Constructions of Sparse Asymmetric Connectors Extended Abstract Andreas Baltz, Gerold J¨ ager, and Anand Srivastav Mathematisches Seminar, Christian-Albrechts-Universit¨ at zu Kiel, Christian-Albrechts-Platz 4, D-24118 Kiel, Germany {aba,gej,asr}@numerik.uni-kiel.de

Abstract. We consider the problem of connecting a set I of n inputs to a set O of N outputs (n ≤ N ) by as few edges as possible, such that for each injective mapping f : I → O there are n vertex disjoint paths from i to f (i) of length k for a given k ∈ . For k = Ω(log N + log2 n) Oru¸c [5] gave the presently best (n, N )-connector with O(N + n · log n) edges. For k = 2 we show by a probabilistic argument that an optimal (n, N )1 connector has Θ(N ) edges, if n ≤ N 2 −ε for some ε > 0. Moreover, we give explicit constructions based on a new number theoretic approach 1 3 1 that need O(N n 2 + N 2 n 2 ) edges.

N

1

Introduction

A major task in the design of communication networks is to establish sparse connections between n inputs and N outputs that allow all the inputs to send information to arbitrary distinct outputs simultaneously. In the usual graph model, the problem can be stated as follows.

N

Given n, N ∈ (n ≤ N ), construct a digraph G = (V, E), where V = I ∪ L ∪ O is partitioned into input vertices, link vertices and output vertices such that • |I| = n, |O| = N , • for every injective mapping f : I → O there are vertex disjoint paths connecting i to f (i) for all i ∈ I, • |E| is small, or even minimum. We call a digraph as above an (n, N )-connector (well-known in literature also as rearrangeable network, permutation network and (N, n)-permuter). An (n, N, k)connector (or (n, N )-connector of depth k) is an (n, N )-connector where any output can be reached by any input via a path of length k. Let e(n, N, k) denote the minimum number of edges sufficient for building an (n, N, k)-connector.

Previous Work The size of e(n, N, k) in the symmetric case N = n is well-studied. Pippenger and Yao [8] proved that e(n, n, k) = Ω(n1+1/k ) and showed by a probabilistic

argument that e(n, n, k) = O(n1+1/k (log n)1/k ). The best explicit construction of sparse symmetric connectors with odd depth is also due to Pippenger [7] who showed how to build (n, n, 2j + 1)-connectors with O(n1+1/(j+1) ) edges. Hwang and Richards [4] gave a construction of an (n, n, 2)-connector with O(n5/3 ) edges that can be extended by Pippenger’s method to yield (n, n, 2j)-connectors with O(n1+2/(3j−1) ) edges (j ≥ 2) [3]. Less results are known for asymmetric connectors. Oru¸c [5] was the first who devised (n, N )-connectors for arbitrary n and N . In particular, he gave constructions of (n, N, Ω(log2 N + log2 n))-connectors and (n, N, Ω(log2 N +log22 n))-connectors with O((N +n) log2 n) and O(N +n log2 n) edges, respectively, relying on recursive Clos networks [1] and concentrators as building blocks. A weak lower bound on e(n, N, k) can be obtained from the minimal crosspoint complexity of sparse crossbar concentrators. A sparse (a, b)-crossbar of capacity c connects an a-element vertex set A with a set B of b vertices in such a way, that every c-element subset of A can be perfectly matched into B. Oru¸c and Huang [6] proved that the minimum number of crosspoints (=edges) in a cascade of k sparse crossbars with capacity c establishing paths of length k between an a-element and a b-element set is at least !' & 1/k a−c+1 −1 . kb + a − b + (c − 1)k b−c+1 Since an (n, N )-connector of depth k is such a k-cascade with a = N and b = c = n we have the following bound: m l e(n, N, k) ≥ N − n + k + k(n − 1)(N − n + 1)1/k

(1)

Oru¸c and Huang showed by an explicit construction that √ for k = 1 their bound is attainable (within a factor of 2) when a − b ≤ c ≤ b. For other choices of parameters the given bound is unlikely to be tight. Our Results We are interested in (n, N, k)-connectors with constant k and n l ≥ |Γ (S)| ≥ |Γ (Γ (y) ∩ S)| ≥ (d + 1)(d − 1) + 1 = d2 , √ contradicting d ≥ n.

Moreover, we can derive a lower bound on the cardinality of L: j k Condition (3) implies that each y ∈ L is the joint neighbor of at most |L|−1 d−1 vertices from O (we count the number of possible completions of y to a d-element neighbor set). So we have k j ≥ Nd |L| |L|−1 d−1 giving

'

|L| ≥

&

1 + 2

r

1 + N (d2 − d) 4

≥

&

1 + 2

r

' √ 1 . +N n− n 4

Interestingly, this bound is attained by combinatorial designs. Definition 1 ([2]). Let l ≥ d ≥ t. A t-(l, d, λ)-design is a pair (L, B), where |L| = l and B ⊆ Ld , and each t-element subset of L is contained in λ elements of B. √ We are interested in 2-(l, n, 1)-designs.

√ Proposition 2. If (L, B) is a 2-(l, n, 1)-design with |B| = N , then l = q √ 1 n). 4 + N (n −

1 2

+

Clearly, 2-(l, d, 1)-designs can exist only for certain choices of parameters. As we do not require each 2-element set to be contained in exactly, √ but in at most one element of B, we are actually looking for combinatorial 2-(l, n, 1)-packings. Definition 2 ([2]). Let l ≥ d ≥ t. A t-(l, d, λ)-packing is a pair (L, B), where |L| = l and B ⊆ Ld , and each t-element subset of L is contained in at most λ elements of B. √ Unfortunately, it is not known how to construct optimal 2-(l, n, 1)-packings for all values of l and n. However, the following number theoretic method approximates the above bound quite closely. The Diagonal Method 1. 2. 3. 4. 5.

Let O√ = {0, . . . , N − 1}. d := ⌈ n⌉. q := min{q ′ ∈ ≥√N | for all p ∈ {2, 3, . . . , d − 1} : p 6 |q ′ }. L := {0, . . . , qd − 1} (we assume L and O to be disjoint, though!). For x ∈ O let x = x1 + q · x2 (x1 , x2 ∈ {0, . . . , q − 1}) be the q-ary decomposition of x. Choose

N

Γ (x) := {(i − 1)(1 + x2 · d) + x1 · d

(mod qd) | i ∈ {1, . . . , d}}.

Theorem 2. The diagonal method yields connectors with (1 + o(1))(N n1/2 + 2N 1/2 n3/2 ) edges. Proof: We have to show that condition (3) is satisfied. Consider x, x′ ∈ O with |Γ (x) ∩ Γ (x′ )| ≥ 2. By step 5 we have (i1 − 1)(1 + x2 d) + x1 d ≡ (i1 − 1)(1 + x′2 d) + x′1 d (mod qd) (i2 − 1)(1 + x2 d) + x1 d ≡ (i2 − 1)(1 + x′2 d) + x′1 d (mod qd) for some i1 6= i2 ∈ {1, . . . , d}. Subtracting these relations we conclude qd | (i2 − i1 ) · (x2 − x′2 ) · d and thus q | (x2 − x′2 ),

since gcd(q, i2 − i1 ) = 1 by the choice of q. This implies x2 = x′2 and so x = x′ . The claim follows by Lemma 1 using that |E| = N d + |L|n√and the fact √ that √ q ≤ 2 N (by Bertrand’s theorem there is a prime between N and 2 N , see [9]).

Remark 1. The diagonal method’s name reflects the fact that the neighborhoods in step 5 can be obtained as rows of the following matrices: arrange the set {0, . . . , qd − 1} into a q × d matrix; build a second matrix by taking diagonals of the first matrix; arrange the diagonals of the second matrix into a third matrix; etc.

0

1

2

3

0

5

10

15

4

5

6

7

4

9

14

19

8

9

10

11

8

13

18

3

12

13

14

15

12

17

2

7

16

17

18

19

16

1

6

11

0

9

18

7

0

13

6

19

4

13

2

11

4

17

10

3

8

17

6

15

8

1

14

7

12

1

10

19

12

5

18

11

16

5

14

3

16

9

2

15

0

17

14

11

4

1

18

15

8

5

2

19

12

9

6

3

16

13

10

7

Fig. 1. Example: d = 4, q = 5

We can slightly improve the construction by including d-element subsets of the “transpose” of one of the above matrices as additional neighborhoods, i.e. for

the example considered we can get the following additional sets: {0, 4, 8, 12}, {1, 5, 9, 13}, {2, 6, 10, 14}, {3, 7, 11, 15}. Remark 2. As mentioned above, the diagonal method can be improved in certain cases by constructing the neighborhoods as optimal combinatorial packings. We will refer to this construction as the packing variant of the diagonal method

5

Explicit Construction II: Copy Method

The following method assigns each element two neighbors in some small subset of L, and subsequently copies this neighborhood several times in order to satisfy the connector condition. For odd d an additional neighbor is added to the last copy. The Copy Method 1. 2. 3. 4. 5.

Let O√ = {0, . . . , N − 1}. d := l⌈ n⌉.m √ N . q := L := {0, . . . , qd − 1} (still, we assume L and O to be disjoint!). For x ∈ O let x = x1 + q · x2S(x1 , x2 ∈ {0, . . . , q − 1}) be the q-ary decomposition of x. Choose Γ (x) := i∈{0,...,⌊d/2−1⌋} Γi (x), where ( d−3 {x1 , x2 + q, (x1 + x2 ) (mod q) + 2q} + d−3 2 · 2q, if i = 2 , d odd, Γi (x) := {x1 , x2 + q} + i · 2q otherwise. To prove the connector condition we need the following lemma.

N

N N N

Lemma 2. Let l ∈ , S ⊆ 0 with |S| = l. √ a) Let f = (f1 , f2 ) : S → 20 be injective, then |f1 (S)| + |f2 (S)| ≥ 2 l. √ b) If f = (f1 , f2 , f3 ) : S → 30 is injective, |f1 (S)| + |f2 (S)| + |f3 (S)| ≥ 3 l. Proof: a) By the pigeon-hole principle, there is an x ∈ f1 (S) with |f1−1 (x)| ≥ As f is injective, we have |f2 (S)| ≥ and thus

l . |f1 (S)| l , |f1 (S)|

p p p √ |f1 (S)| + |f2 (S)| = ( |f1 (S)| − |f2 (S)|)2 + 2 |f1 (S)| · |f2 (S)| ≥ 2 l.

b) follows from a), since |f1 (S)|+|f2(S)|+|f3 (S)| = 12 (|f1 (S)|+|f2 (S)|+|f1(S)|+ |f3 (S)| + |f2 (S)| + |f3 (S)|).

Remark 3. The claim of Lemma 2 b) can be strengthened as follows: |f1 (S)| + |f2 (S)| + |f3 (S)| ≥

3 l √m · 2 l 2

This allows us to decrease d by 1 in certain cases (see the example (n, N ) = (28, 10000) in section 7). Theorem 3. The copy method yields connectors with (1 + o(1))(N n1/2 + N 1/2 · n3/2 ) edges. Proof: We have to show that the neighborhoods constructed by the copy method satisfy Hall’s condition. Let S ⊆ O with |S| ≤ n. The function f = (f1 , f2 ) : S →

N20, x 7→ (x1 , x2 ),

where x1 · q + x2 is the q-ary decomposition of x, is injective. By construction of Γ0 (x), and Lemma 2 a), |Γ0 (S)| = |f1 (S)| + |f2 (S)| ≥ 2 For d even, we obtain: |Γ (S)| =

d 2

· |Γ0 (S)| ≥

For d odd, we have: |Γ (S)| =

d−3 2

d 2

p |S|.

p · 2 |S| ≥ |S|.

p p · |Γ0 (S)| + |Γ d−3 (S)| ≥ (d − 3) |S| + 3 |S| ≥ |S|. 2

The number of edges can be estimated as in the proof of Theorem 2.

6

Iteration of the Methods

Our construction methods can be iterated to yield (n, N, k)-connectors for arbitrary k ≥ 2. Iteration of the Diagonal/Copy Method 1. Let k be the depth of the connector. 2. Let L0 := I and Lk := O. 3. For s = 0, . . . , k − 2 Apply the diagonal or copy method to n := |L0 | and N := |Lk−s |. Define Lk−s−1 := L. 4. Connect L0 and L1 by a complete bipartite graph.

We can reformulate the connector condition for depth k: Proposition 3. A digraph with vertices partitioned into sets L0 , L1 , . . . , Lk−1 , and Lk , |L0 | = n, |Lk | = N is an (n, N, k)-connector, if L0 and L1 are completely connected and |Γ (S) ∩ Li−1 | ≥ |S| for all i ∈ {1, . . . , k − 1} and all S ⊆ Li with |S| ≤ n. For connectors of depth k we can show by induction on k: s−1

s−1

Theorem 4. Let a0 := N 1/2 ·n3/2 − 1/2 for s = 1, 2, . . . , k − 1 and ak−s := 1/2k−1 2 − 1/2k−1 N ·n P Thenk − 1 iterations of the copy method yield connectors k−1 with (1 + o(1)) · s=0 as edges.

For the proof we refer to the full version.

Remark 4. We can get an analogous result for the diagonal method.

7

Comparison of the Methods

The following table compares the diagonal and copy method for several values of n and N . In addition we list values obtained from the packing version of the diagonal method (see Remark 2). Apart from the case k = 2, we include the trivial case k = 1 and the case k = 3 for comparison with the asymmetric 3-stage Clos network [1]. Additionally, we compare the results with the lower bound for e(n, N, k), see (1).

(n, N, k) (16, 2048, 1) (16, 2048, 2) (16, 2048, 3) (24, 6000, 1) (24, 6000, 2) (24, 6000, 3) (28, 10000, 1) (28, 10000, 2) (28, 10000, 3)

Diag 11200 10032 39480 34735 76968 68508

Copy 32768 11136 9824 144000 39360 34350 280000 64000 56500

Pack 10720 9544 38376 33785 75568 66588

Clos Low. bound – 32528 – 3387 16448 2606 – 143448 – 9535 58920 7232 – 279244 – 15367 106835 11719

We see that the Clos method approximately improves the trivial method of depth 1 by a factor of at least 2. The diagonal or copy method nearly gives an additional improvement of a factor of more than 1.5. The copy method is slightly superior to the diagonal method, as can be seen from the Theorems 2 and 3. In the last example the copy method yields better results due to the rounding effect described in Remark 3.

8

Open Questions

1. Can we derandomize the probabilistic proof of Theorem 1? 2. How can we construct a good (n, N, 2)-connector when |L| is prespecified? 3. What are efficient constructions for the multicast case where each of the n inputs is allowed to connect to r ≥ 2 specified outputs?

Acknowledgments We would like to thank Amnon Ta-Shma for the idea of the probabilistic proof of Theorem 1.

References 1. C. Clos, A study of non-blocking switching networks, Bell System Technical Journal, 32 (1953), 406–424. 2. C.J. Colbourn, J.H. Dinitz, The CRC Handbook of Combinatorial Designs, CRC Press, 1996. 3. D.Z. Du, H.Q. Ngo, Notes on the Complexity of Switching Networks, Advances in Switching Networks, Kluwer Academic Publishers (2000), 307–357. 4. F.K. Hwang, G.W. Richards, A two-stage network with dual partial concentrators, Networks, 23 (1992), 53–58. 5. A.Y. Oru¸c, A study of permutation networks: some generalizations and tradeoffs, Journal of Parallel and Distributed Computing (1994), 359–366. 6. A.Y. Oru¸c, H.M. Huang, Crosspoint complexity of sparse crossbar concentrators, IEEE Transactions on Information Theory, 42(5) (1996), 1466–1471. 7. N. Pippenger, On rearrangeable and nonblocking switching networks, Journal on Computer and System Sciences, 17 (1978), 145–162. 8. N. Pippenger, A.C. Yao, Rearrangeable networks with limited depth, SIAM Journal on Algebraic Discrete Methods, 3 (1982), 411–417. 9. H. Scheid, Zahlentheorie, BI Wissenschaftsverlag Mannheim/Wien/Z¨ urich, 1991.

Abstract. We consider the problem of connecting a set I of n inputs to a set O of N outputs (n ≤ N ) by as few edges as possible, such that for each injective mapping f : I → O there are n vertex disjoint paths from i to f (i) of length k for a given k ∈ . For k = Ω(log N + log2 n) Oru¸c [5] gave the presently best (n, N )-connector with O(N + n · log n) edges. For k = 2 we show by a probabilistic argument that an optimal (n, N )1 connector has Θ(N ) edges, if n ≤ N 2 −ε for some ε > 0. Moreover, we give explicit constructions based on a new number theoretic approach 1 3 1 that need O(N n 2 + N 2 n 2 ) edges.

N

1

Introduction

A major task in the design of communication networks is to establish sparse connections between n inputs and N outputs that allow all the inputs to send information to arbitrary distinct outputs simultaneously. In the usual graph model, the problem can be stated as follows.

N

Given n, N ∈ (n ≤ N ), construct a digraph G = (V, E), where V = I ∪ L ∪ O is partitioned into input vertices, link vertices and output vertices such that • |I| = n, |O| = N , • for every injective mapping f : I → O there are vertex disjoint paths connecting i to f (i) for all i ∈ I, • |E| is small, or even minimum. We call a digraph as above an (n, N )-connector (well-known in literature also as rearrangeable network, permutation network and (N, n)-permuter). An (n, N, k)connector (or (n, N )-connector of depth k) is an (n, N )-connector where any output can be reached by any input via a path of length k. Let e(n, N, k) denote the minimum number of edges sufficient for building an (n, N, k)-connector.

Previous Work The size of e(n, N, k) in the symmetric case N = n is well-studied. Pippenger and Yao [8] proved that e(n, n, k) = Ω(n1+1/k ) and showed by a probabilistic

argument that e(n, n, k) = O(n1+1/k (log n)1/k ). The best explicit construction of sparse symmetric connectors with odd depth is also due to Pippenger [7] who showed how to build (n, n, 2j + 1)-connectors with O(n1+1/(j+1) ) edges. Hwang and Richards [4] gave a construction of an (n, n, 2)-connector with O(n5/3 ) edges that can be extended by Pippenger’s method to yield (n, n, 2j)-connectors with O(n1+2/(3j−1) ) edges (j ≥ 2) [3]. Less results are known for asymmetric connectors. Oru¸c [5] was the first who devised (n, N )-connectors for arbitrary n and N . In particular, he gave constructions of (n, N, Ω(log2 N + log2 n))-connectors and (n, N, Ω(log2 N +log22 n))-connectors with O((N +n) log2 n) and O(N +n log2 n) edges, respectively, relying on recursive Clos networks [1] and concentrators as building blocks. A weak lower bound on e(n, N, k) can be obtained from the minimal crosspoint complexity of sparse crossbar concentrators. A sparse (a, b)-crossbar of capacity c connects an a-element vertex set A with a set B of b vertices in such a way, that every c-element subset of A can be perfectly matched into B. Oru¸c and Huang [6] proved that the minimum number of crosspoints (=edges) in a cascade of k sparse crossbars with capacity c establishing paths of length k between an a-element and a b-element set is at least !' & 1/k a−c+1 −1 . kb + a − b + (c − 1)k b−c+1 Since an (n, N )-connector of depth k is such a k-cascade with a = N and b = c = n we have the following bound: m l e(n, N, k) ≥ N − n + k + k(n − 1)(N − n + 1)1/k

(1)

Oru¸c and Huang showed by an explicit construction that √ for k = 1 their bound is attainable (within a factor of 2) when a − b ≤ c ≤ b. For other choices of parameters the given bound is unlikely to be tight. Our Results We are interested in (n, N, k)-connectors with constant k and n l ≥ |Γ (S)| ≥ |Γ (Γ (y) ∩ S)| ≥ (d + 1)(d − 1) + 1 = d2 , √ contradicting d ≥ n.

Moreover, we can derive a lower bound on the cardinality of L: j k Condition (3) implies that each y ∈ L is the joint neighbor of at most |L|−1 d−1 vertices from O (we count the number of possible completions of y to a d-element neighbor set). So we have k j ≥ Nd |L| |L|−1 d−1 giving

'

|L| ≥

&

1 + 2

r

1 + N (d2 − d) 4

≥

&

1 + 2

r

' √ 1 . +N n− n 4

Interestingly, this bound is attained by combinatorial designs. Definition 1 ([2]). Let l ≥ d ≥ t. A t-(l, d, λ)-design is a pair (L, B), where |L| = l and B ⊆ Ld , and each t-element subset of L is contained in λ elements of B. √ We are interested in 2-(l, n, 1)-designs.

√ Proposition 2. If (L, B) is a 2-(l, n, 1)-design with |B| = N , then l = q √ 1 n). 4 + N (n −

1 2

+

Clearly, 2-(l, d, 1)-designs can exist only for certain choices of parameters. As we do not require each 2-element set to be contained in exactly, √ but in at most one element of B, we are actually looking for combinatorial 2-(l, n, 1)-packings. Definition 2 ([2]). Let l ≥ d ≥ t. A t-(l, d, λ)-packing is a pair (L, B), where |L| = l and B ⊆ Ld , and each t-element subset of L is contained in at most λ elements of B. √ Unfortunately, it is not known how to construct optimal 2-(l, n, 1)-packings for all values of l and n. However, the following number theoretic method approximates the above bound quite closely. The Diagonal Method 1. 2. 3. 4. 5.

Let O√ = {0, . . . , N − 1}. d := ⌈ n⌉. q := min{q ′ ∈ ≥√N | for all p ∈ {2, 3, . . . , d − 1} : p 6 |q ′ }. L := {0, . . . , qd − 1} (we assume L and O to be disjoint, though!). For x ∈ O let x = x1 + q · x2 (x1 , x2 ∈ {0, . . . , q − 1}) be the q-ary decomposition of x. Choose

N

Γ (x) := {(i − 1)(1 + x2 · d) + x1 · d

(mod qd) | i ∈ {1, . . . , d}}.

Theorem 2. The diagonal method yields connectors with (1 + o(1))(N n1/2 + 2N 1/2 n3/2 ) edges. Proof: We have to show that condition (3) is satisfied. Consider x, x′ ∈ O with |Γ (x) ∩ Γ (x′ )| ≥ 2. By step 5 we have (i1 − 1)(1 + x2 d) + x1 d ≡ (i1 − 1)(1 + x′2 d) + x′1 d (mod qd) (i2 − 1)(1 + x2 d) + x1 d ≡ (i2 − 1)(1 + x′2 d) + x′1 d (mod qd) for some i1 6= i2 ∈ {1, . . . , d}. Subtracting these relations we conclude qd | (i2 − i1 ) · (x2 − x′2 ) · d and thus q | (x2 − x′2 ),

since gcd(q, i2 − i1 ) = 1 by the choice of q. This implies x2 = x′2 and so x = x′ . The claim follows by Lemma 1 using that |E| = N d + |L|n√and the fact √ that √ q ≤ 2 N (by Bertrand’s theorem there is a prime between N and 2 N , see [9]).

Remark 1. The diagonal method’s name reflects the fact that the neighborhoods in step 5 can be obtained as rows of the following matrices: arrange the set {0, . . . , qd − 1} into a q × d matrix; build a second matrix by taking diagonals of the first matrix; arrange the diagonals of the second matrix into a third matrix; etc.

0

1

2

3

0

5

10

15

4

5

6

7

4

9

14

19

8

9

10

11

8

13

18

3

12

13

14

15

12

17

2

7

16

17

18

19

16

1

6

11

0

9

18

7

0

13

6

19

4

13

2

11

4

17

10

3

8

17

6

15

8

1

14

7

12

1

10

19

12

5

18

11

16

5

14

3

16

9

2

15

0

17

14

11

4

1

18

15

8

5

2

19

12

9

6

3

16

13

10

7

Fig. 1. Example: d = 4, q = 5

We can slightly improve the construction by including d-element subsets of the “transpose” of one of the above matrices as additional neighborhoods, i.e. for

the example considered we can get the following additional sets: {0, 4, 8, 12}, {1, 5, 9, 13}, {2, 6, 10, 14}, {3, 7, 11, 15}. Remark 2. As mentioned above, the diagonal method can be improved in certain cases by constructing the neighborhoods as optimal combinatorial packings. We will refer to this construction as the packing variant of the diagonal method

5

Explicit Construction II: Copy Method

The following method assigns each element two neighbors in some small subset of L, and subsequently copies this neighborhood several times in order to satisfy the connector condition. For odd d an additional neighbor is added to the last copy. The Copy Method 1. 2. 3. 4. 5.

Let O√ = {0, . . . , N − 1}. d := l⌈ n⌉.m √ N . q := L := {0, . . . , qd − 1} (still, we assume L and O to be disjoint!). For x ∈ O let x = x1 + q · x2S(x1 , x2 ∈ {0, . . . , q − 1}) be the q-ary decomposition of x. Choose Γ (x) := i∈{0,...,⌊d/2−1⌋} Γi (x), where ( d−3 {x1 , x2 + q, (x1 + x2 ) (mod q) + 2q} + d−3 2 · 2q, if i = 2 , d odd, Γi (x) := {x1 , x2 + q} + i · 2q otherwise. To prove the connector condition we need the following lemma.

N

N N N

Lemma 2. Let l ∈ , S ⊆ 0 with |S| = l. √ a) Let f = (f1 , f2 ) : S → 20 be injective, then |f1 (S)| + |f2 (S)| ≥ 2 l. √ b) If f = (f1 , f2 , f3 ) : S → 30 is injective, |f1 (S)| + |f2 (S)| + |f3 (S)| ≥ 3 l. Proof: a) By the pigeon-hole principle, there is an x ∈ f1 (S) with |f1−1 (x)| ≥ As f is injective, we have |f2 (S)| ≥ and thus

l . |f1 (S)| l , |f1 (S)|

p p p √ |f1 (S)| + |f2 (S)| = ( |f1 (S)| − |f2 (S)|)2 + 2 |f1 (S)| · |f2 (S)| ≥ 2 l.

b) follows from a), since |f1 (S)|+|f2(S)|+|f3 (S)| = 12 (|f1 (S)|+|f2 (S)|+|f1(S)|+ |f3 (S)| + |f2 (S)| + |f3 (S)|).

Remark 3. The claim of Lemma 2 b) can be strengthened as follows: |f1 (S)| + |f2 (S)| + |f3 (S)| ≥

3 l √m · 2 l 2

This allows us to decrease d by 1 in certain cases (see the example (n, N ) = (28, 10000) in section 7). Theorem 3. The copy method yields connectors with (1 + o(1))(N n1/2 + N 1/2 · n3/2 ) edges. Proof: We have to show that the neighborhoods constructed by the copy method satisfy Hall’s condition. Let S ⊆ O with |S| ≤ n. The function f = (f1 , f2 ) : S →

N20, x 7→ (x1 , x2 ),

where x1 · q + x2 is the q-ary decomposition of x, is injective. By construction of Γ0 (x), and Lemma 2 a), |Γ0 (S)| = |f1 (S)| + |f2 (S)| ≥ 2 For d even, we obtain: |Γ (S)| =

d 2

· |Γ0 (S)| ≥

For d odd, we have: |Γ (S)| =

d−3 2

d 2

p |S|.

p · 2 |S| ≥ |S|.

p p · |Γ0 (S)| + |Γ d−3 (S)| ≥ (d − 3) |S| + 3 |S| ≥ |S|. 2

The number of edges can be estimated as in the proof of Theorem 2.

6

Iteration of the Methods

Our construction methods can be iterated to yield (n, N, k)-connectors for arbitrary k ≥ 2. Iteration of the Diagonal/Copy Method 1. Let k be the depth of the connector. 2. Let L0 := I and Lk := O. 3. For s = 0, . . . , k − 2 Apply the diagonal or copy method to n := |L0 | and N := |Lk−s |. Define Lk−s−1 := L. 4. Connect L0 and L1 by a complete bipartite graph.

We can reformulate the connector condition for depth k: Proposition 3. A digraph with vertices partitioned into sets L0 , L1 , . . . , Lk−1 , and Lk , |L0 | = n, |Lk | = N is an (n, N, k)-connector, if L0 and L1 are completely connected and |Γ (S) ∩ Li−1 | ≥ |S| for all i ∈ {1, . . . , k − 1} and all S ⊆ Li with |S| ≤ n. For connectors of depth k we can show by induction on k: s−1

s−1

Theorem 4. Let a0 := N 1/2 ·n3/2 − 1/2 for s = 1, 2, . . . , k − 1 and ak−s := 1/2k−1 2 − 1/2k−1 N ·n P Thenk − 1 iterations of the copy method yield connectors k−1 with (1 + o(1)) · s=0 as edges.

For the proof we refer to the full version.

Remark 4. We can get an analogous result for the diagonal method.

7

Comparison of the Methods

The following table compares the diagonal and copy method for several values of n and N . In addition we list values obtained from the packing version of the diagonal method (see Remark 2). Apart from the case k = 2, we include the trivial case k = 1 and the case k = 3 for comparison with the asymmetric 3-stage Clos network [1]. Additionally, we compare the results with the lower bound for e(n, N, k), see (1).

(n, N, k) (16, 2048, 1) (16, 2048, 2) (16, 2048, 3) (24, 6000, 1) (24, 6000, 2) (24, 6000, 3) (28, 10000, 1) (28, 10000, 2) (28, 10000, 3)

Diag 11200 10032 39480 34735 76968 68508

Copy 32768 11136 9824 144000 39360 34350 280000 64000 56500

Pack 10720 9544 38376 33785 75568 66588

Clos Low. bound – 32528 – 3387 16448 2606 – 143448 – 9535 58920 7232 – 279244 – 15367 106835 11719

We see that the Clos method approximately improves the trivial method of depth 1 by a factor of at least 2. The diagonal or copy method nearly gives an additional improvement of a factor of more than 1.5. The copy method is slightly superior to the diagonal method, as can be seen from the Theorems 2 and 3. In the last example the copy method yields better results due to the rounding effect described in Remark 3.

8

Open Questions

1. Can we derandomize the probabilistic proof of Theorem 1? 2. How can we construct a good (n, N, 2)-connector when |L| is prespecified? 3. What are efficient constructions for the multicast case where each of the n inputs is allowed to connect to r ≥ 2 specified outputs?

Acknowledgments We would like to thank Amnon Ta-Shma for the idea of the probabilistic proof of Theorem 1.

References 1. C. Clos, A study of non-blocking switching networks, Bell System Technical Journal, 32 (1953), 406–424. 2. C.J. Colbourn, J.H. Dinitz, The CRC Handbook of Combinatorial Designs, CRC Press, 1996. 3. D.Z. Du, H.Q. Ngo, Notes on the Complexity of Switching Networks, Advances in Switching Networks, Kluwer Academic Publishers (2000), 307–357. 4. F.K. Hwang, G.W. Richards, A two-stage network with dual partial concentrators, Networks, 23 (1992), 53–58. 5. A.Y. Oru¸c, A study of permutation networks: some generalizations and tradeoffs, Journal of Parallel and Distributed Computing (1994), 359–366. 6. A.Y. Oru¸c, H.M. Huang, Crosspoint complexity of sparse crossbar concentrators, IEEE Transactions on Information Theory, 42(5) (1996), 1466–1471. 7. N. Pippenger, On rearrangeable and nonblocking switching networks, Journal on Computer and System Sciences, 17 (1978), 145–162. 8. N. Pippenger, A.C. Yao, Rearrangeable networks with limited depth, SIAM Journal on Algebraic Discrete Methods, 3 (1982), 411–417. 9. H. Scheid, Zahlentheorie, BI Wissenschaftsverlag Mannheim/Wien/Z¨ urich, 1991.