The Average Common Substring (ACS) is a popular alignment-free distance measure for phylogeny reconstruction. The ACS can be computed in O(n) space and time, where n=x+y is the input size. The compressed string matching is the study of string matching problems with the following twist: the input data is in a compressed format and the underling task must be performed with little or no decompression. In this paper, we revisit the ACS problem under this paradigm where the input sequences are given in their run-length encoded format. We present an algorithm to compute ACS(X,Y) in O(Nlog N) time using O(N) space, where N is the total length of sequences after run-length encoding.
The Average Common Substring (ACS), proposed by Burstein @cite_12 , is a simple alignment-free sequence comparison method. This measure and its various extensions @cite_17 @cite_11 @cite_16 @cite_9 @cite_2 @cite_3 @cite_6 have proven to be useful in multiple applications @cite_13 @cite_0 @cite_1 @cite_4 @cite_14 @cite_10 . Formally, ACS of a sequence @math w.r.t. another sequence @math , denoted by @math , is @math The @math of two input sequences is the length of their longest common prefix. The (symmetric) distance based on ACS is @cite_12 : @math
@cite_14 @cite_4 @cite_10 @cite_9 @cite_1 @cite_17 @cite_3 @cite_6 @cite_0 @cite_2 @cite_16 @cite_13 @cite_12 @cite_11
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Abstract Alignment-free approaches are gaining persistent interest in many sequence analysis applications such as phylogenetic inference and metagenomic classification clustering, especially for large-scale sequence datasets. Besides the widely used k-mer methods, the average common substring (ACS) approach has emerged to be one of the well-known alignment-free approaches. Two recent works further generalize this ACS approach by allowing a bounded number k of mismatches in the common substrings, relying on approximation (linear time) and exact computation, respectively. Albeit having a good worst-case time complexity aastex amsbsy amsfonts amssymb bm mathrsfs pifont stmaryrd textcomp portland, xspace amsmath, amsxtra empty 10 9 7 6 @math log ^ ... Background Alignment-free sequence comparison approaches have been garnering increasing interest in various data- and compute-intensive applications such as phylogenetic inference for large-scale sequences. While k-mer based methods are predominantly used in real applications, the average common substring (ACS) approach is emerging as one of the prominent alignment-free approaches. This ACS approach has been further generalized by some recent work, either greedily or exactly, by allowing a bounded number of mismatches in the common substrings. Alignment free sequence comparison methods are attracting persistent interest, driven by data-intensive applications in genome-wide molecular taxonomy and phylogentic reconstruction. Among the methods based on substring composition, the Average Common Substring ( ( ACS )) measure proposed by (RECOMB 2005) admits a straightforward linear time sequence comparison algorithm, while yielding impressive results in multiple applications. An important direction of research is to extend the approach to permit a bounded edit hamming distance between substrings, so as to reflect more accurately the evolutionary process. To date, however, algorithms designed to incorporate (k 1 ) mismatches have (O(kn^2) ) worst-case complexity, worse than the (O(n^2) ) alignment algorithms they are meant to replace. On the other hand, accounting for mismatches does show to lead to much improved classification, while heuristics can improve practical performance. In this paper, we close the gap by presenting the first provably efficient algorithm for the (k ) -mismatch average common string ( ( ACS _k )) problem that takes (O(n) ) space and (O(n ^ k+1 n) ) time in the worst case for any constant (k ). Our method extends the generalized suffix tree model to incorporate a carefully selected bounded set of perturbed suffixes, and can be applicable to other complex approximate sequence matching problems. The problems of finding shortest common supersequences (SCS) and longest common subsequences (LCS) are two well-known NP -hard problems that have applications in many areas including computational molecular biology, data compression, robot motion planning and scheduling, text editing, etc. A lot of fruitless effort has been spent in searching for good approximation algorithms for these problems. In this paper, we show that these problems are inherently hard to approximate in the worst case. In particular, we prove that (i) SCS does not have a polynomial time linear approximation algorithm, unless P = NP ; (ii) There exists a constant @math such that, if SCS has a polynomial time approximation algorithm with ratio @math , where @math is the number of input sequences, then NP is contained in DTIME @math ; (iii) There exists a constant @math such that, if LCS has a polynomial time approximation algorithm with performance ratio @math , then P = NP . The proofs utilize the recent results of [Proc. 23rd IEEE Symposium on Foundations of Computer Science, 1992, pp. 14-23] on the complexity of approximation problems. In the second part of the paper, we introduce a new method for analyzing the average-case performance of algorithms for sequences, based on Kolmogorov complexity. Despite the above nonapproximability results, we show that near optimal solutions for both SCS and LCS can be found on the average. More precisely, consider a fixed alphabet @math and suppose that the input sequences are generated randomly according to the uniform probability distribution and are of the same length @math . Moreover, assume that the number of input sequences is polynomial in @math . Then, there are simple greedy algorithms which approximate SCS and LCS with expected additive errors @math and @math for any @math , respectively. Incidentally, our analyses also provide tight upper and lower bounds on the expected LCS and SCS lengths for a set of random sequences, solving a generalization of another well-known open question on the expected LCS length for two random sequences [K. Alexander, The rate of convergence of the mean length of the longest common subsequence, 1992, manuscript],[V. Chvatal and D. Sankoff, J. Appl. Probab., 12 (1975), pp. 306-315], [D. Sankoff and J. Kruskall, eds., Time Warps, String Edits, and Macromolecules: The Theory and Practice of Sequence Comparison, Addison-Wesley, Reading, MA, 1983].