N.G. Bruijn de. A combinatorial problem. In Proceedings of the section of science, pages 758--764. Appl. Mathematical Science, Koninklije Nederlandse Academie van Wetenshapen, 1946.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(15) the parameter k varies between k = 1 (i.e. the direct neighbourhood) and k = M=2. Hence we need a host network which can handle all communication demands that occur in our algorithm. We will use a de Bruijn network which is known to have logarithmic diameter and a constant node degree ( Bru46] Moreover the de Bruijn network can simulate any other bounded degree network with at most logarithmic slowdown ( Lei92] The nodes of a de Bruijn graph are described by strings. The length n of the strings corresponds to the dimension of the de Bruijn graph. The range of the string elements ....

.... is given by V T = f(a m Gamma1 ; a 0 )j0 a i l i 8 0 i mg E T = f(a; b)j9i = 0 : m Gamma 1 : b i = a i 1) mod l i a j = b j 8 j 6= ig: While it is possible to embed 1d tori (Figure 13) which may be seen as hamiltonian cycles, in de Bruijn graphs with dilation one ( Bru46] the theoretical lower limit on the dilation for the embedding of 2d tori in de Bruijn graphs is given by O(log log N) BCH 88] An algorithm to determine hamiltonian cycles in de Bruijn (see Figure 13) graphs is given in [Bru46, Lei92] 4.2 Implementation results Using a hamiltonian ....

[Article contains additional citation context not shown here]

N.G. Bruijn de. A combinatorial problem. In Proceedings of the section of science, pages 758--764. Appl. Mathematical Science, Koninklije Nederlandse Academie van Wetenshapen, 1946.

....In case of an extremely unbalanced distribution of keys it can be advantageous to apply a key balancing step before the first local sort of SRS in order to obtain a balanced workload. When running SRS on a de Bruijn graph, it is possible to use a Hamiltonian path of the de Bruijn graph [24] for the even odd transposition step. For simplicity, consider the case of N keys on N = 2 processors. Let be a permutation of N keys such that i and (i 1) i = 0 : 2 Gamma 2, are neighbors in D 2 (n) Then SRS can be modified such that the odd even transposition sort is performed ....

N.G. Bruijn de , "A combinatorial problem ", in Proceedings of the section of science. Appl. Mathematical Science, 1946, pp. 758--764, Koninklije Nederlandse Academie van Wetenshapen.

....is said to be a free code if no two elements of the set have a common substring of length more than . Given n, the design problem implied in [1] is to construct the largest possible free code. A De Bruijn sequence of order is a cyclic sequence in which each possible mer occurs exactly once [7]. In [2] the authors observe that by parsing a De Bruijn sequence of order , an optimal free code, of size Fragments spanning the polymorphism sites for all the SNPs in the set are extracted. The different shapes denote different variants. Oligonucleotides complementary to the sequences ....

....of a meta string , our construction will ensure that each instance of the is paired with a different pattern in the bit string. For k 2 N , a binary De Bruijn sequence of order k is a cyclic binary sequence of length 2 k in which each possible substring of length k occurs exactly once [7]. Such sequences exist for all k 2 N and can be constructed in linear time. We assume that a fixed De Bruijn sequence of order k is given for each k 2 N . Reading this sequence once, starting from a specific offset i relative to a fixed origin position, we obtain a linear string, a linearization ....

N.G. de Bruijn. A combinatorial problem. Proc. Kon. Ned. Akad. v. Wetensch., 49:758--764, 1946.

....The (b,k) de Bruijn digraph, denoted by B(b; k) has vertices in the set of b adic blocks of length k, and arrows between overlapping blocks. Thus, p(1 : k) 7 q(1 : k) if and only if p(2 : k) q(1 : k 1) Note that B(b; k) is b regular for all k. The family B(b; k) was introduced by de Bruijn [Br] and was already used in the framework of normal numbers by Good [Go] An Eulerian path on B(b; k) de nes a Hamiltonian path in B(b; k 1) by identifying the arrows of the rst digraph with the vertices of the second digraph. We also associate 3 paths in B(b; k) to b adic blocks. The path (a(1 ....

N. G. de Bruijn, A combinatorial problem, Konink. Nederl. Akad. Wetersh. Afd. Naturuk. Eerste Reelss, A49 (1946), 758-764.

.... known that these sequences exist for all n and the standard proof shows that a de Bruijn sequence of order n corresponds to an Euler tour in the de Bruijn digraph whose vertices are the binary n tuples, with one edge (x 1 : x n Gamma1 ) x 2 : x n ) for every binary n tuple x 1 : x n [dB46, Mar34]. This result has been generalized for k ary n tuples [Fre82] for higher dimensions (de Bruijn tori) Coc88, FFMS85] and for k ary tori [HI95] It is known also, for any m satisfying n m 2 n , that there is a cyclic binary sequence of length m in which no n tuple appears more than once ....

N. G. de Bruijn. A combinatorial problem. Proc. Nederl. Akad. Wetensch., 49:758--764, 1946.

....words of order n is still open. Here the F i are the Fibonacci numbers defined by F 1 = F 2 =1and F n = F n 1 F n 2 for n 2. Previously Good [G46] showed that the length of a shortest word containing as subwords all 2 n binary words of length n is 2 n n 1. In the same year de Bruijn [B46] gave a complete enumeration of all such words (see also [B75] The converse problem is usually formulated as finding the function f when given a set of words w. When the words w are the prefixes of some infinite sequence S, the function f is one measure of the complexity of S, and is usually ....

....the subword complexity of S. For related results on subword complexity see the survey article of Allouche [A94] The proof of our results centers on a detailed analysis of a version of the de Bruijn graph which appeared first implicitly in [F94] and explicitly in [R83] Good [G46] and de Bruijn [B46] independently defined a version of these graphs in 1946. See Fredricksen [F82] for more references for the de Bruijn graph. Observe that f(1) F 3 = 2, which means the number of distinct subwords of length 1 is 2. Thus we need only consider binary words over 0, 1 in the rest of this paper. ....

de Bruijn, N. G. A combinatorial problem, Nederl. Akad. Wetensch. Proc. 49 (1946), 758--764.

.... to, for example, 2] and [9] De Bruijn graphs, both directed and undirected, have received considerable attention since de Bruijn graphs can accommodate more vertices while the maximum degree and the diameter are given, in comparison with other prevailing topologies such as hypercubes and meshes [4,6]. In this paper, we focus on study of undirected de Bruijn graphs, and henceforth, undirected de Bruijn graphs are referred to as de Bruijn graphs. The multiprocessor system which can be modeled by a de Bruijn graph is called de Bruijn network. A de Bruijn graph UB(d; n) is a simple graph ....

N.G. de Bruijn, "A combinatorial problem," Proc. Akademe Van Wetenschappen, vol. 49, pp. 758--764, 1946.

....distance E p is a close estimate for the measured average distance of the implemented random graphs. The average distance of the random graphs is smaller compared to a de Bruijn graph of the same size and degree. The de Bruijn graph belongs to the class of graphs with logarithmic diameter [6], 8] In Fig. 4 the expected number of nodes with distance i is compared to measured distance distributions of different graph types. The average number of nodes with a shortest path of length l to a selected node is drawn over the path length. The expected value is close to the measured ....

N.G. de Bruijn. A combinatorial problem. In Proceedings of the section of science, pages 758--764. Appl. Mathematical Science, Koninklije Nederlandse Academie van Wetenshapen, 1946.

....Bos ak studied d diregular digraphs (possibly with loops) satisfying the more general matrix equation A a A a 1 : A b = J (5) where a b and J denotes the matrix with all its entries equal 1. Then he proved that, for d 1, such digraphs exist only if either b = a (De Bruijn digraphs [7]) or b = a 1 (Kautz digraphs [15, 16] As we have a = 1 and b = k 3, no such digraph can exist. Nevertheless, Bos ak s proof is rather long and thus, for the reader s convinience, we give another simpler proof which is based on a proof technique similar to that from [5] The eigenvalues of J ....

N.G. de Bruijn, A combinatorial problem, Nederl. Akad. Wetensch., Proc. Ser. A 49 (1946) 758-764.

....connection costs must grow faster than linear with N . de Bruijn Graph BG(n;b) 7 de Bruijn Graphs have been rediscovered since 1894 [Sain 94] several times using different approaches (see [Rals 82] for a detailed presentation) We want to follow the graph theoretic model proposed by de Bruijn [Brui 46] and further investigated by Pradhan and Reddy [Prad 82] Definition 21 A de Bruijn Graph, denoted by BG(n; b) consists of b n nodes, where n denotes the length of the node address in radix b representation (n 2, b 2) where node u is connected to node v if either ffl the first n Gamma ....

N. F. de Bruijn. "A combinatorial problem". In: Proceedings Vol.49, part 20, Koninklijke Nederlands, Academe van Wetenschappen, 1946.

.... that either A = B = DPP j 0 1 1 I (then we call the pair (A; B) a DPP design) or for some r and s: AJ = JA = rJ ; BJ = JB = sJ ; A T B = BA T = J (rs Gamma n)I (then we call the pair (A; B) an (r; s) design) This result generalizes the earlier results of De Bruijn and Erdos [2] and Ryser [11] and it is one of the main arguments in the proof of Lehman s theorem on minimally non ideal polyhedra [7] In this paper we would like to investigate the d designs a bit more generally. Our main goal is to find sufficient conditions which force a d design to become an (r; ....

....0, the number of zeros in the ith row is equal to the number of zeros in the jth column. It is easy to see that if A is a DE matrix and J Gamma A is connected, then A is totally regular. Here we summarize the information about the matrices having DE property, which we use in this paper. Lemma 2 [2] Let A be an n Theta m f0; 1g matrix without all one columns. If n m and for each pair i and j with a ij = 0, the number of zeros in the ith row is less or equal than the number of zeros in the jth column, then A is a DE matrix. Lemma 3 [9] 14] A DE matrix has an equal number of all one rows ....

N. G. De Bruijn, P. Erdos, On a combinatorial problem, Indag. Math. 10 (1948) 421-423.

....18] However, efficient algorithms do exist for finding the shortest string consistent with the results of a classical sequencing chip experiment. In particular, Pevzner s algorithm for sequencing chip reconstruction [25] is based on finding Eulerian paths in a subgraph of the de Bruijn digraph [5]. For a given alphabet Sigma and length k, the de Bruijn digraph G k ( Sigma) will contain j Sigmaj k Gamma1 vertices, each corresponding to a (k Gamma 1) length string on Sigma. As shown in Figure 2, there will be an edge from vertex u to v labeled oe 2 Sigma if the string associated with v ....

....sequence of length ff m , which contains all the strings of length m exactly once. For example, 0000100110101110 and 0000101001101110 are two distinct de Bruijn sequences of span 4 on the binary alphabet f0; 1g. Let D(ff; m) denote the set of distinct de Bruijn sequences on ff. It is well known [5, 26] that jD(ff; m)j = ff Gamma 1) ff m Gamma1 ff ff m Gamma1 Gammam = 1 n (ff ) n=ff Good [11] demonstrated how to construct the sequences of D(ff; m) by building a directed graph G(ff; m) where the vertex set of G(ff; m) represents each string of length m Gamma 1 on ff. Create a ....

N.G. de Bruijn. A combinatorial problem. Proc. Kon. Ned. Akad. Wetensch, 49:758--764, 1946.

....0 l13 l11 l12 l14 l16 l15 l10 l9 l1 l2 l6 l3 l4 l7 l8 l5 Figure 4: An 8 node (2,2) ShuffleNet. 110 111 101 010 011 001 100 000 l6 l7 l11 l16 l15 l13 l5 l10 l9 l1 l2 l4 l3 l12 l8 l14 0 4 6 3 2 7 5 1 Figure 5: An 8 node de Bruijn Graph. 3. 2 de Bruijn Graph A (ffi, D) de Bruijn Graph (ffi 2; D 2) [13][39] consists of the set of nodes f0; 1; 2; ffi Gamma 1g D with an edge from node a 1 a 2 : a D to node b 1 b 2 : b D if and only if b i = a i 1 where a i ; b i 2 f0; 1; 2; ffi Gamma 1g and 1 i D Gamma 1. Each node has an in degree and out degree of ffi , and there are N = ffi ....

N. G. de Bruijn, "A combinatorial problem," Proc., Koninklijke Nederlandse Acad. van Wetenschappen, Ser. A49, pp. 758-764, 1949.

....(s) F 1 (s) F 0 (s) 2) where F 1 (s) L(s) is the linear span of s. The inequalities in eq. 2) state that the size of the shift register may decrease as we allow the degree of f(X) to increase. We may illustrate this possible reduction in memory by considering de Bruijn sequences [8]. Chan and Games [7] have studied the quadratic span of these sequences, and their results show a large difference between the linear and quadratic spans. For example, the de Bruijn sequences of length 63 = 2 6 Gamma 1 have an expected linear span of at least 62, while the expected quadratic ....

N. G. de Bruijn. A combinatorial problem. Nederl. Akad. Wetensch. Proc, 49:754-- 758, 1946.

.... [HlK88] hmin = p Gamma p D 1 ND(p Gamma 1) 2 D(p Gamma 1) N Gamma 1) p Gamma 1) 2 (8) It is also known that there are no directed Moore graphs for nontrivial values of D and p [BrT80] One of the best known family of graphs which come close to the Moore bound are de Bruijn graphs [Bru46]. A directed de Bruijn graph with connectivity p and diameter D has N = D p nodes. A general class of graphs has been proposed by Imase and Itoh [ImI83, ISO85, HoP88] which contains de Bruijn graphs as a subclass. The design procedure outlined in [ImI83] produces graphs with the upper bound D ....

N.G. de Bruijn, "A Combinatorial Problem ", Koninklijke Netherlands: Academe Van Wetenschappen, Proc, 49, 20 (1946), 758--764.

....on sequences with period 2 n for which an entire period of the sequence is known. While this appears to be an algorithm with limited general applicability the mathematical foundations were used to prove some very interesting results [40, 38] in the study of what are called de Bruijn sequences [15, 35]. Very recently the validity of this algorithm has been extended by Blackburn [9] to sequences with any period, although an entire period of the sequence is still required for its application. 4.2.2 Other measures of complexity As a generalization of the linear complexity a great deal of research ....

N.G. de Bruijn. A combinatorial problem. Nederl. Akad. Wetensch. Proc., 49:758--764, 1946.

....which may belong to different groups. In this case, the ready bits have to be virtualized as well. The details depend on the virtualization strategy, which may be a mixture of looping and context switching. 3. 3 Communications Network The Triton 1 network is based on the generalized De Bruijn Net [2, 3]. The number N of nodes in the network is not limited to powers of two. The (maximum) diameter is dlog d Ne. The average diameter is well below log d N and in practice quite close to the theoretical lower bound, the average diameter of directed Moore graphs. In our implementation we use degree d = ....

N. G. De Bruijn. A combinatorial problem. In Proc. of the Sect. of Science Akademie van Wetenschappen, pages 758--764, Amsterdam, June 29 1946.

....need ffl the hash table to be small, ffl the hash function to be easily computable, and ffl the hash function to produce no collisions, i.e. no two single 1 words x and y should produce hash values such that h(x) h(y) Idea #3: de Bruijn sequences. The final idea uses de Bruijn sequences [3] to satisfy all three criteria. The size of the hash table is exactly n, the minimum possible. The hash function is a typical multiplicative hash function [2, pp. 228 229] that involves a single unsigned integer multiplication of the key by a de Bruijn constant. Surprisingly, no two single 1 ....

Nicolaas G. de Bruijn. A combinatorial problem. In Indagationes Mathematicae, volume VIII, pages 461--467. Koninklije Nederlandsche Akademie van Wetenschappen, 1946.

....an N node network) along with its limited connectivity. The recent quest for massively parallel computing systems is placing a major emphasis on scalable networks with small diameters and bounded node degrees[14] As an alternative to the hypercube and the mesh topologies, the de Bruijn topology[15, 16] has recently been receiving much attention. Its properties and applications have been studied by several researchers[2, 17, 18, 19, 20] Its topological properties show that the de Bruijn network is a good candidate for interconnection networks of the next generation of parallel computers after ....

N. G. de Bruijn, "A Combinatorial Problem," Koninklije Nedderlandse Academie van Wetenshappen Proc, vol. Ser A49, pp. 758--764, 1946.

....has in degree and out degree d, and there are Nd arcs. Whereas in UB(d; D) there exist N Gamma d 2 vertices of degree 2d, d 2 Gamma d vertices of degree 2d Gamma 1 and d vertices of degree 2d Gamma 2. These networks have been discovered by many authors and are named after de Bruijn [7]. They are sometimes also called Good graphs [9] They present many attractive features. In particular they are among the best known networks for a given degree and diameter (see the survey [2] for more details on this problem known as the (d; D) or ( Delta; D) graph problem) They have a good ....

N. G. de Bruijn, "A combinatorial problem", In Koninklijke Nederlands Akademie van Wetenschappen Proceedings, 49-2, pp. 758--764, 1946.

....has in degree and out degree d, and there are Nd arcs. Whereas in UB(d; D) there exist N Gamma d 2 vertices of degree 2d, d 2 Gamma d vertices of degree 2d Gamma 1 and d vertices of degree 2d Gamma 2. These networks have been discovered by many authors and are named after de Bruijn [8]. They are sometimes also called Good graphs [10] They present many attractive features. In particular they are among the best known networks for a given degree and diameter (see the survey [2] for more details on this problem known as the (d; D) or ( Delta; D) graph problem) They have good ....

N. G. de Bruijn, "A combinatorial problem", In Koninklijke Nederlands Akademie van Wetenschappen Proceedings, 49-2, pp. 758--764, 1946.

....National Science Foundation grant No. 20 4035494 and by the grant No. 95 5305 277 of the Slovak Grant Agency. Preprint submitted to Elsevier Preprint 23 February 1 Introduction The de Bruijn and Kautz digraphs were originally studied as asymptotically largest digraphs w.r.t. degree and diameter [4] and were proposed as promising topologies for massively parallel computer architectures [3] Several graphtheoretic properties and algorithmic design problems have been widely studied for these graphs [1,2,13,17] In this paper, we study vertex and edge bisection width of the k ary n ....

N.G. de Bruijn, A combinatorial problem, Neder. Akad. Wetensch. Proc. Ser. A 49 (1946) 758--764.

....Abstract. The number of cycles of length k that can be generated by q ary n stage feedback shift registers is studied. This problem is equivalent to finding the number of cycles of length k in the natural generalization, from binary to q ary digits, of the so called de Bruijn Good graphs [2, 7]. The number of cycles of length k in the q ary graph G (q) n of order n is denoted by fi (q) n; k) Known results about fi (2) n; k) are summarized and extensive new numerical data is presented. Lower and upper bounds on fi (q) n; k) are derived showing that, for large k, virtually ....

....(q) k = 1 for n(k) b2 log q k f(k)c; where b:c denotes the integer part of the enclosed number. Finally, some approximations for fi (q) n; k) are given that make the global behavior of fi (q) n; k) more transparent. 1. Introduction The n th order q ary de Bruijn Good graph G (q) n [7, 11], sometimes in the literature also simply called de Bruijn graph, is the graph of all states and all possible state transitions of a q ary n stage feedback shift register [12] G (q) n is hence a directed graph on q n vertices labelled with q ary n tuples (b 0 ; b 1 ; b n Gamma1 ) b i ....

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N.G. de Bruijn, A combinatorial problem, Proc. K. Ned. Akad. Wet. Ser. A, Vol. 49, pp. 758-764, 1946.

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N.G. De Bruijn, "A combinatorial problem", Koninklijke Nederlandsche Akademie van Wetenschappen Proc., Vol. 49, pp. 758-764, 1946.

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N.G. de Bruijn, A combinatorial problem, Nederl. Akad. Wetensch,. Proc. Ser. A 49 (1946) 758-764.

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