1. Number of nodes = 4 × (M  ×  N  ×  Z)−(M  ×  Z  +  N  ×  Z  +  M  ×  N)
2. Diameter = 2 ⌊(log 2 M)⌋ + 2⌊(log 2 N)⌋ + 2 ⌊(log 2 Z)⌋
3. Bisection width = min(M  ×  Z, N  ×  Z, M  ×  N)
4. Symmetric and recursive structure
       
   
       
       
326
Network-on-Chip
4. The MoT-based 3D NoC also works fine under real benchmark
applications such as dual video object plane decoder (DVOPD)
having 32 cores.
11.3.1 3D Mesh-of-Tree Topology
A 2D M × N MoT topology can be extended to a 3D structure by connecting
multiple M × N 2D MoTs via M × N number of vertical trees. The number
of leaf nodes in each vertical tree of an M × N × Z MoT is Z, where Z is the
number of M × N 2D MoTs. Figure 11.6 shows a 2 × 4 × 4 MoT structure having four layers of 2 × 4 2D MoTs, each of which has two row trees of depth
2 and four column trees of depth 1. For each row tree, RS and RR denote the
row stem and row root nodes, respectively, whereas for each column tree, CR
denotes a column root node. The ZS and ZR are the stem and root nodes in
the vertical trees, respectively. The leaf nodes (L) are common to all the three
types of trees. Two cores (not shown in Figure 11.6) are attached with each
leaf node. The stem (RS and ZS) and root (RR, CR, and ZR) nodes are not having any core attached to them. Thus, in an M × N × Z MoT, 2 × (M × N × Z)
number of cores can be attached. In general, an M × N × Z MoT has the following properties:
In Sections 11.3.1.1 and 11.3.1.2, a general formulation of the number of
directed edges and the average distance for an M × N × Z MoT is presented
where all the trees are complete binary trees.
11.3.1.1 Number of Directed Edges
The number of undirected edges in any complete binary tree having k leaf
nodes is (2k − 2). In an M × N × Z MoT structure, the number of leaf nodes
in each row tree, column tree, and vertical tree are N, M, and Z, respectively.
Thus, the number of edges (É) of an undirected M × N × Z MoT graph can be
formulated as
É = M × × (2N − 2) + N Z (2M − 2) + M × N ×(2Z − 2)]
[ Z
× ×
In the proposed 3D MoT structure, adjacent nodes are connected by two unidirectional opposite edges. Thus, the number of directed edges can be written as
E = 2É = 12 × M × N × Z − 4(M × Z + N × Z + M × N)
(11.1)