scholarly journals On the Comparison Complexity of the String Prefix-Matching Problem

1995 ◽  
Vol 2 (46) ◽  
Author(s):  
Dany Breslauer ◽  
Livio Colussi ◽  
Laura Toniolo
Keyword(s):  
Linear Time ◽  
String Matching ◽  
Random Access ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Matching Problem ◽  
Machine Model ◽  
Worst Case ◽  
On Line ◽  
Sequential Comparison

In this paper we study the exact comparison complexity of the string<br />prefix-matching problem in the deterministic sequential comparison model<br />with equality tests. We derive almost tight lower and upper bounds on<br />the number of symbol comparisons required in the worst case by on-line<br />prefix-matching algorithms for any fixed pattern and variable text. Unlike<br />previous results on the comparison complexity of string-matching and<br />prefix-matching algorithms, our bounds are almost tight for any particular pattern.<br />We also consider the special case where the pattern and the text are the<br />same string. This problem, which we call the string self-prefix problem, is<br />similar to the pattern preprocessing step of the Knuth-Morris-Pratt string-matching<br />algorithm that is used in several comparison efficient string-matching<br />and prefix-matching algorithms, including in our new algorithm.<br />We obtain roughly tight lower and upper bounds on the number of symbol<br />comparisons required in the worst case by on-line self-prefix algorithms.<br />Our algorithms can be implemented in linear time and space in the<br />standard uniform-cost random-access-machine model.

scholarly journals Two-Machine Job-Shop Scheduling Problem to Minimize the Makespan with Uncertain Job Durations

Algorithms ◽  
2019 ◽  
Vol 13 (1) ◽  
pp. 4 ◽  
Author(s):  
Yuri N. Sotskov ◽  
Natalja M. Matsveichuk ◽  
Vadzim D. Hatsura
Keyword(s):  
Relative Error ◽  
Job Shop ◽  
Sufficient Conditions ◽  
Upper Bounds ◽  
Maximum Relative Error ◽  
Scheduling Problems ◽  
Lower And Upper Bounds ◽  
Shop Scheduling ◽  
Job Duration ◽  
On Line

We study two-machine shop-scheduling problems provided that lower and upper bounds on durations of n jobs are given before scheduling. An exact value of the job duration remains unknown until completing the job. The objective is to minimize the makespan (schedule length). We address the issue of how to best execute a schedule if the job duration may take any real value from the given segment. Scheduling decisions may consist of two phases: an off-line phase and an on-line phase. Using information on the lower and upper bounds for each job duration available at the off-line phase, a scheduler can determine a minimal dominant set of schedules (DS) based on sufficient conditions for schedule domination. The DS optimally covers all possible realizations (scenarios) of the uncertain job durations in the sense that, for each possible scenario, there exists at least one schedule in the DS which is optimal. The DS enables a scheduler to quickly make an on-line scheduling decision whenever additional information on completing jobs is available. A scheduler can choose a schedule which is optimal for the most possible scenarios. We developed algorithms for testing a set of conditions for a schedule dominance. These algorithms are polynomial in the number of jobs. Their time complexity does not exceed O ( n 2 ) . Computational experiments have shown the effectiveness of the developed algorithms. If there were no more than 600 jobs, then all 1000 instances in each tested series were solved in one second at most. An instance with 10,000 jobs was solved in 0.4 s on average. The most instances from nine tested classes were optimally solved. If the maximum relative error of the job duration was not greater than 20 % , then more than 80 % of the tested instances were optimally solved. If the maximum relative error was equal to 50 % , then 45 % of the tested instances from the nine classes were optimally solved.

A New Algorithm for Subset Matching Problem Based on Set-String Transformation

2009 ◽  
pp. 607-615
Author(s):  
Yangjun Chen
Keyword(s):  
Programming Languages ◽  
Linear Time ◽  
String Matching ◽  
Special Problem ◽  
Computer Engineering ◽  
Data Types ◽  
Matching Problem ◽  
Matching Problems ◽  
Abstract Data ◽  
Geometric Pattern Matching

In computer engineering, a number of programming tasks involve a special problem, the so-called tree matching problem (Cole & Hariharan, 1997), as a crucial step, such as the design of interpreters for nonprocedural programming languages, automatic implementation of abstract data types, code optimization in compilers, symbolic computation, context searching in structure editors and automatic theorem proving. Recently, it has been shown that this problem can be transformed in linear time to another problem, the so called subset matching problem (Cole & Hariharan, 2002, 2003), which is to find all occurrences of a pattern string p of length m in a text string t of length n, where each pattern and text position is a set of characters drawn from some alphabet S. The pattern is said to occur at text position i if the set p[j] is a subset of the set t[i + j - 1], for all j (1 = j = m). This is a generalization of the ordinary string matching and is of interest since an efficient algorithm for this problem implies an efficient solution to the tree matching problem. In addition, as shown in (Indyk, 1997), this problem can also be used to solve general string matching and counting matching (Muthukrishan, 1997; Muthukrishan & Palem, 1994), and enables us to design efficient algorithms for several geometric pattern matching problems. In this article, we propose a new algorithm on this issue, which needs only O(n + m) time in the case that the size of S is small and O(n + m·n0.5) time on average in general cases.

scholarly journals Time-Varying Vector Norm and Lower and Upper Bounds on the Solutions of Uniformly Asymptotically Stable Linear Systems

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 915
Author(s):  
Robert Vrabel
Keyword(s):  
Linear Systems ◽  
Linear Time ◽  
Upper Bounds ◽  
Time Varying ◽  
Asymptotically Stable ◽  
Lower And Upper Bounds ◽  
Linear Time Invariant ◽  
Vector Norm ◽  
Linear Time Invariant Systems ◽  
Stable Linear

Based on the eigenvalue idea and the time-varying weighted vector norm in the state space R n we construct here the lower and upper bounds of the solutions of uniformly asymptotically stable linear systems. We generalize the known results for the linear time-invariant systems to the linear time-varying ones.

scholarly journals The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functions

2021 ◽  
Author(s):  
Philipp J. di Dio ◽  
Mario Kummer
Keyword(s):  
Moment Problem ◽  
Hankel Matrix ◽  
Upper Bounds ◽  
Moment Problems ◽  
Algebraic Varieties ◽  
Lower And Upper Bounds ◽  
Worst Case ◽  
Flat Extension ◽  
Truncated Moment Problem ◽  
Carathéodory Number

AbstractIn this paper we improve the bounds for the Carathéodory number, especially on algebraic varieties and with small gaps (not all monomials are present). We provide explicit lower and upper bounds on algebraic varieties, $$\mathbb {R}^n$$ R n , and $$[0,1]^n$$ [ 0 , 1 ] n . We also treat moment problems with small gaps. We find that for every $$\varepsilon >0$$ ε > 0 and $$d\in \mathbb {N}$$ d ∈ N there is a $$n\in \mathbb {N}$$ n ∈ N such that we can construct a moment functional $$L:\mathbb {R}[x_1,\cdots ,x_n]_{\le d}\rightarrow \mathbb {R}$$ L : R [ x 1 , ⋯ , x n ] ≤ d → R which needs at least $$(1-\varepsilon )\cdot \left( {\begin{matrix} n+d\\ n\end{matrix}}\right) $$ ( 1 - ε ) · n + d n atoms $$l_{x_i}$$ l x i . Consequences and results for the Hankel matrix and flat extension are gained. We find that there are moment functionals $$L:\mathbb {R}[x_1,\cdots ,x_n]_{\le 2d}\rightarrow \mathbb {R}$$ L : R [ x 1 , ⋯ , x n ] ≤ 2 d → R which need to be extended to the worst case degree 4d, $$\tilde{L}:\mathbb {R}[x_1,\cdots ,x_n]_{\le 4d}\rightarrow \mathbb {R}$$ L ~ : R [ x 1 , ⋯ , x n ] ≤ 4 d → R , in order to have a flat extension.

New lower and upper bounds for on-line scheduling

1994 ◽  
Vol 16 (4) ◽  
pp. 221-230 ◽  
Author(s):  
Bo Chen ◽  
André van Vliet ◽  
Gerhard J. Woeginger
Keyword(s):  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
On Line

scholarly journals New bounds and algorithms for on-line scheduling: two identical processors, known sum and upper bound on the tasks

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Enrico Angelelli ◽  
Maria Grazia Speranza ◽  
Zsolt Tuza
Keyword(s):  
Upper Bound ◽  
Completion Time ◽  
Optimal Algorithm ◽  
Multiprocessor Scheduling ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
International Audience ◽  
On Line ◽  
Maximum Completion Time ◽  
New Algorithms

International audience In this paper we study a semi on-line version of the classical multiprocessor scheduling problem on two identical processors. We assume that the sum of the tasks and an upper bound gamma on the size of each task are known. Each task has to be assigned upon arrival and the assignment cannot be changed later. The objective is the minimization of the maximum completion time on the processors. In this paper we propose new algorithms and improve known lower and upper bounds on the competitive ratio. Algorithms and bounds depend on the value of gamma. An optimal algorithm is obtained for gamma in the interval [ 1/n,2(n+1)/n(2n+1) ] and gamma = (2n-1)/2n(n-1), where n is any integer value larger or equal 2.

scholarly journals Schedule Execution for Two-Machine Job-Shop to Minimize Makespan with Uncertain Processing Times

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1314 ◽  
Author(s):  
Yuri N. Sotskov ◽  
Natalja M. Matsveichuk ◽  
Vadzim D. Hatsura
Keyword(s):  
Job Shop ◽  
Job Shop Scheduling ◽  
Optimal Schedule ◽  
Sufficient Conditions ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Processing Times ◽  
On Line ◽  
Asymptotic Complexity ◽  
The Stability

This study addresses a two-machine job-shop scheduling problem with fixed lower and upper bounds on the job processing times. An exact value of the job duration remains unknown until completing the job. The objective is to minimize a schedule length (makespan). It is investigated how to best execute a schedule, if the job processing time may be equal to any real number from the given (closed) interval. Scheduling decisions consist of the off-line phase and the on-line phase of scheduling. Using the fixed lower and upper bounds on the job processing times available at the off-line phase, a scheduler may determine a minimal dominant set of schedules (minimal DS), which is based on the proven sufficient conditions for a schedule dominance. The DS optimally covers all possible realizations of the uncertain (interval) processing times, i.e., for each feasible scenario, there exists at least one optimal schedule in the minimal DS. The DS enables a scheduler to make the on-line scheduling decision, if a local information on completing some jobs becomes known. The stability approach enables a scheduler to choose optimal schedules for most feasible scenarios. The on-line scheduling algorithms have been developed with the asymptotic complexity O(n2) for n given jobs. The computational experiment shows the effectiveness of these algorithms.

EFFICIENT ALGORITHMS FOR (δ,γ,α) AND (δ, kΔ, α)-MATCHING

2008 ◽  
Vol 19 (01) ◽  
pp. 163-183 ◽  
Author(s):  
KIMMO FREDRIKSSON ◽  
SZYMON GRABOWSKI
Keyword(s):  
Computational Biology ◽  
String Matching ◽  
Efficient Algorithms ◽  
Experimental Results ◽  
Matching Problem ◽  
Worst Case ◽  
Average Case ◽  
Previous Algorithm ◽  
New Algorithms

We propose new algorithms for (δ,γ,α)-matching. In this string matching problem we are given a pattern P = p0p1 … pm−1 and a text T = t0t1 … tn−1 over some integer alphabet Σ = {0…σ − 1}. The pattern symbol pi δ-matches the text symbol tj iff |pi − tj| ≤ δ. The pattern P (δ,γ)-matches some text substring tj … tj+m−1 iff for all i it holds that |pi − tj+i| ≤ δ and Σ |pi − tj+i| ≤ γ. Finally, in (δ,γ,α)-matching we also permit at most α-symbol gaps between each matching text symbol. The only known previous algorithm runs in O(nm) time. We give several algorithms that improve the average case up to O(n) for small α, and the worst case to [Formula: see text] or O(nm log (γ)/w), where [Formula: see text] and w is the number of bits in a machine word. The proposed algorithms can be easily modified to solve several other related problems, we explicitly consider e.g. character classes (instead of δ-matching), (Δ-limited) k-mismatches (instead of γ-matching) and more general gaps, including negative ones. These find important applications in computational biology. We conclude with experimental results showing that the algorithms are very efficient in practice.

ON THE NOTION OF LINEAR TIME COMPUTABILITY

1990 ◽  
Vol 01 (03) ◽  
pp. 295-307 ◽  
Author(s):  
ERICH GRÄDEL
Keyword(s):  
Linear Time ◽  
Random Access ◽  
Rational Numbers ◽  
Turing Machines ◽  
Machine Model ◽  
Class Time ◽  
Complete Problems ◽  
Time Bounds

To capture the informal concept of linear time computability, we propose and discuss the class TIME(n1+), consisting of all functions which are computable by a Successor RAM with exponent at most one; the exponent of a function is the infimum of all rational numbers r such that the function is computable in time O(nr). This class properly contains the class NLT (“Nearly Linear Time”)—proposed by Gurevich and Shelah—which contains all functions computable by a Successor RAM in time O(n(log n)k) for some k∈N. Both classes are very robust under changes of the underlying machine model: using the same time bounds they can be defined by Random Access Computers, Storage Modification Machines, Kolmogorov algorithms, Random Access Turing machines etc.; this distinguishes them from similar classes defined by ordinary Turing machines. However, we show that TIME(n1+) can be defined e.g. by multi-dimensional Turing machines or by Turing machines which can jump; this is probably not true for NLT. Furthermore we consider two notions of completeness for TIME(n1+) and construct complete problems. These problems are based on restrictions of Gurevich’s function logic for P.

scholarly journals Fast circular dictionary-matching algorithm

2015 ◽  
Vol 27 (2) ◽  
pp. 143-156 ◽  
Author(s):  
TANVER ATHAR ◽  
CARL BARTON ◽  
WIDMER BLAND ◽  
JIA GAO ◽  
COSTAS S. ILIOPOULOS ◽  
...  
Keyword(s):  
String Matching ◽  
Synthetic Data ◽  
Experimental Results ◽  
Matching Problem ◽  
Worst Case ◽  
Average Case ◽  
Matching Algorithm ◽  
Dictionary Matching ◽  
Total Length

Circular string matching is a problem which naturally arises in many contexts. It consists in finding all occurrences of the rotations of a pattern of length m in a text of length n. There exist optimal worst- and average-case algorithms for circular string matching. Here, we present a suboptimal average-case algorithm for circular string matching requiring time $\mathcal{O}$(n) and space $\mathcal{O}$(m). The importance of our contribution is underlined by the fact that the proposed algorithm can be easily adapted to deal with circular dictionary matching. In particular, we show how the circular dictionary-matching problem can be solved in average-case time $\mathcal{O}$(n + M) and space $\mathcal{O}$(M), where M is the total length of the dictionary patterns, assuming that the shortest pattern is sufficiently long. Moreover, the presented average-case algorithms and other worst-case approaches were also implemented. Experimental results, using real and synthetic data, demonstrate that the implementation of the presented algorithms can accelerate the computations by more than a factor of two compared to the corresponding implementation of other approaches.

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