Algorithmic Aspects of Dynamic Intelligent Systems Part 3: Page migration in dynamic networks

1. Algorithmic Aspects of Dynamic Intelligent SystemsPart 3: Page migration in dynamic networks Friedhelm Meyer auf der Heide Joint work with Marcin Bienkowski

2. Data management in networks How to store data items in a network, so that arbitrary sequences of accesses to data items can be served efficiently? Widely explored basic online problem A classical, simple, basic variant: Page Migration in static networks New: Page Migration in dynamic networks

3. Page Migration in Static Networks

4. Page Migration Model (1) • processors connected by a network • Cost of communication between pair of nodes = cost of a cheapest path between these nodes. Costs of communication fulfill the triangle inequality. v3 v4 v2 v5 v1 v6 v7

5. Page Migration Model (2) Alternative view: • processors in a metric space • Indivisible memory page of size in the local memory of one processor (initially at ) v3 v4 v2 v5 v1 v6 v7

6. Page Migration Model (3) • Input: sequence of processors, dictated by a request adversary • : processor which wants to access (read or write) one unit of data from the memory page. • After serving a request an algorithm may move the page to a new processor. v3 v4 v2 v5 v1 v6 v7

7. Page Migration (cost model) Cost model: • The page is at node . • Serving a request issued at costs . • Moving the page to node costs .

8. Page Migration Offline : simple optimization problem (dynamic programming) Online : standard competitive analysis – competitive ratio Online randomized:

9. A randomized online algorithm Memoryless coin-flipping algorithm CF [Westbrook 91] Theorem: CF is 3-competitive against an adaptive-online adversary(may see the outcomes of the coinflips) In each step, after serving a request issued at , move page to with probability .

10. Results on static page migration The best known bounds:

11. Page Migration in Dynamic Networks e.g. in mobile ad-hoc networks or in static networks with varying communication bandwidth

12. The model (1) Extensions to the Page Migration model • We model page migration in dynamic networks, where both request sequence and network mobility come up online. • Request sequence is created by a request adversary and network mobility is given by a network adversary. • Various scenarios imposing different restrictions on power of adversaries and their cooperation.

13. The model (2) Page migration, but nodes are mobile • Input sequence: • denotes positions of all the nodes in step • The network adversary can move each processorwithin aball of diameter 1 centered at the current position. • Configuration • Nodes move to configuration • Request is issued at • Algorithm serves the request • Algorithm (optionally) moves the page

14. Cost model Cost model: • The page is at node • Serving a request issued at costs . • Moving the page to node costs . Offline: easy, dynamic programming

15. Static versus dynamic Can we achieve constant competitive ratio also in the dynamic model? No! Even not on a dynamic two-node network!

16. Lower bound for dynamic two-node network • For the deterministic case: • For the oblivious adversary case, at the decision point we toss a coin. time decision point Lower bound of

17. Results for Dynamic Page Migration B : Marcin Bienkowski

18. Randomized algorithm for two nodes Algorithm EDGE • Similar to Coin-Flipping, but probability of movement depends on the distance between two nodes In each step, after serving a request issued at , move page to with probability , where function plot:

19. Competitiveness of EDGE Theorem: EDGE is -competitive • We analyze two events separately (as in case of CF) • Nodes move, request is issued, EDGE and OPT serve the request, EDGE (possibly) moves the page 2. OPT (possibly) moves the page • We define the following potential function where

20. Proof of competitiveness of EDGE Let We show: Note: Thus the are telescopic and cancel out We get the competitive ratio

21. Analysis of EDGE (1) 1a. Request serving request

22. Analysis of EDGE (2) 1b. Request serving request

23. Analysis of EDGE (3) 1c. Request serving request

24. Analysis of EDGE (4) 1d. Request serving request

25. Analysis of EDGE (5) 2. OPT moves the page

26. 2-node networks summary • Algorithm EDGE achieves competitive ratio against adaptive-online adversary • Lower bound against oblivious adversary is EDGE is up to a constant factor optimal online algorithm. Can EDGE be extended to general networks?

27. Randomized algorithm for n nodes • Direct extension of EDGE does not work! • No algorithm which considers only nodes which issued requests as destinations for movescan be better than -competitive (against adaptive adversary).

28. Randomized algorithm for n nodes Algorithm DIST In each step, after serving a request issued at , choose a node uniformly at random from neighborhood of . With probability move the page to . Theorem: DIST is - competitive

29. Deterministic algorithm • … is much more complicated • … is also - competitive • … its „randomization“ is - competitive against oblivious adversaries

30. What did we learn? • Competitive ratio grows with and some function in , this is very much compared to the static case. • Why? We look at very strong models:two adversaries fight against the online algorithm, and may even cooperate! • This does not seem to reflect a realistic scenario! Weaken the power of the adversaries and their coordination! HOW??

31. Relaxation of the model Replace one of the adversaries by a stochastic process. A) Stochastic requests scenario Generate requests randomly with some given frequencies B) Brownian motion scenario Replace the adversarial description of the mobility by random walks of the nodes

32. Stochastic Requests Scenario • In each step is drawn uniformly and independently according to the probability distribution • The mobility is still dictated by an adversary! Performance metric: algorithm is -competitive with prob. if for all configuration sequences and all it holds that Theorem: There exists a (simple) algorithm, which achieves constant competitive ratio with high probability.

33. Brownian Motion Scenario (1) • The request adversary still chooses (obliviously, at the beginning) the requests sequence . • The initial positions of the processors are chosen by network adversary, then each node performs a random walk on a -dimensional torus (or mesh) of diameter . For each dimension: prob:

34. Brownian Motion Scenario (2) Performance metric: Algorithm is -competitive with probabality if there is a constant such that for all request sequences and all initial nodes positions it holds that Results: The competitive ratio is at most

35. Some future research directions • Extend results to file allocation (compare Bartal, Fiat, Rabani 95; Maggs, MadH, Vöcking, Westermann 97; MadH, Vöcking, Westermann 00) • Combine network dynamics and scheduling (compare Leonardi, Marchetti-Spaccamela, MadH 04) • Create more realistic models (that may allow two adversaries that do NOT cooperate), and prove results.

36. Thank you for your attention! Heinz Nixdorf Institute & Computer Science Institute University of Paderborn Fürstenallee 11 33102 Paderborn, Germany Tel.: +49 (0) 52 51/60 64 80 Fax: +49 (0) 52 51/62 64 82 E-Mail: fmadh@upb.de http://www.upb.de/cs/ag-madh