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Broadcasting in Heterogeneous Tree Networks with Edge Weight Uncertainty

Cheng-Hsiao Tsou, Ching-Chi Lin, Gen-Huey Chen

TL;DR

A broadcasting problem in heterogeneous tree networks with edge weight uncertainty under the postal model with an O(n \log n \log \log n)$-time algorithm is proposed for solving the broadcasting problem.

Abstract

A broadcasting problem in heterogeneous tree networks with edge weight uncertainty under the postal model is considered in this paper. The broadcasting problem asks for a minmax-regret broadcast center, which minimizes the worst-case loss in the objective function. Due to the presence of edge weight uncertainty, it is not easy to attack the broadcasting problem. An $O(n \log n \log \log n)$-time algorithm is proposed for solving the broadcasting problem.

Broadcasting in Heterogeneous Tree Networks with Edge Weight Uncertainty

TL;DR

A broadcasting problem in heterogeneous tree networks with edge weight uncertainty under the postal model with an O(n \log n \log \log n)$-time algorithm is proposed for solving the broadcasting problem.

Abstract

A broadcasting problem in heterogeneous tree networks with edge weight uncertainty under the postal model is considered in this paper. The broadcasting problem asks for a minmax-regret broadcast center, which minimizes the worst-case loss in the objective function. Due to the presence of edge weight uncertainty, it is not easy to attack the broadcasting problem. An -time algorithm is proposed for solving the broadcasting problem.

Paper Structure

This paper contains 12 sections, 14 theorems, 11 equations, 6 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finding a minmax-regret broadcast center
  4. Correctness
  5. Proof of Fact \ref{['fact:Tx-is-base']}
  6. Proof of Fact \ref{['fact:Tx-is-base-with-y-prime']}
  7. Proof of Fact \ref{['fact:critical-after']}
  8. Proof of Fact \ref{['fact:critical-before']}
  9. Time complexity
  10. Computation of $\textit{min\_v}(\sigma_{j,1})+ \textit{acc}(\sigma_{j,1}) \cdot \rho$ under $\beta^j_{x, v_i}$
  11. Proofs of Fact \ref{['fact:preprocessing-broadcast-time']} and Fact \ref{['fact:preprocessing-increasing-order']}
  12. Conclusion

Key Result

Lemma 1

Suppose $s \in C$ and $(u,v)\in E(T)$. If ${\textit{b}\_time}^s(u, \bar{B}_{u,v}) \le {\textit{b}\_time}^s(v, \bar{B}_{v, u})$, then the following hold:

Figures (6)

  • Figure 1: A tree $T$.
  • Figure 2: $B_{x,y}$ and $\bar{B}_{x,y}$.
  • Figure 3: proof of Lemma \ref{['lemma:Direct-to-center']}.
  • Figure 4: An illustrative example with $\rho = 1$. (a) A tree $T$. (b) The base scenario $\alpha_{x, v_i}$.
  • Figure 5: Two instances of $\beta_{x, v_i}^j$. (a) $\beta_{x, v_i}^1$. (b) $\beta_{x, v_i}^2$.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Lemma 1: Su2016
  • Lemma 2: Su2016
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Theorem 9
  • Lemma 10
  • ...and 4 more