Ressourcenmanagement in Netzwerken SS06 Vorl. 12,

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1 Ressourcenmanagement in Netzwerken SS06 Vorl. 12, 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 Static model access frequencies: h r : processors x objects R h w : processors x objects R read write Goal: Efficiently compute placements of objects and routing paths, such that congestion is minimized. How many copies per object? Where? Remark: Optimal Solution for each object yields globally optimal solution. Name hinzufügen 6

7 Static strategy for trees T 2 V 1 T 1 V 2 g V d T d W T ) = ( h ( z) + h ( z)) ( [ = Congestion (g, v i ) ] i z If g gets copy: Congestion = T i r Place copy on center of gravity, i.e., on processor g that produces minimum congestion. easy to compute. w max i { w ( T )} i Name hinzufügen 7

8 Static strategy for trees g has copy. Should v get a copy? T 2 T 1 - Gain for Congestion on (v, v): h r (v) - Loss for Congestion on (v, v) : w h w (v) g with w = h (z) Z V w T i V v gets copy, if h r (v) > w h w (v) V Name hinzufügen 8

9 Static strategy for trees Result: For trees T the above greedy-strategy (applied separately to each object) yields optimal congestion on each link. Parallel time: O ( # objects + diam (T) * deg (T)) Name hinzufügen 9

10 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 (and by a factor m more storage per processor (m, c) competitive strategy ) Name hinzufügen 10

11 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 11

12 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 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 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 0 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 0 v 2 Name hinzufügen 15

16 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 16

17 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 17

18 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 18

19 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 19

20 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 20

21 Tree embedding Randomized, locality preserving embedding! Example: nxn-mesh M v leaves: nodes of the mesh link-capacity: # links leaving the submesh Name hinzufügen 21

22 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 22

23 Memory Bounds Our strategy may fail, if the processors have to obey memory bounds. Problem: We can no longer treat the global objects separately! Our approach: Use sequential caching strategy in each processor, e.g., LRU or RANDOM are (2,2)-competitive. Try to use our strategy as far as possible. Apply repair routines in case of ejected copies. Name hinzufügen 23

24 Wir danken für Ihre Aufmerksamkeit! Heinz Nixdorf Institut & Institut für Informatik Fürstenallee Paderborn Tel.: / Fax: / mail@upb.de Name hinzufügen 24