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Greedy Algorithms CS 6030 by Savitha Parur Venkitachalam Outline • • • • • • Greedy approach to Motif searching Genome rearrangements Sorting by Reversals Greedy algorithms for sorting by reversals Approximation algorithms Breakpoint Reversal sort Greedy motif searching • Developed by Gerald Hertz and Gary Stormo in 1989 • CONSENSUS is the tool based on greedy algorithm • Faster than Brute force and Simple motif search algorithms • An approximation algorithm with an unknown approximation ratio Greedy motif search – Psuedocode Greedy motif search – Steps • Input – DNA Sequence , t (# sequences) , n (length of one sequence) , l (length of motif to search) • Output – set of starting points of l-mers • Performs an exhaustive search using hamming distance on first two sequences of the DNA • Forms a 2 x l seed matrix with the two closest l-mers • Scans the rest of t-2 sequences to find the l-mer that best matches the seed and add it to the next row of the seed matrix Complexity • Exhaustive search on first two sequences require l(n-l+1)2 operations which is O(ln2) • The sequential scan on t-2 sequences requires l(n-l+1)(t-2) operations which is O(lnt) • Thus running time of greedy motif search is O(ln2 + lnt) • If t is small compared to n algorithm behaves O(ln2) Consensus tool • Greedy motif algorithm may miss the optimal motif • Consensus tool saves large number of seed matrices • Consensus tool can check sequences in random • Consensus tool is less likely to miss the optimal motif Genome rearrangements • Gene rearrangements results in a change of gene ordering • Series of gene rearrangements can alter genomic architecture of a species • 99% similarity between cabbage and turnip genes • Fewer than 250 genomic rearrangements since divergence of human and mice History of Chromosome X Rat Consortium, Nature, 2004 Types of Rearrangements Reversal 1 2 3 4 5 6 1 2 -5 -4 -3 6 Translocation 1 2 3 45 6 1 26 4 53 Fusion 1 2 3 4 5 6 1 2 3 4 5 6 Fission Greedy algorithms in Gene Rearrangements • Biologists are interested in finding the smallest number of reversals in an evolutionary sequence • gives a lower bound on the number of rearrangements and the similarity between two species • Two greedy algorithms used - Simple reversal sort - Breakpoint reversal sort Gene Order • Gene order is represented by a permutation p: p = p 1 ------ p i-1 p i p i+1 ------ p j-1 p j p j+1 ----- p n Reversal r ( i, j ) reverses (flips) the elements from i to j in p p * r ( i, j ) ↓ p 1 ------ p i-1 p j p j-1 ------ p i+1 p i p j+1 ----- pn Reversal example p=12345678 r(3,5) ↓ 12543678 r(5,6) ↓ 12546378 Reversal distance problem • Goal: Given two permutations, find the shortest series of reversals that transforms one into another • Input: Permutations p and s • Output: A series of reversals r1,…rt transforming p into s, such that t is minimum • t - reversal distance between p and s • d(p, s) - smallest possible value of t, given p and s Sorting by reversal • Goal : Given a permutation , find a shortest series of reversals that transforms it into the identity permutation. • Input: Permutation π • Output : A series of reversals r1,…rt transforming p into identity permutation, such that t is minimum Sorting by reversal - Greedy algorithm • If sorting permutation p = 1 2 3 6 4 5, the first three elements are already in order so it does not make any sense to break them. • The length of the already sorted prefix of p is denoted prefix(p) – prefix(p) = 3 • This results in an idea for a greedy algorithm: increase prefix(p) at every step Simple Reversal sort – Psuedocode • A very generalized approach leads to analgorithm that sorts by moving ith element to ith position SimpleReversalSort(p) 1 for i 1 to n – 1 2 j position of element i in p (i.e., pj = i) 3 if j ≠i 4 p p * r(i, j) 5 output p 6 if p is the identity permutation 7 return Example – SimpleReversalSort not optimal Input – 612345 612345 ->162345 ->126345 ->123645->123465 -> 123456 Greedy SimpleReversalSort takes 5 steps where as optimal solution only takes 2 steps 612345 -> 543216 -> 123456 • An example of SimpleReversalSort is ‘Pancake Flipping problem’ Approximation Ratio • These algorithms produce approximate solution rather than an optimal one • Approximation ratio is of an algorithm A is given by A(p) / OPT(p) – For algorithm A that minimizes objective function (minimization algorithm): • max|p| = n A(p) / OPT(p) – For maximization algorithm: • min|p| = n A(p) / OPT(p) Breakpoints – A different face of greed • In a permutation p = p 1 ----p n - if p i and p i+1 are consecutive numbers it is an adjacency - if p i and p i+1 are not consecutive numbers it is a breakpoint Example: p =1|9|3 4|7 8|2 |6 5 • Pairs (1,9), (9,3), (4,7), (8,2) and (2,6) form breakpoints • Pairs (3,4) (7,8) and (6,5) form adjacencies • b(p) - # breakpoints in permutation p • Our goal is to eliminate all breakpoints and thus forming the identity permutation Breakpoint Reversal Sort – Steps • • • • Put two elements p 0 =0 and p n + 1=n+1 at the ends of p Eliminate breakpoints using reversals Each reversal eliminates at most 2 breakpoints This implies reversal distance ≥ #breakpoints/2 p =2 3 1 4 6 5 0 0 0 0 2 1 1 1 3 3 2 2 1 2 3 3 4 4 4 4 6 6 6 5 57 5 7 5 7 6 7 b(p) = 5 b(p) = 4 b(p) = 2 b(p) = 0 • Not efficient as it may run forever Psuedocode – Breakpoint reversal Sort BreakPointReversalSort(p) 1 while b(p) > 0 2 Among all possible reversals, choose reversal r minimizing b(p • r) 3 p p • r(i, j) 4 output p 5 return Using strips A strip is an interval between two consecutive breakpoints in a permutation • Decreasing strip: strip of elements in decreasing order • Increasing strip: strip of elements in increasing order 0 1 9 4 3 7 8 2 5 6 10 • A single-element strip can be declared either increasing or decreasing. We will choose to declare them as decreasing with exception of the strips with 0 and n+1 Reducing breakpoints • Choose the decreasing strip with the smallest element k in p • Find K-1 in the permutation • Reverse the segment between k and k-1 Eg: p = 1 4 6 5 7 8 3 2 0 1 4 6 5 7 8 3 2 9 b(p) = 5 0 1 2 3 8 7 5 6 4 9 b(p ) = 4 01234 65 789 b(p ) = 2 0123456789 ImprovedBreakpointReversalSort • Sometimes permutation may not contain any decreasing strips • So an increasing strip has to be reversed so that it becomes a decreasing strip • Taking this into consideration we have an improved algorithm ImprovedBreakpointReversalSort(p) 1 while b(p) > 0 2 if p has a decreasing strip 3 Among all possible reversals, choose reversal r that minimizes b(p • r) 4 else 5 Choose a reversal r that flips an increasing strip in p p p • r output p 6 7 8 return Example – ImprovedBreakPointSort • There are no decreasing strips in p, for: p = 0 1 2 | 5 6 7 | 3 4 | 8 b(p) = 3 p • r(6,7) = 0 1 2 | 5 6 7 | 4 3 | 8 b(p) = 3 r(6,7) does not change the # of breakpoints r(6,7) creates a decreasing strip thus guaranteeing that the next step will decrease the # of breakpoints. Approximation Ratio ImprovedBreakpointReversalSort • Approximation ratio is 4 – It eliminates at least one breakpoint in every two steps; at most 2b(p) steps – Approximation ratio: 2b(p) / d(p) – Optimal algorithm eliminates at most 2 breakpoints in every step: d(p) b(p) / 2 – Performance guarantee: • ( 2b(p) / d(p) ) [ 2b(p) / (b(p) / 2) ] = 4 References • An Introduction to Bioinformatics Algorithms - Neil C.Jones and Pavel A.Pevzner • http://bix.ucsd.edu/bioalgorithms/slides.php# Ch5 Questions