Direct interconnection networks I+II

Direct interconnection networks I+II - 5/13/2011 Direct...

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Section#5: Direct interconnection networks I+II (CS838: Topics in parallel computing, CS1221, Tue+Thu, Feb 2+4, 1999, 8:00-9:15 a.m.) The contents 1. Basic notions and terminology 2. Requirements on interconnection networks 3. Mesh-based topologies 4. Hypercubic topologies 5. Tree-based topologies 6. Shuffle-based topologies A direct interconnection network (IN) of a multiprocessor system is represented by a connected graph whose vertices represent processing nodes and edges represent communication links . A processing node (PN) usually consists of one or more processors, local memory, and communication router. This section is devoted to the description and analysis of topologies and properties of important INs. Back to the beginning of the page Back to the CS838 class schedule Basic notions and terminology Alphabets and strings d -ary alphabet is denoted by Z d ={0,1,. .,d-1} . The operation + modulo n + n . n -letter d -ary strings Z d n ={x n-1 ... x 0 ; x i in Z d } , n>= 1 . The length of string x len(x) . The empty string (i.e., of length 0) is denoted by e . The i - fold concatenation of string x x i . Binary alphabet B . The inversion of bit b i is \non(b i )=1-b 1 . If b=b n-1 .. b i+1 b i b i-1 .. b 0 in B n , then \non i (b)=b n-1 .. b \non(b i )b i-1 .. b 0 . Graph theory Vertex and edge set of graph G V(G) and E(G) , respectively. Two adjacent vertices u and v form edge ( u,v) . They are incident with the edge. Edges ( u,v) and ( v,w) are adjacent . H = subgraph of G , H\subset G , if V(H)\subset V(G) and E(H)\subset E(G) . 5/13/2011 Direct interconnection networks I+II pages.cs.wisc.edu/…/Section5.html 1/21
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H = induced subgraph of G if it is a maximal subgraph of G with vertices V(H) . H = spanning subgraph G if V(H)=V(G) . Union of two disjoint graphs G 1 and G 2 , G 1 \cup G 2 , is a graph with vertices V(G 1 )\cup V(G 2 ) and edges E(G 1 )\cup E(G 2 ) . Cartesian product G 1 and G 2 is graph G=G 1 x G 2 with V(G)={(x,y); x in V(G 1 ), y in V(G 2 )} and E(G)={( (x 1 ,y), (x 2 ,y)); ( x 1 ,x 2 ) in E(G 1 )}\cup {( (x,y 1 ),(x,y 2 )); ( y 1 ,y 2 ) in E(G 2 . CAPTION: An example of a cartesian product Degree of vertex u , deg G (u) , is the number of neighbors of u . Degree set of graph G , deg(G) , is the set {deg G (u); u in V(G)} . Maximum degree G is \triangle(G)=max(deg(G)) . Minimum degree G \delta(G)=min(deg(G)) . k -regular graph G has \triangle(G)=\delta(G)=k . Connected graph Every pair u , v of vertices is joined by a path P(u,v) , a sequence of adjacent edges. Path length len(P(u,v)) , is the number of edges P(u,v) . Distance between vertices u and v , dist G (u,v) , is the length of a shortest path joining u and v . Average distance in G , dist G (u,v) , is \Sigma u,v dist G (u,v)/(N(N-1)) . Diameter G , diam(G) , is the maximum distance between any two vertices of G . Cycle is a closed path. Vertex-disjoint paths P 1 (u,v) and P 2 (u,v) have no vertices in common except for u and v Edge-disjoint paths P 1 (u,v) and P 2 (u,v) have no edges in common.
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This note was uploaded on 09/10/2011 for the course CS 6143 taught by Professor Hadimioglu during the Spring '10 term at NYU Poly.

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Direct interconnection networks I+II - 5/13/2011 Direct...

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