Constant Factor Optimal 2-Resilient Local Failover Routing Scheme on Directed Graphs

RESUMO

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.