Birkhoff’s Decomposition Revisited: Sparse Scheduling for High-Speed Circuit Switches

Abstract

Data centers are increasingly using high-speed circuit switches to cope with the growing demand and reduce operational costs. One of the fundamental tasks of circuit switches is to compute a sparse collection of switching configurations to support a traffic demand matrix. Such a problem has been addressed in the literature with variations of the approach proposed by Birkhoff in 1946 to decompose a doubly stochastic matrix exactly. However, the existing methods are heuristic and do not have theoretical guarantees on how well a collection of switching configurations (i.e., permutations) can approximate a traffic matrix (i.e., a scaled doubly stochastic matrix). In this paper, we revisit Birkhoff's approach and make three contributions. First, we establish the first theoretical bound on the sparsity of Birkhoff's algorithm (i.e., the number of switching configurations necessary to approximate a traffic matrix). In particular, we show that by using a subset of the admissible permutation matrices, Birkhoff's algorithm obtains an -approximate decomposition with at most O(log(1/)) permutations. Second, we propose a new algorithm, Birkhoff+, which combines the wealth of Frank-Wolfe with Birkhoff's approach to obtain sparse decompositions in a fast manner. And third, we evaluate the performance of the proposed algorithm numerically and study how this affects the performance of a circuit switch. Our results show that Birkhoff+ is superior to previous algorithms in terms of throughput, running time, and number of switching configurations.

CacheNet: Leveraging the principle of locality in reconfigurable network design

“…) and the uniform matrix (i.e., M (n − 1)( as special cases. We believe this family of matrices can capture some of the complexities of real network traffic in a more concise framework and is similar in spirit to the traffic models used in [18], [20], [27], and others.…”

“…We use the demand completion time (DCT) explained below as a proxy to study the throughput. While previous analytical works studied separately topology and traffic, i.e., either systems with given static topologies [32], or dynamic topologies with naive traffic scheduling [18], [27], or only one specific kind of system [13], we extend the model to support the integration of both topology and traffic schedules. To the best of our knowledge, this is the first work that formalizes the joint optimization of both the topology and the traffic scheduling, enabling us to compare different systems: dynamic, static, demand-aware, or demand-oblivious, and with different parameters, on a simple model.…”

“…In BVN-SYS we assume that S top is generated through a BVN decomposition, and for each i in the decomposition we simply set the switch configuration C i = P i . The specific BVN algorithm can vary, but the goal is to minimize the number of reconfigurations (in the simulation, we use the state-of-the-art algorithm [27]). Since S top is based on a BVN decomposition and M is doubly stochastic the following will hold, |Stop| i=1 β i P i = M and the sum of coefficients is one, that is, i=1 β i = 1.…”

“…Our traffic generation model is based on the model used in Eclipse [18] and in [27] and denoted as TM(t l , n f , c l , n) A demand matrix is generated using the following four parameters. n f : the total number of flows per node, t l : the fraction of large flows per node, where 0 ≤ t l ≤ 1, and c l : the load of large flow traffic, e.g., where if c l = 0.7, then the large flows have 70% of all traffic in terms of bits.…”

“…Furthermore, they need to calculate a novel schedule, which may be computationally expensive. Indeed, improving the run time of the BvN decomposition is the subject of some recent works [18], [26], [27].…”