Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

Communication Complexity: Basics

What is discrete math? • The real numbers are continuous in the senses that: * between any two real numbers there is a real number • The integers do not share this property. In this sense the integers are lumpy, or " discrete " So discrete math is the study of mathematical objects that are discrete. " It's all the math that counts " Some discrete mathematical concepts: • Integers: Between two integers there is not another integer. • Propositions: Either true or false, there are no 1/2 truths (in math) • Sets: An item is either in a set or not in a set, never partly in and partly out. • Relations: A pair of items are related or not. • Networks (graphs): Between two terminals of a network connection there are no terminals. Propositional Logic Propositions are statements that are either true or false. Principles: Substituting an equivalent statement. Replacing a logic variable in a tautology. Defn algebraic proof

Discrete math = 離散数学

The Internet of People (IoP): A new wave in pervasive mobile computing

We present two algorithms for dynamically maintaining a spanning forest of a graph undergoing edge insertions and deletions. Our algorithms guarantee worst-case update time and work against an adaptive adversary, meaning that an edge update can depend on previous outputs of the algorithms. We provide the first polynomial improvement over the long-standing O(√n) bound of [Frederickson STOC'84, Eppstein, Galil, Italiano and Nissenzweig FOCS'92] for such type of algorithms. The previously best improvement was O(√n (loglogn)2/logn) [Kejlberg-Rasmussen, Kopelowitz, Pettie and Thorup ESA'16]. We note however that these bounds were obtained by deterministic algorithms while our algorithms are randomized. Our first algorithm is Monte Carlo and guarantees an O(n0.4+o(1)) worst-case update time, where the o(1) term hides the O(√loglogn/logn) factor. Our second algorithm is Las Vegas and guarantee an O(n0.49306) worst-case update time with high probability. Algorithms with better update time either needed to assume that the adversary is oblivious (e.g. [Kapron, King and Mountjoy SODA'13]) or can only guarantee an amortized update time. Our second result answers an open problem by Kapron et al. To the best of our knowledge, our algorithms are among a few non-trivial randomized dynamic algorithms that work against adaptive adversaries. The key to our results is a decomposition of graphs into subgraphs that either have high expansion or sparse. This decomposition serves as an interface between recent developments on (static) flow computation and many old ideas in dynamic graph algorithms: On the one hand, we can combine previous dynamic graph techniques to get faster dynamic spanning forest algorithms if such decomposition is given. On the other hand, we can adapt flow-related techniques (e.g. those from [Khandekar, Rao and Vazirani STOC'06], [Peng SODA'16], and [Orecchia and Zhu SODA'14]) to maintain such decomposition. To the best of our knowledge, this is the first time these flow techniques are used in fully dynamic graph algorithms.

Dynamic spanning forest with worst-case update time: adaptive, Las Vegas, and O(n1/2 - ε)-time

We undertake a systematic study of sketching a quadratic form: given an n x n matrix A, create a succinct sketch sk(A) which can produce (without further access to A) a multiplicative (1+e)-approximation to xT A x for any desired query x ∈ Rn. While a general matrix does not admit non-trivial sketches, positive semi-definite (PSD) matrices admit sketches of size θ(e{-2n), via the Johnson-Lindenstrauss lemma, achieving the "for each" guarantee, namely, for each query x, with a constant probability the sketch succeeds. (For the stronger "for all" guarantee, where the sketch succeeds for all x's simultaneously, again there are no non-trivial sketches.)We design significantly better sketches for the important subclass of graph Laplacian matrices, which we also extend to symmetric diagonally dominant matrices. A sequence of work culminating in that of Batson, Spielman, and Srivastava (SIAM Review, 2014), shows that by choosing and reweighting O(e{-2n) edges in a graph, one achieves the "for all" guarantee. Our main results advance this front. For the "for all" guarantee, we prove that Batson et al.'s bound is optimal even when we restrict to "cut queries" x ∈ (0,1}n. Specifically, an arbitrary sketch that can (1+e)-estimate the weight of all cuts (S, bar S) in an n-vertex graph must be of size Ω(e{-2n) bits. Furthermore, if the sketch is a cut-sparsifier (i.e., itself a weighted graph and the estimate is the weight of the corresponding cut in this graph), then the sketch must have Ω(e{-2n) edges.In contrast, previous lower bounds showed the bound only for spectral-sparsifiers.For the "for each" guarantee, we design a sketch of size O(e{-1n) bits for "cut queries" x ∈{0,1}n. We apply this sketch to design an algorithm for the distributed minimum cut problem. We prove a nearly-matching lower bound of Ω(e{-1n) bits. For general queries x ∈ Rn, we construct sketches of size O(e{-1.6n) bits.Our results provide the first separation between the sketch size needed for the "for all" and "for each" guarantees for Laplacian matrices.

/pdf/on-sketching-quadratic-forms-59c086xr5d.pdf

On Sketching Quadratic Forms

This paper presents the design, analysis, and implementation of a novel data center network architecture, named NovaCube. Based on regular Torus topology, NovaCube is constructed by adding a number of most beneficial jump-over links, which offers many distinct advantages and practical benefits. Moreover, in order to enable NovaCube to achieve its maximum theoretical performance, a probabilistic oblivious routing algorithm PORA is carefully designed. PORA is a both deadlock and livelock free routing algorithm, which achieves near-optimal performance in terms of average routing path length with better load balancing thus leading to higher throughput. Theoretical derivation and mathematical analysis further prove the good performance of NovaCube and PORA.

NovaCube: A low latency Torus-based network architecture for data centers

The data center network connecting the servers in a data center plays a crucial role in orchestrating the infrastructure to deliver peak performance to users. In order to meet high performance and reliability requirements, the data center network is usually constructed of a massive number of network devices and links to achieve 1:1 oversubscription for peak workload. However, traffic rarely ever hits the peak capacity in practice and the links are underutilized most of the time, which results in an enormous waste of energy. Therefore, aiming to achieve an energy proportional data center network without compromising throughput and fault tolerance too much, in this paper we propose two efficient schemes from the perspective of resource allocation, routing and flow scheduling. We mathematically formulate the energy optimization problem as a multi-commodity minimum cost flow problem, and prove its NP-hardness. Then we propose a heuristic solution with high computational efficiency by applying an AI resource abstraction technique. Additionally, we design a practical topology-based solution with the benefit of Random Packet Spraying consistent with multipath routing protocols. Both simulations and theoretical analysis have been conducted to demonstrate the feasibility and convincing performance of our frameworks.

/pdf/towards-bandwidth-guaranteed-energy-efficient-data-center-18aomul31u.pdf

Towards bandwidth guaranteed energy efficient data center networking

Towards cost-effective and low latency data center network architecture

Embedding multiple virtual networks (VNs) onto a shared substrate by allocating substrate resources to virtual nodes and virtual links of VN requests under a collection of constrains is known to be an NP-hard problem even for the offline VN embedding. To deal with this issue, this paper formulates the VN embedding problem as a multiple objective linear programming optimization program, and solves it in a preemptive strategy by decomposing the problem into node mapping and link mapping phases. Furthermore, based on an AI model, named Blocking Island, we propose an efficient online heuristic VN embedding framework called Presto. Presto operates with quite low computation complexity and greatly reduces the search space, which far outperforms other candidates. The goal of Presto is to maximize the economic revenue of infrastructure providers while minimizing the embedding cost. The extensive simulation results further prove the feasibility and good performance of Presto.

Bo Qin

Papers

On Sketching Quadratic Forms

NovaCube: A low latency Torus-based network architecture for data centers

Towards bandwidth guaranteed energy efficient data center networking

Towards cost-effective and low latency data center network architecture

An efficient framework for online virtual network embedding in virtualized cloud data centers