有向图上的常数因子最优2-弹性本地故障切换路由方案
Constant Factor Optimal 2-Resilient Local Failover Routing Scheme on Directed Graphs
- The University of Osaka（大阪大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究有向图上容忍至多k条弧故障的本地故障切换路由，提出改进的位复杂度上界与下界，其中k=2时上下界相差至多3位，达到常数因子最优。
AI中文摘要：
本地故障切换路由仅使用预计算的每节点转发规则以及一个小的可重写数据包头，将数据包从源节点 $s$ 传送到目的节点 $t$，即使在网络中多达 $k$ 条弧发生故障时仍能保持正确性。我们在具有 $n$ 个节点的有向图上研究此问题，以包头中携带的可重写位数衡量效率，并关注 $k \ge 2$ 次故障的情况。我们的第一个贡献是改进了一般 $k$ 的上界。我们设计了一种使用至多 $\lceil \log\\!\binom{2n+k-4}{k-1} \rceil+1$ 位的方案，对于每个 $k < 2n-3$，改进了先前已知的最佳 $\lceil \log \binom{2n+k-3}{k} \rceil$ 位方案。对于特殊情况 $k=2$，我们进一步将其改进为使用至多 $\left\lceil \log\\!\left(1+2\lfloor 2(n-1)/3 \rfloor\right)\right\rceil$ 位的方案。我们的第二个贡献是通过一种新的路径小工具构造得到的一系列改进下界。对于 $k=2$，我们证明了 $\left\lceil \log \lfloor n/3 \rfloor \right\rceil$ 位的下界；对于一般 $k$，记 $k_2 = \lfloor k/2 \rfloor$，下界为 $\left\lceil k_2 \log \left\lfloor (n-1+k_2)/(3k_2) \right\rfloor \right\rceil$ 位，改进了先前已知的最佳下界。特别地，对于 $k=2$，我们的上界 $\left\lceil \log\\!\left(1+2\lfloor 2(n-1)/3 \rfloor\right)\right\rceil$ 位与下界 $\left\lceil \log \lfloor n/3 \rfloor \right\rceil$ 位相差至多 $3$ 位。
英文摘要：
Local failover routing delivers a packet from a source $s$ to a destination $t$ using only pre-computed, per-node forwarding rules together with a small rewritable packet header, remaining correct even when up to $k$ arcs of the network fail. We study this problem on directed graphs with $n$ nodes, measuring efficiency by the number of rewritable bits carried in the header, and focus on the case of $k \ge 2$ failures. Our first contribution is an improved upper bound for general $k$. We design a scheme using at most $\lceil \log\!\binom{2n+k-4}{k-1} \rceil+1$ bits, improving the best previously known $\lceil \log \binom{2n+k-3}{k} \rceil$ bit scheme for every $k < 2n-3$. For the special case $k=2$, we further improve this to a scheme using at most $\left\lceil \log\!\left(1+2\lfloor 2(n-1)/3 \rfloor\right)\right\rceil$ bits. Our second contribution is a family of improved lower bounds, obtained via a new path-gadget construction. For $k=2$, we prove a lower bound of $\left\lceil \log \lfloor n/3 \rfloor \right\rceil$ bits, and for general $k$, writing $k_2 = \lfloor k/2 \rfloor$, a lower bound of $\left\lceil k_2 \log \left\lfloor (n-1+k_2)/(3k_2) \right\rfloor \right\rceil$ bits, improving on the best previously known bound. In particular, for $k=2$, our upper bound of $\left\lceil \log\!\left(1+2\lfloor 2(n-1)/3 \rfloor\right)\right\rceil$ bits and our lower bound of $\left\lceil \log \lfloor n/3 \rfloor \right\rceil$ bits differ by at most $3$ bits.
