Bisection (Band)Width of Product Networks with Application to Data Centers

The bisection width of interconnection networks has always been important in parallel computing, since it bounds the amount of information that can be moved from one side of a network to another, i.e., the bisection bandwidth. Finding its exact value has proven to be challenging for some network families. For instance, the problem of finding the exact bisection width of the multidimensional torus was posed by Leighton and has remained open for almost 20 years. In this paper we provide the exact value of the bisection width of the torus, as well as of several d-dimensional classical parallel topologies that can be obtained by the application of the Cartesian product of graphs. To do so, we first provide two general results that allow to obtain upper and lower bounds on the bisection width of a product graph as a function of some properties of its factor graphs. We also apply these results to obtain bounds for the bisection bandwidth of a d-dimensional BCube network, a recently proposed topology for data centers.

💡 Research Summary

The paper investigates the bisection width (BW) and bisection bandwidth (BBW) of interconnection networks that can be expressed as Cartesian products of simpler factor graphs, a class commonly referred to as product networks. The motivation stems from the fact that BW measures the minimum number of links that must be cut to split a network into two equal halves, and BBW translates this structural property into a performance metric (the amount of data that can be transferred across the cut when all links operate at full capacity). While BW has been extensively used to evaluate parallel architectures, NoC designs, and more recently data‑center topologies, exact values are known only for a few families; many classic networks remain unresolved.

The authors first introduce two graph‑theoretic parameters that capture essential properties of a factor graph: (i) normalized congestion β_r(G), defined as the minimum, over all embeddings of an r‑complete multigraph rK_n onto G, of the maximum number of embedded paths that traverse any edge, divided by a dimension‑specific normalization factor σ_n; and (ii) central cut CC(G), the size of a minimum edge cut that bisects G. Using these, they prove two general theorems: a lower‑bound theorem that expresses BW of a product graph in terms of the smallest β among its factors, and an upper‑bound theorem that expresses BW as a sum of the central cuts of the factors weighted by the sizes of the other dimensions. These results reduce the problem of evaluating BW for high‑dimensional product networks to computing simple quantities on the one‑dimensional factors.

Armed with the general bounds, the paper derives exact BW formulas for several important families:

• d‑dimensional arrays (grids) A(d){k1,…,kd} = P{k1}×…×P_{kd}. The exact BW equals Ψ(α)=∏{i=1}^{α}∏{j=i+1}^{d}k_j, where α is the smallest dimension index with an even number of nodes (α = d if none).

• d‑dimensional tori T(d){k1,…,kd} = C{k1}×…×C_{kd}. The long‑standing open problem posed by Leighton is solved: BW(T) = 2·Ψ(α). The factor 2 reflects the two opposite “rings” that cross any separating hyperplane.

• Products of complete binary trees (CBT) and extended CBTs (CBTs whose leaves are linked by a path). Their BW also follows the Ψ(α) pattern, with constant multipliers 1/4 or 1/2 depending on whether the factor is a pure CBT or an extended CBT, and whether the product includes paths or rings.

• Mixed products such as CBT×Path and extended CBT×Ring are handled similarly, yielding BW = (constant)·Ψ(α).

The authors then turn to the BCube topology, a recent data‑center architecture built as a Cartesian product of “k‑port switches” connecting k servers per dimension. Because switches are not counted as vertices, the simple relation BBW = 2·T·BW (where T is link capacity) no longer holds. The paper models two extreme bottleneck scenarios: (a) link‑limited, where the total traffic crossing the cut is constrained solely by the aggregate link capacity, and (b) switch‑limited, where the per‑switch switching capacity s dominates. For both even and odd values of k, they derive upper and lower bounds on BBW that differ by at most a factor of two. For example, in the even‑k, link‑limited case BBW lies between (k−1)·k^{d−1}·T/4 and 2·k^{d}·T, while in the switch‑limited case BBW is bounded by (k−1)·k^{d−1}·s/2 ≤ BBW ≤ k^{d−1}·s. These results give designers a clear quantitative picture of how many servers can be supported across a cut, depending on whether the network is switch‑ or link‑constrained.

The paper concludes by emphasizing that the two parameters β and CC provide a unified, tractable framework for evaluating BW (and consequently BBW) across a wide spectrum of product networks. The exact solution for torus BW resolves a two‑decade‑old open problem, and the BCube analysis offers practical guidance for data‑center architects. Future work is suggested on heterogeneous factor graphs, dynamic traffic patterns, and experimental validation on real hardware.