View PDF HTML (experimental) Abstract:The $k$-server conjecture states that a deterministic online algorithm can achieve competitive ratio $k$ on every metric space. We prove the conjecture. Specifically, we show that the work function algorithm satisfies it. Our proof uses a natural algebraic representation of the work function as a matrix, which encodes all feasible paths to reach a configuration. In this representation, the minimum and addition operations arising in the definition of optimal ...