A Toolbox for Online Algorithms Marcin Bieńkowski U niversity of Wroc ław phd open 2011

A Toolbox for Online Algorithms

Marcin BieńkowskiUniversity of Wrocław

phd open 2011

A Toolbox for Online Algorithms 2Marcin Bieńkowski

A (short) story

about an angry student

who was (eventually) right.


What you need

are smart tools

and not smart problems

This course:

quite good tools

for very simple problems


What’s in the box?

10 minute crash course into online algorithms

Toolbox Potential functions Work functions Linear programming Classify and randomly select


Online algorithms


Online algorithms (1)

Optimization problemse.g. set cover, independent set, facility location, TSP, …

Approximate solutionsfor any instance we want

Online problems = input is revealed graduallyfor any instance we want

scarce computational resources

scarce computational resources+ we don’t know the future

offline optimum


Online algorithms (2)

Deterministic algorithmsfor any instance we want

Randomized algorithmsfor any instance we want

We say that is -competitive


Online algorithms example

Ski Rental Problem (SRP)

Each day, in the morning, a skier may either borrow skis for 1$ or buy them for B$

Input: in the evening a skier may break his/her leg

Objective: minimize the total cost


Online algorithms example: SRP (1)

1-approximation algorithm is trivial: is the day when skier breaks leg If T · B, rent all the time If T > B, buy at the first day

For any instance ,

What about online solutions?



Bad (but natural) strategies:

Skier always rentsFor the instance I where the leg is never broken and

Skier buys at the first dayFor the instance I where the leg is broken at the first day

and



Best online strategy: rent for B-1 days and buy at day B.

Analysis:If T < B, then If T ¸ B, then and this strategy is -competitive

This is the best deterministic online algorithm



Best randomized online strategy: Choose purchase day randomly:

day k · B with probability proportional to For large B the such strategy is -competitive

But WHY this probability distribution?! Make several observations and solve the system of equations… … or wait till tomorrow till you see LP-based approach!


World is online

Many real life problems have online nature: network protocols routing scheduling exploration cache organization some data structures financial decisions and many more


Toolbox Outline

1. Potential functions (PF)

2. Work functions

3. Linear programming

4. Classify and randomly select


PF: a toy problem

Problem: List reorganization (singly-linked)

Input: sequence of the following operationsinsert(x): inserts x at the end of the list, cost L+1search(x): finds x on the list, cost = position of xdelete(x): deletes x from the list, cost as for search.

Model: After search or insert, ALG may move x towards the front of the

list for free. Afterwards, ALG may swap arbitrary adjacent elements paying 1.

C F A D G H B J


PF: a natural algorithm

Algorithm Move To Front (MTF)After each insert(x) or search(x) operation, move x to the beginning of the list (for free).


PF: costs (1)

Typical online setting: the problem consists of many rounds.In round : there is a request the algorithm serves it and pays

What is OPT doing?

Casual approach: simulate both ALG and OPT (actually any other algorithm) on the same sequence and relate their costs.

OPT? Do I look I care?Neither should you!

Besides, computing OPT is NP-complete


PF: costs (2)

Typical online setting: the problem consists of many rounds.In round : there is a request the algorithm serves it and pays

Casual approach: simulate both ALG and OPT (actually any other algorithm) on the same sequence and relate their costs.

We want:

Showing for all steps would be sufficient, but it is not the case!


PF: introduction to potential

We want:

Solution: take a potential function , such that1. 2.

where

Does it help? Also known as: How the heck can I guess the correct potential function?

There is a rule of the thumb that USUALLY works.

PROOF


PF: states

For many online problems, algorithms are in states and the costdepends on request and state only.

State space:

ALGOPT

Rule of the thumb:Define as a constant times the cost of changing state from ALG to OPT.

Why does it help (in the bad case)?When is small and is large…… then ALG changes state towards OPT, decreasing the potential!

Access(x) request x is at the beginning of OPT x is at the end of MTF

MTF moves x to the beginning of the list


PF: potential for MTF

Algorithm Move To Front (MTF):After each insert(x) or search(x) operation, move x to the beginning of the list (for free).

Potential function:Inversion: a pair (x,y) such that: x is before y in MTF list x is after y in OPT list

Potential function = number of inversions

Theorem:MTF is 2-competitive. PROOF


Toolbox Outline

1. Potential functions

2. Work functions (WF)




WF: a toy problem (1)

Problem: File migration A graph, distances on the edges. One indivisible file of size D (e.g., shared database) placed

at one node.

Input: sequence of requests (accesses to the database).

In one step t: ALG serve the request paying ALG may move the file to any node

paying

Note that d satisfies triangle inequality


WF: a toy problem (2)

Let’s make it even simpler Just two nodes connected by an edge of length 1 File is initially at node a

a b

Elementary, dear Watson


WF: randomization

Our algorithm will be randomized!

Behavioral definition:``At the end of step t, move the file to some random node.’’

Distributional definition:``At the end of step t, choose probability distribution over nodes, i.e., file is at v with probability .’’

Lemma:We may emulate distributional definition of ALG by behavioral one. The cost of changing from to is

a b

PROOF


WF: work functions (1)

After t requests, we may compute work function :The optimal cost of serving sequence up to step t and ending with file at node x.

Computing the work functionCan be computed efficiently by a simple dynamic programming We assume that ALG may move the file before the input seq. starting conditions: and


WF: work functions (2)

After t requests, we may compute work function :The optimal cost of serving sequence up to step t and ending with file at node x.

Relation to OPT It does not mean that OPT has the file at such minimizer! However, we may assume that (the amortized) OPT cost in

step t is


WF: work function evolution

Define gradientObservation: . Moreover, if g = D (or –D), then the algorithm should have the file at a (or at b) with probability 1.

In this example, we assume D = 2

0

1234567

DFor such configuration, request at a does not change the work function (or OPT cost)

Good guidance! Let’s extrapolate it to intermediate values!


GradientRecall:

Work Function AlgorithmAt step t, choose distribution , such that

WF: the algorithm

TheoremWork function algorithm is -competitive.

0

1234567

D

and

PROOF


Thank you for your attention!(and see you tomorrow)


Toolbox Outline


2. Work functions

3. Linear programming (LP)



LP: a toy problem (1)

Problem: set cover n elements family of m sets covering all elements

is the cost of set

Input: sequence of elements to coverOutput: at each step, output the subset of that covers all

elements seen so far.Online factor: removing already chosen sets not possible!

12

4

4

2

In this example:


LP: a toy problem (2)

Make the problem easier: FRACTIONAL set cover n elements family of m sets covering all elements

is the cost of set

Input: sequence of elements to coverOutput: at each step, output the function

which covers each already seen element , i.e.,

Online factor: removing already chosen fractions of sets not possible!

12

4

4

2


LP: offline solution (1)

What about offline (fractional) solution after step k? We have to cover elements

Yes, you guessed right, use linear programming!

minimize:

subject to: (for all 1 · i · k)

(for all )

12

4

4

2


LP: offline solution (2)

Do you recall?

For potential functions we did not care about OPT at all

For work functions, our constructionrelied heavily on OPT

Are we going to use it now? We may compute it!For the algorithm: NO. For the analysis: YES.


Primal program (Pk)

min.:

subject to:

(for all )

(for all )

LP: dual program

``If you have an LP, write a dual program and stare at it long enough’’ (quote from a random Stanford professor)

Dual program (Dk)

max.:

subject to:

(for all )

(for all )

Strong duality theorem: OPT(Pk) = OPT(Dk)


LP: online solution (1)

We generate our online feasible solutions to primal and dual programs.

Online = monotonic for analysis only

Observation: implies competitiveratio at most PROOF

Feasible dual solution is sometimes hard to

achieve…


LP: online solution (2)

We generate: feasible solutions to primal program solutions to dual program in which each constraint is violated at

most by factor H.

Corollary: implies competitiveratio at most

This one is feasible!


Dual program (Dk)

maximize: subject to:

(for all )

(for all )

LP: the algorithm (1)Primal program (Pk)

minimize: subject to:

(for all )

(for all )

Assume we have a solution for Pk-1

Pk contains one extra constraint

Algorithm for step k: While :

For each containing let

Observation 1: Generated primal solution is feasible TRIVIAL

Observation 2: Observation 3: Each constraint of the dual solution is violated at

most by factor O(log m)

PROOF

PROOF


LP: the algorithm (2)

Observation 1: Generated primal solution is feasibleObservation 2: Observation 3: Each constraint of the dual solution is violated at

most by factor O(log m)

Corollary: ALG is O(2 log m) = O(log m)-competitive


Toolbox Outline


2. Work functions


4. Classify and randomly select (CRS)


CRS: a toy problem (1)

Problem: Call admission A graph, capacities of the edges.

Input: sequence of call requests: pairs (si, ti).

For each call request: decide whether to admit this call (INFINITE DURATION!) if so choose a routing path for it not violating capacities

Goal: maximize the number of accepted calls

23 1

232

2132

Way too complicated, dear

Watson


CRS: a toy problem (2)

Problem: Call admission on line A line graph of n nodes all edges have capacity 1

Input: sequence of call requests: pairs (si, ti).

For each call request: decide whether to admit this call (INFINITE DURATION!)

without violating capacities if so choose a routing path

Goal: maximize the number of accepted calls


CRS: deterministic lower bound

Lemma:Any deterministic algorithm on n-node line graph has the competitive ratio at least n-1.

For a nontrivial competitive ratio, we need randomization!

PROOF


CRS: randomized solution (1)

We assume that n = 2k. We divide all edges into k classes.

i-th level call = contains edges from , but not edges from

Algorithm CRS: Choose j randomly from Accept greedily all calls from level j, and reject all other calls

Theorem: CRS is (log n)-competitive.

2-nd level call

PROOF


Thank you for your attention!

A Toolbox for Online Algorithms Marcin Bieńkowski U niversity of Wroc ław phd open 2011

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