A better lower bound on the competitive ratio of the randomized 2-server problem

1997 ◽  
Vol 63 (2) ◽  
pp. 79-83 ◽  
Author(s):  
Marek Chrobak ◽  
Lawrence L. Larmore ◽  
Carsten Lund ◽  
Nick Reingold
Keyword(s):  
Lower Bound ◽  
Competitive Ratio ◽  
Server Problem

The Infinite Server Problem

2021 ◽  
Vol 17 (3) ◽  
pp. 1-23
Author(s):  
Christian Coester ◽  
Elias Koutsoupias ◽  
Philip Lazos
Keyword(s):  
Lower Bound ◽  
Competitive Ratio ◽  
Metric Spaces ◽  
Bounded Interval ◽  
Weighted Trees ◽  
Infinite Server ◽  
Tight Connection ◽  
Server Problem ◽  
Double Coverage ◽  
Special Case

We study a variant of the k -server problem, the infinite server problem, in which infinitely many servers reside initially at a particular point of the metric space and serve a sequence of requests. In the framework of competitive analysis, we show a surprisingly tight connection between this problem and the resource augmentation version of the k -server problem, also known as the (h,k) -server problem, in which an online algorithm with k servers competes against an offline algorithm with h servers. Specifically, we show that the infinite server problem has bounded competitive ratio if and only if the (h,k) -server problem has bounded competitive ratio for some k = O ( h ). We give a lower bound of 3.146 for the competitive ratio of the infinite server problem, which holds even for the line and some simple weighted stars. It implies the same lower bound for the (h,k) -server problem on the line, even when k/h → ∞, improving on the previous known bounds of 2 for the line and 2.4 for general metrics. For weighted trees and layered graphs, we obtain upper bounds, although they depend on the depth. Of particular interest is the infinite server problem on the line, which we show to be equivalent to the seemingly easier case in which all requests are in a fixed bounded interval. This is a special case of a more general reduction from arbitrary metric spaces to bounded subspaces. Unfortunately, classical approaches (double coverage and generalizations, work function algorithm, balancing algorithms) fail even for this special case.

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.

ONLINE SCHEDULING OF DYNAMIC TREES

1995 ◽  
Vol 05 (04) ◽  
pp. 635-646 ◽  
Author(s):  
MICHAEL A. PALIS ◽  
JING-CHIOU LIOU ◽  
SANGUTHEVAR RAJASEKARAN ◽  
SUNIL SHENDE ◽  
DAVID S.L. WEI
Keyword(s):  
Lower Bound ◽  
Competitive Ratio ◽  
Scheduling Algorithm ◽  
Optimal Schedule ◽  
Online Scheduling ◽  
The Other ◽  
Static Case ◽  
Scheduling Problem ◽  
Dynamic Trees ◽  
Dynamic Tree

The scheduling problem for dynamic tree-structured task graphs is studied and is shown to be inherently more difficult than the static case. It is shown that any online scheduling algorithm, deterministic or randomized, has competitive ratio Ω((1/g)/ log d(1/g)) for trees with granularity g and degree at most d. On the other hand, it is known that static trees with arbitrary granularity can be scheduled to within twice the optimal schedule. It is also shown that the lower bound is tight: there is a deterministic online tree scheduling algorithm that has competitive ratio O((1/g)/ log d(1/g)). Thus, randomization does not help.

Online-List Scheduling on a Single Bounded Parallel-Batch Machine to Minimize Makespan

2015 ◽  
Vol 32 (04) ◽  
pp. 1550028
Author(s):  
Wenhua Li ◽  
Jie Gao ◽  
Jinjiang Yuan
Keyword(s):  
Lower Bound ◽  
Competitive Ratio ◽  
Online Algorithm ◽  
Positive Root ◽  
List Scheduling ◽  
Batch Machine

In this paper, we consider the online-list scheduling on a single bounded parallel-batch machine to minimize makespan. In the problem, the jobs arrive online over list. The first unassigned job in the list should be assigned to a batch before the next job is released. Each batch can accommodate up to b jobs. For b = 2, we establish a lower bound 1 + γ of competitive ratio and provide an online algorithm with a competitive ratio of [Formula: see text], where γ is the positive root of γ(γ + 1)2 = 1. For b = 3, we establish a lower bound 1 + α of competitive ratio and provide an online algorithm with a competitive ratio of 2, where α is the positive root of the equation (1 + α)(1 + α2) = 2.

scholarly journals On-line Search in Two-Dimensional Environment

2019 ◽  
Vol 63 (8) ◽  
pp. 1819-1848
Author(s):  
Dariusz Dereniowski ◽  
Dorota Osula
Keyword(s):  
Lower Bound ◽  
Competitive Ratio ◽  
Mobile Agents ◽  
Line Search ◽  
Search Strategy ◽  
A Priori ◽  
Two Dimensional ◽  
Pursuit Evasion ◽  
On Line ◽  
Evasion Problem

Abstract We consider the following on-line pursuit-evasion problem. A team of mobile agents called searchers starts at an arbitrary node of an unknown network. Their goal is to execute a search strategy that guarantees capturing a fast and invisible intruder regardless of its movements using as few searchers as possible. We require that the strategy is connected and monotone, that is, at each point of the execution the part of the graph that is guaranteed to be free of the fugitive is connected and whenever some node gains a property that it cannot be occupied by the fugitive, the strategy must operate in such a way to keep this property till its end. As a way of modeling two-dimensional shapes, we restrict our attention to networks that are embedded into partial grids: nodes are placed on the plane at integer coordinates and only nodes at distance one can be adjacent. Agents do not have any knowledge about the graph a priori, but they recognize the direction of the incident edge (up, down, left or right). We give an on-line algorithm for the searchers that allows them to compute a connected and monotone strategy that guarantees searching any unknown partial grid with the use of $O(\sqrt {n})$ O ( n ) searchers, where n is the number of nodes in the grid. As for a lower bound, there exist partial grids that require ${\varOmega }(\sqrt {n})$ Ω ( n ) searchers. Moreover, we prove that for each on-line searching algorithm there is a partial grid that forces the algorithm to use ${\varOmega }(\sqrt {n})$ Ω ( n ) searchers but $O(\log n)$ O ( log n ) searchers are sufficient in the off-line scenario. This gives a lower bound on ${\varOmega }(\sqrt {n}/\log n)$ Ω ( n / log n ) in terms of achievable competitive ratio of any on-line algorithm.

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.

LOWER BOUNDS FOR STREETS AND GENERALIZED STREETS

2001 ◽  
Vol 11 (04) ◽  
pp. 401-421 ◽  
Author(s):  
ALEJANDRO LÓPEZ-ORTIZ ◽  
SVEN SCHUIERER
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real Line ◽  
Upper Bound ◽  
Competitive Ratio ◽  
Search Strategy ◽  
The Real ◽  
Simple Polygons ◽  
On Line ◽  
Special Classes

We present lower bounds for on-line searching problems in two special classes of simple polygons called streets and generalized streets. In streets we assume that the location of the target is known to the robot in advance and prove a lower bound of [Formula: see text] on the competitive ratio of any deterministic search strategy—which can be shown to be tight. For generalized streets we show that if the location of the target is not known, then there is a class of orthogonal generalized streets for which the competitive ratio of any search strategy is at least [Formula: see text] in the L2-metric—again matching the competitive ratio of the best known algorithm. We also show that if the location of the target is known, then the competitive ratio for searching in generalized streets in the L1-metric is at least 9 which is tight as well. The former result is based on a lower bound on the average competitive ratio of searching on the real line if an upper bound of D to the target is given. We show that in this case the average competitive ratio is at least 9-O(1/ log D).

A SEMI-ON-LINE SCHEDULING PROBLEM OF TWO PARALLEL MACHINES WITH COMMON MAINTENANCE TIME

2012 ◽  
Vol 29 (04) ◽  
pp. 1250020 ◽  
Author(s):  
YUHUA CAI ◽  
QI FENG ◽  
WENJIE LI
Keyword(s):  
Lower Bound ◽  
Competitive Ratio ◽  
Parallel Machines ◽  
Time Interval ◽  
Scheduling Problem ◽  
Maintenance Time ◽  
Identical Machines ◽  
On Line

In this paper, we consider a semi-on-line scheduling problem of two identical machines with common maintenance time interval and nonresumable availability. We prove a lower bound of 2.79129 on the competitive ratio and give an on-line algorithm with competitive ratio 2.79633 for this problem.

scholarly journals COMPETITIVE ALGORITHMS FOR ONLINE PRICING

2012 ◽  
Vol 04 (02) ◽  
pp. 1250015 ◽  
Author(s):  
YONG ZHANG ◽  
YUXIN WANG ◽  
FRANCIS Y. L. CHIN ◽  
HING-FUNG TING
Keyword(s):  
Lower Bound ◽  
Competitive Ratio ◽  
Online Algorithm ◽  
Value Function ◽  
Unit Price ◽  
Competitive Algorithms ◽  
Online Pricing

Given a seller with m items, a sequence of users {u1, u2, …} come one by one, the seller must set the unit price and assign some items to each user on his/her arrival. Items can be sold fractionally. Each ui has his/her value function vi(⋅) such that vi(x) is the highest unit price ui is willing to pay for x items. The objective is to maximize the revenue by setting the price and number of items for each user. In this paper, we have the following contributions: if the highest value h among all vi(x) is known in advance, we first show the lower bound of the competitive ratio is ⌊ log h⌋/2, then give an online algorithm with competitive ratio 4⌊ log h⌋ + 6; if h is not known in advance, we give an online algorithm with competitive ratio 2⋅h log -1/2 h + 8⋅h3 log -1/2 h.

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