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On Completely Edge-Independent Spanning Trees in Locally Twisted Cubes

Xiaorui Li, Baolei Cheng, Jianxi Fan, Yan Wang, Dajin Wang

TL;DR

The paper tackles embedding the maximum number of completely edge-independent spanning trees (CEISTs) in the locally twisted cube LTQ_n to enhance fault-tolerant broadcasting. It introduces a recursive construction, CEISTs_$LTQ$, composed of Odd_CEISTs and Even_CEISTs, to produce $\left\lfloor \frac{n}{2} \right\rfloor$ CEISTs in LTQ_n and proves their correctness. The authors establish an $O(n \cdot 2^n)$ time algorithm and validate it through implementation and broadcasting simulations, showing improved latency metrics over single-tree broadcasting. They also discuss how the approach generalizes to hypercube variants like $Q_n$ and $CQ_n$, underscoring practical implications for robust network communications in bijective connection networks.

Abstract

A network can contain numerous spanning trees. If two spanning trees $T_i,T_j$ do not share any common edges, $T_i$ and $T_j$ are said to be pairwisely edge-disjoint. For spanning trees $T_1, T_2, ..., T_m$, if every two of them are pairwisely edge-disjoint, they are called completely edge-independent spanning trees (CEISTs for short). CEISTs can facilitate many network functionalities, and constructing CEISTs as maximally allowed as possible in a given network is a worthy undertaking. In this paper, we establish the maximal number of CEISTs in the locally twisted cube network, and propose an algorithm to construct $\lfloor \frac{n}{2} \rfloor$ CEISTs in $LTQ_n$, the $n$-dimensional locally twisted cube. The proposed algorithm has been actually implemented, and we present the outputs. Network broadcasting in the $LTQ_n$ was simulated using $\lfloor\frac{n}{2}\rfloor$ CEISTs, and the performance compared with broadcasting using a single tree.

On Completely Edge-Independent Spanning Trees in Locally Twisted Cubes

TL;DR

The paper tackles embedding the maximum number of completely edge-independent spanning trees (CEISTs) in the locally twisted cube LTQ_n to enhance fault-tolerant broadcasting. It introduces a recursive construction, CEISTs_, composed of Odd_CEISTs and Even_CEISTs, to produce CEISTs in LTQ_n and proves their correctness. The authors establish an time algorithm and validate it through implementation and broadcasting simulations, showing improved latency metrics over single-tree broadcasting. They also discuss how the approach generalizes to hypercube variants like and , underscoring practical implications for robust network communications in bijective connection networks.

Abstract

A network can contain numerous spanning trees. If two spanning trees do not share any common edges, and are said to be pairwisely edge-disjoint. For spanning trees , if every two of them are pairwisely edge-disjoint, they are called completely edge-independent spanning trees (CEISTs for short). CEISTs can facilitate many network functionalities, and constructing CEISTs as maximally allowed as possible in a given network is a worthy undertaking. In this paper, we establish the maximal number of CEISTs in the locally twisted cube network, and propose an algorithm to construct CEISTs in , the -dimensional locally twisted cube. The proposed algorithm has been actually implemented, and we present the outputs. Network broadcasting in the was simulated using CEISTs, and the performance compared with broadcasting using a single tree.
Paper Structure (10 sections, 5 theorems, 9 figures)

This paper contains 10 sections, 5 theorems, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Terminology and notation
  4. Locally twisted cubes
  5. Completely edge-independent spanning trees in locally twisted cubes
  6. Construction Algorithm of CEISTs for locally twisted cubes
  7. Correctness of CEISTs_$LTQ$
  8. CEISTs in $LTQ$ vs. in hypercubes/crossed cubes
  9. Implementation and simulation
  10. Conclusion

Key Result

Lemma 1

[21] Let $u = u_{n-1}u_{n-2}...u_0$ and $v = v_{n-1}v_{n-2}...v_0$ be two adjacent vertices in $LTQ_n ( n \geq 2)$ with $u > v$. Then, the following statements hold. (1) If $|u|$ is even, then $|u| - |v|$ = $2^i$ for some $0 \leq i \leq {n-1}$. (2) If $|u|$ is odd, then either $|u| - |v|$ = $2^i$ fo

Figures (9)

  • Figure 1: (a) $LTQ_3$; (b) $LTQ_4$.
  • Figure 2: CEIST $T_1$ in $LTQ_2$.
  • Figure 3: CEIST $T_1$ in $LTQ_3$.
  • Figure 4: Two CEISTs $T_1$ and $T_2$ in $LTQ_5$.
  • Figure 5: Two CEISTs $T_1$ and $T_2$ in $LTQ_4$.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Lemma 1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4