Slide 1

Yossi Azar Tel Aviv University

Joint work with Ilan Cohen Serving in the Dark1 1Oblivious AlgorithmsExperts all past information is knownCompetitive analysis the past is knownBandit only the value of the action taken chose is knownWhat if you do not see the input and do not get any feedback ?Can one design meaningful algorithms which do not see the input ?

2Visiting a Doctor

3Balls and Binsm2 balls to be assigned into m bins m out of the balls are of unit weight & all other balls are of zero weightGoal: minimize the max loadMax load of 1 is trivial

4Deterministic Oblivious AlgorithmsWhat if the algorithm does not see the weights (oblivious) ?Any deterministic algorithm: some bin has at least m ballsThey can all be of unit weight Max load is m (in contrast to 1)

5Randomized Oblivious AlgorithmsAssign balls uniformally at randomMaximum load is logm/loglogmAlgorithm is oblivious of the input (weights)6Scheduling on idenitical machinesm parallel identical machinesJobs arrive over time job i has weight wiEach job should be dispatched to some machineGoal: minimize the total flow timeClearly each machine should run SRPT7

Dispatching AlgorithmSend a job to a random machineconstant competitive for flow time [ChekuriGoelKhannaKumar, SchulzSkutella]Dispatching algorithm is oblivious (does not see the input)Processing jobs on machines are not oblivious (SRPT)8Routing in GraphsGiven undirected graph G=(V,E)Request (si , ti ) with demand diAllocate a path (flow) for request iLoad on edge = flow through edgeMinimize the maximum load Offline/Online ratios exact or log|V|9

Routing in GraphsWhat if the algorithm does not see the demands (oblivious) ?Routes between any two nodes should be chosen in advance (independent of demand) [Racke] seminal result: can be done up to log|V| factorPath = Randomized (Flow = deterministic)10Price of ObliviousnessRatio between serving in the dark (unknown input) vs optimal solution (in the light)11Returning back to Balls and BinsAlgorithm should extract balls from bins (instead of assign balls into bins)Unit balls arrive to arbitrary bins at arbitrary times

12Serving In The Dark GameB bins Sequence of Arrival & Extraction EventsArrivalnew ball is assigned to empty bin If all bins are full the ball is thrownExtraction Player chooses a bin, clears it and gains its contentGoal: max the gain = # of extracted ballsThe Player does not get any input during the sequence!1313

Serving Game14The Optimal AlgorithmThe optimal player, in each extraction step, chooses a bin with a ball (if one exists)

15

Serving In The Dark Game16The Competitive ratioPrice of obliviousnessWe compare (expected) # of extracted balls by the Player vs.an optimal player which serves in the light on the same sequence

17Deterministic AlgorithmsThe round-robin algorithm has a competitive ratio of 2-1/B2-1/B is a lower bound for any deterministic algorithm18

Serving Game Round Robin Player

Round Robin- Optimal-19Randomized AlgorithmsSuggestion: choose a bin uniformly at random at every extraction Better or worse than RoundRobin ?Competitive ratio about 1.69 [AzarCohenGamzu2013]Is there a better randomized algorithm?20New Results: Breaking the Uniformity! Order Based algorithm:Order bins by their last extraction eventExtraction: choose a bin according to a fixed non-decreasing probabilities, permuted according to the order above Note: extraction events may (or may not) have extracted a ball

21

ACDBEProbability:

`2222

CBDAEProbability:

`23Breaking the Uniformity!Round-Robin &Uniform are order basedWe provide probability distribution which achieves a competitive ratio of about 1.56Lower bound of 1.5 for randomized algorithm24Proof SketchAdversary places a ball in the bin whose extraction is least probableSequences for which the adversary has no underflow & no overflowFractional algorithm: extract a fraction from each bin (which sum up to 1)

25

Proof sketch:Grouping bins together:

Extraction Event:

26

Proof SketchDefine a deterministic fractional block process:

Extraction Event:

27Proof Sketch (fractional):Characterizing the worst sequence:Alg has overflows balls arrive until opt is fullNo overflow no balls arrives unless opt has underflowSequence defined up to one parameter the number of steps between two overflows! (which depend on the probability function)

28Proof SketchThe sequence between two overflows:

The number of 0s is B-1, the number of 1s depends on the probability function. Define , the expected number of balls extracted after d steps, starting with full load.The competitive ratio:

29Proof SketchAnalyze for B >> 1, and smooth p.Define as a portion of the load after steps, i.e. , we have

Where . The competitive ratio

30Proof SketchAnalyze the randomized versus the fractional:Analyze a single fractional block versus a single integer block.Define an hybrid process where the first blocks are integer and the rest are fractional. Replace the first fractional block by an integer one, prove that for sequences that would not overflow the first would not overflow the new one. 31SummaryAlgorithms can be oblivious and meaningfulCost of Serving in the dark 2 1.69 1.56Lower bound (randomized) = 1.5

32Open problemsIs the best bound 1.5 or is it larger ???Explore oblivious algorithmsCan we give an oblivious talk ?Can we write an oblivious paper ?33 Thanks34