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Exact Wirelength of Embedding 3-Ary n-Cubes into certain Cylinders and Trees

Rajeshwari S, M Rajesh

TL;DR

This work solves the exact wirelength problem for embedding the 3-ary n-cube $Q_n^3$ into a cylinder and several tree topologies. By leveraging edge isoperimetric results, the Congestion Lemma, and the Partition Lemma, it develops lexicographic embeddings for cylinder targets and lexicographic-plus-preorder schemes for trees, deriving explicit, closed-form wirelength expressions. The main contributions are the precise WL formulas for $C_3\times P_{3^{n-1}}$ and for caterpillars, Firecracker graphs, and banana trees, along with proof of optimal embeddings. The findings have practical implications for designing efficient interconnection networks and NoC mappings, enabling predictable, optimized communication layouts for parallel architectures that use $Q_n^3$-based topologies.

Abstract

Graph embeddings play a significant role in the design and analysis of parallel algorithms. It is a mapping of the topological structure of a guest graph G into a host graph H, which is represented as a one-to-one mapping from the vertex set of the guest graph to the vertex set of the host graph. In multiprocessing systems the interconnection networks enhance the efficient communication between the components in the system. Obtaining minimum wirelength in embedding problems is significant in the designing of network and simulating one architecture by another. In this paper, we determine the wirelength of embedding 3-ary n-cubes into cylinders and certain trees.

Exact Wirelength of Embedding 3-Ary n-Cubes into certain Cylinders and Trees

TL;DR

This work solves the exact wirelength problem for embedding the 3-ary n-cube into a cylinder and several tree topologies. By leveraging edge isoperimetric results, the Congestion Lemma, and the Partition Lemma, it develops lexicographic embeddings for cylinder targets and lexicographic-plus-preorder schemes for trees, deriving explicit, closed-form wirelength expressions. The main contributions are the precise WL formulas for and for caterpillars, Firecracker graphs, and banana trees, along with proof of optimal embeddings. The findings have practical implications for designing efficient interconnection networks and NoC mappings, enabling predictable, optimized communication layouts for parallel architectures that use -based topologies.

Abstract

Graph embeddings play a significant role in the design and analysis of parallel algorithms. It is a mapping of the topological structure of a guest graph G into a host graph H, which is represented as a one-to-one mapping from the vertex set of the guest graph to the vertex set of the host graph. In multiprocessing systems the interconnection networks enhance the efficient communication between the components in the system. Obtaining minimum wirelength in embedding problems is significant in the designing of network and simulating one architecture by another. In this paper, we determine the wirelength of embedding 3-ary n-cubes into cylinders and certain trees.
Paper Structure (9 sections, 20 theorems, 7 equations, 5 figures)

This paper contains 9 sections, 20 theorems, 7 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $3$-Ary $n$-cube, $Q_n^3$
  4. Embedding of $Q_n^3$ into cylinder $C_{3}\times P_{3^{n-1}}$
  5. Embedding of $Q_n^3$ into certain trees
  6. Wirelength of embedding $Q_n^3$ in caterpillar
  7. Wirelength of embedding $Q_n^3$ in Firecracker graph
  8. Wirelength of embedding $Q_n^3$ in banana tree
  9. Conclusion

Key Result

Lemma 2.6

(miller2015minimum, Congestion Lemma) Let $f$ be an embedding of an arbitrary graph $G$ into $H$. Let $S$ be an edge cut of $H$ such that the removal of edges of $S$ separates $H$ into two components $H_1$ and $H_2$ and let $G_1=f^{-1} (H_1 )$ and $G_2=f^{-1} (H_2 )$. Also $S$ satisfies the followin Then, $c_f(S)=\sum_{v\in V(G_1)}deg_{G}(v)-2|E(G_1)| = \sum_{v\in V(G_2)}deg_{G}(v)-2|E(G_2)|$ and

Figures (5)

  • Figure 1: 3-ary 3-cube, $Q_3^{3}$.
  • Figure 2: (a) Vertical edge cuts $X_{i}^{t}$, $1\leq i \leq 8$, $1\leq t \leq 2$ of Cylinder $C_3\times P_{9}$ with lexicographic ordering. (b)Horizontal edge cuts $Y_{j}$, $1\leq j \leq 2$ of Cylinder $C_3\times P_{9}$ with lexicographic ordering.
  • Figure 3: Edge cuts of Caterpillar.
  • Figure 4: Edge cuts of $F_{3^{n-1},3}$.
  • Figure 5: Edge cuts of $B_{2,\lfloor\frac{3^n}{2}\rfloor}$.

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • Remark 2.7
  • Lemma 2.8
  • Definition 3.1
  • Definition 3.2
  • ...and 23 more