scholarly journals AN OPTIMAL CONSTRUCTION METHOD FOR GENERALIZED CONVEX LAYERS

1993 ◽  
Vol 03 (03) ◽  
pp. 245-267
Author(s):  
HANS-PETER LENHOF ◽  
MICHIEL SMID
Keyword(s):  
Data Structure ◽  
Optimal Solution ◽  
Construction Method ◽  
Query Time ◽  
Query Point ◽  
Dynamic Data ◽  
Time Optimal ◽  
New Construction ◽  
On Line ◽  
Dynamic Data Structure

Let P be a set of n points in the Euclidean plane and let C be a convex figure. In 1985, Chazelle and Edelsbrunner presented an algorithm, which preprocesses P such that for any query point q, the points of P in the translate C+q can be retrieved efficiently. Assuming that constant time suffices for deciding the inclusion of a point in C, they provided a space and query time optimal solution. Their algorithm uses O(n) space. A query with output size k can be solved in O( log n+k) time. The preprocessing step of their algorithm, however, has time complexity O(n2). We show that the usage of a new construction method for layers reduces the preprocessing time to O(n log n). We thus provide the first space, query time and preprocessing time optimal solution for this class of point retrieval problems. Besides, we present two new dynamic data structures for these problems. The first dynamic data structure allows on-line insertions and deletions of points in O(( log n)2) time. In this dynamic data structure, a query with output size k can be solved in O( log n+k( log n)2) time. The second dynamic data structure, which allows only semi-online updates, has O(( log n)2) amortized update time and O( log n+k) query time.

scholarly journals THREE PROBLEMS ABOUT DYNAMIC CONVEX HULLS

2012 ◽  
Vol 22 (04) ◽  
pp. 341-364 ◽  
Author(s):  
TIMOTHY M. CHAN
Keyword(s):  
Data Structure ◽  
Convex Hull ◽  
Query Time ◽  
Convex Hulls ◽  
Lower Envelope ◽  
Dynamic Data ◽  
Line Segments ◽  
3 Dimensional ◽  
Small Constant ◽  
Dynamic Data Structure

We present three results related to dynamic convex hulls: • A fully dynamic data structure for maintaining a set of n points in the plane so that we can find the edges of the convex hull intersecting a query line, with expected query and amortized update time O( log 1+εn) for an arbitrarily small constant ε > 0. This improves the previous bound of O( log 3/2n). • A fully dynamic data structure for maintaining a set of n points in the plane to support halfplane range reporting queries in O( log n+k) time with O( polylog n) expected amortized update time. A similar result holds for 3-dimensional orthogonal range reporting. For 3-dimensional halfspace range reporting, the query time increases to O( log 2n/ log log n + k). • A semi-online dynamic data structure for maintaining a set of n line segments in the plane, so that we can decide whether a query line segment lies completely above the lower envelope, with query time O( log n) and amortized update time O(nε). As a corollary, we can solve the following problem in O(n1+ε) time: given a triangulated terrain in 3-d of size n, identify all faces that are partially visible from a fixed viewpoint.

scholarly journals An Optimal Dynamic Data Structure for Stabbing-Semigroup Queries

2012 ◽  
Vol 41 (1) ◽  
pp. 104-127 ◽  
Author(s):  
Pankaj K. Agarwal ◽  
Lars Arge ◽  
Haim Kaplan ◽  
Eyal Molad ◽  
Robert E. Tarjan ◽  
...  
Keyword(s):  
Data Structure ◽  
Dynamic Data ◽  
Optimal Dynamic ◽  
Dynamic Data Structure

scholarly journals A dynamic data structure for flexible molecular maintenance and informatics

2010 ◽  
Vol 27 (1) ◽  
pp. 55-62 ◽  
Author(s):  
C. Bajaj ◽  
R. A. Chowdhury ◽  
M. Rasheed
Keyword(s):  
Data Structure ◽  
Dynamic Data ◽  
Dynamic Data Structure

scholarly journals Metannot: A succinct data structure for compression of colors in dynamic de Bruijn graphs

2017 ◽  
Author(s):  
Harun Mustafa ◽  
André Kahles ◽  
Mikhail Karasikov ◽  
Gunnar Rätsch
Keyword(s):  
Data Structure ◽  
High Throughput Sequencing ◽  
Sequence Data ◽  
Data Sets ◽  
Sequencing Data ◽  
Dynamic Data ◽  
De Bruijn Graphs ◽  
Dna And Rna ◽  
Succinct Data Structure ◽  
Dynamic Data Structure

AbstractMuch of the DNA and RNA sequencing data available is in the form of high-throughput sequencing (HTS) reads and is currently unindexed by established sequence search databases. Recent succinct data structures for indexing both reference sequences and HTS data, along with associated metadata, have been based on either hashing or graph models, but many of these structures are static in nature, and thus, not well-suited as backends for dynamic databases.We propose a parallel construction method for and novel application of the wavelet trie as a dynamic data structure for compressing and indexing graph metadata. By developing an algorithm for merging wavelet tries, we are able to construct large tries in parallel by merging smaller tries constructed concurrently from batches of data.When compared against general compression algorithms and those developed specifically for graph colors (VARI and Rainbowfish), our method achieves compression ratios superior to gzip and VARI, converging to compression ratios of 6.5% to 2% on data sets constructed from over 600 virus genomes.While marginally worse than compression by bzip2 or Rainbowfish, this structure allows for both fast extension and query. We also found that additionally encoding graph topology metadata improved compression ratios, particularly on data sets consisting of several mutually-exclusive reference genomes.It was also observed that the compression ratio of wavelet tries grew sublinearly with the density of the annotation matrices.This work is a significant step towards implementing a dynamic data structure for indexing large annotated sequence data sets that supports fast query and update operations. At the time of writing, no established standard tool has filled this niche.

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