# Locally Self-Adjusting Skip Graphs

## TL;DR

This paper introduces a decentralized self-adjusting skip graph algorithm that optimizes routing costs based on communication patterns, with provable bounds and efficient implementation in the CONGEST model.

## Contribution

It presents the first fully decentralized, self-adjusting skip graph algorithm with theoretical bounds and a new computational model for evaluating such algorithms.

## Key findings

- Routing cost within a constant factor of the optimal amortized cost.
- Expected transformation cost is logarithmically bounded relative to the optimal.
- Algorithm conforms to the CONGEST model with O(log n) message complexity.

## Abstract

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.