Toward Self-Adjusting k-ary Search Tree Networks

TL;DR

The paper tackles optimizing programmable datacenter topologies under traffic patterns by proposing self-adjusting k-ary search tree networks. It combines an offline DP approach for static optimal topologies with online self-adjusting methods, including a novel k-ary SplayNet and a centroid-based variant, to adapt to changing demand. The authors establish complexity bounds, present linear-time centroid constructions for uniform workloads, and demonstrate through experiments that k-ary SANs often outperform classic SplayNets, especially at moderate locality levels. The work advances practical SAN design by enabling higher-degree, locally routable topologies with persistent node identifiers and adaptive rotations, offering meaningful routing-cost improvements in real and synthetic traces.

Abstract

Datacenter networks are becoming increasingly flexible with the incorporation of new networking technologies, such as optical circuit switches. These technologies allow for programmable network topologies that can be reconfigured to better serve network traffic, thus enabling a trade-off between the benefits (i.e., shorter routes) and costs of reconfigurations (i.e., overhead). Self-Adjusting Networks (SANs) aim at addressing this trade-off by exploiting patterns in network traffic, both when it is revealed piecewise (online dynamic topologies) or known in advance (offline static topologies). In this paper, we take the first steps toward Self-Adjusting k-ary tree networks. These are more powerful generalizations of existing binary search tree networks (like SplayNets), which have been at the core of SAN designs. k-ary search tree networks are a natural generalization offering nodes of higher degrees, reduced route lengths for a fixed number of nodes, and local routing in spite of reconfigurations. We first compute an offline (optimal) static network for arbitrary traffic patterns in $O(n^3 \cdot k)$ time via dynamic programming, and also improve the bound to $O(n^2 \cdot k)$ for the special case of uniformly distributed traffic. Then, we present a centroid-based topology of the network that can be used both in the offline static and the online setting. In the offline uniform-workload case, we construct this quasi-optimal network in linear time $O(n)$ and, finally, we present online self-adjusting k-ary search tree versions of SplayNet. We evaluate experimentally our new structure for $k=2$ (allowing for a comparison with existing SplayNets) on real and synthetic network traces. Our results show that this approach works better than SplayNet in most of the real network traces and in average to low locality synthetic traces, and is only little inferior to SplayNet in all remaining traces.

Figures (12)

• Figure 1: Node in $k$-ary search tree
• Figure 2: A centroid tree after the reposition of subtrees. Lined rectangles represent leaves.
• Figure 3: The initial state for $k$-semi-splay.
• Figure 4: Example state before $k$-splay.
• Figure 5: $k$-splay rotation. The first case.

Theorems & Definitions (35)

• Definition 1
• Theorem 2
• Remark 3
• Theorem 4
• Definition 5
• Theorem 6
• Remark 7
• Theorem 8
• Lemma 9
• Remark 10