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   Problem Statement  

 Problem Statement for PalindromicSubseq

Problem Statement

    

A palindrome is a string that reads the same forwards and backwards. For example, "abba" and "racecar" are palindromes.



Limak has a String s consisting of N lowercase English letters.



For each valid i, let X[i] be the number of palidromic subsequences of s that contain the i-th character of s.



In other words: For a fixed i, there are exactly 2^(N-1) ways to erase some characters of s other than the character s[i]. X[i] is the number of ways of erasing for which the string that remains is a palindrome.



For each i, compute the value Y[i] = ((i+1) * X[i]) modulo 1,000,000,007. Then, compute and return the bitwise XOR of all the values Y[i].

 

Definition

    
Class:PalindromicSubseq
Method:solve
Parameters:String
Returns:int
Method signature:int solve(String s)
(be sure your method is public)
    
 

Constraints

-s will contain exactly N characters.
-N will be between 1 and 3000, inclusive.
-Each character in s will be a lowercase English letter 'a' - 'z'.
 

Examples

0)
    
"aaba"
Returns: 30

For this string we have X = {5, 5, 3, 6}. Thus, you should return the value (1*5) XOR (2*5) XOR (3*3) XOR (4*6) = 5 XOR 10 XOR 9 XOR 24 = 30.



Here is an explanation why X[0] = 5: Given that we want to keep the character s[0], there are 8 possible subsequences of s:

a...  *
a..a  *
a.b.
a.ba  *
aa..  *
aa.a  *
aab.
aaba

Five of these subsequences (marked by stars) are palindromes. Note that there are two different ways to produce the palindrome "aa".

1)
    
"abcd"
Returns: 4
X[i] = 1 for all i. You should return (1*1) XOR (2*1) XOR (3*1) XOR (4*1) = 1 XOR 2 XOR 3 XOR 4 = 4.
2)
    
"tcoct"
Returns: 60
X[] = {7, 6, 4, 6, 7}
3)
    
"zyzyzzzzxzyz"
Returns: 717
4)
    
"aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa"
Returns: 1025495382

If all characters are the same, all 2^N subsequences are palindromes. For every index i all 2^(N-1) subseqeunces that contain it are palindromes. It means that X[i] = 2^(N-1) for all i.



Note that in this case the answer exceeds the modulo value, because we return the XOR of modulo-ed values, without computing the modulo of that XOR.

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This problem was used for:
       Single Round Match 708 Round 1 - Division I, Level Two