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Fixed-Parameter Algorithms for CLOSEST STRING and Related Problems Algorithmica(2003) Jens Gramm, Rolf Niedermeier, Peter Rossmanith

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Outline Introduction Preliminaries Linear-Time solution for constant d Related Problems Linear-Time solution for fixed k Conclusion

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Intro : Problem Definition Input: String s 1, s 2, …, s k over alphabet Σ of length L each, and a nonnegative integer d. Question: Is there a string s of length L such that d H (s, s i )≤d for all i=1,…,k d H (s 1, s 2 ) = |{i|s 1 [i]≠s 2 [i]}|, |s 1 |=|s 2 |

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NP-completeness CLOSEST STRING is NP-complete d is usually small in biological applications O(kL+kd*d d ) result in this paper PTAS by Li et al

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Extended problems d-MISMATCH DISTINGUISHING STRING SELECTION DISTINGUISHING SUBSTRING SELECTION

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Preliminaries Given a set of string S={s 1, …,s k }, each of length L s is optimal center string iff no s ’ such that max i=1, …,k d H (s ’,s i )

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Given a set of k strings of length L, think of this string as k x L matrix Optimal median string : a c c a s1abcd s2aadb s3bcda s4accc

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Main idea Search! Fixed-parameter tractibility Reduction to problem kernel

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LEMMA 1. Given a set of strings S={s 1, …,s k }, each of length L, and a permutationσ:{1,…,L} {1,…,L}. Then s is an optimal center string for {s 1,…,s k } iff σ(s) is an optimal center string for {σ(s 1 ), σ(s 2 ), …, σ(s k )}

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LEMMA 2. To compute an optimal center string, it is sufficient to solve a normalized and reordered instance. From this, the solution of the original instance can be derived in linear time s1abcd s2aadb s3bcda s4accc s1abaa s2acbb s3babc s4aaad s1baaa s2cabb s3abbc s4aaad

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LEMMA 3. A CLOSEST STRING instance with arbitrary alphabet Σ, |Σ|>k, isomorphic to a CLOSEST STRING instance with alphabet Σ’, |Σ’|=k. By normalization

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LEMMA 4. Given a CLOSTEST STRING instance s 1, …,s k of length L and d. If the resulting k x L matrix has more than kd dirty dirty columns, then there is no string s with max i=1, …,k d H (s,s i )≤d A column is dirty iff it contains at least two different symbols from alphabet Σ By pigeon theorem

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A Linear-Time solution for constant d Bounded search tree algorithm LEMMA 5. Given a set of strings S={s 1, …,s k } and a positive integer d. If there are i, j {1, …,k} with d H {s i,s j }>2d, then there is no string s with max i=1, …,k d H (s, s i )≤d

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Theorem 1. Given a set of string S={s 1, …,s k } and d, Algorithm D determines in O(kL+kd*d d ) time. By lemma 4, reduced the input instance to O(kd) in O(kL) time Depth=d, Time(D0+D1+D2+D3)=kd by building a table containing the distances of candidate s 1 to all other given strings

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correctness Show only the correctness of first step If s 1 is not a solution but there exists a center string s P :={p|s 1 [p]≠s i [p]}, |P|=d+1 P s1≠s=s i := {p|s 1 [p]≠s[p]=s i [p]} goal! P s1≠s=si =P s≠si ∪ P (disjoint), |P s≠si |≤d So d+1 subcases is sufficient

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Related Problems d-MISMATCH problem S i,p,L denote the length L substring of a given string s i starting at position p Whether there is a string of length L and a position p with 1≤p≤n-L+1, such that d H (s,s i,p,L )≤d, for all I Stojanvoic et al give a linear time algorithm fo 1-MISMATCH Theorem 2. d-MISMATCH is solvable in O(kL+(n- L)kd*d d ) time which O(n*k) for fixed d Naively: O(n*(KL+kd*d d )) Maintain the queue of dirty columns Considering only the first L columns, we can build a FIFO queue in O(kL) Update at each position in O(k) time

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DSS problem DISTINGUISHING STRING SELECTION Given S={s 1, …,s k1 }, S ’ ={s ’ 1, …,s ’ k2 } all of the same length L, and d 1,d 2 ≥0, is there a s such that LEMMA 6. Given two set of strings S 1 ={s 1,…,s k1 } and S 2 ={s’ 1,…,s’ k2 } and positive d1,d2. If there are i {1, …,k 1 } and j {1, … k 2 } with d H (s i,s ’ j )

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A Linear-Time Solution for Fixed k Is CLOSEST STRING fixed parameter tractable? Use integer linear programming (ILP) Lenstra: ILP with a fixed number of variables can be solved in linear time(exponential space)

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CLOSEST STRING in ILP Column types for k For k=3: (a,a,a) t, (a,a,b) t, (a,b,a) t, (b,a,a) t, (a,b,c) t |column types|=B(k)≤k! X t,φ, t: column type, φ Σ Number of column type t whose corresponding character in the desired solution string of CLOSEST STRING is set to φ B(k)*k Variables needed Minimize Φ t,i denates the alphabet symbol at the i th entry of column type t

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Conclusion Fixed parameter tractability for CLOSEST STRING in d, k Improve previous work in d-MISMATCH DSS CLOSEST SUBSTRING ?

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