D K H (1, D, s) by Duality in Software Compose Denso QR Bar Code in Software D K H (1, D, s) by Duality

D K H (1, D, s) by Duality use software qr barcode development toembed qr codes on software Planet The duality con Denso QR Bar Code for None struction described earlier for undirected hypergraphs is extended easily to the directed case. Starting with a Kautz digraph K (s, D 1), we obtain a dual directed Kautz hypergraph D K H (1, D, s) of order s D + s D 1 , gaining a factor of s in number of nodes while saving a factor of s in nodal degree. Figure 7.

20 illustrates the construction based on K (2, 3) and yielding D K H (1, 4, 2). The labeling convention for the undirected case is followed here as well. The assignment of hypergraph vertices to hyperarc in- or out-sets is inherited from the orientations of their dual arcs in the digraph and is indicated by arrows pointing in or out of each hyperarc.

For example, the vertices 2101, 0101, 1012, and 1010 in the hyperedge E 101 are duals of the four arcs incident on vertex 101 in the digraph. The rst two are in E 101 because their dual arcs are incident to vertex 101 in the digraph, and the last two are in + E 101 because their dual arcs are incident from vertex 101. This construction also suggests a multistar realization of D K H (1, D, s) based on K (s, D 1)21 (see Section 7.


The hypergraph D K H (1, D, s) can also be de ned directly, using alphabets, following the same general procedure as for K H (2, D , r ).. Multiwavelength Optical Networks 121 012. 212 120 201. K(2, 3) Duality 210 1210 121. 2101 101 012 2102. 0210 1212 2121 Software qr barcode 212 120 2120 0120 2012 1201 021 0212 1202 2020 2021 0202 1020 020 1021 0201 0102 2010 201 0121 1012 0101 1010. KH(2, 3, 4). Figure 7.19 Duality construction. Logically-Routed Networks 121 012. 212 120 201. K(2, 3) Duality 210 1210 121. 2101 101 012 21 Software QR Code JIS X 0510 02 0101 1010 2010 201 010. 0210 1212 2121 212 120 2120 0120 2012 1201 0201 2020 2021 202 1021 0202 1020 020 0121 1012. 0212 1202. D K H(1, 4, 2). Figure 7.20 Directed hypergraph construction via duality. Multiwavelength Optical Networks 121 012. 212 120 201. 010 Edge Grouping K(2, 3). 012 010. 202 021 D K H(1, 3, 2). 020 102. Figure 7.21 Directed hypergraph construction via edge grouping. Edge Grouping The duality con struction of Section 7.4.3.

1 is limited to undirected hypergraphs of degree 2 and directed hypergraphs of out-degree 1. For more general cases, we must turn to other techniques. One that is not limited in degree is edge grouping.

This technique can be used to construct a directed Kautz hypergraph from an underlying Kautz or generalized Kautz digraph. Using edge grouping, the vertex set of the derived hypergraph is the same as that of the underlying digraph..

Logically-Routed Networks The edge groupi ng construction was illustrated for Shuf eNets in Section 7.4.2.

Figure 7.21 shows an application of edge grouping to Kautz hypergraphs. The edges of the digraph K (2, 3) are grouped into dicliques, with each diclique replaced by a hyperarc in the resultant hypergraph D K H (1, 3, 2).

For example, in the diclique with the shaded vertices, the four arcs 201 010, 201 012, 101 012, and 101 + 010 correspond to the hyperarc E 01 , with in-set E 01 = {201, 101} and out-set E 01 = {010, 012}. Although this example is of degree 1, edge grouping also yields hypergraphs of higher degrees..

Generalized Kautz Hypergraphs The generalized Kautz hypergraphs G K H (d, n, s, m) were de ned in Section 6.5.7.

The objective there was to nd dense LCHs of diameter 1. These were implemented as purely optical multistar networks (LLNs) con gured to provide connectivity among all pairs of nodes in one logical hop. We use the same family of hypergraphs here in the more general context of LRNs.

The de nition is repeated here for convenience. Let n be the number of vertices and d be the vertex out-degree. Choose the number of hyperarcs m and the hyperarc out-size s such that dn 0 (mod m) sm 0 (mod n).

(7.31). The vertices ar Software QR Code e labeled as integers modulo n and the hyperarcs are labeled as integers modulo m. The incidence rules are as follows. Vertex v is incident to the hyperarcs e dv + (mod m), 0 <d (7.

32). and the out-set of the hyperarc e consists of the vertices u se (mod n), 1 s. (7.33).

The diameter of Denso QR Bar Code for None G K H (d, n, s, m) is at most logds n . If n = (ds) D + (ds) D k for a positive odd integer k, then the diameter is D. Thus for k = 1 and D = 2, the orders of these hypergraphs come within 1 of the Moore bound.

In general the in- and out-size of a hyperarc of G K H (d, n, s, m) will differ, as will the in- and out-degree of a vertex. However, there is a symmetric case, which is of particular interest here: The in- and out-size of all hyperarcs equals s, and the in- and out-degree of all vertices equals d if and only if dn = sm. To simplify the discussion that follows, we focus on the symmetric case of G K H , assuming henceforth that dn = sm.

Adding the condition that n = (ds) D + (ds) D 1 , G K H reduces to the directed Kautz hypergraph D K H (d, D, s), which has the maximum possible order for its diameter D. (Recall that D K H [1, D, s] was derived in Section 7.4.

3.1 using duality.) To illustrate, the tripartite representation of G K H (2, 42, 3, 28) is shown in Figure 7.

22. Because it obeys the previous symmetry conditions and has n = (ds)2 + ds, this is also D K H (2, 2, 3). Its order is 42 and the Moore bound for these parameters is 43.

Note how it forms a 28-fold multistar structure. Each star provides full connectivity.
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