scholarly journals Partially Persistent Data Structures of Bounded Degree with Constant Update Time

1994 ◽  
Vol 1 (35) ◽  
Author(s):  
Gerth Stølting Brodal
Keyword(s):  
Lower Bound ◽  
Data Structures ◽  
Upper Bound ◽  
Constant Time ◽  
Machine Model ◽  
Worst Case ◽  
Bounded Degree ◽  
New Strategy ◽  
Persistent Data ◽  
Pebble Game

The problem of making bounded in-degree and out-degree data structures partially persistent is considered. The node copying method of Driscoll <em>et al.</em> is extended so that updates can be performed in <em>worst-case</em> constant time on the pointer machine model. Previously it was only known to be possible in amortised constant time [Driscoll89]. The result is presented in terms of a new strategy for Dietz and Raman's dynamic two player pebble game on graphs. It is shown how to implement the strategy, and the upper bound on the required number of pebbles is improved from 2b + 2d + O(sqrt(b)) to d + 2b, where b is the bound of the in-degree and d the bound of the out-degree. We also give a lower bound that shows that the number of pebbles depends on the out-degree d.

scholarly journals On Compact Encoding of Pagenumber $k$ Graphs

2008 ◽  
Vol Vol. 10 no. 3 ◽  
Author(s):  
Cyril Gavoille ◽  
Nicolas Hanusse
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Constant Time ◽  
Worst Case ◽  
Encoding Scheme ◽  
Information Theoretic ◽  
Auxiliary Table ◽  
Minimum Number ◽  
International Audience

International audience In this paper we show an information-theoretic lower bound of kn - o(kn) on the minimum number of bits to represent an unlabeled simple connected n-node graph of pagenumber k. This has to be compared with the efficient encoding scheme of Munro and Raman of 2kn + 2m + o(kn+m) bits (m the number of edges), that is 4kn + 2n + o(kn) bits in the worst-case. For m-edge graphs of pagenumber k (with multi-edges and loops), we propose a 2mlog2k + O(m) bits encoding improving the best previous upper bound of Munro and Raman whenever m ≤ 1 / 2kn/log2 k. Actually our scheme applies to k-page embedding containing multi-edge and loops. Moreover, with an auxiliary table of o(m log k) bits, our coding supports (1) the computation of the degree of a node in constant time, (2) adjacency queries with O(logk) queries of type rank, select and match, that is in O(logk *minlogk / loglogm, loglogk) time and (3) the access to δ neighbors in O(δ) runs of select, rank or match;.

scholarly journals Approximation Ratios of RePair, LongestMatch and Greedy on Unary Strings

Algorithms ◽  
2021 ◽  
Vol 14 (2) ◽  
pp. 65
Author(s):  
Danny Hucke ◽  
Carl Philipp Reh
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Word Length ◽  
Approximation Ratio ◽  
Input Word ◽  
Worst Case ◽  
Context Free Grammar ◽  
Single Input ◽  
Context Free

A grammar-based compressor is an algorithm that receives a word and outputs a context-free grammar that only produces this word. The approximation ratio for a single input word is the size of the grammar produced for this word divided by the size of a smallest grammar for this word. The worst-case approximation ratio of a grammar-based compressor for a given word length is the largest approximation ratio over all input words of that length. In this work, we study the worst-case approximation ratio of the algorithms Greedy, RePair and LongestMatch on unary strings, i.e., strings that only make use of a single symbol. Our main contribution is to show the improved upper bound of O((logn)8·(loglogn)3) for the worst-case approximation ratio of Greedy. In addition, we also show the lower bound of 1.34847194⋯ for the worst-case approximation ratio of Greedy, and that RePair and LongestMatch have a worst-case approximation ratio of log2(3).

scholarly journals Scatter Search on a Disk

2021 ◽  
Author(s):  
Nisha Chopra
Keyword(s):  
Lower Bound ◽  
Wireless Communication ◽  
Unit Disk ◽  
Lower Bounds ◽  
Upper Bound ◽  
Scatter Search ◽  
Search Time ◽  
Communication Model ◽  
Worst Case ◽  
First Case

Consider a unit disk with two objects at unidentified locations. We examine the problem of two or more robots in search of both objects in the wireless communication model. We begin with two robots and both are needed to carry an object. Subsequently, we design several algorithms that describe robots trajectories in search of the objects. We were able to achieve a minimum worst-case search time of 6.7518 and a lower bound of 3 + π 2 . Additionally, we define two general cases and bound the worst-case search time for both. The first of the cases is for n ≥ 3 robots and an object can be moved by one robot. The second case is where we have n ≥ 3 robots and two robots are needed to carry an object. We achieve an upper bound of 1 + 2π n + sin (⌊n 2 ⌋ π n ) for the first case and an upper bound of 3 + 2π n + sin π n for the second case, with lower bounds of 2 + π n and 3 + π n respectively.

scholarly journals Scatter Search on a Disk

2021 ◽  
Author(s):  
Nisha Chopra
Keyword(s):  
Lower Bound ◽  
Wireless Communication ◽  
Unit Disk ◽  
Lower Bounds ◽  
Upper Bound ◽  
Scatter Search ◽  
Search Time ◽  
Communication Model ◽  
Worst Case ◽  
First Case

Consider a unit disk with two objects at unidentified locations. We examine the problem of two or more robots in search of both objects in the wireless communication model. We begin with two robots and both are needed to carry an object. Subsequently, we design several algorithms that describe robots trajectories in search of the objects. We were able to achieve a minimum worst-case search time of 6.7518 and a lower bound of 3 + π 2 . Additionally, we define two general cases and bound the worst-case search time for both. The first of the cases is for n ≥ 3 robots and an object can be moved by one robot. The second case is where we have n ≥ 3 robots and two robots are needed to carry an object. We achieve an upper bound of 1 + 2π n + sin (⌊n 2 ⌋ π n ) for the first case and an upper bound of 3 + 2π n + sin π n for the second case, with lower bounds of 2 + π n and 3 + π n respectively.

scholarly journals On the Average Case of MergeInsertion

2020 ◽  
Vol 64 (7) ◽  
pp. 1197-1224
Author(s):  
Florian Stober ◽  
Armin Weiß
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Sorting Algorithm ◽  
Worst Case ◽  
Average Case ◽  
Information Theoretic ◽  
Johnson Algorithm ◽  
The Impact ◽  
Combined Algorithm ◽  
Worst Case Behavior

AbstractMergeInsertion, also known as the Ford-Johnson algorithm, is a sorting algorithm which, up to today, for many input sizes achieves the best known upper bound on the number of comparisons. Indeed, it gets extremely close to the information-theoretic lower bound. While the worst-case behavior is well understood, only little is known about the average case. This work takes a closer look at the average case behavior. In particular, we establish an upper bound of $n \log n - 1.4005n + o(n)$ n log n − 1.4005 n + o ( n ) comparisons. We also give an exact description of the probability distribution of the length of the chain a given element is inserted into and use it to approximate the average number of comparisons numerically. Moreover, we compute the exact average number of comparisons for n up to 148. Furthermore, we experimentally explore the impact of different decision trees for binary insertion. To conclude, we conduct experiments showing that a slightly different insertion order leads to a better average case and we compare the algorithm to Manacher’s combination of merging and MergeInsertion as well as to the recent combined algorithm with (1,2)-Insertionsort by Iwama and Teruyama.

scholarly journals Tight Bounds on the Round Complexity of Distributed 1-Solvable Tasks

1991 ◽  
Vol 20 (377) ◽  
Author(s):  
Ofer Biran ◽  
Shlomo Moran ◽  
Shmuel Zaks
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Worst Case ◽  
Crash Failure ◽  
Round Complexity ◽  
Precise Characterization

<p>A distributed task T is 1-solvable if there exists a protocol that solves it in the presence of (at most) one crash failure. A precise characterization of the 1-solvable tasks was given by the authors in 1990.</p><p>In this paper we determine the number of rounds of communication that are required, in the worst case, by a protocol which 1-solves a given 1-solvable task T for <em>n</em> processors. We define the radius R(T) of T, and show that if R(T) is finite, then this number is Theta (log_n R(T)) ; more precisely, we give a lower bound of log_(n-1) R(T), and an upper bound of 2+|log_(n-1)R(T)| . The upper bound implies, for example, that each of the following tasks: renaming, order preserving renaming and binary monotone consensus can be solved in the presence of one fault in 3 rounds of communications. All previous protocols that 1-solved these tasks required Omega(n) rounds. The result is also generalized to tasks whose radii are not bounded, e.g., the approximate consensus and its variants.</p>

scholarly journals The Provable Virtue of Laziness in Motion Planning

2019 ◽  
Author(s):  
Nika Haghtalab ◽  
Simon Mackenzie ◽  
Ariel D. Procaccia ◽  
Oren Salzman ◽  
Siddhartha Srinivasa
Keyword(s):  
Lower Bound ◽  
Motion Planning ◽  
Shortest Path ◽  
Upper Bound ◽  
Probabilistic Model ◽  
Shortest Paths ◽  
Worst Case ◽  
Asymptotically Optimal ◽  
Cluttered Environments ◽  
Running Time

The Lazy Shortest Path (LazySP) class consists of motion-planning algorithms that only evaluate edges along candidate shortest paths between the source and target. These algorithms were designed to minimize the number of edge evaluations in settings where edge evaluation dominates the running time of the algorithm such as manipulation in cluttered environments and planning for robots in surgical settings; but how close to optimal are LazySP algorithms in terms of this objective? Our main result is an analytical upper bound, in a probabilistic model, on the number of edge evaluations required by LazySP algorithms; a matching lower bound shows that these algorithms are asymptotically optimal in the worst case.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

scholarly journals Multiple closed orbits for N-body-type problems

1998 ◽  
Vol 58 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Shiqing Zhang
Keyword(s):  
Lower Bound ◽  
Periodic Solutions ◽  
Upper Bound ◽  
Critical Value ◽  
Body Type ◽  
Fixed Energy ◽  
Closed Orbits

Using the equivariant Ljusternik-Schnirelmann theory and the estimate of the upper bound of the critical value and lower bound for the collision solutions, we obtain some new results in the large concerning multiple geometrically distinct periodic solutions of fixed energy for a class of planar N-body type problems.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

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