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A Toolbox for Online Algorithms Marcin Bieńkowski University of Wrocław phd open 2011 A Toolbox for Online Algorithms Marcin Bieńkowski University of Wrocław phd open 2011

A (short) story about an angry student who was (eventually) right. Marcin Bieńkowski A A (short) story about an angry student who was (eventually) right. Marcin Bieńkowski A Toolbox for Online Algorithms 2

What you need course: This are smart tools good tools quite and not smart What you need course: This are smart tools good tools quite and not smart problemssimple problems for very Marcin Bieńkowski A Toolbox for Online Algorithms 3

What’s in the box? § 10 minute crash course into online algorithms § Toolbox What’s in the box? § 10 minute crash course into online algorithms § Toolbox § Potential functions § Work functions § Linear programming § Classify and randomly select Marcin Bieńkowski A Toolbox for Online Algorithms 4

Online algorithms Marcin Bieńkowski A Toolbox for Online Algorithms 5 Online algorithms Marcin Bieńkowski A Toolbox for Online Algorithms 5

Online algorithms (1) Optimization problems e. g. set cover, independent set, facility location, TSP, Online algorithms (1) Optimization problems e. g. set cover, independent set, facility location, TSP, … Approximate solutions for any instance we want scarce computational resources Online problems = input is revealed gradually for any instance we want scarce computational resources + we don’t know the future offline optimum Marcin Bieńkowski A Toolbox for Online Algorithms 6

Online algorithms (2) Deterministic algorithms for any instance we want Randomized algorithms for any Online algorithms (2) Deterministic algorithms for any instance we want Randomized algorithms for any instance we want We say that Marcin Bieńkowski is -competitive A Toolbox for Online Algorithms 7

Online algorithms example Ski Rental Problem (SRP) § Each day, in the morning, a 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 Marcin Bieńkowski A Toolbox for Online Algorithms 8

Online algorithms example: SRP (1) 1 -approximation algorithm is trivial: § is the day 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? Marcin Bieńkowski A Toolbox for Online Algorithms 9

Online algorithms example: SRP (2) Bad (but natural) strategies: § Skier always rents For Online algorithms example: SRP (2) Bad (but natural) strategies: § Skier always rents For the instance I where the leg is never broken and § Skier buys at the first day For the instance I where the leg is broken at the first day and Marcin Bieńkowski A Toolbox for Online Algorithms 10

Online algorithms example: SRP (3) Best online strategy: rent for B-1 days and buy Online algorithms example: SRP (3) Best online strategy: rent for B-1 days and buy at day B. Analysis: If T < B, then If T ¸ B, then this strategy is and -competitive This is the best deterministic online algorithm Marcin Bieńkowski A Toolbox for Online Algorithms 11

Online algorithms example: SRP (4) Best randomized online strategy: § Choose purchase day randomly: Online algorithms example: SRP (4) 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! Marcin Bieńkowski A Toolbox for Online Algorithms 12

World is online Many real life problems have online nature: § network protocols § 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 Marcin Bieńkowski A Toolbox for Online Algorithms 13

Toolbox Outline 1. Potential functions (PF) 2. Work functions 3. Linear programming 4. Classify Toolbox Outline 1. Potential functions (PF) 2. Work functions 3. Linear programming 4. Classify and randomly select Marcin Bieńkowski A Toolbox for Online Algorithms 14

PF: a toy problem Problem: List reorganization (singly-linked) C F A D G H PF: a toy problem Problem: List reorganization (singly-linked) C F A D G H B J Input: sequence of the following operations insert(x): inserts x at the end of the list, cost L+1 search(x): finds x on the list, cost = position of x delete(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. Marcin Bieńkowski A Toolbox for Online Algorithms 15

PF: a natural algorithm Algorithm Move To Front (MTF) After each insert(x) or search(x) 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). Marcin Bieńkowski A Toolbox for Online Algorithms 16

PF: costs (1) Typical online setting: the problem consists of many rounds. In round 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? Besides, computing OPT is OPT? Do I look I care? NP-complete Neither should you! Casual approach: simulate both ALG and OPT (actually any other algorithm) on the same sequence and relate their costs. Marcin Bieńkowski A Toolbox for Online Algorithms 17

PF: costs (2) Typical online setting: the problem consists of many rounds. In round 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 sufficient, but it is not the case! Marcin Bieńkowski for all steps would be A Toolbox for Online Algorithms 18

PF: introduction to potential We want: Solution: take a potential function 1. 2. where PF: introduction to potential We want: Solution: take a potential function 1. 2. where , such that PROOF 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. Marcin Bieńkowski A Toolbox for Online Algorithms 19

PF: states For many online problems, algorithms are in states and the cost depends PF: states For many online problems, algorithms are in states and the cost depends on request and state only. State space: ALG OPT Rule of the thumb: Define as a constant times the cost of changing state from ALG to OPT. § Access(x) request § x is at the beginning of OPT § x is at the end of MTF Why does it help (in the bad case)? When is small and is large… … then ALG changes state towards OPT, MTF moves x to the decreasing the potential! beginning of the list Marcin Bieńkowski A Toolbox for Online Algorithms 20

PF: potential for MTF Algorithm Move To Front (MTF): After each insert(x) or search(x) 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. Marcin Bieńkowski PROOF A Toolbox for Online Algorithms 21

Toolbox Outline 1. Potential functions 2. Work functions (WF) 3. Linear programming 4. Classify Toolbox Outline 1. Potential functions 2. Work functions (WF) 3. Linear programming 4. Classify and randomly select Marcin Bieńkowski A Toolbox for Online Algorithms 22

WF: a toy problem (1) Problem: File migration § A graph, distances on the 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. Note that d satisfies triangle inequality 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 Marcin Bieńkowski A Toolbox for Online Algorithms 23

WF: a toy problem (2) Let’s make it even simpler § Just two nodes 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 Marcin Bieńkowski b Elementary, dear Watson A Toolbox for Online Algorithms 24

WF: randomization Our algorithm will be randomized! a b Behavioral definition: ``At the end WF: randomization Our algorithm will be randomized! a b 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 nodes, i. e. , file is at v with probability. ’’ over PROOF Lemma: We may emulate distributional definition of ALG by behavioral one. The cost of changing from to is Marcin Bieńkowski A Toolbox for Online Algorithms 25

WF: work functions (1) After t requests, we may compute work function : The 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 function Can be computed efficiently by a simple dynamic programming § We assume that ALG may move the file before the input seq. § starting conditions: and Marcin Bieńkowski A Toolbox for Online Algorithms 26

WF: work functions (2) After t requests, we may compute work function : The 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 Marcin Bieńkowski A Toolbox for Online Algorithms 27

WF: work function evolution 7 6 5 4 3 2 1 In this example, WF: work function evolution 7 6 5 4 3 2 1 In this example, we assume D = 2 D For such configuration, request at a does not change the work function (or OPT cost) 0 Good guidance! Let’s extrapolate it to intermediate values! Define gradient Observation: . Moreover, if g = D (or –D), then the algorithm should have the file at a (or at b) with probability 1. Marcin Bieńkowski A Toolbox for Online Algorithms 28

WF: the algorithm 7 6 5 4 3 2 1 Gradient Recall: D Work WF: the algorithm 7 6 5 4 3 2 1 Gradient Recall: D Work Function Algorithm At step t, choose distribution such that and 0 Theorem Work function algorithm is Marcin Bieńkowski , -competitive. A Toolbox for Online Algorithms PROOF 29

Thank you for your attention! (and see you tomorrow) Marcin Bieńkowski A Toolbox for Thank you for your attention! (and see you tomorrow) Marcin Bieńkowski A Toolbox for Online Algorithms 30

Toolbox Outline 1. Potential functions 2. Work functions 3. Linear programming (LP) 4. Classify Toolbox Outline 1. Potential functions 2. Work functions 3. Linear programming (LP) 4. Classify and randomly select Marcin Bieńkowski A Toolbox for Online Algorithms 31

LP: a toy problem (1) Problem: set cover § n elements § family of LP: a toy problem (1) Problem: set cover § n elements § family of m sets covering all elements is the cost of set 2 4 1 4 2 In this example: § § Input: sequence of elements to cover Output: at each step, output the subset of that covers all elements seen so far. Online factor: removing already chosen sets not possible! Marcin Bieńkowski A Toolbox for Online Algorithms 32

LP: a toy problem (2) Make the problem easier: FRACTIONAL set cover § n 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 2 4 1 4 2 Input: sequence of elements to cover Output: at each step, output the function which covers each already seen element , i. e. , Online factor: removing already chosen fractions of sets not possible! Marcin Bieńkowski A Toolbox for Online Algorithms 33

LP: offline solution (1) What about offline (fractional) solution after step k? § We LP: offline solution (1) What about offline (fractional) solution after step k? § We have to cover elements 1 2 4 Yes, you guessed right, use linear programming! minimize: subject to: (for all 1 · i · k) (for all Marcin Bieńkowski A Toolbox for Online Algorithms ) 34

LP: offline solution (2) Do you recall? § For potential functions we did not LP: offline solution (2) Do you recall? § For potential functions we did not care about OPT at all § For work functions, our construction relied heavily on OPT § Are we going to use it now? We may compute it! For the algorithm: NO. For the analysis: YES. Marcin Bieńkowski A Toolbox for Online Algorithms 35

LP: dual program ``If you have an LP, write a dual program and stare LP: dual program ``If you have an LP, write a dual program and stare at it long enough’’ (quote from a random Stanford professor) Primal program (Pk) Dual program (Dk) min. : max. : subject to: (for all ) ) Strong duality theorem: OPT(Pk) = OPT(Dk) Marcin Bieńkowski A Toolbox for Online Algorithms 36

LP: online solution (1) We generate our online feasible solutions to primal and dual LP: online solution (1) We generate our online feasible solutions to primal and dual programs. Online = monotonic for analysis only Feasible dual solution is sometimes hard to achieve… Observation: ratio at most Marcin Bieńkowski implies competitive PROOF A Toolbox for Online Algorithms 37

LP: online solution (2) We generate: § feasible solutions to primal program § solutions 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. This one is feasible! Corollary: ratio at most Marcin Bieńkowski implies competitive A Toolbox for Online Algorithms 38

LP: the algorithm (1) minimize: Primal program (Pk) subject to: (for all maximize: ) LP: the algorithm (1) minimize: Primal program (Pk) subject to: (for all maximize: ) Dual program (Dk) subject to: (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 2: PROOF Observation 3: Each constraint of the dual solution is violated at most by 1: Generated primal solution is feasible Observationfactor O(log m) TRIVIAL PROOF Marcin Bieńkowski A Toolbox for Online Algorithms 39

LP: the algorithm (2) Observation 1: Generated primal solution is feasible Observation 2: Observation LP: the algorithm (2) Observation 1: Generated primal solution is feasible Observation 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 Marcin Bieńkowski A Toolbox for Online Algorithms 40

Toolbox Outline 1. Potential functions 2. Work functions 3. Linear programming 4. Classify and Toolbox Outline 1. Potential functions 2. Work functions 3. Linear programming 4. Classify and randomly select (CRS) Marcin Bieńkowski A Toolbox for Online Algorithms 41

CRS: a toy problem (1) Problem: Call admission § A graph, capacities of the CRS: a toy problem (1) Problem: Call admission § A graph, capacities of the edges. 3 2 2 3 1 Way too complicated, dear Watson 1 2 3 2 2 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 Marcin Bieńkowski A Toolbox for Online Algorithms 42

CRS: a toy problem (2) Problem: Call admission on line § A line graph 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 Marcin Bieńkowski A Toolbox for Online Algorithms 43

CRS: deterministic lower bound Lemma: Any deterministic algorithm on n-node line graph has the CRS: deterministic lower bound Lemma: Any deterministic algorithm on n-node line graph has the competitive ratio at least n-1. PROOF For a nontrivial competitive ratio, we need randomization! Marcin Bieńkowski A Toolbox for Online Algorithms 44

CRS: randomized solution (1) We assume that n = 2 k. We divide all CRS: randomized solution (1) We assume that n = 2 k. We divide all edges into k classes. i-th level call = contains edges from 2 -nd level call , 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. Marcin Bieńkowski A Toolbox for Online Algorithms PROOF 45

Thank you for your attention! Marcin Bieńkowski A Toolbox for Online Algorithms 46 Thank you for your attention! Marcin Bieńkowski A Toolbox for Online Algorithms 46