Self-stabilizing Overlays for high-dimensional Monotonic Searchability

We extend the concept of monotonic searchability for self-stabilizing systems from one to multiple dimensions. A system is self-stabilizing if it can recover to a legitimate state from any initial illegal state. These kind of systems are most often used in distributed applications. Monotonic searchability provides guarantees when searching for nodes while the recovery process is going on. More precisely, if a search request started at some node $u$ succeeds in reaching its destination $v$, then all future search requests from $u$ to $v$ succeed as well. Although there already exists a self-stabilizing protocol for a two-dimensional topology and an universal approach for monotonic searchability, it is not clear how both of these concepts fit together effectively. The latter concept even comes with some restrictive assumptions on messages, which is not the case for our protocol. We propose a simple novel protocol for a self-stabilizing two-dimensional quadtree that satisfies monotonic searchability. Our protocol can easily be extended to higher dimensions and offers routing in $\mathcal O(\log n)$ hops for any search request.