International Graduate School of Dynamic Intelligent Systems, University of Paderborn Improved Algorithms for Dynamic Page Migration Marcin Bieńkowski Mirosław Dynia Mirosław Korzeniowski

2 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Improved Algorithms for DPM / M. Bienkowski  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 ) Problem description v1v1 v2v2 v3v3 v4v4 v7v7 v6v6 v5v5

3 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Improved Algorithms for DPM / M. Bienkowski 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. v1v1 v2v2 v3v3 v4v4 v7v7 v6v6 v5v5

4 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Improved Algorithms for DPM / M. Bienkowski 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 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Improved Algorithms for DPM / M. Bienkowski 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 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Improved Algorithms for DPM / M. Bienkowski 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]: AlgorithmLower bound Deterministic: Randomized: Adaptive-online adversary Randomized: Oblivious adversary

7 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Improved Algorithms for DPM / M. Bienkowski Our contribution New results for Dynamic Page Migration: AlgorithmLower bound Deterministic: Randomized: Adaptive-online adversary Randomized: Oblivious adversary

8 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Improved Algorithms for DPM / M. Bienkowski Marking scheme  We divide input sequence into intervals of length.  Marking scheme: Epoch 1 : a cost in current epoch of an algorithm which remains at If, then becomes marked Epoch ends when all nodes are marked  Marking and epochs are independent from the algorithm  Any algorithm in one epoch has cost

9 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Improved Algorithms for DPM / M. Bienkowski Deterministic algorithm MARK MARK remains at one node till becomes marked, then it chooses not yet marked node and moves to. Epoch 1 Phase 1Phase 2Phase 3Phase 4 There are at most n phases in one epoch

10 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Improved Algorithms for DPM / M. Bienkowski 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 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Improved Algorithms for DPM / M. Bienkowski 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 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Improved Algorithms for DPM / M. Bienkowski Analysis of MARK (3) Closer look at one phase :  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!

13 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Improved Algorithms for DPM / M. Bienkowski 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 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Improved Algorithms for DPM / M. Bienkowski 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. Epoch 1  In the worst case we still have phases  But on average –  In each phase worst-case bounds apply R-MARK is -competitive

International Graduate School of Dynamic Intelligent Systems, University of Paderborn Thank you for your attention.