1. International Graduate School of Dynamic Intelligent Systems, University of Paderborn Fighting Against Two Adversaries: Page Migration in Dynamic Networks Marcin Bieńkowski Mirosław Korzeniowski Friedhelm Meyer auf der Heide

2. 2 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Page Migration in Dynamic Networks  An online problem  processors in a metric space  Indivisible memory page of size in the local memory of one processor (initially at ) Page Migration v1v1 v2v2 v3v3 v4v4 v7v7 v6v6 v5v5

3. 3 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Page Migration in Dynamic Networks Page Migration (cont.)  Input: sequence of processor numbers, 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. v1v1 v2v2 v3v3 v4v4 v7v7 v6v6 v5v5

4. 4 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Page Migration in Dynamic Networks Page Migration (cost model) Cost model:  The page is at node.  Serving a request issued at costs.  Moving the page to node costs.  We measure the efficiency of an algorithm by standard competitive analysis – competitive ratio

5. 5 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Page Migration in Dynamic Networks Our contribution Extensions to the 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.  We design and analyse online algorithm for various scenarios imposing different restrictions on power of adversaries and their cooperation.

6. 6 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Page Migration in Dynamic Networks Dynamic Page Migration Input divided in rounds, in each round:  the network adversary can move each processor within a ball of diameter 1.  the requests adversary issues a request at one of the processors.  Servicing request cost  Moving page cost

7. 7 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Page Migration in Dynamic Networks Previous work  Algorithms achieving constant competitive ratios for Page Migration (PM) problem. AlgorithmLower bound Deterministic [Bartal, Charikar, Indyk ‘96][Chrobak, Larmore, Reingold, Westbrook ‘94] Randomized: Oblivious adversary [Westbrook ‘91][Chrobak, Larmore, Reingold, Westbrook ‘94] Randomized: Adaptive-online adversary [Westbrook ‘91]

8. 8 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Page Migration in Dynamic Networks Our results Different results for various scenarios 1) If network and request adversaries can cooperate, then the competitive ratio is: , if both adversaries are adaptive , if both adversaries are oblivious 2) Hybrid scenario: If we replace network adversary with a some random walk of nodes then for constant the competitive ratio is (on expectation).

9. 9 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Page Migration in Dynamic Networks Lower bound for oblivious scenario  For the deterministic case:    For the oblivious adversary case at the decision point we toss a coin. The proof follows from Yao min-max theorem. time decision point

10. 10 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Page Migration in Dynamic Networks Hybrid scenario Motivation: the network does not play against our algorithm.  The request adversary still chooses (obliviously, at the beginning) the requests sequence.  The initial positions of the processors are chosen by network adversary, then the changes are dictated by the random walk – each node performs a random walk on a -dimensional torus (or mesh) of diameter. For each dimension: ppb:

11. 11 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Page Migration in Dynamic Networks ALG MAJ : a deterministic algorithm which after each time steps moves to the node that issued majority of the requests in the last steps. Hybrid scenario (cont.) Performance measure: expected competitive ratio Theorem : For and constant number of nodes it is possible to construct an algorithm with expected competitive ratio of The proof sketch (for two processors and ).

12. 12 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Page Migration in Dynamic Networks Analysis of ALG MAJ We divide time into phases of length. 1)For each red interval we could find a hard subinterval of length such that 2) Two consecutive hard subintervals are separated by at least time steps  mixing  probabilities in (1) are (almost) independent. In the long run, we use Hoeffding bound to prove the ratio. time

13. 13 International Graduate School of Dynamic Intelligent Systems, University of Paderborn Page Migration in Dynamic Networks Extensions and future work  Explore the hybrid model; prove the similar bounds for other values of diameter. We can prove the expected competitive ratio of for (a ring or a linear array), constant number of nodes and any diameter.  Prove similar bound for any number of nodes.  Add the network dynamics to other data management problems like file allocation or distributed paging.

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