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An Introduction to Online Computation
Authors Komm, Dennis

文章目录

1 Introduction
- 1.1 Offline Algorithms
- 1.2 Online Algorithms and Paging
- - Online Problem
  - Online Algorithms
  - Competitive Ratio
  - Theorem 1.2，1.3
  - Paging
  - pageingproblempageing\; problempageingproblem
- 1.3 An Upper Bound for Paging
- - k−k-k−Phase Partition
  - Theorem 1.4 Fifo is strictly k-competitive for paging
- 1.4 A Lower Bound for Paging：k-compititive
- - Theorem 1.5 No online algorithm for paging is better than k-compititive
  - Theorem 1.6 LIFO is not competitive for paging
  - Theorem 1.7 LFU（使用频率最低） is not competitive for paging
- 1.5 Marking Algorithms
- - Theorem 1.8. Every marking algorithm is strictly kkk-competitive for paging
  - Theorem 1.9. LRU is a marking algorithm
- 1.6 Refined Competitive Analysis
- - lookahead
  - - No online algorithm with lookahead ℓ\ellℓ for paging is better than kkk competitive.
  - resource augmentation
  - - (h,k)-paging problem
  - Every marking algorithm is k/(k-h+1)-competitive for (h,k)-paging
  - Every marking algorithm is k/(k-h+1)-competitive for (h,k)-paging

1 Introduction

Online computation, that is, algorithms that work on problem instances which they do not know from the start, but that get revealed piece by piece. These algorithms are called online algorithms, and they are commonly analyzed using so-called competitive analysis, which compares their solution to an optimal one.

在线计算，也就是在问题实例上工作的算法，这些问题实例他们从一开始并不知道，但逐渐显露出来。这些算法被称为在线算法，通常使用所谓的竞争分析来分析它们，将它们的解与最优解进行比较。

1.1 Offline Algorithms

停机问题，P,NP,NPC,NP-hard,优化问题的定义

近似算法：

对于最大化问题：

gain(OPT(I))≤r⋅gain(gain(I))gain(OPT(I))\le r\cdot gain(gain(I))gain(OPT(I))≤r⋅gain(gain(I)),

$\frac {gain(OPT(I)) } { gain(ALG(I))} \le r $

对于最小化问题：

cost(ALG(I))≤r⋅cost(ALG(I))cost(ALG(I))\le r\cdot cost(ALG(I))cost(ALG(I))≤r⋅cost(ALG(I)),

$\frac {cost(ALG(I)) } { cost(OPT(I))}\le r $

背包问题

1.2 Online Algorithms and Paging

Online Problem

problem:Π\PiΠ

instances:I\mathcal{I}I

solutions:O\mathcal{O}O

instance: I∈II\in\mathcal{I}I∈I is a sequence of requestsrequestsrequests I=(x1,x2,...,xn)I=(x_1,x_2,...,x_n)I=(x1,x2,...,xn)

output: $O\in\mathcal{O} $is a sequence of answersanswersanswers O=(y1,y2,...,yn)O=(y_1,y_2,...,y_n)O=(y1,y2,...,yn)

n∈N+n\in\mathbb{N^+}n∈N+,i.e. all instances and solutions are finite

onlineminimizationproblemonline \; minimization \; problemonlineminimizationproblem:

cost(I,OPT(I))=mincost(I,O)∣O∈sol(I)cost(I,OPT(I))=min{cost(I,O)|O\in sol(I)}cost(I,OPT(I))=mincost(I,O)∣O∈sol(I)

onlinemaximizationproblemonline \; maximization \; problemonlinemaximizationproblem:

gain(I,OPT(I))=maxgain(I,O)∣O∈sol(I)gain(I,OPT(I))=max{gain(I,O)|O\in sol(I)}gain(I,OPT(I))=maxgain(I,O)∣O∈sol(I)

only knows a prefix of the input in every given time step,只知道之前的输入
makes decisions that are based only on this knowledge, and，只基于已经知道的知识做决策
may not revoke any decision it already made.不能改变之前的决策

Online Algorithms

Online problem : Π\PiΠ

An instance of Π\PiΠ: I=(x1,x2,...,xn)I=(x_1,x_2,...,x_n)I=(x1,x2,...,xn)

An onlinealgorithmonline \; algorithmonlinealgorithm ALG for Π\PiΠ computes the output ALG(I)=(y1,y2,...,yn)ALG(I)=(y_1,y_2,...,y_n)ALG(I)=(y1,y2,...,yn), where yiy_iyi only depends on x1,x2,...,xix_1,x_2,...,x_ix1,x2,...,xi,and y1,y2,...,yi−1y_1,y_2,...,y_{i-1}y1,y2,...,yi−1, $ALG(I) $is a feasible solution for III,that is , ALG(I)∈sol(I)ALG(I) \in sol(I)ALG(I)∈sol(I).

Competitive Ratio

Online problem : Π\PiΠ

Online algorithm : ALGALGALG

For c≥1c \ge 1c≥1,ALGALGALG is c−competitivec-competitivec−competitive for Π\PiΠ if there is a non-negative constant α\alphaα such that, for every instance I∈II \in \mathcal{I}I∈I,

for an online maximization problem,

gain(OPT(I))≤r⋅gain(gain(I))+αgain(OPT(I))\le r\cdot gain(gain(I))+\alphagain(OPT(I))≤r⋅gain(gain(I))+α,

for an online minimization problem,

cost(ALG(I))≤r⋅cost(ALG(I))+αcost(ALG(I))\le r\cdot cost(ALG(I))+\alphacost(ALG(I))≤r⋅cost(ALG(I))+α,

if α=0\alpha=0α=0, we call ALGstrictlyc−competitiveALG \;strictly\; c-competitiveALGstrictlyc−competitive ,ALGALGALG is called if it is strictly 1-competitive.

cALGc_{ALG}cALG

stronglycALG−competitivestrongly \;c_{ALG}-competitivestronglycALG−competitive

近似比和竞争比考虑的都是当前算法的worst−caseworst-caseworst−case

competitiveanalysiscompetitive\;analysiscompetitiveanalysis

As with the approximation ratio, the competitive ratio is not necessarily constant, but may be a function of the input length nnn .We use the following terminology. 竞争比不一定是一个常数，可能是与输入长度n有关的函数

notcompetitivenot\; competitivenotcompetitive :如果在线算法不具有与输入长度无关的竞争比的上界???

即，算法的上界，应该与输入长度无关，这样才能是competitive的

α\alphaα这个常数是必不可少的

Theorem 1.2，1.3

对于最小化/最大化问题，there is no online algorithm that is "better than c−competitvec-competitvec−competitve"

Paging

phasesphasesphases:consist of a number of consecutive(连续的) time steps.

c是任何分页在线算法竞争比的下界

pageingproblempageing\; problempageingproblem

minimization

In memory: p1,...,pmp_1,...,p_mp1,...,pm

Instance: I=(x1,x2,...,xn)I=(x_1,x_2,...,x_n)I=(x1,x2,...,xn), such that xi∈{p1,...,pm}x_i\in \{p_1,...,p_m\}xi∈{p1,...,pm}

xix_ixiis requested in time Step TiT_iTi,

ALGALGALG for paging maintains a cache memory of size kkk with k<mk < mk<m .

time step TiT_iTi,cache 中存的东西:Bi={pj1,pj2,...,pjk}B_i=\{p_{j1},p_{j2},...,p_{jk}\}Bi={pj1,pj2,...,pjk}

B0={p1,p2,...,pk}B_0=\{p_{1},p_{2},...,p_{k}\}B0={p1,p2,...,pk}

如果TiT_iTi请求的page在Bi−1B_{i-1}Bi−1中，命中，cost=0，如果不在，选一个yiy_iyi置换出去，ALGALGALG outputs yi=pjy_i=p_jyi=pj

cost(ALG(I)ALG(I)ALG(I))=|{i∣yi≠0i|y_i\ne0i∣yi=0}|,下标是集合的大小？

the goal is to minimize this number.

demandpagingalgorithmsdemand \;paging \;algorithmsdemandpagingalgorithms:每次只替换出一页，请求分页算法

Strategies：FIFO，LIFO，LFU(使用频率最低)，LRU，FWF（flush when full）,LFD(离线算法，remove 之后最晚被请求的)

1.3 An Upper Bound for Paging

这里是在算上界，也就是说，求cost(ALG(I))cost(ALG(I))cost(ALG(I))max,和cost(OPT(I))cost(OPT(I))cost(OPT(I))min

先介绍些东西，以便于证明上界。

在上一节中，我们定义了一个在线算法是竞争的，如果它的竞争比c是与输入长度相关的一个常数。

在分页问题中，c与k,m相关？

k−k-k−Phase Partition

k步划分，针对输入，而不针对算法

instance：I=(x1,x2,...,xn)I=(x_1,x_2,...,x_n)I=(x1,x2,...,xn)

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