1.

Using suffix trees, give an algorithm to nd a longest common substring shared among three input strings. s₁ of length n₁, s₂ of length n₂ and s₃ of length n₃.

2.

A non-empty string  is called a minimal unique substring of s if and only if it satises.

(i) occurs exactly once in s (uniqueness),
(ii) all proper prexes of  occur at least twice in s (mimimality), and
(iii) l for some constant l.

Give an optimal algorithm to enumerate all minimal unique substrings of s.

3.

Redundant sequence identication. Given a set of k DNA sequences, S = {s₁, s₂, . . . , s_k}, give an optimal algorithm to identify all sequences that are completely contained in (i.e., substrings of) at least one other sequence in S.

4.

Let S = {s₁, s₂, . . . , s_k} denote a set of k genomes. The problem of ngerprinting is the task of identifying a shortest possible substring i from each string si such that i is unique to si | i.e., no other genome in the set S has i. Such an i will be called a ngerprint of si.

Give an algorithm to enumerate a ngerprint for each input genome, if one exists. Assume that no two input genomes are identical.

5.
A string s is said to be periodic with a period α, if s is α^k for some k ≥2. A DNA sequence s is called a tandem repeat if it is periodic. Given a DNA sequence s, determine if it is periodic, and if so, the values for and k. Note that there could be more than one period for a periodic string. In such a case, you need to report the shortest period.

6.

A non-empty string  is called a repeat prex of a string s if  is a prex of s. Give a linear time algorithm to nd the longest repeat prex of s.

7.

Given strings s₁ and s₂ of lengths m and n respectively, a minimum cover of s₁ by s₂ is a decomposition s₁ = w₁w₂ . . .w_k, where each wi is a non-empty substring of s2 and k is minimized. Eg., given s₁ = accgtatct and s₂ = cgtactcatc, there are several covers of s₁ by s₂ possible, two of which are. (i) cover1. s₁ = w₁w₂w₃w₄ (where w₁ = ac, w₂ = cgt, w₃ = atc, w₄ = t) , and (ii) cover2. s₁ =  w₁w₂w₃w₄w₅ (where w₁ = ac, w₂ = c, w₃ = gt, w₄ = atc, w₅ = t). However, only cover1 is a minimal cover. Give an algorithm to compute a minimum cover (if one exists) in O(m + n) time and space. If a minimum cover does not exist your algorithm should state so and terminate within the same time and space bounds. Give a brief justication of why you think your algorithm is correct | meaning, how it guarantees nding the minimial cover.