# The Princes in the Tower

May 8th, 2015
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One of England's greatest scandals was the disappearance and murder of two princes in 1483. The children of King Edward IV, Edward V and Richard, Duke of York were allegedly "imprisoned" shortly after his death. King Edward's brother, Richard, Duke of Gloucester, secretly wanted the throne for himself.

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PrefixPrefix and Suffix* A string w is a prefix of a string x, denoted w [ x, if x = wy for some string y in * * In this case, |w| |x|* Example: ab [ abcca* A string w is a suffix of a string x, denoted w ] x, if x = yw for some y in * * Also in this case, |w| |x|* Example: ca ] abcca* The empty string is both a prefix and suffix of every string Nave String Matching Algorithm* Loops to check if P[1..m] = T[s+1..s+m] for each of the n-m+1 possible shifts s * Keeps sliding the pattern along the text and compares corresponding characters * Runtime: O((n-m+1)m)NAVE-STRING-MATCHER(T, P)* 1 n = T.length* 2 m = P.length* 3 for s = 0 to n-m* 4 if P[1..m] == T[s+1..s+m]* 5 print pattern occurs with shift sRabin-Karp Algorithm* To compute p:* p = P[m] + 10(P[m-1] + 10(P[m-2] + + 10(P[2] + 10P[1]))) * To compute t1, t2, , tn-m:* ts+1 = 10(ts 10m-1T[s+1]) + T[s+m+1]* ts+1 = (d(ts T[s+1]h) + T[s+m+1]) mod q* where h dm-1 (mod q)RABIN-KARP-MATCHER(T, P, d, q)* 1 n = T.length* 2 m = P.length* 3 h = dm-1 mod q* 4 p = 0* 5 t0 = 0* 6 for i = 1 to m // preprocessing* 7 p = (dp + P[i]) mod q* 8 t0 = (dt0 + T[i]) mod q* 9 for s = 0 to n-m // matching* 10 if p == ts* 11 if P[1..m] == T[s+1..s+m]* 12 print pattern occurs with shift s* 13 if s < n m* 14 ts+1 = (d(ts T[s+1]h) + T[s+m+1]) mod q Runtime* Preprocessing: (m)* Matching: ((n-m+1)m) in worst case* Usually only have a few valid shift

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