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

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 January 9 to 13, 2006

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS 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.

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS Motivation 2 1 7 1 4 5 2 5 3 1 8 5 3 9 10 9 12 6 3 13 9 10 16 13 Problem Given a graph, decide whether all demand patterns are routable. If yes, route each pattern within strict time limits.

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS Motivation 2 1 5 1 4 5 2 5 3 1 8 5 3 9 10 9 12 3 3 13 9 10 16 13 Problem Given a graph, decide whether all demand patterns are routable. If yes, route each pattern within strict time limits.

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS Motivation 1 6 16 4 3 3 5 3 8 13 3 9 13 12 6 12 13 16 2 Problem Given a graph, decide whether all demand patterns are routable. If yes, route each pattern within strict time limits.

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS Motivation 1 6 16 4 3 3 5 3 8 13 3 9 13 12 6 12 13 16 2 Problem Given a graph, decide whether all demand patterns are routable. If yes, route each pattern within strict time limits.

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS Naive Approach • Enumerate all patterns • Use a MIP solver to determine each routing • Total running time for a 16 × 16 network: 16 16 · 0 . 01 seconds ≈ 5 . 85 · 10 9 years for a 32 × 32 network: 32 32 · 0 . 05 seconds ≈ 2 . 31 · 10 35 years

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS Naive Approach • Enumerate all patterns • Use a MIP solver to determine each routing • Total running time for a 16 × 16 network: 16 16 · 0 . 01 seconds ≈ 5 . 85 · 10 9 years for a 32 × 32 network: 32 32 · 0 . 05 seconds ≈ 2 . 31 · 10 35 years

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS 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 .

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS Unicast Request 1 1 8 12 4 15 5 10 11 13 8 14 9 2 3 5 12 6 13 4 7 9 16 16 Definition A unicast request is a partial one-one function from the set of outlets to the set of inlets. Definition A routing for a unicast request a in a switching network G is a

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS Unicast Request 1 1 8 12 4 15 5 10 11 13 8 14 9 2 3 5 12 6 13 4 7 9 16 16 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 .

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS Multicast Request 2 1 7 1 4 5 2 5 3 1 8 5 3 9 10 9 12 6 3 13 9 10 16 13 Definition A multicast request is a partial function from the set of outlets to the set of inlets.

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS Multicast Routing 2 1 5 1 4 5 2 5 3 1 8 5 3 9 10 9 12 3 3 13 9 10 16 13 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 .

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS Objective Function Components of the network Multiplexer Switch Objective function Minimize the number of components subject to guarantee routability.

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS 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)

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS A 16 × 32 Trivial Network � c DEV Systemtechnik GmbH

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS Trivial Network with Test Station c � DEV Systemtechnik GmbH

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS Clos Networks Definition A (symmetric) Clos network C ( n, r, m ) is a network composed of trivial networks (called crossbars) 1 1 1 1 1 arranged in three stages such that 1 1 r r 1 1 • stage 1 consists of r many n n m m n × m crossbar, 1 1 n n • stage 2 consists of m many r × r crossbar, 1 1 n n • stage 3 consists of r many m 1 1 m × n crossbar, r r n n • every crossbar in stage i is connected to every crossbar in stage i + 1 by exactly one link.

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS 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, C (4 , 4 , 4) • 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 C (4 , 4 , 5) stage i + 1 by exactly one link.

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS 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 ) | = 2 r ( n ( m − 1) + m ( n − 1)) + m (2 r ( r − 1)) = 2 r ( m (2 n + r − 2) − n ) • Objective: For given n and r minimize | C ( n, r, m ) | , that is m 1 1 1 1 1 1 1 r r 1 1 n n m m 1 1 n n 1 1 n n m 1 1 r r n n

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS 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) ⌈ log 2 r ⌉ + 2 n − 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, ...”

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS Mathematical Model of Clos Networks Model Every request for a Clos network can be described by a binary 1 1 1 1 6 matrix with r rows (= output 7 n 8 crossbars), n · r columns (= inlets), 2 1 arranged into r blocks of n 6 7 n 8 columns, such that the sum of each row is less than or equal to n . 3 1 6 7 n 8 1 r 5 1 1 6 r r 7 n 8 r 1 n 1 n 1 n 1 n

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS Modelling Routability Routability A given request is routable if and only if one can assign a color to 1 1 1 every nonzero entry of the matrix 6 1 7 n 8 1 such that 2 1. every color occurs at most 1 6 7 n 8 once in each row, 2. every color occurs in at most 3 1 6 7 one column in each block. n 8 m 5 1 r 1 6 r r 7 1 n 8 r 1 n 1 n 1 n 1 n

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS 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

M OTIVATION S WITCHING N ETWORKS L ITERATURE M ATHEMATICAL M ODEL S OLUTION A PPROACH R ESULTS Ad 1: Mathematical Theorems K¨ onig’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.