The minimum distance diagram and diameter of undirected double-loop networks

This thesis proposes a minimum distance diagram of undirected double-loop, which makes use of the cyclic graph's excellent properties on the node number and path length to achieve high performance. Main issues include the method to build the minimum distance diagram of undirected double-loop networks, the algorithm to calculate its diameter, and the step to build optimal undirected double-loop networks, etc. We prove that the diameter of the undirected double-loop network is equal to the height of its tree structure, and propose a rapid algorithm to calculate the diameter, and find that there are lots of optimal undirected double-loop networks in some infinite clusters. Finally, the lower bound proposed by Yebra is verified by experiment. According to these results above, the transmission performance of undirected double-loop will be optimized. Introduction The double-loop network which has symmetry, simple structure, easy scalability and strong fault tolerance, is easy to be constructed and widely used. High reliability of double-loop Networks is closely related to the network transmission delay, which is relevant to the network diameter and its routing algorithm, the shorter diameter brings about the less transmission delay, and the better routing algorithm will get the shortest path between any two nodes. So, how to choose the proper steps to decrease the diameter, how to design the optimal routing algorithm, these issues are worth to study. According to the principle of communication that shorter diameter means less delay. Suppose an undirected double-loop network G(N; ±r, ±s) has a limited diameter d(N; ±r, ±s), denote D(N) = min{ d(N; ±r, ±s): 1 ≤ r ≠s < N}, Wong et al. [1] gave the lower bound of D(N), that is D(N) ≥ 2 / ) 3 2 ( − N . Later, Yebra et al. [2] adjust D(N) ≥       − − 2 / ) 1 1 2 ( N . If D(N) obtains the minimum value (the lower bound), then we call G(N; ±r, ±s) as optimal undirected double-loop network. A problem is that for a double-loop network with given N, how to calculate its diameter according to step r and s? In this field, Chen [3–5] and Fang et al. [6–8] have done a lot of work and have made plenty of valuable achievements. Our further study is on the basis of results which researched by Chen et al. who had described the diameter upper bound of the undirected network G(N; ±1, ±s), they also presented the formula of diameter about the undirected double-loop network G(N; ±1, ±s ), which can compute the diameter of undirected double-loop network whose the first step is 1, but it did not resolve the information routing problem, the same applies to Fang , they are not universal. The method of this thesis is mapping the space structure of the undirected double-loop network to a plane tree structure and gives its minimum distance diagram, according to witch, we not only can compute its diameter but also can find the shortest path between any two nodes rapidly. 3rd International Conference on Materials Engineering, Manufacturing Technology and Control (ICMEMTC 2016) © 2016. The authors Published by Atlantis Press 1682 The minimum distance diagram Undirected double-loop network graph theory model is an undirected graph G(N; ±r, ±s) , whose each vertex i is denoted as 0, 1, ..., N – 1, and there are four undirected edges i →i + r (mod N) , i →i + s (mod N), i →i + N – r (mod N) and i →i + N – s (mod N) denoted as [+r] edge, [+s] edge, [–r] edge and [–s] edge respectively, r and s are two natural numbers, and satisfy 1≤ r ≠s < N. Fig. 1 shows the topology of undirected double-loop network G(14; ±3, ±4 ). For a given N, r and s can decide the structure of the undirected double-loop networks G(N; ±r, ±s), it also can determines its diameter and routing strategy. According to the symmetry of the double-loop network, the distance of node u to node v equals to the distance of node 0 to node v–u, therefore, we just pay close attention to routing strategies from node 0 to other nodes, and then we can know the whole routing table about this network. According to the above principles, the spatial structure of the undirected double-loop network G(N; ±r, ±s) can be mapped plane structure as Definition 1. Fig. 1 The topology of undirected double-loop network G(14; ±3, ±4). Fig. 2 The tree structure of undirected double-loop network T(14; ±3, ±4). Definition 1 Step1. Let node 0 as the root node of a tree put node 0 on layer 0, write it and its child nodes r, s, N–s, N–r to set U, put them on layer1 and construct the tree with five nodes as Fig.2 ; Step2. Take each child node i respectively to create its child nodes with the value of i + r (mod N) , i + s (mod N), i + N – r (mod N) and i + N – s (mod N), if new child node have not appear in set U, then put it into U, otherwise abandon it, put these new nodes on layer 2 and construct child tree with these new nodes; Step3. Repeat Step 2 above until all nodes appears in this tree. The corresponding tree structure created by Definition 1 is denoted as T(N; ±r, ±s), which is the minimum distance diagram of undirected double-loop network. But no each G(N; ±r, ±s) has its T(N; ±r, ±s) beside gcd(N , r , s)= 1. 13 6 7 1 4 0