Improved Algorithms for Dynamic Page Migration

1. Improved Algorithms for Dynamic Page Migration • Marcin Bieńkowski • Mirosław Dynia • Mirosław Korzeniowski

2. Problem description • An online problem (of data management in a network) • processors in a metric space • One indivisible memory page of size in the local memory of one processor (initially at ) v3 v4 v2 v5 v1 v6 v7

3. Page Migration • Discrete time steps • Input: a sequence of processor numbers, dictated by an 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

4. Dynamic Page Migration Page migration, but additionally nodes are mobile • Input sequence: • denotes positions of all the nodes in step • The adversary can move each processor only within a ball of diameter 1 centered at the current position. • Configuration • Nodes are moved to configuration • Request is issued at • Algorithm serves the request • Algorithm (optionally) moves the page

5. Cost model Goal: Compute (online) a schedule of page movements to minimize total cost of communication Cost model: • The page is at node • Serving a request issued at costs . • Moving the page to node costs . Performance metric: We measure the efficiency of an algorithm by standard competitive analysis – competitive ratio

6. Previous work • For Page Migration there existed algorithms attaining competitive ratio (with almost matching lower bounds) Awerbuch, Bartal, Charikar, Chrobak, Indyk, Fiat, Larmore, Lund, Reingold, Westbrook, Yan, ... • For Dynamic Page Migration [BKM04]:

7. Our contribution New results for Dynamic Page Migration:

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

9. 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

10. Analysis of MARK (1) Technique: • We run OPT and MARK “in parallel” on an input sequence. • We define a potential in time step : • For each epoch we will prove: MARK is - competitive.

11. Analysis of MARK (2) Closer look at one phase : In all but last interval: • Lemma: Intuition: almost all requests are close to If is large at the end of , it means that is far away from , and thus far away from the requests.

12. Analysis of MARK (3) Closer look at one phase : 1 3 • We compute statistics in • Gravity center (GC) – the node optimizing communication cost if requests were issued at • Jump set – a ball of diameter centered at GC • Lemma: If node is outside jump set, then • In fact, MARK chooses some node from not marked nodes of jump set! 2 1

13. Analysis of MARK (4) If an algorithm at the end of phase moves to any node from jump set, then we can show: Crucial Lemma: (In the proof we use standard techniques from page migration algorithm analysis + worst-case analysis of node movement) • Each epoch has at most phases and

14. Randomized algorithm R-MARK MARK remains at one node till becomes marked, then it chooses not yet marked node and moves to . 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

15. Thank you for your attention.