exactly once in increasing lexicographic order, ending at z. In other words, the chain must form the following sequence:
\('abcdefghijklmnopqrstuvwxyz'\)
To join a box \(b_i\) and \(b_j\) with chain, the length of the chain to be used is \(|i - j|\).
Help Bob to minimize the total length of the chain used to make the desired sequence.
Input Format:
First line of input contains of a single integer T - the number of testcases.
Each testcase consists of a single line, which contains a string made up of lowercase Latin Characters - a to z.
The string may have one or more occurrences of each character.
Output Format:
For each testcase, print a single integer - the minimum length of chain required to make the required sequence.
Input Constraints:
\(1 \le T \le 10\)
\(26 \le |S| \le 10^6\)
All 26 characters ('a' - 'z') are present in S at least once.
Testcase 1:
There is only one way to connect the chain for this sequence. The chain starts from \(S_0\), then connects to \(S_1\), then \(S_2\), ... , \(S_{25}\), which creates the sequence \('abcdefghijklmnopqrstuvwxyz'\). Hence, the answer for testcase 1 is \(25\) as the total length of the chain used is:
\((b_0 -> b_1) + (b_1 -> b_2) + ... + (b_{24} -> b_{25}) = |0 - 1| + |1 - 2| + ... + |24 - 25| = 1 + 1 + ... + 1 = 25.\)
Testcase 2:
There are several chain sequences that can possibly be made from the given sequence.
A few of them are:
1. \(S_2 -> S_4 -> S_0 -> S_6 -> S_7 -> S_8 -> ... -> S_{25}\)
2. \(S_3 -> S_4 -> S_5 -> S_6 -> ... -> S_{25}\)
Among the possible chain sequences, the sequence 2. listed above provides the minimum length of chain used.
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Math > Number Theory
