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

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

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

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  1. A Toolbox for Online AlgorithmsMarcin BieńkowskiUniversity of Wrocławphd open 2011

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

  3. What you need • are smart tools • and not smart problems • This course: • quite good tools • for very simple problems A Toolbox for Online Algorithms

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

  5. Online algorithms A Toolbox for Online Algorithms

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

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

  8. 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 A Toolbox for Online Algorithms

  9. 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? A Toolbox for Online Algorithms

  10. 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 A Toolbox for Online Algorithms

  11. 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 and this strategy is -competitive This is the best deterministic online algorithm A Toolbox for Online Algorithms

  12. 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! A Toolbox for Online Algorithms

  13. 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 A Toolbox for Online Algorithms

  14. Toolbox Outline • Potential functions (PF) • Work functions • Linear programming • Classify and randomly select A Toolbox for Online Algorithms

  15. PF: a toy problem Problem: List reorganization (singly-linked) 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. G H C F A D B J A Toolbox for Online Algorithms

  16. 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). A Toolbox for Online Algorithms

  17. 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 A Toolbox for Online Algorithms

  18. 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! A Toolbox for Online Algorithms

  19. PF: introduction to potential We want: Solution: take a potential function , such that 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 A Toolbox for Online Algorithms

  20. PF: states For many online problems, algorithms are in states and the cost depends on request and state only. State space: Rule of the thumb: Define as a constant times the cost of changing state from ALG to OPT. OPT ALG • 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, decreasing the potential! MTF moves x to the beginning of the list A Toolbox for Online Algorithms

  21. 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 A Toolbox for Online Algorithms

  22. Toolbox Outline • Potential functions • Work functions (WF) • Linear programming • Classify and randomly select A Toolbox for Online Algorithms

  23. 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 A Toolbox for Online Algorithms

  24. 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 Elementary, dear Watson a b A Toolbox for Online Algorithms

  25. 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 A Toolbox for Online Algorithms

  26. 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 A Toolbox for Online Algorithms

  27. 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 A Toolbox for Online Algorithms

  28. WF: work function evolution Definegradient Observation: . 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 7 6 For such configuration, request at a does not change the work function (or OPT cost) 5 D 4 3 2 1 0 Good guidance! Let’s extrapolate it to intermediate values! A Toolbox for Online Algorithms

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

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

  31. Toolbox Outline • Potential functions • Work functions • Linear programming (LP) • Classify and randomly select A Toolbox for Online Algorithms

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

  33. 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 1 2 4 2 4 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! A Toolbox for Online Algorithms

  34. 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 ) 1 2 4 2 4 A Toolbox for Online Algorithms

  35. 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. A Toolbox for Online Algorithms

  36. 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) • min.: • subject to: • (for all ) • (for all ) • Dual program (Dk) • max.: • subject to: • (for all ) • (for all ) Strong duality theorem: OPT(Pk) = OPT(Dk) A Toolbox for Online Algorithms

  37. 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: implies competitive ratio at most PROOF A Toolbox for Online Algorithms

  38. 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: implies competitive ratio at most A Toolbox for Online Algorithms

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

  40. 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 A Toolbox for Online Algorithms

  41. Toolbox Outline • Potential functions • Work functions • Linear programming • Classify and randomly select (CRS) A Toolbox for Online Algorithms

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

  43. 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 A Toolbox for Online Algorithms

  44. 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 A Toolbox for Online Algorithms

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

  46. Thank you for your attention! A Toolbox for Online Algorithms

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