  We present a distributed self-adjusting algorithm for skip graphs that
minimizes the average routing costs between arbitrary communication pairs by
performing topological adaptation to the communication pattern. Our algorithm
is fully decentralized, conforms to the $\mathcal{CONGEST}$ model (i.e. uses
$O(\log n)$ bit messages), and requires $O(\log n)$ bits of memory for each
node, where $n$ is the total number of nodes. Upon each communication request,
our algorithm first establishes communication by using the standard skip graph
routing, and then locally and partially reconstructs the skip graph topology to
perform topological adaptation. We propose a computational model for such
algorithms, as well as a yardstick (working set property) to evaluate them. Our
working set property can also be used to evaluate self-adjusting algorithms for
other graph classes where multiple tree-like subgraphs overlap (e.g. hypercube
networks). We derive a lower bound of the amortized routing cost for any
algorithm that follows our model and serves an unknown sequence of
communication requests. We show that the routing cost of our algorithm is at
most a constant factor more than the amortized routing cost of any algorithm
conforming to our computational model. We also show that the expected
transformation cost for our algorithm is at most a logarithmic factor more than
the amortized routing cost of any algorithm conforming to our computational
model.
