scholarly journals Chasing Convex Bodies with Linear Competitive Ratio

2021 ◽  
Vol 68 (5) ◽  
pp. 1-10
Author(s):  
C. J. Argue ◽  
Anupam Gupta ◽  
Ziye Tang ◽  
Guru Guruganesh
Keyword(s):  
Competitive Ratio ◽  
Convex Bodies ◽  
Competitive Algorithm ◽  
Sum Of Distances

We study the problem of chasing convex bodies online: given a sequence of convex bodies the algorithm must respond with points in an online fashion (i.e., is chosen before is revealed). The objective is to minimize the sum of distances between successive points in this sequence. Bubeck et al. (STOC 2019) gave a -competitive algorithm for this problem. We give an algorithm that is -competitive for any sequence of length .

ON THE LOWER BOUND OF THE COMPETITIVE RATIO FOR THE WEIGHTED ONLINE ROOMMATES PROBLEM

2010 ◽  
Vol 02 (02) ◽  
pp. 257-262
Author(s):  
SATYAJIT BANERJEE
Keyword(s):  
Lower Bound ◽  
Competitive Ratio ◽  
Deterministic Algorithm ◽  
Weighted Version ◽  
Worst Case ◽  
Competitive Algorithm

We show that the best possible worst case competitive ratio of any deterministic algorithm for weighted online roommates problem is arbitrarily close to 4. This proves that the 4-competitive algorithm proposed by Bernstein and Rajagopalan [3] for the weighted version of the online roommates problem actually attains the best possible competitive ratio.

ONLINE SCHEDULING OF MIXED CPU-GPU JOBS

2014 ◽  
Vol 25 (06) ◽  
pp. 745-761 ◽  
Author(s):  
LIN CHEN ◽  
DESHI YE ◽  
GUOCHUAN ZHANG
Keyword(s):  
Competitive Ratio ◽  
Completion Time ◽  
Online Algorithm ◽  
Online Scheduling ◽  
Scheduling Problem ◽  
Competitive Algorithm ◽  
Processing Times ◽  
Special Cases ◽  
Identical Machines ◽  
The One

We consider the online scheduling problem in a CPU-GPU cluster. In this problem there are two sets of processors, the CPU processors and the GPU processors. Each job has two distinct processing times, one for the CPU processor and the other for the GPU processor. Once a job is released, a decision should be made immediately about which processor it should be assigned to. The goal is to minimize the makespan, i.e., the largest completion time among all the processors. Such a problem could be seen as an intermediate model between the scheduling problem on identical machines and unrelated machines. We provide a 3.85-competitive online algorithm for this problem and show that no online algorithm exists with competitive ratio strictly less than 2. We also consider two special cases of this problem, the balanced case where the number of CPU processors equals to that of GPU processors, and the one-sided case where there is only one CPU or GPU processor. For the balanced case, we first provide a simple 3-competitive algorithm, and then a better algorithm with competitive ratio of 2.732 is derived. For the one-sided case, a 3-competitive algorithm is given.

ON THE k-TRUCK SCHEDULING PROBLEM

2004 ◽  
Vol 15 (01) ◽  
pp. 127-141 ◽  
Author(s):  
WEIMIN MA ◽  
YINFENG XU ◽  
JANE YOU ◽  
JAMES LIU ◽  
KANLIANG WANG
Keyword(s):  
Lower Bound ◽  
Greedy Algorithm ◽  
Competitive Ratio ◽  
Scheduling Problem ◽  
Competitive Algorithm ◽  
Truck Scheduling ◽  
New Variant ◽  
On Line ◽  
Server Problem ◽  
Better Than

In this paper, some results concerning the k-truck problem are produced. Firstly, the algorithms and their complexity concerning the off-line k-truck problem are discussed. Following that, a lower bound of competitive ratio (1+θ)·k/(θ·k+2) for the on-line k-truck problem is given, where θ is the ratio of cost of the loaded truck and the empty truck on the same distance, and a relevant lower bound for the on-line k-taxi problem followed naturally. Thirdly, based on the Position Maintaining Strategy (PMS), some new results which are slightly better than those of [11] for general cases are obtained. For example, a c-competitive or (c/θ+1/θ+1)-competitive algorithm for the on-line k-truck problem depending on the value of θ, where c is the competitive ratio of some algorithm to a relevant k-server problem, is developed. The Partial-Greedy Algorithm (PG) is used as well to solve this problem on a line with n nodes and is proved to be a (1+(n-k)/θ)-competitive algorithm for this case. Finally, the concepts of the on-line k-truck problem are extended to obtain a new variant: Deeper On-line k-Truck Problem (DTP). We claim that results of PMS for the STP (Standard Truck Problem) hold for the DTP.

scholarly journals Chasing Convex Bodies with Linear Competitive Ratio

2020 ◽  
pp. 1519-1524 ◽  
Author(s):  
C.J. Argue ◽  
Anupam Gupta ◽  
Guru Guruganesh ◽  
Ziye Tang
Keyword(s):  
Competitive Ratio ◽  
Convex Bodies

WIN Algorithm for Discrete Online TSP

2011 ◽  
Vol 15 (9) ◽  
pp. 1199-1202
Author(s):  
Yonghua Wu ◽  
◽  
Guohun Zhu ◽  
Huaying Chen ◽  
Jucun Qin ◽  
...  
Keyword(s):  
Traveling Salesman Problem ◽  
Metric Space ◽  
Competitive Analysis ◽  
Competitive Ratio ◽  
Traveling Salesman ◽  
Complete Problem ◽  
Competitive Algorithm ◽  
Np Complete

Traveling Salesman Problem (TSP) which is proved as an NP-Complete problem is solved by many algorithms. In this paper, we propose online TSP which is based on general discrete metric space. A Waiting-If-Necessary (WIN) algorithm is proposed that involves with increasing cost caused by zealous algorithms and unnecessary waiting caused by cautious algorithms. We measure the performance of the WIN algorithm using competitive analysis and found that it is a 2-competitive algorithm. The competitive ratio of theWIN algorithm can be improved by setting parameterT0.

ONE-SPACE BOUNDED ALGORITHMS FOR TWO-DIMENSIONAL BIN PACKING

2010 ◽  
Vol 21 (06) ◽  
pp. 875-891 ◽  
Author(s):  
FRANCIS Y. L. CHIN ◽  
HING-FUNG TING ◽  
YONG ZHANG
Keyword(s):  
Lower Bound ◽  
Competitive Ratio ◽  
Online Algorithm ◽  
Bin Packing ◽  
Two Dimensional ◽  
Unit Square ◽  
Competitive Algorithm ◽  
Previous Time ◽  
Bounded Space ◽  
Over Time

In this paper, we study the bounded space variation, especially one-space bounded, of two-dimensional bin packing. A sequence of rectangular items arrive over time, and the following item arrives after the packing of the previous one. The height and width of each item are no more than 1, we need to pack these items into unit square bins of size 1 × 1 where rotation of 90° is allowed and our objective is to minimize the number of used bins. Once an item is packed into a square bin, the position of this item is fixed and it cannot be shifted within this bin. At any time, there is at most one active bin; the current unpacked item can be only packed into the active bin and the inactive bins (closed at some previous time) cannot be used for any future items. We first propose an online algorithm with a constant competitive ratio 12, then improve the competitive ratio to 8.84 by the some complicated analysis. Our results significantly improve the previous best known O(( log log m)2)-competitive algorithm [10], where m is the width of the square bin and the size of each item is a × b, where a, b are integers no more than m. Furthermore, the lower bound for the competitive ratio is also improved to 2.5.

Excess entropy of the systems of molecules differing in size and shape

1983 ◽  
Vol 48 (1) ◽  
pp. 192-198 ◽  
Author(s):  
Tomáš Boublík
Keyword(s):  
Equation Of State ◽  
Hard Spheres ◽  
Excess Entropy ◽  
Convex Bodies ◽  
Pure Substance ◽  
Liquid Equilibrium ◽  
Simple Rules ◽  
Different Shapes ◽  
The Mean

The excess entropy of mixing of mixtures of hard spheres and spherocylinders is determined from an equation of state of hard convex bodies. The obtained dependence of excess entropy on composition was used to find the accuracy of determining ΔSE from relations employed for the correlation and prediction of vapour-liquid equilibrium. Simple rules were proposed for establishing the mean parameter of nonsphericity for mixtures of hard bodies of different shapes allowing to describe the P-V-T behaviour of solutions in terms of the equation of state fo pure substance. The determination of ΔSE by means of these rules is discussed.

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