Optimal Color Range Reporting in One Dimension

Color (or categorical) range reporting is a variant of the orthogonal range reporting problem in which every point in the input is assigned a \emph{color}. While the answer to an orthogonal point reporting query contains all points in the query range $Q$, the answer to a color reporting query contains only distinct colors of points in $Q$. In this paper we describe an O(N)-space data structure that answers one-dimensional color reporting queries in optimal $O(k+1)$ time, where $k$ is the number of colors in the answer and $N$ is the number of points in the data structure. Our result can be also dynamized and extended to the external memory model

Scheduling Jobs in Flowshops with the Introduction of Additional Machines in the Future

This is the author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by Elsevier and can be found at: http://www.journals.elsevier.com/expert-systems-with-applications/.The problem of scheduling jobs to minimize total weighted tardiness in flowshops,\ud with the possibility of evolving into hybrid flowshops in the future, is investigated in\ud this paper. As this research is guided by a real problem in industry, the flowshop\ud considered has considerable flexibility, which stimulated the development of an\ud innovative methodology for this research. Each stage of the flowshop currently has\ud one or several identical machines. However, the manufacturing company is planning\ud to introduce additional machines with different capabilities in different stages in the\ud near future. Thus, the algorithm proposed and developed for the problem is not only\ud capable of solving the current flow line configuration but also the potential new\ud configurations that may result in the future. A meta-heuristic search algorithm based\ud on Tabu search is developed to solve this NP-hard, industry-guided problem. Six\ud different initial solution finding mechanisms are proposed. A carefully planned\ud nested split-plot design is performed to test the significance of different factors and\ud their impact on the performance of the different algorithms. To the best of our\ud knowledge, this research is the first of its kind that attempts to solve an industry-guided\ud problem with the concern for future developments

Efficient LZ78 factorization of grammar compressed text

We present an efficient algorithm for computing the LZ78 factorization of a text, where the text is represented as a straight line program (SLP), which is a context free grammar in the Chomsky normal form that generates a single string. Given an SLP of size $n$ representing a text $S$ of length $N$, our algorithm computes the LZ78 factorization of $T$ in $O(n\sqrt{N}+m\log N)$ time and $O(n\sqrt{N}+m)$ space, where $m$ is the number of resulting LZ78 factors. We also show how to improve the algorithm so that the $n\sqrt{N}$ term in the time and space complexities becomes either $nL$, where $L$ is the length of the longest LZ78 factor, or $(N - \alpha)$ where $\alpha \geq 0$ is a quantity which depends on the amount of redundancy that the SLP captures with respect to substrings of $S$ of a certain length. Since $m = O(N/\log_\sigma N)$ where $\sigma$ is the alphabet size, the latter is asymptotically at least as fast as a linear time algorithm which runs on the uncompressed string when $\sigma$ is constant, and can be more efficient when the text is compressible, i.e. when $m$ and $n$ are small.Comment: SPIRE 201

Dictionary Matching with One Gap

The dictionary matching with gaps problem is to preprocess a dictionary $D$ of $d$ gapped patterns $P_1,\ldots,P_d$ over alphabet $\Sigma$, where each gapped pattern $P_i$ is a sequence of subpatterns separated by bounded sequences of don't cares. Then, given a query text $T$ of length $n$ over alphabet $\Sigma$, the goal is to output all locations in $T$ in which a pattern $P_i\in D$, $1\leq i\leq d$, ends. There is a renewed current interest in the gapped matching problem stemming from cyber security. In this paper we solve the problem where all patterns in the dictionary have one gap with at least $\alpha$ and at most $\beta$ don't cares, where $\alpha$ and $\beta$ are given parameters. Specifically, we show that the dictionary matching with a single gap problem can be solved in either $O(d\log d + |D|)$ time and $O(d\log^{\varepsilon} d + |D|)$ space, and query time $O(n(\beta -\alpha )\log\log d \log ^2 \min \{ d, \log |D| \} + occ)$, where $occ$ is the number of patterns found, or preprocessing time and space: $O(d^2 + |D|)$, and query time $O(n(\beta -\alpha ) + occ)$, where $occ$ is the number of patterns found. As far as we know, this is the best solution for this setting of the problem, where many overlaps may exist in the dictionary.Comment: A preliminary version was published at CPM 201

Suffix Tree of Alignment: An Efficient Index for Similar Data

We consider an index data structure for similar strings. The generalized suffix tree can be a solution for this. The generalized suffix tree of two strings $A$ and $B$ is a compacted trie representing all suffixes in $A$ and $B$. It has $|A|+|B|$ leaves and can be constructed in $O(|A|+|B|)$ time. However, if the two strings are similar, the generalized suffix tree is not efficient because it does not exploit the similarity which is usually represented as an alignment of $A$ and $B$. In this paper we propose a space/time-efficient suffix tree of alignment which wisely exploits the similarity in an alignment. Our suffix tree for an alignment of $A$ and $B$ has $|A| + l_d + l_1$ leaves where $l_d$ is the sum of the lengths of all parts of $B$ different from $A$ and $l_1$ is the sum of the lengths of some common parts of $A$ and $B$. We did not compromise the pattern search to reduce the space. Our suffix tree can be searched for a pattern $P$ in $O(|P|+occ)$ time where $occ$ is the number of occurrences of $P$ in $A$ and $B$. We also present an efficient algorithm to construct the suffix tree of alignment. When the suffix tree is constructed from scratch, the algorithm requires $O(|A| + l_d + l_1 + l_2)$ time where $l_2$ is the sum of the lengths of other common substrings of $A$ and $B$. When the suffix tree of $A$ is already given, it requires $O(l_d + l_1 + l_2)$ time.Comment: 12 page

A Minimal Periods Algorithm with Applications

Kosaraju in ``Computation of squares in a string'' briefly described a linear-time algorithm for computing the minimal squares starting at each position in a word. Using the same construction of suffix trees, we generalize his result and describe in detail how to compute in O(k|w|)-time the minimal k-th power, with period of length larger than s, starting at each position in a word w for arbitrary exponent $k\geq2$ and integer $s\geq0$. We provide the complete proof of correctness of the algorithm, which is somehow not completely clear in Kosaraju's original paper. The algorithm can be used as a sub-routine to detect certain types of pseudo-patterns in words, which is our original intention to study the generalization.Comment: 14 page

Practical methods for constructing suffix trees

Sequence datasets are ubiquitous in modern life-science applications, and querying sequences is a common and critical operation in many of these applications. The suffix tree is a versatile data structure that can be used to evaluate a wide variety of queries on sequence datasets, including evaluating exact and approximate string matches, and finding repeat patterns. However, methods for constructing suffix trees are often very time-consuming, especially for suffix trees that are large and do not fit in the available main memory. Even when the suffix tree fits in memory, it turns out that the processor cache behavior of theoretically optimal suffix tree construction methods is poor, resulting in poor performance. Currently, there are a large number of algorithms for constructing suffix trees, but the practical tradeoffs in using these algorithms for different scenarios are not well characterized.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47869/1/778_2005_Article_154.pd