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An Algorithm for Computing the Restriction ... - McGill University

An Algorithm for Computing the Restriction ... - McGill University

0 2 7 12 0 27 121 4 11 1

0 2 7 12 0 27 121 4 11 1 4 11(a)(b)Figure 1: (a) A surjection between the two one dimensional sets S = {0, 2, 7, 12} and T = {1, 4, 11}. (b) Aminimal surjection between S and T .Furthermore, we may consider the more general input consisting of two sets of numbers on the real line ratherthan binary sequences. Here the real numbers play the role of the indices of the 1’s in the binary sequence. Inthis setting, if both sets have equal cardinalities, the simple algorithm described in the preceding for binarysequences may still be used after sorting the sets, thus yielding an O(nlogn) time algorithm.An alternate way of viewing the directed swap distance is as a surjection, ψ, between two sets of elementsS (the source) and T (the target) on the interval (0,X) where |S| ≥ |T |. This mapping is bound by theconstraint that each element of T must have at least one element of S mapped to it. More formally thedirected swap distance may be expressed as a surjection as follows:min ∑ |s − ψ(s)|. (1)ψs∈SAny surjection that satisfies the preceding equation we call a minimal surjection. Figure 1 depicts twodifferent surjections between two sets of points on the line, one of which is minimal. Note that all the pointsactually have zero y-coordinates; they are shown in this way merely for the purpose of clarity.In 1979 the philosopher Graham Oddie proposed using surjections to measure the distance between twotheories expressed in a logical language [7]. In 1997 Eiter and Mannila extended this idea by expressingtheories as models, and thus as points in a metric space [5]. This gave them a new distance measure in ametric space which they called the surjection distance. The surjection distance between two sets S and T isdefined as follows:min ∑ δ(s,ψ(s)), (2)ψs∈Swhere δ is a distance metric on the space, and ψ is a surjection between S and T . They also proposed analgorithm for computing the surjection distance in O(n 3 ) time, where n = |S|, by reducing the problem tofinding a minimum-weight perfect matching in an appropriate graph.In 2003, Ben-Dor et al. [2], in the context of the shotgun sequencing problem in computational biology,introduced a modified assignment problem similar to the directed swap problem where the points are realnumbers on the line rather than bits in a binary string: the restriction scaffold assignment problem. Theyalso presented an O(nlogn) algorithm to compute this assignment problem. Their result relies heavily ona result of Karp and Li [6] which provides a linear time algorithm (after sorting) for computing the one-tooneassignment problem in the special case where all the points lie on a line. In the one-to-one assignmentproblem between S and T some elements of S remain unassigned.In this note we give a counter-example to the algorithm of Ben-Dor et al., [2] for computing the restrictionscaffold assignment problem, and show that the problem may be solved in O(n 2 ) time.2

1x 1 x 2 x 3 x 41 1 1 62 94 73 113 481 1u 1 u 2 u 3 v 1Figure 2: The bipartite graph created from S = {0, 2, 7, 12}, T = {1, 4, 11}. The bold lines represent theminimum perfect bipartite matching, which has a cost of 6.2 The Algorithm of Eiter and MannilaSince our O(n 2 ) algorithm is inspired by, and based on, some results of Eiter and Mannila [5], we describein this section their main results for the sake of clarity and completion. The proofs of these results may befound in [5]. Their algorithm is based on a reduction to finding a minimum weight perfect matching in asuitable bipartite graph. The reduction makes use of the following observation.Lemma 2.1. Let ψ be a minimal surjection from S to T . Then for any s i ≠ s j if ψ(s i ) = ψ(s j ) the distancefrom s i to ψ(s i ) is not more than the distance from s i to any other element of T.Taking advantage of this result their algorithm creates k = |S|−|T| auxiliary vertices. These vertices serveas dummy nodes, for which the distance to any s ∈ S is made equal to that of the shortest distance from s toany element of T . This allows the graph to effectively simulate the situation where two or more vertices aremapped to the same element of T .This construction is carried out as follows. For two sets, S and T in a metric space, create a completebipartite weighted graph G = (X ∪Y,E,w). For each s i ∈ S construct a vertex x i ∈ X. For each t j ∈ T createa vertex u j ∈ U. Finally, for k = |S| − |T|, create a set of nodes V = {v 1 ,....,v k }, and let Y = (U ∪V). Theweight function w is defined as follows:{ e = (xi ,uw(e) =j ), δ(s i ,t j )(3)e = (x i ,v j ), min t∈T δ(s i ,t)where s i ,t j are the set elements corresponding to the vertices x i ,u j , respectively.Figure 2 illustrates this construction with the sets S = {0, 2, 7, 12} and T = {1, 4, 11}. The bold linesrepresent the minimum perfect bipartite matching, and correspond to the minimal surjection depicted in Figure1 (b). Let w(M) denote the minimum-weight perfect matching of the resulting graph, and let c(ψ) denotethe cost of the minimum surjection distance between S and T . Eiter and Mannila also prove the followingtheorem.Theorem 2.2. c(ψ) = w(M).Letting n equal to |S|, Eiter and Mannila note that this reduction yields an O(n 3 ) time algorithm sincethe construction takes O(n 2 ) time and their matching algorithm of choice takes O(n 3 ) time for a completegraph [3].3 A Counter-Example to the Algorithm of Ben-Dor et al.Ben-Dor et al. [2] presented an O(nlogn) time algorithm for computing the restriction scaffold assignmentproblem. In this section we describe their algorithm and exhibit an example on which the algorithm does not3

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