Ressourcenmanagement in Netzwerken SS06 Vorl. 13,

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1 Ressourcenmanagement in Netzwerken SS06 Vorl. 13, Friedhelm Meyer auf der Heide Name hinzufügen 1

2 Prüfungstermine Dienstag, Montag, und Freitag, Bitte melden sie sich bis zum bzw bei Olaf Bonorden an. Name hinzufügen 2

3 Scenario II Networks have low bandwidth, global objects are small, access is fine grained. typical for parallel processor networks, partially also for the internet. bottleneck: link-congestion task: distribute global objects (maybe dynamically) among processors such that an application (sequence of read/write access to global variables) can be executed using small link-congestion storage overhead is small. - Exploit Locality - Name hinzufügen 3

4 Scenario II, models network Static Model: Dynamic Model: characteristics of application are represented by read and write frequencies. No à-priori-knowledge about application. Name hinzufügen 4

5 Basic Strategy Design strategy for trees Produce strategy for target-network by tree embedding Name hinzufügen 5

6 Dynamic Model Application: Sequence of read / write requests from processors to objects. Each processor decides solely based on its local knowledge. distributed online-strategy Goal: Develop strategy that produces only by a factor c more congestion than an optimal offline strategy. c-competitive strategy Name hinzufügen 6

7 Dynamic strategy for trees v writes to x : v creates (or updates) copy of x in v, and invalidates all other copies (consistency!) v reads x: v reads the closest copy of x and creates copies in every processor on the path back to v. (Remark: Data Tracking in trees is easy!) Name hinzufügen 7

8 Example and Analysis Consider phase write (v 0 ), read (v 1 ), read (v 2 ),..., read (v k-1 ), write (v k ) v 0 Name hinzufügen 8

9 Example and Analysis Consider phase write (v 0 ), read (v 1 ), read (v 2 ),..., read (v k-1 ), write (v k ) v 1 v 0 Name hinzufügen 9

10 Example and Analysis Consider phase write (v 0 ), read (v 1 ), read (v 2 ),..., read (v k-1 ), write (v k ) v 1 v 0 Name hinzufügen 10

11 Example and Analysis Consider phase write (v 0 ), read (v 1 ), read (v 2 ),..., read (v k-1 ), write (v k ) v 1 v 0 v 2 Name hinzufügen 11

12 Example and Analysis Consider phase write (v 0 ), read (v 1 ), read (v 2 ),..., read (v k-1 ), write (v k ) v 1 v 0 v 2 Name hinzufügen 12

13 Example and Analysis Consider phase write (v 0 ), read (v 1 ), read (v 2 ),..., read (v k-1 ), write (v k ) v 1 v 0 v 2 v 3 Name hinzufügen 13

14 Example and Analysis Consider phase write (v 0 ), read (v 1 ), read (v 2 ),..., read (v k-1 ), write (v k ) v 1 v k v 0 v 2 v 3 Name hinzufügen 14

15 Example and Analysis Consider phase write (v 0 ), read (v 1 ), read (v 2 ),..., read (v k-1 ), write (v k ) v 1 v k v 0 v 2 v 3 Each strategy has to use each link of the red subtree at least once. Our strategy uses each of these links at most three times. Strategy is 3-competitive for trees Name hinzufügen 15

16 Other networks Idea: Simulate suitable tree in target-network M. tree embedding: Goals: - small dilation (in order to reduce overall load) - randomized embedding (in order to reduce congestion) Goals contradict?!? Name hinzufügen 16

17 Tree embedding Randomized, locality preserving embedding! Example: nxn-mesh M v leaves: nodes of the mesh link-capacity: # links leaving the submesh Name hinzufügen 17

18 Result for meshes The static and dynamic strategy are O (log(n))-competitive in nxn-meshes, w.h.p. Finding an optimal static placement for several variable is NP-hard already on 3x3-meshes. Name hinzufügen 18

19 Name hinzufügen 19

20 Page Migration Model (1) Page migration Classical online problem processors connected by a network There are costs of communication associated with each edge. Cost of communication between pair of nodes = cost of the cheapest path between these nodes. Costs of communication fulfill the triangle inequality. Name hinzufügen 20

21 Page Migration Model (2) Alternative view: processors in a metric space Indivisible memory page of size in the local memory of one processor (initially at ) Name hinzufügen 21

22 Page Migration Model (3) Input: sequence of processors, 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. Name hinzufügen 22

23 Page Migration (cost model) Cost model: The page is at node. Serving a request issued at costs. Moving the page to node costs. Name hinzufügen 23

24 Page Migration (goal) Goal: Exploit the topological locality of the requests in order to compute a schedule of page movements to minimize the total cost of communication. Offline : simple optimization problem Online : standard competitive analysis competitive ratio Online randomized: Name hinzufügen 24

25 A randomized algorithm Memoryless coin-flipping algorithm CF [Westbrook 92] Theorem: CF is 3-competitive against an adaptive-online adversary Remark: This ratio is optimal against adaptive-online Adversary (may see the outcomes of the coinflips) Name hinzufügen 25

26 Proof of competitiveness of CF We run CF and OPT in parallel on the input sequence We define potential function There are two events to consider in each step Request occurs at a node, CF and OPT serve the requests, part 1 CF optionally moves the page OPT optionally moves the page } part 2 For each part separately, we prove that Name hinzufügen 26

27 Proof of competitiveness of CF Note: Thus the cancel out and we get the competitive ratio 3. Name hinzufügen 27

28 Competitiveness of CF, a step Page in and resp. Request occurs at CF and OPT serve the requests CF optionally moves the page to part 1 OPT optionally moves the page part 2 to Name hinzufügen 28

29 Competitiveness of CF part 1 Request occurs at Cost of serving requests: in CF :, in OPT : Expected cost of moving the page: Potential before: Exp. potential after: Exp. change of the potential: Name hinzufügen 29

30 Competitiveness of CF part 2 OPT moves to Name hinzufügen 30

31 Deterministic algorithm Algorithm Move-To-Min (MTM) [Awerbuch, Bartal, Fiat 93] Theorem: MTM is 7-competitive Remark: The currently best deterministic algorithm achieves competitive ratio of Name hinzufügen 31

32 Results on static page migration The best known bounds: Deterministic Randomized: Oblivious adversary Randomized: Adaptive-online adversary Algorithm [Bartal, Charikar, Indyk 96] [Westbrook 91] [Westbrook 91] Lower bound [Chrobak, Larmore, Reingold, Westbrook 94] [Chrobak, Larmore, Reingold, Westbrook 94] [Westbrook 91] Name hinzufügen 32

33 Ich wünsche Ihnen schöne Ferien und erfolgreiche Prüfungen! Name hinzufügen 33

34 Wir danken für Ihre Aufmerksamkeit! Heinz Nixdorf Institut & Institut für Informatik Fürstenallee Paderborn Tel.: / Fax: / mail@upb.de Name hinzufügen 34