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An $O(n\log \log n)$-Time Algorithm for Triangulating a Simple Polygon

Robert E. Tarjan, Christopher J. Van Wyk
1988 SIAM journal on computing (Print)  
We propose an O(n log logn)-time algorithm for this problem, improving on the previously best bound of O (n log n) and showing that triangulation is not as hard as sorting.  ...  Given a simple n-vertex polygon, the triangulation problem is to partition the interior of the polygon into n-2 triangles by adding n-3 nonintersecting diagonals.  ...  We thank Brenda Baker for her many helpful comments on an earlier draft of this paper, and Bernard Chazelle for his thorough reading of this version of the paper.  ... 
doi:10.1137/0217010 fatcat:h7x4xw7xuzbx3mdhz776gcj3ce

An In-Place Sorting with O(n log n) Comparisons and O(n) Moves [article]

Gianni Franceschini, Viliam Geffert
2003 arXiv   pre-print
We present the first in-place algorithm for sorting an array of size n that performs, in the worst case, at most O(n log n) element comparisons and O(n) element transports.  ...  Raman, Sorting with minimum data movement, J.  ...  The corresponding heap tree is thus of constant height, which results in an algorithm sorting with O(n) moves, O(1) storage, and O(n 1+ε ) comparisons.  ... 
arXiv:cs/0305005v1 fatcat:zc5qqdgpyre4bohnzxjlb4xqqi

Building a Balanced k-d Tree in O(kn log n) Time [article]

Russell A. Brown
2022 arXiv   pre-print
The original description of the k-d tree recognized that rebalancing techniques, such as are used to build an AVL tree or a red-black tree, are not applicable to a k-d tree.  ...  This paper discusses an alternative algorithm that builds a balanced k-d tree by presorting the data in each of k dimensions prior to building the tree.  ...  Quicksort [14] finds the median in O (n) time in the best case, but in O n 2 time in the worst case [23] .  ... 
arXiv:1410.5420v33 fatcat:ymoqzbr2mvcq3podmfeapft5kq

QuickXsort: Efficient Sorting with n log n - 1.399n +o(n) Comparisons on Average [article]

Stefan Edelkamp, Armin Weiß
2013 arXiv   pre-print
A worst case of n log n + O(n) comparisons can be achieved without significantly affecting the average case. Furthermore, we describe an implementation of MergeInsertion for small n.  ...  Taking MergeInsertion as a base case for QuickMergesort, we establish a worst-case efficient sorting algorithm calling for n log n - 1.3999n + o(n) comparisons on average.  ...  Table 1 : 1 Constant-factor-optimal sorting with n log n + κn + o(n) comparisons. Mem. Other κ Worst κ Avg. κ Exper.  ... 
arXiv:1307.3033v1 fatcat:ksej3rtoxzcmropysxdhuqagzy

An $O(n\log ^2 n)$ Algorithm for the kth Longest Path in a Tree with Applications to Location Problems

N. Megiddo, A. Tamir, E. Zemel, R. Chandrasekaran
1981 SIAM journal on computing (Print)  
It is desirable in these cases to have selection algorithms that run in sublinear time in terms of the cardinality of the set. This paper presents a successful development in this direction.  ...  The methods developed here are applied to improve the previously known upper bounds for the time complexity of various location problems.  ...  Thus, the searching stage takes O((n + n log n ) log IRl) time, and hence the location problem is solved in O(n log'' n ) time. B . A / N / p .  ... 
doi:10.1137/0210023 fatcat:ixaxf22jvfcern7e7nrdcjpthq

Fully Dynamic Connectivity in O(log n(loglog n)^2) Amortized Expected Time [article]

Shang-En Huang, Dawei Huang, Tsvi Kopelowitz, Seth Pettie, Mikkel Thorup
2022 arXiv   pre-print
We present a randomized Las Vegas dynamic connectivity data structure with O(log n(loglog n)^2) amortized expected update time and O(log n/logloglog n) worst case query time, which comes very close to  ...  in worst case O(log n/ log log log n) time.  ...  A buffer tree is implemented by an off-the-shelf mergeable binary tree with O(log log n) worst case time for each attach, detach, and merge operation. 9 However, in order to support updates to the vector  ... 
arXiv:1609.05867v2 fatcat:avm5glsex5hlbgzoigxhvrip5a

Efficient Range ORAM with 핆(log 2 N) Locality [article]

Anrin Chakraborti, Adam J. Aviv, Seung Geol Choi, Travis Mayberry, Daniel S. Roche, Radu Sion
2018 IACR Cryptology ePrint Archive  
log N ).  ...  Oblivious RAM protocols (ORAMs) allow a client to access data from an untrusted storage device without revealing to that device any information about their access pattern.  ...  rORAM answers affirmatively and provides a highly efficient range query mechanism with locality, with O(log 2 N ) seek and O(r • log 2 N ) non-amortized communication complexity, O(log N ) times more efficient  ... 
dblp:journals/iacr/ChakrabortiACMR18 fatcat:es7anaoqxzhnbfhg4hiqwyffjy

A Simple Forward Algorithm to Solve General Dynamic Lot Sizing Models with n Periods in 0(n log n) or 0(n) Time

Awi Federgruen, Michal Tzur
1991 Management science  
One can therefore argue that our procedure is in practice of linear time even in those rare settings where the worst case bound is 0(n log n).  ...  The n log n complexity term in the most general procedure is obtained from the effort to insert or delete an element in this list which in the worst case may be of size n, but in practice is very small  ...  The access time in a balanced binary tree is 0 (log n). We can also rebalance such a tree after an insertion or deletion in 0(1og n) time, see Lemma 4.1 in Tarjan ( 1983).  ... 
doi:10.1287/mnsc.37.8.909 fatcat:5mjvm7uwwnekdlisdciqhqa6ly

An in-place sorting with O(nlog n) comparisons and O(n) moves

Gianni Franceschini, Viliam Geffert
2005 Journal of the ACM  
We present the first in-place algorithm for sorting an array of size n that performs, in the worst case, at most O(n log n) element comparisons and O(n) element transports.  ...  The heapsort [Floyd 1964; Williams 1964] was the first in-place sorting algorithm with a total running time bounded by O(n · log n) in the worst case.  ...  This leaves us with a fascinating question: Does there exist an algorithm operating in-place and performing, in the worst case, at most O(n · log n) comparisons, O(n) moves, O(n · log n) arithmetic operations  ... 
doi:10.1145/1082036.1082037 fatcat:n36ky7vl6jfsdacyh3zc6dmcma

The Snapshot Index: An I/O-Optimal access method for timeslice queries

Vassilis J. Tsotras, Nickolas Kangelaris
1995 Information Systems  
We present an access method for timeslice queries that reconstructs a past state s(t) of a time-evolving collection of objects, in O(log" n + Is(t)l/b) I/O '8, where Is(t)1 denotes the size of the collection  ...  This is the first I I O-optimal access method for this problem using O(n/b) space and O(1) updating (in the expected amortized sense due to the use of hashing.)  ...  Sellis for kindly providing us with access to the computing facilities of his laboratory at the National TechnicaI University of Athens, where some of the simulations were performed while the second author  ... 
doi:10.1016/0306-4379(95)00011-r fatcat:j5rns3b7yrfcvivncffwizmzka

Incremental discovery of the irredundant motif bases for all suffixes of a string in O(n2logn) time

Alberto Apostolico, Claudia Tagliacollo
2008 Theoretical Computer Science  
Given a sequence s of n characters drawn from an alphabet Σ, the problem of extracting such a base from s had been previously solved in time O(n 2 log n log | Σ |) and O(| Σ | n 2 log 2 n log log n), respectively  ...  This problem was solved in a previous work in time O(n 3 ). A much faster incremental algorithm is described here, which takes time O(n 2 log n) for binary strings.  ...  This problem was solved in previous work in time O(n 3 ) [3] . A faster incremental algorithm is described here, which takes time O(n 2 log n) on a binary string.  ... 
doi:10.1016/j.tcs.2008.08.002 fatcat:wjvrtw4pbrel7epltf3an5xxoi

An I/O-efficient Distance Oracle for Evolving Real-World Graphs [chapter]

Deepak Ajwani, Ulrich Meyer, David Veith
2014 2015 Proceedings of the Seventeenth Workshop on Algorithm Engineering and Experiments (ALENEX)  
Furthermore, in a batch setting, one would like to answer O(n) such distance queries inÕ(n/B) I/Os.  ...  iii) answers batched shortest path queries using HDDs with an average time per query of a few microseconds, (iv) results in a highly accurate shortest path estimate and (v) uses space linear in the number  ...  Complete Binary Tree We first observe that for complete binary trees, merely keeping the inorder number (O(log n) bits) with each node allows us to answer distance queries with O(1) instructions.  ... 
doi:10.1137/1.9781611973754.14 dblp:conf/alenex/AjwaniMV15 fatcat:ydy7dz3jjjer3lqujwgui27bqy

Should Static Search Trees Ever Be Unbalanced? [chapter]

Prosenjit Bose, Karim Douïeb
2010 Lecture Notes in Computer Science  
This suggests that for most applications, a balanced tree is always a better option than an unbalanced one since the balanced tree has similar average access time and much better worst case access time  ...  At the same time it is possible to guarantee that the average access time P (R) in tree R is no more than the average access time P (T ) in tree T plus O(log k P (T )).  ...  Related work Optimal search trees Knuth [12] showed that an optimal binary search tree can be built in O(n 2 ) time using O(n 2 ) space.  ... 
doi:10.1007/978-3-642-17517-6_12 fatcat:y66tuj7qyfa2zi62adyx76qkey

Should Static Search Trees Ever Be Unbalanced? [article]

Prosenjit Bose, Karim Douïeb
2010 arXiv   pre-print
This suggests that for most applications, a balanced tree is always a better option than an unbalanced one since the balanced tree has similar average access time and much better worst case access time  ...  We present several methods to restructure an unbalanced k-ary search tree T into a new tree R that preserves many of the properties of T while having a height of _k n +1 which is one unit off of the optimal  ...  Related work Optimal search trees Knuth [12] showed that an optimal binary search tree can be built in O(n 2 ) time using O(n 2 ) space.  ... 
arXiv:1006.3715v1 fatcat:aqejvypclvfyfpsauo54roc33u

An O(log2N) Fully-Balanced Resampling Algorithm for Particle Filters on Distributed Memory Architectures

Alessandro Varsi, Simon Maskell, Paul G. Spirakis
2021 Algorithms  
In this paper, we propose a novel parallel redistribution for DM that achieves an O(log2N) time complexity.  ...  As the models become more complex and accurate, the run-time of PF applications becomes increasingly slow. Parallel computing can help to address this.  ...  data-dependent run-time and an O(N) time complexity in the worst-case.  ... 
doi:10.3390/a14120342 fatcat:tu5ytxcuzbc3lacy4qh5do376q
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