Universal Bufferless Routing

Transcription

1 Universal Bufferless Routing Marios Mavronicolas Costas Busch University of Cyprus Rensselaer Polytechnic Inst. Malik Magdon Ismail Rensselaer Polytechnic Inst. 1

2 Routing Problem Arbitrary network Arbitrary set of packets 2

3 Network Model Synchronous network (time steps) One packet per time step Bi-directional links 3

4 Network Model Buffer-less nodes Time 0 Packets are always moving 4

5 Buffer-less nodes Time 1 Packets are always moving 5

6 Buffer-less nodes Time 2 Packets are always moving 6

7 Buffer-less nodes Time 3 Packets are always moving 7

8 Buffer-less nodes Time 4 Packets are always moving 8

9 Bufferless routing is interesting: Optical networks Simple hardware implementations Works well in practice: Bartzis et al.: EUROPAR 2000 Maxemchuck: INFOCOM

10 Routing Time: the time until the last Packet is absorbed Objective: Minimize Routing Time 10

11 Congestion C 3 Maximum number of packets that cross an edge 11

12 Dilation D 6 Maximum path length 12

13 Lower bound on Routing Time: ( C D) Congestion Dilation We want to find algorithms with routing time close to lower bound 13

14 Our contribution: A buffer-less algorithm with routing time O (( C D) log 3 ( n N )) nodes packets 14

15 Related Work For routing with buffers there exist optimal O ( C D) algorithms: [Leighton - Maggs - Rao, Combinatorica 94] [BS99, LMR99, MV99, OR97, RT96] What about buffer-less routing? 15

16 Related Work Buffer-less Routing has been studied for specific topologies Mesh [BRST93, BES97, BHS98, BU96, BHW00] Hypercube [BH85, BC95, FR92, H91] Trees [BMMW04, RSW00] Leveled [BBPRRS96, B02] Vertex-symmetric [MS95] No previous work for O ( C D) time on arbitrary networks 16

17 Presentation Outiline Algorithm Analysis Conclusion 17

18 Partition the Network into Regions Each edge belongs to only one region Regions share nodes 18

19 Construct Region Graph R 1 R 2 R1 R2 R 3 R 3 R 5 R 4 Region Graph R 5 R 4 Regions are connected if they share a node in the original graph 19

20 Translate Packet Paths R1 R2 R 1 R 2 R 3 R 3 R 5 R 4 Region Graph R 5 R 4 20

21 Translate Packet Paths R1 R2 R 1 R 2 R 3 R 3 R 5 R 4 Region Graph R 5 R 4 21

22 Execute a Buffered Algorithm on Region Graph Packet Buffer 22

23 Buffered Algorithm Simulate Buffered Algorithm R 1 R 2 R1 R2 R 3 R 3 R 5 R 4 R 5 R 4 Buffers No Buffers 23

24 Region Graph Original Graph Buffered Algorithm Bufferless Algorithm Action for a Packet Simulation Action 1 time step 1 phase Buffer in node R Circulare in region R i i Move from node Ri to R j Move from region R i to R j Inject/Absorb at node R i Inject/Absorb at source/destination in region R i 24

25 Packet Buffer Simulation- Circulation R 1 R 1 Follow repeatetly an Euler tour 25

26 Packet Buffer Simulation- Circulation R 1 R 1 Euler tour (abstract) 26

27 Packet Buffer Simulation- Circulation R 1 R 1 Multiple packets Pipeline packets on Euler Tour 27

28 Packet Move Simulation of move R 1 R 2 R1 R2 Packet moves to new region when it reaches common node 28

29 Packet Move Simulation of move R 1 R 2 R1 R2 Packet moves to new region when it reaches common node 29

30 Packet Move Simulation of move R 1 R 2 R1 R2 Packet moves to new region when it reaches common node 30

31 Packet Move Simulation of move R 1 R 2 R1 R2 multiple packets in R2 Insert packet when there is an R empty spot in 2 31

32 Packet Move Oscillate here until empty slot is found R 1 R 2 R1 R2 Slow Circulation Sending Mode Fast Circulation Receiving Mode 32

33 Color the Region Graph R1 R R R 5 R 4 Let k be the maximum color 33

34 Change Colors R1 R R R 5 R 4 Replace each color with 2logk bits b logk b b 2 1 b logk b 2 b 1 complement original color 34

35 Each phase consists of 2logk rounds At round i a node is in sending mode i if its color bit is equal to 1 35

36 Round 0 R1 R sending Round 1 R1 R R 3 12 R R 5 R 4 R 5 R 4 Round 2 R1 R Round 3 R1 R R 3 12 R 3 9 R 5 6 R

37 Ri R j There is a round that Ri R j is receiving is sending And vice-versa 37

38 Presentation Outiline Algorithm Analysis Conclusion 38

39 We give a buffered algorithm with Buffer size: 6log( n N ) Routing time: O C B D Time Steps n, C, D : #nodes, node congestion, dilation in region graph 39

40 We partition original graph so that each region has () edges Buffer size R 1 () edges R 1 40

41 It holds: n n C C Edges in region Edge congestion D D 41

42 Each simulation phase has duration 2 2log k O (log 3 ( n N )) duration of round number of rounds 42

43 43 Total Routing Time )) ( log ) (( ) ( log 3 3 N n D C O N n D B C O Number of phases Phase Duration

44 Presentation Outiline Algorithm Analysis Conclusion 44

45 We presented the first bufferless algorithm for arbitrary networks within logarithmic factors from optimal Future research: Improve logarithmic factors Maintain original paths 45