Minimizing Clos Networks Alexander Martin and Peter Lietz Darmstadt - - PowerPoint PPT Presentation

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Minimizing Clos Networks Alexander Martin and Peter Lietz Darmstadt - - PowerPoint PPT Presentation

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M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS Minimizing Clos Networks Alexander Martin and Peter Lietz Darmstadt University of Technology Workshop on Combinatorial Optimization Aussois

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Minimizing Clos Networks

Alexander Martin and Peter Lietz

Darmstadt University of Technology

Workshop on “Combinatorial Optimization” Aussois January 9 to 13, 2006

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Motivation

1 4 5 8 9 12 13 16

Problem Given a graph, decide whether all demand patterns are

  • routable. If yes, route each pattern within strict time limits.
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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Motivation

5 1 4 8 9 12 13 16 2 7 1 5 2 3 1 5 3 10 9 6 3 9 10 13

Problem Given a graph, decide whether all demand patterns are

  • routable. If yes, route each pattern within strict time limits.
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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Motivation

1 4 5 8 9 12 13 16 2 5 5 1 2 3 1 5 3 10 9 3 3 9 10 13

Problem Given a graph, decide whether all demand patterns are

  • routable. If yes, route each pattern within strict time limits.
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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Motivation

5 1 4 8 9 12 13 16 3 3 6 6 3 3 13 16 2 12 13

Problem Given a graph, decide whether all demand patterns are

  • routable. If yes, route each pattern within strict time limits.
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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Motivation

5 1 4 8 9 12 13 16 3 3 6 6 3 3 13 16 2 12 13

Problem Given a graph, decide whether all demand patterns are

  • routable. If yes, route each pattern within strict time limits.
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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Naive Approach

  • Enumerate all patterns
  • Use a MIP solver to determine each routing
  • Total running time

for a 16 × 16 network: 1616 · 0.01 seconds ≈ 5.85 · 109 years for a 32 × 32 network: 3232 · 0.05 seconds ≈ 2.31 · 1035 years

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Naive Approach

  • Enumerate all patterns
  • Use a MIP solver to determine each routing
  • Total running time

for a 16 × 16 network: 1616 · 0.01 seconds ≈ 5.85 · 109 years for a 32 × 32 network: 3232 · 0.05 seconds ≈ 2.31 · 1035 years

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Definitions

1 4 5 8 9 12 13 16

Switching Network An N × M switching network is a directed graph together with a distinguished set of N vertices called inlets and a distinguished set of M vertices called outlets.

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Unicast Request

1 4 5 8 9 12 13 16 1 8 12 15 10 11 13 14 2 3 5 6 4 7 9 16

Definition A unicast request is a partial one-one function from the set of

  • utlets to the set of inlets.

Definition A routing for a unicast request a in a switching network G is a

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Unicast Request

1 4 5 8 9 12 13 16 1 8 12 15 11 13 14 2 3 5 6 4 7 10 16 9

Definition A routing for a unicast request a in a switching network G is a set of directed, vertex-disjoint paths such that a(w) = v iff w is the end of a path with beginning v.

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Multicast Request

5 1 4 8 9 12 13 16 2 7 1 5 2 3 1 5 3 10 9 6 3 9 10 13

Definition A multicast request is a partial function from the set of outlets to the set of inlets.

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Multicast Routing

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Definition A routing for a multicast request a in a switching network G is a set of vertex-disjoint directed Steiner trees such that a(w) = v iff w is a leaf of a tree with root v.

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Objective Function

Components of the network Multiplexer Switch Objective function Minimize the number of components subject to guarantee routability.

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Trivial Networks

Definition An N × M network is called trivial if each inlet is multiplexed into M nodes and each outlet is separately connected to N of these nodes belonging to mutually disjoint inlets. Number of Components N (M − 1) + M (N − 1)

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

A 16 × 32 Trivial Network

c

  • DEV Systemtechnik GmbH
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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Trivial Network with Test Station

c

  • DEV Systemtechnik GmbH
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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Clos Networks

Definition A (symmetric) Clos network C(n, r, m) is a network composed of trivial networks (called crossbars) arranged in three stages such that

  • stage 1 consists of r many

n × m crossbar,

  • stage 2 consists of m many

r × r crossbar,

  • stage 3 consists of r many

m × n crossbar,

  • every crossbar in stage i is

connected to every crossbar in stage i + 1 by exactly one link.

1 n 1 n 1 n 1 n

1 r 1 m 1 r

1 n 1 n 1 n 1 n 1 1 1 m r r 1 m

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Clos Networks

Definition A (symmetric) Clos network C(n, r, m) is a network composed of trivial networks (called crossbars) arranged in three stages such that

  • stage 1 consists of r many

n × m crossbar,

  • stage 2 consists of m many

r × r crossbar,

  • stage 3 consists of r many

m × n crossbar,

  • every crossbar in stage i is

connected to every crossbar in stage i + 1 by exactly one link.

C(4, 4, 4) C(4, 4, 5)

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Clos Networks

Remarks

  • An N × N trivial network is a C(1, N, 1) Clos network
  • The number of components of a Clos network are

|C(n, r, m)| = 2r(n(m − 1) + m(n − 1)) + m(2r(r − 1)) = 2r(m(2n + r − 2) − n)

  • Objective: For given n and r minimize |C(n, r, m)|, that is m

1 n 1 n 1 n 1 n

1 r 1 m 1 r

1 n 1 n 1 n 1 n 1 1 1 m r r 1 m

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Literature on Multicast Clos Networks

  • Charles Clos (1953)
  • Slepian-Duguid (1959)

m = n (unicast)

  • Masson & Jordan (1972)

m ≤ r · n

  • Hwang (1998)

m ≤ (n − 1)⌈log2 r⌉ + 2n − 1

For a 32 × 32 switching network: m ≤ 29

  • Hwang (2003): “A Survey on Nonblocking Multicast

Three-Stage Clos Networks” “... necessary and sufficient conditions for rearrangebly nonblocking are not known for model 0, ...”

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Mathematical Model of Clos Networks

Model Every request for a Clos network can be described by a binary matrix with r rows (= output crossbars), n · r columns (= inlets), arranged into r blocks of n columns, such that the sum of each row is less than or equal to n.

r 1 1 r 1 n 1 n 1 n 1 n 1 1 n 1 n n 1 n

1 1

1 6 7 8 2 6 7 8 3 6 7 8 5 6 7 8

r r

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Modelling Routability

Routability A given request is routable if and

  • nly if one can assign a color to

every nonzero entry of the matrix such that

  • 1. every color occurs at most
  • nce in each row,
  • 2. every color occurs in at most
  • ne column in each block.

1

r 1 r n 1 1 n 1 n 1 n n 1 n 1 n 1 n 1

m r r 1 1 1

1 6 7 8 2 6 7 8 3 6 7 8 5 6 7 8

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Critical Requests

Reduce the number of requests to be checked by

  • 1. applying mathematical theorems
  • 2. ignoring requests which are implied by harder requests
  • 3. restricting to one representative of each symmetry class

with respect to permutations of rows, permutations of columns within one block, and permutations of entire blocks

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Ad 1: Mathematical Theorems

K¨

  • nig’s edge coloring theorem

There must be one block that constains at least m + 1 nonzeros.

Proof Consider bipartite G = (A ∪ B, E) with A the set of rows and B the set of blocks. Each nonzero entry in the matrix yields one edge. A B We have

  • 1. deg(i) ≤ n for each i ∈ A,
  • 2. deg(j) ≤ m for each j ∈ B.

Then there exists an edge-coloring with at most m colors.

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Ad 1: Mathematical Theorems

Hall’s theorem There must be at least three different rows.

Proof Let G = (A ∪ B, E) be bipartite. A B To show |Γ(X)| ≥ |X| for all X ⊆ A.

  • 1. X is split over at least two blocks ⇒ |Γ(X)| = n
  • 2. Otherwise let l be the number of nodes in B outside that block.

Then |Γ(X)| ≥ l + max{n − l + |X| − n, 0} ≥ |X|

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Ad 2: Merge blocks

Merge blocks by matching columns of different blocks if the nonzeros do not intersect in some row.

  • ✁
✁ ✁ ✁ ✁ ✁ ✂ ✂ ✂ ✂ ✂ ✂ ✄ ✄ ✄ ✄ ✄ ✄ ☎ ☎ ☎ ☎ ☎ ☎ ✆ ✆ ✆ ✆ ✆ ✆ ✝ ✝ ✝ ✝ ✝ ✝ ✞ ✞ ✞ ✞ ✞ ✞ ✟ ✟ ✟ ✟ ✟ ✟ ✠ ✠ ✠ ✠ ✠ ✠ ✡ ✡ ✡ ✡ ✡ ✡ ☛ ☛ ☛ ☛ ☛ ☛ ☞ ☞ ☞ ☞ ☞ ☞ ✌ ✌ ✌ ✌ ✌ ✌ ✍ ✍ ✍ ✍ ✍ ✍ ✎ ✎ ✎ ✎ ✎ ✎
  • ✁
✁ ✁ ✁ ✁ ✁ ✂ ✂ ✂ ✂ ✂ ✂ ✄ ✄ ✄ ✄ ✄ ✄ ☎ ☎ ☎ ☎ ☎ ☎ ✆ ✆ ✆ ✆ ✆ ✆ ✝ ✝ ✝ ✝ ✝ ✝ ✞ ✞ ✞ ✞ ✞ ✞ ✟ ✟ ✟ ✟ ✟ ✟ ✠ ✠ ✠ ✠ ✠ ✠ ✡ ✡ ✡ ✡ ✡ ✡ ☛ ☛ ☛ ☛ ☛ ☛ ☞ ☞ ☞ ☞ ☞ ☞ ✌ ✌ ✌ ✌ ✌ ✌ ✍ ✍ ✍ ✍ ✍ ✍ ✎ ✎ ✎ ✎ ✎ ✎
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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Ad 3: Enumerate all critical requests

  • Define a lexicographical order on the set of binary matrices
  • Efficiently generate a set of minimal representatives of

each symmetry class with respect to permutations of rows, permutations of columns within one block, and permutations of entire blocks

  • Note: Properties 1 and 2 are invariant under the group

action Reduction for a 32 × 32 network 3232 = 2160 → 150346 critical instances in less than 2 hours.

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Ad 3: Enumerate all critical requests

  • Define a lexicographical order on the set of binary matrices
  • Efficiently generate a set of minimal representatives of

each symmetry class with respect to permutations of rows, permutations of columns within one block, and permutations of entire blocks

  • Note: Properties 1 and 2 are invariant under the group

action Reduction for a 32 × 32 network 3232 = 2160 → 150346 critical instances in less than 2 hours.

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

The Algorithm

Routing Algorithm

  • Backtracking

too slow

  • Formulating as a MIP (various models) and use CPLEX

in general fast with exceptions

  • Formulating as a SAT and use ZCHAFF

about 5 times faster than CPLEX

Overall Algorithm

  • Determine all critical cases
  • Store hard cases in a table
  • For the rest use SAT solver

       ⇒ fast routing time guaranteed

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

The Algorithm

Routing Algorithm

  • Backtracking

too slow

  • Formulating as a MIP (various models) and use CPLEX

in general fast with exceptions

  • Formulating as a SAT and use ZCHAFF

about 5 times faster than CPLEX

Overall Algorithm

  • Determine all critical cases
  • Store hard cases in a table
  • For the rest use SAT solver

       ⇒ fast routing time guaranteed

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

The Algorithm

Routing Algorithm

  • Backtracking

too slow

  • Formulating as a MIP (various models) and use CPLEX

in general fast with exceptions

  • Formulating as a SAT and use ZCHAFF

about 5 times faster than CPLEX

Overall Algorithm

  • Determine all critical cases
  • Store hard cases in a table
  • For the rest use SAT solver

       ⇒ fast routing time guaranteed

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MOTIVATION SWITCHING NETWORKS LITERATURE MATHEMATICAL MODEL SOLUTION APPROACH RESULTS

Results

Matrix n r m |C(n, r, m)| Remark 2 × 2 1 2 1 4 trivial 4 × 4 1 4 1 24 2 2 2 24 Hall 16 × 16 1 16 1 480 trivial 4 4 4 288 infeasible 4 5 4 420

  • pen

8 2 8 480 Hall 32 × 32 1 32 1 1984 trivial 8 4 10 1376 infeasible 8 4 11 1520 feasible 64 × 64 1 64 1 8064 trivial 32 2 32 8064 Hall ? ? ? ?