Optimal Pattern Matching in LZW Compressed Strings

For kge 3, a k-rollercoaster is a sequence of numbers whose every maximal contiguous subsequence, that is increasing or decreasing, has length at least k; 3-rollercoasters are called simply rollercoasters. Given a sequence of distinct real numbers, we are interested in computing its maximum-length (not necessarily contiguous) subsequence that is a k-rollercoaster. Biedl et al. (in: ICALP, volume 107 of LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, pp 18:1–18:15, 2018) have shown that each sequence of n distinct real numbers contains a rollercoaster of length at least lceil n/2rceil for n>7, and that a longest rollercoaster contained in such a sequence can be computed in O(nlog n)-time (or faster, in O(n log log n) time, when the input sequence is a permutation of {1,ldots ,n}). They have also shown that every sequence of ngeqslant (k-1)^2+1 distinct real numbers contains a k-rollercoaster of length at least frac{n}{2(k-1)}-frac{3k}{2}, and gave an O(nklog n)-time (respectively, O(n klog log n)-time) algorithm computing a longest k-rollercoaster in a sequence of length n (respectively, a permutation of {1,ldots ,n}). In this paper, we give an O(nk^2)-time algorithm computing the length of a longest k-rollercoaster contained in a sequence of n distinct real numbers; hence, for constant k, our algorithm computes the length of a longest k-rollercoaster in optimal linear time. The algorithm can be easily adapted to output the respective k-rollercoaster. In particular, this improves the results of Biedl et al. (2018), by showing that a longest rollercoaster can be computed in optimal linear time. We also present an algorithm computing the length of a longest k-rollercoaster in O(n log ^2 n)-time, that is, subquadratic even for large values of kle n. Again, the rollercoaster can be easily retrieved. Finally, we show an Omega (n log k) lower bound for the number of comparisons in any comparison-based algorithm computing the length of a longest k-rollercoaster.

A simple polygon is said to be weakly internally visible from a line segment lying inside it (chord, if the endpoints of the segment lie on the boundary of the polygon), if every point on the boundary of the polygon is visible from some point of this line segment. In this paper we describe a simple linear time algorithm for computing such a chord, when it exists. We also computed all non-redundant components of P which must be intersected by any chord of P from which P is wiv. The most interesting aspects of this algorithm are that we have been able to dispense with the complicated triangulation algorithm of Chazelle [2] as well as the algorithm in [7] for computing all shortest paths from a given vertex in a triangulated polygon. As a consequence of this, we can now use this algorithm to derive an optimal algorithm for triangulating a weakly internally visible polygon. Also these results can now be used in [3] to find a shortest line segment from which a polygon is weakly internally visible in optimal linear time.

Recently, there has been a growing focus in solving approximate pattern matching problems in the streaming model. Of particular interest are the pattern matching with k-mismatches (KMM) problem and the pattern matching with w-wildcards (PMWC) problem. Motivated by reductions from these problems in the streaming model to the dictionary matching problem, this paper focuses on designing algorithms for the dictionary matching problem in the multi-stream model where there are several independent streams of data (as opposed to just one in the streaming model), and the memory complexity of an algorithm is expressed using two quantities: (1) a read-only shared memory storage area which is shared among all the streams, and (2) local stream memory that each stream stores separately. In the dictionary matching problem in the multi-stream model the goal is to preprocess a dictionary D={P_1,P_2,...,P_d} of d=|D| patterns (strings with maximum length m over alphabet Sigma) into a data structure stored in shared memory, so that given multiple independent streaming texts (where characters arrive one at a time) the algorithm reports occurrences of patterns from D in each one of the texts as soon as they appear. We design two efficient algorithms for the dictionary matching problem in the multi-stream model. The first algorithm works when all the patterns in D have the same length m and costs O(d log m) words in shared memory, O(log m log d) words in stream memory, and O(log m) time per character. The second algorithm works for general D, but the time cost per character becomes O(log m+log d log log d). We also demonstrate the usefulness of our first algorithm in solving both the KMM problem and PMWC problem in the streaming model. In particular, we obtain the first almost optimal (up to poly-log factors) algorithm for the PMWC problem in the streaming model. We also design a new algorithm for the KMM problem in the streaming model that, up to poly-log factors, has the same bounds as the most recent results that use different techniques. Moreover, for most inputs, our algorithm for KMM is significantly faster on average.

Reduction of OBDDs in linear time

Let P be a simple planar polygon. We present a linear worst-case time algorithm for constructing the bounded Voronoi diagram of P in the Manhattan metric, where each point z in P belongs to the region of the closest vertex of P that is visible from z. Among other consequences, the minimal spanning tree of the vertices in the Manhattan metric that is contained in P can be computed within optimal linear time.

Let P be a simple planar polygon. We present a linear time algorithm for constructing the bounded Voronoi diagram of P in the Manhattan metric, where each point z in P belongs to the region of the closest vertex of P that is visible from z. Among other consequences, the minimum spanning tree of the vertices in the Manhattan metric that is contained in P can be computed within optimal linear time. The same results hold for the L∞-metric.α

We consider almost d-regular graphs, i.e., graphs having all vertex degrees equal to either d or d ? 1. It is known that for each d? ? d every d-regular graph contains an almost d?-regular spanning subgraph. We give an algorithm for selecting such a subgraph in optimal linear time.

To protect sensitive information in a cross tabulated table, it is a common practice to suppress some of the cells in the table. A linear combination of suppressed cells is called a linear invariant if the combination has a unique feasible value. Intuitively, the information contained in an invariant is not protected even though the values of the suppressed cells are not disclosed. This paper gives a surprisingly efficient algorithm for testing whether a linear combination of suppressed cells is an invariant. In sequential computation, the algorithm runs in optimal linear time. In parallel computation, the algorithm runs in polylogarithmic time using a polynomial number of processors on a parallel random access machine. The algorithm exploits a linear algebraic structure of directed and undirected cycles in a mixed graph induced by a given table. This new structure also plays a crucial role in subsequent papers on other aspects of detecting and protecting sensitive information in a cross tabulated table.

In a 2-dimensional (2D) pattern matching problem, the text is arranged as a matrix 𝖬[1..n, 1..n] and consists of N = n × n symbols drawn from alphabet set Σ of size σ. The query consists of a m × m square matrix 𝖯[1..m, 1..m] drawn from the same alphabet set Σ and the task is to find all the locations in 𝖬 where 𝖯 appears as a (contiguous) submatrix. The patterns can be of any size, but as long as they are square in shape data structures like suffix trees and suffix array exist [Raffaele Giancarlo, 1995; Dong Kyue Kim et al., 1998] for the task of efficient pattern matching. These are essentially 2D counterparts of classic suffix trees and arrays known for traditional 1-dimensional (1D) pattern matching. They work based on linearization of 2D suffixes which would preserve the prefix match property (i.e., every pattern match is a prefix of some suffix). The main limitation of the suffix trees and the suffix arrays (in 1D) was their space utilization of O(N log N) bits, where N is the size of the text. This was suboptimal compared to Nlog σ bits of space, which is information theoretic optimal for the text. With the advent of the field of succinct/compressed data structures, it was possible to develop compressed variants of suffix trees and array based on Burrows-Wheeler Tansform and LF-mapping (or Φ function) [Roberto Grossi and Jeffrey Scott Vitter, 2005; Paolo Ferragina and Giovanni Manzini, 2005; Kunihiko Sadakane, 2007]. These data structures indeed achieve O(N log σ) bits of space or better. This gives rise to the question: analogous to 1D case, can we design a succinct or compressed index for 2D pattern matching? Can there be a 2D compressed suffix tree? Are there analogues of Burrows-Wheeler Transform or LF-mapping? The problem has been acknowledged for over a decade now and there have been a few attempts at applying Φ function [Ankur Gupta, 2004] and achieving entropy based compression [Veli Mäkinen and Gonzalo Navarro, 2008]. However, achieving the complexity breakthrough akin to 1D case has yet to be found. In this paper, we still do not know how to answer suffix array queries in O(N log σ) bits of space - which would have led to efficient pattern matching. However, for the first time, we show an interesting result that it is indeed possible to compute inverse suffix array (ISA) queries in near compact space in O(polylog n) time. Our 2D succinct text index design is based on two 1D compressed suffix trees and it takes O(N log log N + N logσ) bits of space which is much smaller than its naive design that takes O(N log N) bits. Although the main problem is still evasive, this index gives a hope on the existence of a full 2D succinct index with all functionalities similar to that of 1D case.

Efficient protocols for private wildcards pattern matching

We propose a protocol for the problem of secure two-party pattern matching, where Alice holds a text t?{0,1}? of length n, while Bob has a pattern p?{0,1}? of length m. The goal is for Bob to (only) learn where his pattern occurs in Alice's text, while Alice learns nothing. Private pattern matching is an important problem that has many applications in the area of DNA search, computational biology and more. Our construction guarantees full simulation in the presence of malicious, polynomial-time adversaries (assuming the hardness of DDH assumption) and exhibits computation and communication costs of O(n+m) group elements in a constant round complexity. This improves over previous work by Gennaro et al. (Public Key Cryptography, pp. 145---160, 2010) whose solution requires overhead of O(nm) group elements and exponentiations in O(m) rounds. In addition to the above, we propose a collection of protocols for important variations of the secure pattern matching problem that are significantly more efficient than the current state of art solutions: First, we deal with secure pattern matching with wildcards. In this variant the pattern may contain wildcards that match both 0 and 1. Our protocol requires O(n+m) communication and O(1) rounds using O(nm) computation. Then we treat secure approximate pattern matching. In this variant the matches may be approximated, i.e., have Hamming distance less than some threshold, ?. Our protocol requires O(n?) communication in O(1) rounds using O(nm) computation. Third, we have secure pattern matching with hidden pattern length. Here, the length, m, of Bob's pattern remains a secret. Our protocol requires O(n+M) communication in O(1) rounds using O(n+M) computation, where M is an upper bound on m. Finally, we have secure pattern matching with hidden text length. Finally, in this variant the length, n, of Alice's text remains a secret. Our protocol requires O(N+m) communication in O(1) rounds using O(N+m) computation, where N is an upper bound on n.

Modern MDCT and micro-CT scanners are able to produce high-resolution three-dimensional (3D) images of anatomical trees, such as the airway tree and the heart and liver vasculature. An important problem arising in many contexts is the matching of trees depicted in two different images. Three basic steps are used in order to match two trees: (1) image segmentation, to extract the raw trees from a given pair of 3D images; (2) axial-analysis, to define the underlying centerline structure of the trees; and (3) tree matching, to match the centerline structures of the trees. We focus on step (3). This task is complicated by several problems associated with current segmentation and axial-analysis methods, including missing branches, false branches, and other topological errors in the extracted trees. We propose a model-based approach in which the extracted trees are assumed to arise from an initially unknown common structure corrupted by a sequence of modelled topological deformations. We employ a novel mathematical framework to directly incorporate this model into the matching problem. Under this framework, it is possible to define the set of matches that are consistent with a given deformation model. The optimal match is the member of this set that maximizes a user-definable similarity measure. We present several such similarity measures based upon geometrical attributes (e.g., branch lengths, branching angles, and relative branchpoint locations as measured from the 3D image data). We locate the globally optimal match via an efficient dynamic programming algorithm. Our primary analytical result is a set of sufficient conditions on the user-definable similarity measure such that our dynamic programming algorithm is guaranteed to locate an optimal match. Experimental results have been generated for 3D human CT chest scans and micro-CT coronary arterial-tree images of mice. The resulting matches are in good agreement with correspondences defined by human experts.

Pattern matching is an important task, which is widely used in many fields, such as information retrieval and bioinformatics. Recently, a much more flexible pattern matching problem with wildcards has been proposed. Chen et al. introduced local constraints, global constraints and the one-off condition into the task of pattern matching, and the most representative algorithm SAIL was designed. However, the performance of SAIL is not analyzed well, which affects its application. Therefore, this paper analyzes the performance of SAIL in-depth, and discovers that the matching result is closely related to the features of patterns. Meanwhile, the completeness of SAIL in the pattern matching with no-recurring characters is proved, and an improved algorithm, named RSAIL, is proposed for pattern matching with recurring tail characters. Extensive experiments demonstrate that RSAIL improves the number of matches by 2.2% compared to SAIL.

Presents an algorithm for designing optimal unfalsified discrete linear time invariant (LTI) output feedback controllers directly from measured data. The approach uses a performance specification that corresponds to minimizing the time domain error between the desired and unfalsified closed loop transfer functions. No assumptions about the plant are explicitly required the only information needed is a plant input-output time history of length n. In particular, no assumptions are made about the linearity of the system. The identified controller is unfalsified with respect to the performance specification by all observed data. With some minor assumptions, the design problem can be written as a linear program that can be solved very efficiently to find the global optimum. The approach is demonstrated with a laboratory experiment.

Generation of Pattern-Matching Algorithms by Extended Regular Expressions