Reference no: EM13728
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, . . . , sk}, 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, . . . , sk} 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 αk 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 . . .wk, 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.