TopCoder problem "OverlappingConcatenation" used in TCO10 Round 5 (Division I Level Three)

Problem Statement

    An overlapping concatenation of an ordered pair of Strings (A, B) is a String S such that A is a prefix of S and B is a suffix of S. A shortest overlapping concatenation (SOC) of (A, B) is an overlapping concatenation whose length is as small as possible. For example, if A is "cabab" and B is "ababc", the shortest overlapping concatenation is "cababc" (all quotes for clarity). You are given two Strings A and B, in which some of the characters are known and some are unknown. Each character in A and B will be a lowercase letter ('a' - 'z') if it is known or '*' if it is unknown.

Assuming that each unknown character will be selected uniformly at random from the characters contained in alphabet, return the expected length of the shortest overlapping concatenation of A and B.


Parameters:String, String, String
Method signature:double shortestExpected(String A, String B, String alphabet)
(be sure your method is public)


-alphabet will contain between 1 and 26 lowercase letters ('a' - 'z'), inclusive.
-The characters in alphabet will be distinct.
-A and B will contain between 1 and 32 characters, inclusive.
-A and B will contain the same number of characters.
-Each character in A and B will either be contained in alphabet or it will be '*'.


Returns: 7.0
The 4 equal-probability possibilities for the shortest overlapping concatenation (SOC) are as follows:

A     | B     | SOC
aaaaa | aaaaa | aaaaa
aabaa | aaaaa | aabaaaaa
aaaaa | abaaa | aaaaabaaa
aabaa | abaaa | aabaaa
The result is therefore (5 + 8 + 9 + 6) / 4 = 7.
Returns: 3.125
There are 16 possibilities for A and B. Of these, 4 give a SOC of length 2; 6 give a SOC of length 3; and 6 give a SOC of length 4. The answer is therefore (4*2 + 6*3 + 6*4) / 16 = 3.125.
Returns: 10.0
It makes no difference what the unknown characters are; the SOC is always the same length.
Returns: 10.555555555555557

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PabloGilberto , ivan_metelsky , zhuzeyuan

Problem categories:

Dynamic Programming, Math