Augmenting the algebraic connectivity of graphs

For any undirected graph G=(V,E) and a set EW of candidate edges with E∩EW=∅, the (k,γ)-spectral augmentability problem is to find a set F of k edges from EW with appropriate weighting, such that the algebraic connectivity of the resulting graph H=(V,E∪F) is least γ. Because of a tight connection between the algebraic connectivity and many other graph parameters, including the graph's conductance and the mixing time of random walks in a graph, maximising the resulting graph's algebraic connectivity by adding a small number of edges has been studied over the past 15 years. In this work we present an approximate and efficient algorithm for the (k,γ)-spectral augmentability problem, and our algorithm runs in almost-linear time under a wide regime of parameters. Our main algorithm is based on the following two novel techniques developed in the paper, which might have applications beyond the (k,γ)-spectral augmentability problem. (1) We present a fast algorithm for solving a feasibility version of an SDP for the algebraic connectivity maximisation problem from [GB06]. Our algorithm is based on the classic primal-dual framework for solving SDP, which in turn uses the multiplicative weight update algorithm. We present a novel approach of unifying SDP constraints of different matrix and vector variables and give a good separation oracle accordingly. (2) We present an efficient algorithm for the subgraph sparsification problem, and for a wide range of parameters our algorithm runs in almost-linear time, in contrast to the previously best known algorithm running in at least Ω(n2mk) time [KMST10]. Our analysis shows how the randomised BSS framework can be generalised in the setting of subgraph sparsification, and how the potential functions can be applied to approximately keep track of different subspaces

SCDP: systematic rateless coding for efficient data transport in data centres

In this paper we propose SCDP, a general-purpose data transport protocol for data centres that, in contrast to all other protocols proposed to date, supports efficient one-to-many and many-to-one communication, which is extremely common in modern data centres. SCDP does so without compromising on efficiency for short and long unicast flows. SCDP achieves this by integrating RaptorQ codes with receiver-driven data transport, packet trimming and Multi-Level Feedback Queuing (MLFQ); (1) RaptorQ codes enable efficient one-to-many and many-to-one data transport; (2) on top of RaptorQ codes, receiver-driven flow control, in combination with in-network packet trimming, enable efficient usage of network resources as well as multi-path transport and packet spraying for all transport modes. Incast and Outcast are eliminated; (3) the systematic nature of RaptorQ codes, in combination with MLFQ, enable fast, decoding-free completion of short flows. We extensively evaluate SCDP in a wide range of simulated scenarios with realistic data centre workloads. For one-to-many and many-to-one transport sessions, SCDP performs significantly better compared to NDP and PIAS. For short and long unicast flows, SCDP performs equally well or better compared to NDP and PIAS