Online Routing on the Mesh and Offline Routing on the Benes Network - - PowerPoint PPT Presentation

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Online Routing on the Mesh and Offline Routing on the Benes Network - - PowerPoint PPT Presentation

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Online Routing on the Mesh and Offline Routing on the Benes Network M. Protsenko Ferienakademie im Sarntal 2008 FAU Erlangen-Nrnberg, TU Mnchen, Uni Stuttgart September 2008 Overview Introduction 1 Parallel computation models Notation

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SLIDE 1

Online Routing on the Mesh and Offline Routing on the Benes Network

  • M. Protsenko

Ferienakademie im Sarntal 2008 FAU Erlangen-Nürnberg, TU München, Uni Stuttgart

September 2008

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SLIDE 2

Overview

1

Introduction Parallel computation models Notation and definitions

2

Permutation routing on the mesh networks Online routing on linear array Online routing on 2D array

3

Permutation networks Congestion in butterfly network Offline routing in beneˇ s network

  • M. Protsenko (Ferienakademie’08, Sarntal)

Routing on Mesh and Benes Networks

  • Sept. 2008

2 / 81

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SLIDE 3

Overview

1

Introduction Parallel computation models Notation and definitions

2

Permutation routing on the mesh networks Online routing on linear array Online routing on 2D array

3

Permutation networks Congestion in butterfly network Offline routing in beneˇ s network

  • M. Protsenko (Ferienakademie’08, Sarntal)

Routing on Mesh and Benes Networks

  • Sept. 2008

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SLIDE 4

Parallel machine

A parallel machine: a set of processors P = {P0, ..., Pn−1} a communication graph G = (P, E)

  • M. Protsenko (Ferienakademie’08, Sarntal)

Routing on Mesh and Benes Networks

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SLIDE 5

Overview

1

Introduction Parallel computation models Notation and definitions

2

Permutation routing on the mesh networks Online routing on linear array Online routing on 2D array

3

Permutation networks Congestion in butterfly network Offline routing in beneˇ s network

  • M. Protsenko (Ferienakademie’08, Sarntal)

Routing on Mesh and Benes Networks

  • Sept. 2008

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Notation

[n] := {0, ..., n − 1} bind(n): binary representation of n using d bits

  • E. g., bin4(3) = 0011

(a)n = k for a ∈ [n]d: base-n representation of k

  • E. g., (201)3 = 19
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Hamming distance

Let a = (ad−1, ..., a0), b = (bd−1, ..., b0) ∈ [n]d The Hamming distance between a and b: Hamming(a, b) :=

d−1

  • i=0

|ai − bi| Example: a = 01101 b = 10111 hamming(a, b) = 3

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Shortest path, diameter

G = (V, E) - graph and x, y ∈ V then the shortest path distG(x, y): the minimum number of edges in path between x and y; the diameter of G: diam(G) = max{distG(x, y)|x, y ∈ V} Example: distG(A, B) = 2, diam(G) = 2

  • M. Protsenko (Ferienakademie’08, Sarntal)

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SLIDE 9

Routing

We have: a network M = (P, E), P = [N] a function f : [N] × [p] → [N] messages x0,0, ..., x0,p−1, x1,0, ..., x1,p−1, ..., xN−1,0, ..., xN−1,p−1

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Function routing

Function routing M routs x0,0, ..., xN−1,p−1 according to f if:

in the beginning processor i stores the package (i, f(i, k), xi,k) in the end processor f(i,k) stores a copy of a message xi,k

If p=1: f is a permutation on [N] ⇒ permutation routing

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Routing protocol

A synchronous routing protocol for M consists of protocols for all processors i. Each processor i has a buffer to store packages. A routing step for every processor i: i chooses one package from his buffer i selects one of his neighbors j i sends the package to j All processors start the t-th routing step simultaneously

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Routing time & buffer size

The routing time is the number of routing steps The buffer size is the maximum amount of packages that can be stored in a buffer at the same time

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Definitions

Routing with preprocessing (also off-line routing): routing protocol depends on f At the beginning of routing some computation is performed depending on f to generate protocols for processors i routing without preprocessing (on-line routing): the routing protocol is independent from f

  • M. Protsenko (Ferienakademie’08, Sarntal)

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SLIDE 14

Mesh networks

The n-dimensional mesh with edge length n, M(n,d): Set of processors P = {a|a ∈ [n]d} Communication graph G = (P, E), E = {{a, b}|a, b ∈ [n]d, hamming(a, b) = 1} Edge in Dimension i: the edge connecting two vertices a and b with |ai − bi| = 1

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Mesh networks: examples

Figure: Some examples of M(n,d)

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Mesh networks: properties

Properties of M(n, d)

1 M(n, d) has nd nodes and dnd − dnd−1 edges. 2 dist(a, b) = hamming(a, b); 3 diam(M(n, d)) = (n − 1) · d 4 M(n, d) |{a|ai=l}∼

= M(n, d − 1) for d > 0 and fixed i, l

5 M(n, d) |{ib|i∈[n]}∼

= M(n, 1) for fixed b ∈ [n]d−1

  • M. Protsenko (Ferienakademie’08, Sarntal)

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SLIDE 17

Overview

1

Introduction Parallel computation models Notation and definitions

2

Permutation routing on the mesh networks Online routing on linear array Online routing on 2D array

3

Permutation networks Congestion in butterfly network Offline routing in beneˇ s network

  • M. Protsenko (Ferienakademie’08, Sarntal)

Routing on Mesh and Benes Networks

  • Sept. 2008

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Linear array

Online routing on linear array Permutation routing without preprocessing in M(n, 1) can be performed with routing time 2 · (n − 1) and buffer size 3

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Linear array

The algorithm works in two phases: 1st phase: send packages which destination is to the left 2nd phase: send all another packages

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Linear array

The buffer size is 3 because any processor i stores at most 3 packages:

it’s own package package addressed to it some other package that must be transferred

The routing time is obvious

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Example

1st phase:

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Example

1st phase:

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Example

1st phase:

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Routing on Mesh and Benes Networks

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SLIDE 24

Example

1st phase:

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Routing on Mesh and Benes Networks

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SLIDE 25

Example

1st phase:

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Routing on Mesh and Benes Networks

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SLIDE 26

Example

1st phase:

  • M. Protsenko (Ferienakademie’08, Sarntal)

Routing on Mesh and Benes Networks

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SLIDE 27

Example

2nd phase:

  • M. Protsenko (Ferienakademie’08, Sarntal)

Routing on Mesh and Benes Networks

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SLIDE 28

Example

2nd phase:

  • M. Protsenko (Ferienakademie’08, Sarntal)

Routing on Mesh and Benes Networks

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SLIDE 29

Overview

1

Introduction Parallel computation models Notation and definitions

2

Permutation routing on the mesh networks Online routing on linear array Online routing on 2D array

3

Permutation networks Congestion in butterfly network Offline routing in beneˇ s network

  • M. Protsenko (Ferienakademie’08, Sarntal)

Routing on Mesh and Benes Networks

  • Sept. 2008

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2D array

Online routing on 2D array Permutation routing without preprocessing in M(n, 2) can be performed with routing time at most 4 · (n − 1) and buffer size at most n

  • M. Protsenko (Ferienakademie’08, Sarntal)

Routing on Mesh and Benes Networks

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SLIDE 31

2D array

Routing in 2 phases:

routing in rows routing in columns

Packages with farthermost destination have higher priority Routing time for each dimension 2 · (n − 1) A node can become within first phase at most n packages

  • M. Protsenko (Ferienakademie’08, Sarntal)

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2D array: example

Rows:

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SLIDE 33

2D array: example

Rows:

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SLIDE 34

2D array: example

Rows:

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SLIDE 35

2D array: example

Rows:

  • M. Protsenko (Ferienakademie’08, Sarntal)

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SLIDE 36

2D array: example

Columns:

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2D array: example

Columns:

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SLIDE 38

2D array: example

Columns:

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Permutation network

The Permutation network (n,d)-PN: Processors set P = {(l, a)|l ∈ {−d, ..., −1, 1, ..., d}, a ∈ [n]d} Communication graph (P, E) The (2,d)-PN: Beneˇ s or Waksman network

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Butterfly network

The d-dimensional butterfly network BF(d) (2,d)-PN with reduced set of processors The levels: 0..d

  • M. Protsenko (Ferienakademie’08, Sarntal)

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SLIDE 41

Properties of permutation and butterfly networks

Properties of (n,d)-PN and BF(d) (a) (n,d)-PN has 2d · nd processors; (b) BF(d) has (d + 1) · 2d processors;

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Shortest path on butterfly network

For every processor (0, a) and (d, b) there is exact one shortest path from (0, a) to (d, b).

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Recursive decomposition of (n,d)-PN

Permutation networks have recursive structure.

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Overview

1

Introduction Parallel computation models Notation and definitions

2

Permutation routing on the mesh networks Online routing on linear array Online routing on 2D array

3

Permutation networks Congestion in butterfly network Offline routing in beneˇ s network

  • M. Protsenko (Ferienakademie’08, Sarntal)

Routing on Mesh and Benes Networks

  • Sept. 2008

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Bit reversal permutation

Example: bit reversal permutation This permutation reverses the bits of a number. E.g., 1011 ⇒ 1101

  • M. Protsenko (Ferienakademie’08, Sarntal)

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SLIDE 46

Bit reversal permutation on BF(4)

0000 ⇒ 0000

  • M. Protsenko (Ferienakademie’08, Sarntal)

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SLIDE 47

Bit reversal permutation on BF(4)

0001 ⇒ 1000

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SLIDE 48

Bit reversal permutation on BF(4)

0010 ⇒ 0100

  • M. Protsenko (Ferienakademie’08, Sarntal)

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SLIDE 49

Bit reversal permutation on BF(4)

0011 ⇒ 1100

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Overview

1

Introduction Parallel computation models Notation and definitions

2

Permutation routing on the mesh networks Online routing on linear array Online routing on 2D array

3

Permutation networks Congestion in butterfly network Offline routing in beneˇ s network

  • M. Protsenko (Ferienakademie’08, Sarntal)

Routing on Mesh and Benes Networks

  • Sept. 2008

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Disjoint paths in (2,d)-PN

For any permutation π there is in (2,d)-PN a set of disjoint paths from sources to sinks with length 2d − 1

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Disjoint paths in (2,d)-PN

Construct disjoint paths recursive (2,1)-PN is obviously (2,d)-PN consists of two (2,d-1)-PN-partitions Ensure that any two paths are connected to different sources and sinks of partitions ⇒ the job is done

  • M. Protsenko (Ferienakademie’08, Sarntal)

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SLIDE 53

Routing graph

For permutation π construct a routing graph G: vertices are paths numbered after their start points edges connect paths that can crossover on sinks or sources of partitions

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Type 1 edges

Pathes that potentially cross on sources: we connect with type 1 edges start nodes differ in most significant bit only each path is incident to exact one type 1 edge

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Type 2 edges

Pathes that potentially cross on sinks: we connect with type 2 edges end nodes differ in most significant bit only each path is incident to exact one type 2 edge

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Routing graph

Each vertex of routing graph is incident to

exact one type 1 edge exact one type 2 edge

⇒ Routing graph G is 2-regular Note: two vertices can be connected with both type 1 and type 2 edges

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Avoiding crossovers

How we avoid crossing of paths: color both partitions in different colors (0 and 1) routing graph:

assign to each vertex ( path ) one of two colors no two vertices with same color may be adjacent ⇒ (vertex coloring)

lay each path through partitions of same color

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Vertex coloring

Routing graph G is 2-vertex-colorable: start with some vertex, assign color 0 a adjacent vertex become color 1 vertex adjacent to this become color 0 and so on ...

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SLIDE 59

Vertex coloring

Routing graph G:

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Vertex coloring

Routing graph G:

  • M. Protsenko (Ferienakademie’08, Sarntal)

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SLIDE 61

Vertex coloring

Routing graph G:

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SLIDE 62

Vertex coloring

Routing graph G:

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SLIDE 63

Vertex coloring

Routing graph G:

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SLIDE 64

Vertex coloring

Routing graph G:

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SLIDE 65

Vertex coloring

Routing graph G:

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SLIDE 66

Vertex coloring

Routing graph G:

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SLIDE 67

Vertex coloring

Routing graph G:

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Result

Result: routing graph is 2-vertex-colorable ⇒ we can lay paths avoiding crossovers ⇒ we can route any permutation on (2,d)-PN without congestion ⇒ routing time: 2d-1 and buffer size: 1

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Example: permutation

Permutation: dec bin 0 → 4 000 → 100 1 → 1 001 → 001 2 → 0 010 → 000 3 → 3 011 → 011 4 → 2 100 → 010 5 → 6 101 → 110 6 → 5 110 → 101 7 → 7 111 → 111

  • M. Protsenko (Ferienakademie’08, Sarntal)

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Example: routing graph

Routing graph 000 → 100 001 → 001 010 → 000 011 → 011 100 → 010 101 → 110 110 → 101 111 → 111

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Example: vertex coloring

Vertex coloring 000 → 100 001 → 001 010 → 000 011 → 011 100 → 010 101 → 110 110 → 101 111 → 111

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Example: vertex coloring

Vertex coloring 000 → 100 001 → 001 010 → 000 011 → 011 100 → 010 101 → 110 110 → 101 111 → 111

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Example: vertex coloring

Vertex coloring 000 → 100 001 → 001 010 → 000 011 → 011 100 → 010 101 → 110 110 → 101 111 → 111

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Example: vertex coloring

Vertex coloring 000 → 100 001 → 001 010 → 000 011 → 011 100 → 010 101 → 110 110 → 101 111 → 111

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Example: vertex coloring

Vertex coloring 000 → 100 001 → 001 010 → 000 011 → 011 100 → 010 101 → 110 110 → 101 111 → 111

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Example: vertex coloring

Vertex coloring 000 → 100 001 → 001 010 → 000 011 → 011 100 → 010 101 → 110 110 → 101 111 → 111

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Example: vertex coloring

Vertex coloring 000 → 100 001 → 001 010 → 000 011 → 011 100 → 010 101 → 110 110 → 101 111 → 111

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Example: vertex coloring

Vertex coloring 000 → 100 001 → 001 010 → 000 011 → 011 100 → 010 101 → 110 110 → 101 111 → 111

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Example: partitions

Partitions 000 → 100 001 → 001 010 → 000 011 → 011 100 → 010 101 → 110 110 → 101 111 → 111

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Example: pathes

Pathes 000 → 100 001 → 001 010 → 000 011 → 011 100 → 010 101 → 110 110 → 101 111 → 111

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Conclusion

The permutation networks is a very important and wide used family of networks There is an efficient offline routing algorithm for permutation networks

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