Compact Routing Messages in Self-Healing Trees

Abstract

Existing compact routing schemes, e.g., Thorup and Zwick [SPAA 2001] and Chechik [PODC 2013], often have no means to tolerate failures, once the system has been setup and started. This paper presents, to our knowledge, the first self-healing compact routing scheme. Besides, our schemes are developed for low memory nodes, i.e., nodes need only $O(\log^2 n)$ memory, and are thus, compact schemes. We introduce two algorithms of independent interest: The first is CompactFT, a novel compact version (using only $O(\log n)$ local memory) of the self-healing algorithm Forgiving Tree of Hayes et al. [PODC 2008]. The second algorithm (CompactFTZ) combines CompactFT with Thorup-Zwick's tree-based compact routing scheme [SPAA 2001] to produce a fully compact self-healing routing scheme. In the self-healing model, the adversary deletes nodes one at a time with the affected nodes self-healing locally by adding few edges. CompactFT recovers from each attack in only $O(1)$ time and $\Delta$ messages, with only +3 degree increase and $O(log \Delta)$ graph diameter increase, over any sequence of deletions ($\Delta$ is the initial maximum degree). Additionally, CompactFTZ guarantees delivery of a packet sent from sender s as long as the receiver t has not been deleted, with only an additional $O(y \log \Delta)$ latency, where $y$ is the number of nodes that have been deleted on the path between $s$ and $t$. If $t$ has been deleted, $s$ gets informed and the packet removed from the network.