Page Migration in Dynamic Networks

1. Page Migration in Dynamic Networks Marcin Bieńkowski Joint work with: Jarek Byrka (Centrum voor Wiskunde en Informatica, NL) Mirek Dynia (University of Paderborn, DE) Mirek Korzeniowski (Technical University of Wroclaw, PL) Friedhelm Meyer auf der Heide (University of Paderborn, DE)

2. Data management problem How to store and manage data items in a network, so that arbitrarysequencesof accesses to (parts of) data items can be served efficiently?

3. Rich engineer’s solution • Build a large data center • Not scalable (building larger storage does not help) • Fixed place for data is always bad!

4. Poor CS’s solution • Use the memory of the network nodes • Replicate and remove copies of data on demand • Use locality of requests Widely explored problem, many variants. A classical, most basic variant: Page Migration

5. Page Migration (1) v3 v4 v2 v5 v1 v6 v7 • nodes in a metric space • One copy of one indivisiblememory page of size at the local memory of one node • Each pair of nodes can communicate directly, cost of communication ~ distance

6. Page Migration (2) Problem: nodes want to access the shared object (page) In one step t: • wants to read / write one unit of data from the page • After serving a request an algorithm may optionally move the whole page to a new processor cost = Input: sequence Output: sequence of page migrations minimizing total cost Decisions have to be made online! v3 v2 v4 v5 v1 v6 v7 movement cost =

7. Page Migration (competitive analysis) • Input sequence is created by a request adversary • Performance metric = competitive analysis: competitive ratio • Previous research -> -competitive algorithms

8. Page migration: randomized algorithm Algorithm CF (coin-flipping) [Westbrook ‘92] Observation: CF exploits the locality of requests Theorem: CF is 3-competitive In each step after serving a request issued at , move page to with probability .

9. CF competitiveness (1) General idea • We run CF and OPT “in parallel” on the same input • Define a potential • In each step, we show

10. CF competitiveness (2) • Request occurs at • Assumption: OPT does not move the page

11. Page migration in static networks is EASY What about dynamic ones?

12. What network dynamics can we allow? • node failures? • link failures? OK, what is the weakest possible model of network changes? Allow small changes in the costs of communication no chance for algorithm! no chance for algorithm!

13. Page Migration in Dynamic Networks Page Migration, but with mobile nodes In one step t: • The network adversary may move each processor only within a ball of diameter 1 centered at the current position • Configuration in step t-1 • Nodes are moved • Request is issued at • Algorithm serves the request • Algorithm (optionally) moves the page

14. Can any algorithm be O(1)-competitive in dynamic model? Not even close.

15. Lower bound for twonodes For the deterministic case: Movement is fixed time decision point Lower bound of

16. Our results Deterministic algorithms competitive ratio = [SPAA 04, STACS 05, MFCS 05] Randomized algorithms competitive ratio = [SPAA 04, ESA 05]

17. Marking scheme • We divide input sequence into intervals of length . • Marking scheme: : a cost in current epoch of an algorithm which remains at If , then becomes marked Epoch ends when all nodes are marked Epoch 1 • Marking and epochs are independent from the algorithm • Any algorithm in one epoch has cost at least

18. Deterministic algorithm MARK MARK remains at one node till becomes marked, then it chooses not yet marked node and moves to . There are at most n phases in one epoch Phase 1 Phase 2 Phase 3 Phase 4 Epoch 1

19. Analysis of MARK (1) • We define a potential function: • For each phase , we prove: • Fix any epoch • MARK is -competitive.

20. Analysis of MARK (2) Closer look at one phase : • Consider , but with all nodes at positions from step • Gravity center (GC) – the nodeoptimizing cost in • Jump set – a ball of diameter centered at GC Nothing interesting here, For these nodes these nodes are marked MARK chooses a node from Jump set … to other nodes from JumpSet AND nodes are moving If MARK moves to GC

21. Randomized algorithm R-MARK R-MARK remains at one node till becomes marked, then it chooses randomly not yet marked node and moves to . • In the worst case we still have phases • But on average – • In each phase worst-case bounds apply Epoch 1 R-MARK is -competitive

22. Outlook Good news: we provided optimal algorithms Bad news: optimal competitive ratios grow with and some function of

23. Outlook (2) • Our weak model appeared to be very difficult: two adversaries (requests and network) fight against the online algorithm, and may even cooperate • Is it a realistic scenario? Probably not. How can we weaken the cooperation between adversaries? Possible solution: replace one of the adversaries by a stochastic process. Competitive ratios are greatly reduced!

24. Thank you for your attention!

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

26. Randomized algorithm for two nodes Algorithm EDGE [ -competitive ] In each step after serving a request issued at , move page to with probability , where function plot