Time-Optimal $k$-Server

TL;DR

This paper introduces the time-optimal $k$-server problem, where the goal is to minimize the total time by summing the maximum per-step server movement rather than total distance. It establishes strong lower bounds for deterministic and randomized online algorithms across diverse metric spaces, including $2k-1$ on general graphs for deterministic algorithms and $k+O(\log k)$ for randomized cases on a diameter-$3$ construction, while also proving $k+1$ lower bounds on Euclidean spaces and $k$ on uniform metrics. An upper bound is provided via the Robin algorithm, achieving $A k$-competitiveness on spaces with aspect ratio $A$, and a matching $k$-competitive deterministic algorithm on uniform metric spaces, highlighting a tight alignment in that setting. The results reveal a substantial gap between upper and lower bounds in the time model and point toward leveraging time-based analyses to gain new insights into the classical $k$-server problem, including potential extensions of the work-function framework to time-costs.

Abstract

The time-optimal $k$-server problem minimizes the time spent serving all requests instead of the distances traveled. We give a lower bound of $2k-1$ on the competitive ratio of any deterministic online algorithm for this problem, which coincides with the best known upper bound on the competitive ratio achieved by the work-function algorithm for the classical $k$-server problem. We provide further lower bounds of $k+1$ for all Euclidean spaces and $k$ for uniform metric spaces. For the latter, we give a matching $k$-competitive deterministic algorithm. Our most technical result, proven by applying Yao's principle to a suitable instance distribution on a specifically constructed metric space, is a lower bound of $k+\mathcal{O}(\log k)$ that holds even for randomized algorithms, which contrasts with the best known lower bound for the classical problem that remains polylogarithmic. With this paper, we hope to initiate a further study of this natural yet neglected problem.

Figures (5)

• Figure 1: Example of the subdivision of phase-initial server positions and the corresponding ranges used in the proof of \ref{['thm:kplusoneline']}. (Note that the figure does not show the entire space, which is a cycle or the infinite line, but only a segment.)
• Figure 2: The double cycle metric space used in the proof of \ref{['lem:doublecircle']}.
• Figure 3: The double cycle chain of length $k/2$ used in \ref{['thm:threehalfk']}.
• Figure 4: A small part of the construction used to prove \ref{['thm:lowertwokminusonefinite']} for $k=3$. One of the $k(k-1)^{k-1}=12$ blocks of one layer at the top (with the hub point on the left and the fringe points in $k$ groups of $k-1$ to the right), two blocks of the next layer at the bottom left and bottom right, and the edges induced by the corresponding choices of $k-1$ groups and one fringe point per chosen group in the block above; with dashed edges for the choice represented by the bottom right block.
• Figure 5: The graph describing the metric space used to prove \ref{['thm:lowertwokminusonefinite']} for $k=2$ and $k=3$. The layers $L_1$, $L_2$, and $L_3$ are arranged counterclockwise with $L_1$ at the top. Edges from $L_1$ to $L_2$ are shown in blue, those from $L_2$ to $L_3$ in red, and those from $L_3$ to $L_1$ in green.

Theorems & Definitions (38)

• Conjecture 1: k-Server Conjecture
• Conjecture 2: Randomized k-Server Conjecture
• Definition 1: Algorithm Robin
• Theorem 1
• proof
• Corollary 1
• Theorem 2
• proof
• Theorem 3
• proof