(Almost) Perfect Discrete Iterative Load Balancing

TL;DR

This paper addresses discrete, iterative load balancing on arbitrary graphs via matchings, proving that a simple local scheme achieves discrepancy at most 3 with high probability in a time matching the continuous model’s spectral bound. The authors introduce a height-sensitive process and a negative-association property for token movements, coupled with a novel Hoeffding-type concentration bound for convex load combinations, to tightly relate discrete and continuous balancing. The main theorem applies simultaneously to random matchings, fixed balancing-circuit sequences, and the asynchronous single-edge model, and yields explicit, small constants on both runtime and discrepancy, extending prior results that were limited to regular graphs or lacked explicit guarantees. This work thus demonstrates that discrete load balancing on general graphs can be as efficient as continuous balancing under a broad, unified framework, with strong probabilistic guarantees and practical distributed applicability.

Abstract

We consider discrete, iterative load balancing via matchings on arbitrary graphs. Initially each node holds a certain number of tokens, defining the load of the node, and the objective is to redistribute the tokens such that eventually each node has approximately the same number of tokens. We present results for a general class of simple local balancing schemes where the tokens are balanced via matchings. In each round the process averages the tokens of any two matched nodes. If the sum of their tokens is odd, the node to receive the one excess token is selected at random. Our class covers three popular models: in the matching model a new matching is generated randomly in each round, in the balancing circuit model a fixed sequence of matchings is applied periodically, and in the asynchronous model the load is balanced over a randomly chosen edge. We measure the quality of a load vector by its discrepancy, defined as the difference between the maximum and minimum load across all nodes. As our main result we show that with high probability our discrete balancing scheme reaches a discrepancy of $3$ in a number of rounds which asymptotically matches the spectral bound for continuous load balancing with fractional load. This result improves and tightens a long line of previous works, by not only achieving a small constant discrepancy (instead of a non-explicit, large constant) but also holding for arbitrary instead of regular graphs. The result also demonstrates that in the general model we consider, discrete load balancing is no harder than continuous load balancing.

Figures (3)

• Figure 2.1: Illustration of the height-sensitive process and the effect on the tokens, which are labelled from $1$ to $9$. $(a)$: The configuration of tokens before the averaging; $(b)$: the configuration after the moving step and $(c)$: the configuration after the shuffling step.
• Figure 3.1: Illustration of Phases $1$ and $2$ in the proof of \ref{['lem:NtokensTo25Height']}. Phase 1 decreases $Y^{(t)}$, the number of tokens with height at least $L+1$, from $n \cdot L$ to $\frac{n}{\log(n)}$. Then Phase 2 decreases $\widehat{Y}^{(t)}$, the number of tokens with height at least $L+2$, to $0$. Both phases first use $\tau_{\mathrm{global}}$ rounds for a "global mixing", and then $\ell:=\frac{\log(n)}{\log \log(n)}$ short epochs of length $\tau_{\mathrm{local}}$.
• Figure 6.1: Illustration of the events $\textcolor{red!80!black}{\Gamma_{\ell}^{(e(k)}(u)}$, $\textcolor{green!80!black}{\Gamma_{g}^{(e(k-1))}}$ and $\textcolor{blue}{\mathcal{L}^{(e(k))}(u)}$, where $L=2$. The token $i$, marked in red, is located at node $u$ in round $e(k-1)$ and has height $L+1=3$. Its location $v$ in round $e(k)$ is random and chosen according to $\mathbf{M}_{u,.}^{E(k)}$. In order to keep its height $L+1=3$, there must be at least $L=2$ other tokens on $v$ in round $e(k)$. Roughly speaking, the event $\textcolor{blue}{\mathcal{L}^{(e(k))}(u)}$ corresponds to a certain upper bound on a convex combination of loads at round $e(k-1)$ holding; this event depends both on the matchings and shuffling decisions in $[0,e(k-1)]$ and the matchings in $E(k)$. The coefficients of the load vector are collision probabilities so that the left-hand side equals the expected number of tokens that reach the (random) location of token $i$ in round $e(k)$. We prove in \ref{['lem:crucialfirststatement']} that if $\textcolor{red!80!black}{\Gamma_{\ell}^{(e(k))}(u)}$ and $\textcolor{green!80!black}{\Gamma_{g}^{(e(k-1)}}$ occur, then also $\textcolor{blue}{\mathcal{L}^{(e(k))}(u)}$ occurs (with high probability). Then, conditional on this implication, we prove in our key lemma (\ref{['lem:generallemmaone:RM']}) that the number of tokens with height $L+1=3$ drops within $E(k)$ by a suitable factor.

Theorems & Definitions (64)

• Definition 2.1: DBLP:conf/focs/SauerwaldS12
• Definition 2.2
• Theorem 2.3
• Lemma 3.1: simplified version of \ref{['lem:height']}
• Lemma 3.2: simplified version of \ref{['ChernoffBound']}
• Lemma 3.3: simplified version of \ref{['lem:toNToken']}
• Proposition 3.4: simplified and informal version of \ref{['lem:NtokensTo25Height']}
• Lemma 3.5: simplified and informal version of \ref{['lem:generallemmaone:RM']} (Key Lemma)
• Lemma 5.1: Height-Sensitive Negative Association
• Proof 1