1. Efficient Bufferless Routing on Leveled Networks Costas Busch Shailesh Kelkar Malik Magdon-Ismail Rensselaer Polytechnic Institute

2. Talk Outline Introduction Centralized Algorithm Distributed Algorithm
Conclusion

3. Leveled Networks
Level: 1 2 3
L-1 L

4. Examples of Leveled Networks1 2 3 3 4 5 6 2 1
Butterfly
Mesh

5. Network Model Synchronous network (time steps) Bi-directional links
One packet per direction, per time step

6. Buffer-less nodes
Time 0
Packets are always moving

7. Buffer-less nodes
Time 1
Packets are always moving

8. Buffer-less nodes
Time 2
Packets are always moving

9. Buffer-less nodes
Time 3
Packets are always moving

10. Buffer-less nodes
Time 4
Packets are always moving

11. Bufferless routing is interesting:
Optical networks
Simple hardware implementations
Works well in practice:
Bartzis et al.: EUROPAR 2000
Maxemchuck: INFOCOM 1989

12. Routing Time:
the time until the last
packet is absorbed
Objective:
Minimize Routing Time

13. Each packet has a pre-selected path
source
destination
Packet path is from left to right

14. The packet follows the pre-selected path
source
destination

15. The packet follows the pre-selected path
source
destination

16. The packet follows the pre-selected path
source
destination

17. There are packets
Each packet has its own path

18. Dilation D: The maximum length
of any path
Routing time:

19. Congestion C: The maximum number of
packets traversing
any edge
Routing time:

20. Lower bound on Routing Time:
Congestion
Dilation
We want algorithms with Routing Time
close to:

21. Our Contributions Centralized Algorithm: Distributed Algorithm:
Both algorithms are randomized
Results hold with high probability
: number of packets

22. Related Work Networks with buffers Leveled networks:
Leighton, Maggs, Ranade, Rao: J. Algorithms 1992
Arbitrary networks:
[Leighton - Maggs - Rao, Combinatorica 94]
[BS99, LMR99, MV99, OR97, RT96]

23. Bufferless networks Mesh [BRST93, BES97, BHS98, BU96, BHW00]
Hypercube [BH85, BC95, FR92, H91]
Trees [BMMW04, RSW00, BMMW]
Leveled [BBPRRS96, B02]
Vertex-symmetric [MS95]
Arbitrary networks [BMM04]

24. Most related work Arbitrary networks: Leveled Networks:
Busch, Magdon-Ismail, Mavronicolas WAOA’04
Leveled Networks:
Busch TCS’04
Leveled networks in different routing model:
Bhatt, Bilardi, Pucci, Ranade, Rosenberg, Scwabe TC’96

25. Talk Outline Introduction Centralized Algorithm Distributed Algorithm
Conclusion

26. Centralized Algorithm
A central node knows all the
parameters of the problem
and computes a packet schedule

28. Packet Grouping A Packet Grouping B Group 1 Group 2 Group 3 Group 4

29. Send packets of Grouping A Send packets of Grouping B
Increases routing time by only a factor of 2
Packet Grouping A
Group 1
Group 2
Group 3
Packets in different groups
can be sent simultaneously
Group 1
We focus only in one group

30. Group 1
set of packets
Congestion
Dilation

31. Partition the packets randomly and uniformly into sets
#of packets

32. Benefit: congestion drops
packets
New Congestion
w.h.p.

33. Before partitioning
Edge
Congestion

34. After partitioning
Expected one packet
from each packet set

35. Expected Congestion 1
(Congestion w.h.p.)

36. We partition the levels into frames
# number of frames:

37. We send packets from frame to frame
Wave

38. We send packets from frame to frame
Wave Duration:
Wave

39. We send packets from frame to frame
Wave

40. We send packets from frame to frame
Wave

41. We send packets from frame to frame
Wave

42. A packet follows its path from source to Destination along the wave
Injection
wave
A packet follows its path from source to
Destination along the wave

43. A packet follows its path from source to
Destination along the wave

44. A packet follows its path from source to
Destination along the wave

45. A packet follows its path from source to Destination along the wave
Absorption
wave
A packet follows its path from source to
Destination along the wave

46. A packet follows its path from source to Destination along the wave
Absorption
wave
A packet follows its path from source to
Destination along the wave

47. Sending packets of different packet sets simultaneously
Wave 1

48. Sending packets of different packet sets simultaneously
Wave 1

49. Sending packets of different packet sets simultaneously
Wave 2
Wave 1

50. Sending packets of different packet sets simultaneously
Wave 2
Wave 1

51. Sending packets of different packet sets simultaneously
Wave 3
Wave 2
Wave 1

52. Sending packets of different packet sets simultaneously
Wave 3
Wave 2
Wave 1

53. Sending packets of different packet sets simultaneously
Wave C
Wave 3
Wave 2

54. Sending packets of different packet sets simultaneously
Wave C
Wave 3
Wave 2

55. Sending packets of different packet sets simultaneously
Wave C
Wave 3

56. Sending packets of different packet sets simultaneously
Wave C
Wave 3

57. Sending packets of different packet sets simultaneously
Wave C

58. Sending packets of different packet sets simultaneously
Wave C

59. Sending packets of different packet sets simultaneously
Wave C

60. All packets have been absorbed!

61. Routing Time = Time until last wave C leaves the network Time when
duration
Wave
duration
#frames
Time when
wave C enters
the network
Time that wave C needs
to traverse the network

62. Oscilation
Simulates buffering
Frame
Time

63. Oscilation
Simulates buffering
Frame
Time

64. Oscilation
Simulates buffering
Frame
Time

65. Oscilation
Simulates buffering
Frame
Time

66. Packet propagation during a wave
Frame
Frame
Wave

67. Packet propagation during a wave
Frame
Frame
Wave

68. Conflict Graph Each node is a packet Two packets are adjacent
if their paths use a common edge in
the frames and

69. Example
Frame
Frame
Share edge
Conflict graph

70. Thus the conflict graph can be colored with
The degree of any node
is bounded by 1 3
(A consequence of packet partitioning) 2 2 1 1 3 2 2 1
Thus the conflict graph
can be colored with
colors

71. We send packets of each color seperately Frame Frame3 3 1 2 2 2 1 1 1 2 1 3
Wave

72. First send packets of color 1 Frame Frame3 3 1 1 1 2 2 2 1 1 1 1 1 2 1 3 1
Wave
Packet paths don’t conflict
Time needed:

73. Similarly, send packets of color 2 Frame Frame3 3 2 1 2 1 2 2 2 1 2 1 2 2 3 1
Wave
Packet paths don’t conflict

74. Similarly, send packets of color 3 Frame Frame2 1 2 1 3 1 2 1 2 3 3 1
Wave
Packet paths don’t conflict

75. All packets have been delivered Frame Frame3 2 1 2 1 3 1 2 1 2 3 1
Wave

76. Wave time:
Colors X 2 Frame size
We can speed up the process
by pipelining different colors:

77. Pipelining using Boats Frame Frame3 3 1 2 2 2 1 1 1 2 1 3
Boat 1
Packets of color follow boat

78. Pipelining using Boats Frame Frame3 3 1 2 2 2 1 1 1 2 1 3
Boat 1
Packets of color follow boat

79. Pipelining using Boats Frame Frame3 3 1 2 2 2 1 1 1 2 1 3
Boat 1
Packets of color follow boat

80. Pipelining using Boats Frame Frame3 3 1 2 2 2 1 1 1 2 3 1
Boat 1
deflected
Packets of color follow boat

81. Pipelining using Boats Frame Frame3 3 1 2 2 2 1 1 1 2 3 1
Back
In position
Boat 2
Boat 1
Packets of color follow boat

82. Pipelining using Boats Frame Frame3 1 3 2 2 1 2 1 2 1 3 1
Boat 2
Boat 1
Packets of color follow boat

83. Pipelining using Boats Frame Frame1 3 3 2 2 2 2 3
Boat 2
Boat 1
Packets of color follow boat

84. Pipelining using Boats Frame Frame1 3 2 3 2 2 2 3
Boat 2
Boat 1
Packets of color follow boat

85. Pipelining using Boats Frame Frame1 3 2 3 2 2 2 3
Boat 3
Boat 2
Boat 1
Packets of color follow boat

86. Pipelining using Boats Frame Frame1 3 3 2 2 2 3 2
Boat 3
Boat 2
Boat 1
Packets of color follow boat

87. Pipelining using Boats Frame Frame2 1 3 3 3
Boat 3
Boat 2
Boat 1
Packets of color follow boat

88. Pipelining using Boats Frame Frame2 1 3 3 3
Boat 3
Boat 2
Boat 1
Packets of color follow boat

89. Pipelining using Boats Frame Frame2 3 1 3 3
Boat 3
Boat 2
Packets of color follow boat

90. Pipelining using Boats Frame Frame2 1 3 3 3
Boat 3
Boat 2
Packets of color follow boat

91. Pipelining using Boats Frame Frame3
Boat 3 1 2 2 2 2
Packets of color follow boat

92. Pipelining using Boats Frame Frame3
Boat 3 1 2 2 2 2
Packets of color follow boat

93. Pipelining using Boats Frame Frame1 3 2 2 3 2 3 2
Packets of color follow boat

94. Wave time:
Time until last boat reaches
target level
Number of colors
Frame size

95. Talk Outline Introduction Centralized Algorithm Distributed Algorithm
Conclusion

96. In the distributed version,
we assume that every node knows
parameters
Nodes do not know the packet paths,
except for the packets in them.

97. The distributed algorithm
is the same with the centralized,
except for one thing:
The conflict graph is colored in a
distributed manner

98. Distributed coloring – Basic Idea
During a wave:
each packet chooses a random color
Between 0 and 2log(DN)
2. each packet assumes the color is
correct and follows the respective boat
3. For packets that conflict, the
process repeats

99. This process repeats a logarithmic
number of times, thus it gives an extra
logarithmic factor in the performance

100. Talk Outline Introduction Centralized Algorithm Distributed Algorithm
Conclusion

101. Conclusion Centralized algorithm Uses conflict graph coloring
Distributed Algorithm
Worse by a logarithmic factor
Implements distributed coloring