Improved Algorithms for Local Failover Routing on Directed Graphs

The local failover routing is a mechanism that routes a packet from a source to a destination only using pre-calculated routing tables, even when several edges fail. In this paper, we study local failover schemes that minimize the number of rewritable bits in the packet header on directed graphs with $k$-arc failures. There are many studies of failover routing on undirected graphs, and it has been investigated whether routing is possible depending on the number of bits in the packet header, the type of failure, the graph properties, etc. In contrast, there is not much research on directed graphs. Van et al.~first showed the upper and lower bounds of rewritable bits in the packet header on directed graphs. However, their results showed a large gap between the upper and lower bounds. The main contribution of this paper is to close the gap between the upper and lower bounds. Specifically, we show that our scheme can route packets with $k$ faulty arcs if the packet header has $\min(k \log ( \frac{e(2n+k-3)}{k}, 2n \log ( \frac{e(2n+k-3)}{2n})))$ rewritable bits, where $n$ is the number of nodes. Moreover, any local failover routing scheme needs $\Omega(k\lceil\log\frac{n}{k}\rceil)$ rewritable bits when the number of faulty arcs is equal to or less than $\frac{3(n-1)}{8}$ and $\frac{n-1}{4}$ rewritable bits when the number of faulty arc is more than $\frac{3(n-1)}{8}$. This result means our scheme is nearly optimal when the number of faulty arcs is approximately less than the number of nodes.