Perturbation-Resilient Trades for Dynamic Service Balancing

TL;DR

This work advances the theory of dynamic load-balancing trades for distributed storage by deriving near-tight analytic bounds on the worst-case block disbalance under limited-magnitude popularity swaps. It introduces a graph-based analytic framework with swap and potential graphs, enabling precise lower bounds that scale linearly with the trade parameter $t$ and swap magnitude $p$, and pairs this with constructive recursive designs that achieve close-to-optimal upper bounds. For $p=1$, the authors establish a recursive construction yielding $D^*(t,1) \le \frac{8t}{5}-\frac{2}{5}$ and a matching-like lower bound $D^*(t,1) \ge \frac{3t-2}{2}$, differing by only a constant factor $1.07$, and extend the methodology to $p\ge 2$ to obtain $D^*(t,p) \ge \frac{p[3t-2(p-1)]}{2}$ with tighter results for $p=2$. They also present an upper bound construction for $p=2$ giving $D^*(t,2) \le \frac{9t-1}{2}$ and extend the analysis to larger defining-set cardinalities, showing the framework's robustness and potential applicability to more general stability criteria in dynamic storage balancing.

Abstract

A combinatorial trade is a pair of sets of blocks of elements that can be exchanged while preserving relevant subset intersection constraints. The class of balanced and swap-robust minimal trades was proposed in [1] for exchanging blocks of data chunks stored on distributed storage systems in an access- and load-balanced manner. More precisely, data chunks in the trades of interest are labeled by popularity ranks and the blocks are required to have both balanced overall popularity and stability properties with respect to swaps in chunk popularities. The original construction of such trades relied on computer search and paired balanced sets obtained through iterative combining of smaller sets that have provable stability guarantees. To reduce the substantial gap between the results of prior approaches and the known theoretical lower bound, we present new analytical upper and lower bounds on the minimal disbalance of blocks introduced by limited-magnitude popularity ranking swaps. Our constructive and near-optimal approach relies on pairs of graphs whose vertices are two balanced sets with edges/arcs that capture the balance and potential balance changes induced by limited-magnitude popularity swaps. In particular, we show that if we start with carefully selected balanced trades and limit the magnitude of rank swaps to one, the new upper and lower bound on the maximum block disbalance caused by a swap only differ by a factor of $1.07$. We also extend these results for larger popularity swap magnitudes.

Figures (3)

• Figure 1: An example of the recursive construction for $z=4$ and $t=19$. We use the companion sets in \ref{['eq:comp16']}, denoted by $B_{16}$, as building blocks. Integers within the same companion sets have the same color. The companion sets have cardinality $2$.
• Figure 2: Two graphs $G_{\text{swp}}(I_{3,1}^*)$ and $G_{\text{pot}}(I_{3,1}^*)$ defined for the worst-case popularity swaps $I_{3,1}^*=\{(2,3),(8,9),(11,12)\}$ (i.e., $p=1$, $t=3$) obtained by computer s. Each gray circle contains $4$ elements, which constitute a pair of companion sets, and represents the same node in both graphs. The three edges in $G_{\text{swp}}(I_{3,1}^*)$ correspond to the three swaps (marked in green in $G_{\text{swp}}(I_{3,1}^*)$) in $I_{3,1}^*$. The three arcs correspond to the remaining swaps (marked in blue in $G_{\text{pot}}(I_{3,1}^*)$) not included in $I_{3,1}^*$ that cause potential discrepancy increase for different pairs of companion sets.
• Figure 3: An illustration of the recursive partition from \ref{['eq:compsestsp2']}. Integers within the same companion sets have the same color.

Theorems & Definitions (25)

• Definition 1
• Theorem 1
• Claim 1
• Claim 2
• Definition 2
• Theorem 2
• proof
• Theorem 3
• Theorem 4
• Theorem 5