scholarly journals Improved Approximation for Fréchet Distance on c-Packed Curves Matching Conditional Lower Bounds

2017 ◽  
Vol 27 (01n02) ◽  
pp. 85-119 ◽  
Author(s):  
Karl Bringmann ◽  
Marvin Künnemann
Keyword(s):  
Lower Bounds ◽  
Time Complexity ◽  
Linear Time ◽  
High Dimensions ◽  
Fréchet Distance ◽  
Worst Case ◽  
One Dimensional ◽  
Dimension Formula ◽  
Frechet Distance ◽  
Special Case

The Fréchet distance is a well studied and very popular measure of similarity of two curves. The best known algorithms have quadratic time complexity, which has recently been shown to be optimal assuming the Strong Exponential Time Hypothesis (SETH) [Bringmann, FOCS'14]. To overcome the worst-case quadratic time barrier, restricted classes of curves have been studied that attempt to capture realistic input curves. The most popular such class are [Formula: see text]-packed curves, for which the Fréchet distance has a [Formula: see text]-approximation in time [Formula: see text] [Driemel et al., DCG'12]. In dimension [Formula: see text] this cannot be improved to [Formula: see text] for any [Formula: see text] unless SETH fails [Bringmann, FOCS'14]. In this paper, exploiting properties that prevent stronger lower bounds, we present an improved algorithm with time complexity [Formula: see text]. This improves upon the algorithm by Driemel et al. for any [Formula: see text]. Moreover, our algorithm's dependence on [Formula: see text], [Formula: see text] and [Formula: see text] is optimal in high dimensions apart from lower order factors, unless SETH fails. Our main new ingredients are as follows: For filling the classical free-space diagram we project short subcurves onto a line, which yields one-dimensional separated curves with roughly the same pairwise distances between vertices. Then we tackle this special case in near-linear time by carefully extending a greedy algorithm for the Fréchet distance of one-dimensional separated curves.

scholarly journals Fast Fréchet Distance Between Curves with Long Edges

2019 ◽  
Vol 29 (02) ◽  
pp. 161-187
Author(s):  
Joachim Gudmundsson ◽  
Majid Mirzanezhad ◽  
Ali Mohades ◽  
Carola Wenk
Keyword(s):  
Data Structure ◽  
Approximation Algorithm ◽  
Special Class ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Fréchet Distance ◽  
Frechet Distance ◽  
Polygonal Curves

Computing the Fréchet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fréchet distance computations become easier. Let [Formula: see text] and [Formula: see text] be two polygonal curves in [Formula: see text] with [Formula: see text] and [Formula: see text] vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fréchet distance between them: (1) a linear-time algorithm for deciding the Fréchet distance between two curves, (2) an algorithm that computes the Fréchet distance in [Formula: see text] time, (3) a linear-time [Formula: see text]-approximation algorithm, and (4) a data structure that supports [Formula: see text]-time decision queries, where [Formula: see text] is the number of vertices of the query curve and [Formula: see text] the number of vertices of the preprocessed curve.

scholarly journals d-PBWT: dynamic positional Burrows-Wheeler transform

2020 ◽  
Author(s):  
Ahsan Sanaullah ◽  
Degui Zhi ◽  
Shaojie Zhang
Keyword(s):  
Data Structure ◽  
Time Complexity ◽  
Linear Time ◽  
Genotype Imputation ◽  
Worst Case ◽  
Average Case ◽  
Insertion And Deletion ◽  
Static Data ◽  
Efficient Retrieval ◽  
Burrows Wheeler Transform

AbstractDurbin’s PBWT, a scalable data structure for haplotype matching, has been successfully applied to identical by descent (IBD) segment identification and genotype imputation. Once the PBWT of a haplotype panel is constructed, it supports efficient retrieval of all shared long segments among all individuals (long matches) and efficient query between an external haplotype and the panel. However, the standard PBWT is an array-based static data structure and does not support dynamic updates of the panel. Here, we generalize the static PBWT to a dynamic data structure, d-PBWT, where the reverse prefix sorting at each position is represented by linked lists. We developed efficient algorithms for insertion and deletion of individual haplotypes. In addition, we verified that d-PBWT can support all algorithms of PBWT. In doing so, we systematically investigated variations of set maximal match and long match query algorithms: while they all have average case time complexity independent of database size, they have different worst case complexities, linear time complexity with the size of the genome, and dependency on additional data structures.

scholarly journals Lyndon Factorization Algorithms for Small Alphabets and Run-Length Encoded Strings

Algorithms ◽  
2019 ◽  
Vol 12 (6) ◽  
pp. 124
Author(s):  
Sukhpal Ghuman ◽  
Emanuele Giaquinta ◽  
Jorma Tarhio
Keyword(s):  
Time Complexity ◽  
Linear Time ◽  
Experimental Results ◽  
The Other ◽  
Worst Case ◽  
Run Length ◽  
Original Algorithm ◽  
Factorization Algorithms

We present two modifications of Duval’s algorithm for computing the Lyndon factorization of a string. One of the algorithms has been designed for strings containing runs of the smallest character. It works best for small alphabets and it is able to skip a significant number of characters of the string. Moreover, it can be engineered to have linear time complexity in the worst case. When there is a run-length encoded string R of length ρ , the other algorithm computes the Lyndon factorization of R in O ( ρ ) time and in constant space. It is shown by experimental results that the new variations are faster than Duval’s original algorithm in many scenarios.

scholarly journals Fine-Grained Complexity of Temporal Problems

2020 ◽  
Author(s):  
Konrad K. Dabrowski ◽  
Peter Jonsson ◽  
Sebastian Ordyniak ◽  
George Osipov
Keyword(s):  
Time Complexity ◽  
Unit Length ◽  
Solution Space ◽  
Worst Case ◽  
Exponential Time ◽  
Fine Grained ◽  
Novel Algorithms ◽  
Open Question ◽  
Special Case ◽  
Single Exponential

Expressive temporal reasoning formalisms are essential for AI. One family of such formalisms consists of disjunctive extensions of the simple temporal problem (STP). Such extensions are well studied in the literature and they have many important applications. It is known that deciding satisfiability of disjunctive STPs is NP-hard, while the fine-grained complexity of such problems is virtually unexplored. We present novel algorithms that exploit structural properties of the solution space and prove, assuming the Exponential-Time Hypothesis, that their worst-case time complexity is close to optimal. Among other things, we make progress towards resolving a long-open question concerning whether Allen's interval algebra can be solved in single-exponential time, by giving a 2^{O(nloglog(n))} algorithm for the special case of unit-length intervals.

Merging Multi-Version Texts: a Generic Solution to the Overlap Problem

2009 ◽  
Author(s):  
Desmond Schmidt
Keyword(s):  
Time Complexity ◽  
Linear Time ◽  
Hard Problem ◽  
Alignment Quality ◽  
Digital Text ◽  
Multiple Sequence ◽  
Worst Case ◽  
Alignment Problem ◽  
Alignment Process ◽  
Simple Format

Multi-Version Documents or MVDs, as described in Schmidt and Colomb (Schm09), provide a simple format for representing overlapping structures in digital text. They permit the reuse of existing technologies, such as XML, to encode the content of individual versions, while allowing overlapping hierarchies (separate, partial or conditional) and textual variation (insertions, deletions, alternatives and transpositions) to exist within the same document. Most desired operations on MVDs may be performed by simple algorithms in linear time. However, creating and editing MVDs is a much harder and more complex operation that resembles the multiple-sequence alignment problem in biology. The inclusion of the transposition operation into the alignment process makes this a hard problem, with no solutions known to be both optimal and practical. However, a suitable heuristic algorithm can be devised, based in part on the most recent biological alignment programs, whose time complexity is quadratic in the worst case, and is often much faster. The results are satisfactory both in terms of speed and alignment quality. This means that MVDs can be considered as a practical and editable format suitable for representing many cases of overlapping structure in digital text.

scholarly journals Approximating the Fréchet Distance for Realistic Curves in Near Linear Time

2012 ◽  
Vol 48 (1) ◽  
pp. 94-127 ◽  
Author(s):  
Anne Driemel ◽  
Sariel Har-Peled ◽  
Carola Wenk
Keyword(s):  
Linear Time ◽  
Fréchet Distance ◽  
Frechet Distance

Linear-Time Approximation Algorithms for the Max Cut Problem

1993 ◽  
Vol 2 (2) ◽  
pp. 201-210 ◽  
Author(s):  
Nguyen van Ngoc ◽  
Zsolt Tuza
Keyword(s):  
Lower Bound ◽  
Approximation Algorithms ◽  
Lower Bounds ◽  
Connected Graph ◽  
Linear Time ◽  
Worst Case ◽  
Time Approximation ◽  
Bipartite Subgraph ◽  
Max Cut Problem ◽  
Multiple Edges

Let G be a connected graph with n vertices and m edges (multiple edges allowed), and let k ≥ 2 be an integer. There is an algorithm with (optimal) running time of O(m) that finds(i) a bipartite subgraph of G with ≥ m/2 + (n − 1)/4 edges,(ii) a bipartite subgraph of G with ≥ m/2 + 3(n−1)/8 edges if G is triangle-free,(iii) a k-colourable subgraph of G with ≥ m − m/k + (n−1)/k + (k − 3)/2 edges if k ≥ 3 and G is not k-colorable.Infinite families of graphs show that each of those lower bounds on the worst-case performance are best possible (for every algorithm). Moreover, even if short cycles are excluded, the general lower bound of m − m/k cannot be replaced by m − m/k + εm for any fixed ε > 0; and it is NP-complete to decide whether a graph with m edges contains a k-colorable subgraph with more than m − m/k + εm edges, for any k ≥ 2 and ε> 0, ε < 1/k.

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