Dynamic Edge Orientation via Random Walks: From Trees to Outerplanar Graphs and Beyond
We study the \emph{fully dynamic edge orientation problem}, focusing on \emph{worst-case} time bounds. An undirected graph undergoes edge insertions and deletions, and the goal is to maintain an orientation with small {\em maximum outdegree} (hereafter, outdegree) and small worst-case update time. The outdegree of any orientation is at least $α-1$, where $α$ is the graph's \emph{arboricity}, i.e., the minimum number of forests into which its edge set can be partitioned. When $α= O(1)$, it is long known that both the outdegree and the worst-case update time can be bounded by $O(\log n)$. Despite numerous follow-ups, no $o(\log^3 n)$ worst-case update time is known for maintaining constant outdegree, even for very basic graph families---with a notable exception, \emph{forests}. For forests, a \emph{simple folklore} algorithm maintains outdegree 2 via \emph{random walks}: When an insertion creates a vertex of outdegree 3, the algorithm repeatedly chooses a uniformly random outgoing edge until reaching a vertex of outdegree at most 1, and then flips the resulting directed path. As the underlying graph is cycle-free, the path length is easily shown to be $O(\log n)$ in expectation, and also with high probability for polynomially long update sequences. We prove that this simple random walk paradigm extends to \emph{outerplanar graphs}. Our algorithm maintains constant outdegree with $O(\log n)$ worst-case update time, where the time bound holds in expectation, and also with high probability for polynomially long update sequences. We give a \emph{tight analysis}: outdegree 4 is achievable with $O(\log n)$-length paths, while outdegree 3 incurs $\mathtt{poly}(n)$-length paths. We also extend the argument to $K_{2,t}$-minor-free graphs, for any $t \ge 2$, with the outdegree bound depending only on $t$ and with the same update time guarantees. The locality of [...]