Parallel Graph Connectivity in Log Diameter Rounds

We study graph connectivity problem in MPC model. On an undirected graph with $n$ nodes and $m$ edges, $O(\log n)$ round connectivity algorithms have been known for over 35 years. However, no algorithms with better complexity bounds were known. In this work, we give fully scalable, faster algorithms for the connectivity problem, by parameterizing the time complexity as a function of the diameter of the graph. Our main result is a $O(\log D \log\log_{m/n} n)$ time connectivity algorithm for diameter-$D$ graphs, using $\Theta(m)$ total memory. If our algorithm can use more memory, it can terminate in fewer rounds, and there is no lower bound on the memory per processor. We extend our results to related graph problems such as spanning forest, finding a DFS sequence, exact/approximate minimum spanning forest, and bottleneck spanning forest. We also show that achieving similar bounds for reachability in directed graphs would imply faster boolean matrix multiplication algorithms. We introduce several new algorithmic ideas. We describe a general technique called double exponential speed problem size reduction which roughly means that if we can use total memory $N$ to reduce a problem from size $n$ to $n/k$, for $k=(N/n)^{\Theta(1)}$ in one phase, then we can solve the problem in $O(\log\log_{N/n} n)$ phases. In order to achieve this fast reduction for graph connectivity, we use a multistep algorithm. One key step is a carefully constructed truncated broadcasting scheme where each node broadcasts neighbor sets to its neighbors in a way that limits the size of the resulting neighbor sets. Another key step is random leader contraction, where we choose a smaller set of leaders than many previous works do

Numerical simulation of the optimal two-mode attacks for two-way continuous-variable quantum cryptography in reverse reconciliation

We analyze the security of the two-way continuous-variable quantum key distribution protocol in reverse reconciliation against general two-mode attacks, which represent all accessible attacks at fixed channel parameters. Rather than against one specific attack model, the expression of secret key rates of the two-way protocol are derived against all accessible attack models. It is found that there is an optimal two-mode attack to minimize the performance of the protocol in terms of both secret key rates and maximal transmission distances. We identify the optimal two-mode attack, give the specific attack model of the optimal two-mode attack and show the performance of the two-way protocol against the optimal two-mode attack. Even under the optimal two-mode attack, the performances of two-way protocol are still better than the corresponding one-way protocol, which shows the advantage of making a double use of the quantum channel and the potential of long-distance secure communication using two-way protocol.Comment: 14 pages, 8 figure

Improvement of two-way continuous-variable quantum key distribution with virtual photon subtraction

We propose a method to improve the performance of two-way continuous-variable quantum key distribution protocol by virtual photon subtraction. The Virtual photon subtraction implemented via non-Gaussian post-selection not only enhances the entanglement of two-mode squeezed vacuum state but also has advantages in simplifying physical operation and promoting efficiency. In two-way protocol, virtual photon subtraction could be applied on two sources independently. Numerical simulations show that the optimal performance of renovated two-way protocol is obtained with photon subtraction only used by Alice. The transmission distance and tolerable excess noise are improved by using the virtual photon subtraction with appropriate parameters. Moreover, the tolerable excess noise maintains a high value with the increase of distance so that the robustness of two-way continuous-variable quantum key distribution system is significantly improved, especially at long transmission distance.Comment: 15 pages, 6 figure

Distributed Resource Allocation Assisted by Intercell Interference Mitigation in Downlink Multicell MC DS-CDMA Systems

This paper investigates the allocation of resources, including subcarriers and spreading codes, as well as intercell interference (ICI) mitigation for multicell downlink multicarrier direct-sequence code division multiple-access systems, which aim to maximize the system's spectral efficiency (SE). The analytical benchmark scheme for resource allocation and ICI mitigation is derived by solving or closely solving a series of mixed integer non-convex optimization problems. Based on the optimization objectives the same as the benchmark scheme, we propose a novel distributed resource allocation assisted by ICI mitigation scheme referred to as resource allocation assisted by ICI mitigation (RAIM), which requires very low implementation complexity and demands little backhaul resource. Our RAIM algorithm is a fully distributed algorithm, which consists of the subcarrier allocation (SA) algorithm named RAIM-SA, spreading code allocation (CA) algorithm called RAIM-CA and the ICI mitigation algorithm termed RAIM-IM. The advantages of the RAIM are that its CA only requires limited binary ICI information of intracell channels, and it is able to make mitigation decisions without any knowledge of ICI information. Our simulation results show that the proposed RAIM scheme, with very low complexity required, achieves significantly better SE performance than other existing schemes, and its performance is very close to that obtained by the benchmark scheme

Distributed Resource Allocation Assisted by Intercell Interference Mitigation in Downlink Multicell MC DS-CDMA Systems

This paper investigates the allocation of resources, including subcarriers and spreading codes, as well as intercell interference (ICI) mitigation for multicell downlink multicarrier direct-sequence code division multiple-access systems, which aim to maximize the system's spectral efficiency (SE). The analytical benchmark scheme for resource allocation and ICI mitigation is derived by solving or closely solving a series of mixed integer non-convex optimization problems. Based on the optimization objectives the same as the benchmark scheme, we propose a novel distributed resource allocation assisted by ICI mitigation scheme referred to as resource allocation assisted by ICI mitigation (RAIM), which requires very low implementation complexity and demands little backhaul resource. Our RAIM algorithm is a fully distributed algorithm, which consists of the subcarrier allocation (SA) algorithm named RAIM-SA, spreading code allocation (CA) algorithm called RAIM-CA and the ICI mitigation algorithm termed RAIM-IM. The advantages of the RAIM are that its CA only requires limited binary ICI information of intracell channels, and it is able to make mitigation decisions without any knowledge of ICI information. Our simulation results show that the proposed RAIM scheme, with very low complexity required, achieves significantly better SE performance than other existing schemes, and its performance is very close to that obtained by the benchmark scheme