Institute of Computer Science University of Wroclaw 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)

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 2 Data management problem How to store and manage data items in a network, so that arbitrary sequences of accesses to (parts of) data items can be served efficiently?

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 3  Build a large data center  Not scalable (building larger storage does not help)  Fixed place for data is always bad! Rich engineer’s solution

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 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

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 5  nodes in a metric space  One copy of one indivisible memory page of size at the local memory of one node  Each pair of nodes can communicate directly, cost of communication ~ distance Page Migration (1) v1v1 v2v2 v3v3 v4v4 v7v7 v6v6 v5v5

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 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 Input: sequence Output: sequence of page migrations minimizing total cost Decisions have to be made online! v1v1 v2v2 v3v3 v4v4 v7v7 v6v6 v5v5 cost = movement cost =

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 7 Page Migration (competitive analysis)  Input sequence is created by a request adversary  Performance metric = competitive analysis: competitive ratio  Previous research -> -competitive algorithms

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 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.

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 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

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 10 CF competitiveness (2)  Request occurs at  Assumption: OPT does not move the page

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 11 Page migration in static networks is EASY What about dynamic ones?

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 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!

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 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

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 14 Can any algorithm be O(1)-competitive in dynamic model? Not even close.

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 15 Lower bound for two nodes For the deterministic case:   time decision point Lower bound of Movement is fixed

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 16 Our results Deterministic algorithms competitive ratio = [SPAA 04, STACS 05, MFCS 05] Randomized algorithms competitive ratio = [SPAA 04, ESA 05]

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 17 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 at least

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 18 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

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 19 Analysis of MARK (1)  We define a potential function:  For each phase, we prove:  Fix any epoch  MARK is -competitive.

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 20 Nothing interesting here,  Consider, but with all nodes at positions from step  Gravity center (GC) – the node optimizing cost in  Jump set – a ball of diameter centered at GC For these nodes these nodes are marked MARK chooses a node from Jump set Analysis of MARK (2) Closer look at one phase : If MARK moves to GC … to other nodes from JumpSet AND nodes are moving

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 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. Epoch 1  In the worst case we still have phases  But on average –  In each phase worst-case bounds apply R-MARK is -competitive

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 22 Outlook Good news: we provided optimal algorithms Bad news: optimal competitive ratios grow with and some function of

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 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!

Institute of Computer Science University of Wroclaw Thank you for your attention!

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 25 Results on static page migration The best known bounds: 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]

Institute of Computer Science University of Wroclaw Page Migration in Dynamic Networks / M. Bienkowski 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