Serving Online Requests with Mobile Servers
Abdolhamid Ghodselahi University of Freiburg, Germany
joint work withFabian Kuhn (University of Freiburg)
Presented at ISAAC 2015, Nagoya, JapanDecember 9-11, 2015
Our online problem:
๐ points are given
๐ mobile servers
Online requests
service cost = ๐
service cost = ๐
service cost = ๐service cost = ๐
service cost = ๐service cost = ๐
Goal: Minimize #movements
service cost = ๐
#movements = 1#movements = 2
๐ฎ = 1๐ฎ = 2๐ฎ = 4๐ฎ = 1๐ฎ = 3๐ฎ = 1
What if some algorithm moves no server at all?!
Feasible Configuration:
Any algorithm that solves the problem must satisfy the following condition at all time steps :
Problem condition: ๐ฎ < ๐ผ โ ๐ฎโ + ๐ฝ
๐ฎ โ Current service cost of any algorithmโข Service cost is not cumulative over time
๐ฎโ โ Optimal current service cost =Minimum service cost among all configurations
๐ผ โฅ 1 and ๐ฝ โฅ 0 are two given parameters
Recap:
๐ points are given ๐ mobile servers Online requests
Requests need to be servedโข At the requested pointโข By a remote server
A request has to be served atall time steps after it is issuedโข Reassignment is allowed
Problem condition must be satisfied at all time steps
Goal: Minimize #movements
service cost = ๐
service cost = ๐
service cost = ๐
service cost = ๐
Our Model VS. ๐-Server/Paging
Our Model
Requests are served
to serve However, some on the current service cost
๐-Server/Paging
Requests are served when they are issued
Servers serve No service cost
Known Results for ๐-Server & Paging Deterministic
โข ๐-Server conjecture: Competitive factor is ๐[Manasse, McGeoch, & Sleator 1990]
โข Competitive factor of 2๐ โ 1 for ๐-Server[Koutsoupias & Papadimitriou 1995]
โข Any deterministic algorithm is ฮฉ ๐ -competitive[Sleator & Tarjan 1985]
โข Least recently used (LRU) algorithm is ๐-competitive[Sleator & Tarjan 1985]
Randomized
โข Competitive factor of ฮ log ๐ 2 log ๐ 3
[Bansal, Buchbinder, Madry, & Naor 2011]
Outline
1. Motivation & Model
2. Minimizing #Movements
a. Lower-Bound
3. Minimizing #Movements + Service Cost
a. Upper-Bound
b. Lower-Bound
4. Future Work
Any deterministic online algorithm is
ฮฉ ๐ -competitive
2. Minimizing #movements
Proof Sketch ๐ โถ Any deterministic online algorithm (ALG)
๐ช โถ Any optimal offline algorithm (OPT)
Two cases:
๐ > ๐/2 :
โข Competitive factor is โฅ ๐
๐ โค ๐/2 :
โข Competitive factor is โฅ ๐ โ ๐
โฅ max ๐, ๐ โ ๐
โฅ ๐ 2 โ ฮฉ(๐)
๐ โค ๐/2 โถ Main Idea
large enough #requests
๐ฎ๐ โฎ ๐ผ โ ๐ฎโ + ๐ฝ
ALG must move some server(s)
OPT moves to a point where#requests is large at all time steps
points without servers
๐ = 3 , ๐ = 1
Assume ๐ผ = 1
Problem condition:โ๐ก โถ ๐ฎ๐(๐ก) < ๐ฎโ(๐ก) + ๐ฝ
Repeat for ๐ , ๐ = 1 โโฅ ๐ โ 1 #movements
๐ฎโ = ๐ฝ๐ฎ๐ = 2๐ฝ
๐ฎ๐ = ๐ฎโ + ๐ฝ
๐ โค ๐/2 : Simple Example#Movements by ALG
๐ฝ
๐ฝ
๐ฎโ = ๐ฝ๐ฎ๐ = ๐ฝ
๐ฎ๐ < ๐ฎโ + ๐ฝ
2๐ฝ
2๐ฝ
๐ฎโ = 3๐ฝ๐ฎ๐ = 4๐ฝ
๐ฎ๐ = ๐ฎโ + ๐ฝ
๐ฎโ = 3๐ฝ๐ฎ๐ = 3๐ฝ
๐ฎ๐ < ๐ฎโ + ๐ฝ
๐ = 3 , ๐ = 1
Assume ๐ผ = 1
OPT knows the sequence in advance
Problem condition:โ๐ก โถ ๐ฎ๐ช(๐ก) < ๐ฎโ(๐ก) + ๐ฝ
Repeat for ๐ , ๐ = 1 โโค 1 #movements
๐ฎโ = ๐ฝ๐ฎ๐ช = 2๐ฝ
๐ฎ๐ = ๐ฎโ + ๐ฝ
๐ โค ๐/2 : Simple Example#Movements by OPT
๐ฝ
๐ฝ
๐ฎโ = ๐ฝ๐ฎ๐ช = ๐ฝ
๐ฎ๐ < ๐ฎโ + ๐ฝ
2๐ฝ
2๐ฝ
๐ฎโ = 3๐ฝ๐ฎ๐ช = 3๐ฝ
๐ฎ๐ช < ๐ฎโ + ๐ฝ
๐ โค ๐ 2 โถ Reduction to any ๐, ๐
โฏ
๐
โฏ
๐ โ 1โฏ โฏ
๐ โ 1
๐
โ
All algorithms can only move this server
โฅ ๐ โ ๐ #movements by ALG and โค 1 by OPT
3. Minimizing Combined CostThe objective is to minimize the
Current service cost+ #Movements
This modification in the objective helps us to be more competitive against OPT
A natural greedy algorithm (denoted by ๐) is introduced which provides an almost tight bound
Minimizing combined cost is closer to an online variant of
mobile facility location problem [Friggstad & Salavatipour FOCSโ08]
The algorithm does nothing as long as ๐ฎ๐ < ๐ผ โ ๐ฎโ + ๐ฝ
It greedily moves some server(s) as soon as ๐ฎ๐ โฎ ๐ผ โ ๐ฎโ + ๐ฝ
Greedy Approach:
Decrease current service cost as much as possible
Maximal improvement = 6๐ฝ
Greedy Algorithm
5๐ฝ
2๐ฝ
6๐ฝ
8๐ฝ
๐ฎโ = 7๐ฝ๐ฎ๐ = 14๐ฝ
๐ฎ๐ < 2๐ฎโ + ๐ฝ
๐ฎโ = 7๐ฝ๐ฎ๐ = 15๐ฝ
๐ฎ๐ = 2๐ฎโ + ๐ฝ
7๐ฝ๐ฎโ = 7๐ฝ๐ฎ๐ = 9๐ฝ
๐ฎ๐ < 2๐ฎโ + ๐ฝ
Our online algorithm is 1 + ๐ -competitive
for every constant ๐ > 0,
at the cost of an additional additive term
Results
Any deterministic online algorithm cannot get
a better competitive factor than almost similar
above upper-bound
Upper-Bound: Proof Sketch
Goal: Minimize the combined cost
๐ฎ๐ < ๐ผ โ ๐ฎโ + ๐ฝ ๐๐ โค ?
๐ฎ๐ช +๐๐ช โฅ ๐ฎโ
๐๐ โค ๐ โ ๐ฎโ + ฮ(๐ log ๐)
General Service Cost Function Recall:
๐๐ฃ ๐ฆ โ 0, ๐ ๐๐๐ฃ๐๐ ๐๐ก ๐ฃ๐ฆ, ๐๐กโ๐๐๐ค๐๐ ๐
Generalization:๐๐ฃ ๐ฅ, ๐ฆ โ Service cost of ๐ฃ if ๐ฅ servers and ๐ฆ requests at ๐ฃ
The function has to satisfy some natural properties: Monotonicity (in ๐ฅ and ๐ฆ)
Effect of adding additional servers to a node ๐ฃ
โข should become smaller (convexity in ๐ฅ)
โข should not decrease if #requests gets larger
The upper-bound result holds for this generalization
Both lower-bound results even hold for the previous service cost
4. Future Work
With respect to minimizing the #movements:
โข Study randomized online algorithms
With respect to minimizing the combined cost:
โข Study the online variant of mobile facility location problem (OMFLP) in general metrics
OMFLP definition
our lower-bound already holds for any det. online algorithm that solves OMFLP
Thanks for your attention