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On-Chip Bus vs. Network-on-Chip
Published in Marcello Coppola, Miltos D. Grammatikakis, Riccardo Locatelli, Giuseppe Maruccia, Lorenzo Pieralisi, Design of Cost-Efficient Interconnect Processing Units, 2020
Marcello Coppola, Miltos D. Grammatikakis, Riccardo Locatelli, Giuseppe Maruccia, Lorenzo Pieralisi
Several currently proposed NoC interconnects target homogeneous Multicore processors. Similar to parallel interconnects, homogeneous Multicore processors focus on regular topologies, usually 2-d mesh [251]; see also the NoC instances discussed later in this Section. Other innovative NoC architectures are based on common topologies, as illustrated in Figure 2.8. More complex network topologies, such as expander graphs, e.g. multi-butterflies and multi-Benes networks, are very powerful in terms of global communication capabilities, especially for large number of nodes [208]. However, they are not practical for current NoC network sizes.
The last chapter
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
Let Γ have n left points and m right points, where n > m. Assume that each left point has r neighbors and each right point has l neighbors. Let be an [l, k0, d0]q code, the auxiliary code. We construct a code = (Γ ) as follows: for each right point Rj, j = 1, ...,m, fix an order of the neighbors of Rj. A tuple (x1, ..., xn) ∈ Fqn is in if the projection to the neighbors of Rj is in , for all j. As each right point imposes l − k0 linear conditions, it follows that C has dimension k ≥ n − m(l − k0). Which graph-theoretic condition must be satisfied to guarantee a high minimum distance? Assume has a codeword of weight w. Let W be the corresponding set of left points. There are wr edges emanating from points of W. Let N(W) be the set of neighbors of elements from W. There must be some R ∈ N(W) which is on at most wr/|N(W)| of those edges. This implies that contains a nonzero codeword of weight ≤ wr/|N(W)|. It follows that |N(W)| ≤ wr/d0. This shows which graph theoretic property must be satisfied in order to guarantee that the minimum distance of is at least d. We must have that any set of w ≤ d left points has more than wr/d0 neighbors. In other words, if W is a set of left points and |W | = w ≤ d, then |N (W)|/|W | > c, where c = r/d0. This is an expansion property: every set of left points (provided it is not too large) has many neighbors. Loosely speaking, expander graphs are graphs with good expansion properties. There are many variants of this concept and there is a broad range of applications. What is more, strong algebraic and number-theoretic mechanisms have been developed to construct graphs with good expansion properties, making this area one of the best developed fields of discrete mathematics. For an introduction see P. Sarnak’s Some Applications of Modular Forms [179]. The resulting graphs are known as magnifiers, expanders, dispersers but there are even more variations of this concept. The starting point is a link to Eigenvalues.
A PageRank-like measure for evaluating process flexibility
Published in IISE Transactions, 2022
Fengming Cui, Chen Wang, Lefei Li
The expander graph, as a type of highly connected graph, has been extensively studied. The expansion capability of an expander graph is measured by the expansion ratios for any subset of resources with a size where is the ratio of the size of the neighborhood to the size of the given subset By Definition 2, is the minimum expansion ratio of any subset with a size of no larger than γn. Existing studies have shown that the expander graph with a higher (implying higher expansion ratios) is nearly as effective as a full-flexibility graph, in both production systems (Chou et al., 2011) and service systems (Tsitsiklis and Xu, 2017). We next elucidate the connection between and the second largest eigenvalue λ2 of M.