Fast Routing Table Construction Using Small Messages

We describe a distributed randomized algorithm computing approximate distances and routes that approximate shortest paths. Let n denote the number of nodes in the graph, and let HD denote the hop diameter of the graph, i.e., the diameter of the graph when all edges are considered to have unit weight. Given 0 < eps <= 1/2, our algorithm runs in weak-O(n^(1/2 + eps) + HD) communication rounds using messages of O(log n) bits and guarantees a stretch of O(eps^(-1) log eps^(-1)) with high probability. This is the first distributed algorithm approximating weighted shortest paths that uses small messages and runs in weak-o(n) time (in graphs where HD in weak-o(n)). The time complexity nearly matches the lower bounds of weak-Omega(sqrt(n) + HD) in the small-messages model that hold for stateless routing (where routing decisions do not depend on the traversed path) as well as approximation of the weigthed diameter. Our scheme replaces the original identifiers of the nodes by labels of size O(log eps^(-1) log n). We show that no algorithm that keeps the original identifiers and runs for weak-o(n) rounds can achieve a polylogarithmic approximation ratio. Variations of our techniques yield a number of fast distributed approximation algorithms solving related problems using small messages. Specifically, we present algorithms that run in weak-O(n^(1/2 + eps) + HD) rounds for a given 0 < eps <= 1/2, and solve, with high probability, the following problems: - O(eps^(-1))-approximation for the Generalized Steiner Forest (the running time in this case has an additive weak-O(t^(1 + 2eps)) term, where t is the number of terminals); - O(eps^(-2))-approximation of weighted distances, using node labels of size O(eps^(-1) log n) and weak-O(n^(eps)) bits of memory per node; - O(eps^(-1))-approximation of the weighted diameter; - O(eps^(-3))-approximate shortest paths using the labels 1,...,n.Comment: 40 pages, 2 figures, extended abstract submitted to STOC'1

On Efficient Distributed Construction of Near Optimal Routing Schemes

Given a distributed network represented by a weighted undirected graph $G=(V,E)$ on $n$ vertices, and a parameter $k$, we devise a distributed algorithm that computes a routing scheme in $(n^{1/2+1/k}+D)\cdot n^{o(1)}$ rounds, where $D$ is the hop-diameter of the network. The running time matches the lower bound of $\tilde{\Omega}(n^{1/2}+D)$ rounds (which holds for any scheme with polynomial stretch), up to lower order terms. The routing tables are of size $\tilde{O}(n^{1/k})$, the labels are of size $O(k\log^2n)$, and every packet is routed on a path suffering stretch at most $4k-5+o(1)$. Our construction nearly matches the state-of-the-art for routing schemes built in a centralized sequential manner. The previous best algorithms for building routing tables in a distributed small messages model were by \cite[STOC 2013]{LP13} and \cite[PODC 2015]{LP15}. The former has similar properties but suffers from substantially larger routing tables of size $O(n^{1/2+1/k})$, while the latter has sub-optimal running time of $\tilde{O}(\min\{(nD)^{1/2}\cdot n^{1/k},n^{2/3+2/(3k)}+D\})$

A Density-Based General Greedy Channel Routing Algorithm in VLSI Design Automation.

One of the most important forms of routing strategies is called channel routing . This approach allows us to reduce the extremely difficult VLSI layout problem to a collection of simpler subproblems. For channel routing problems, most frequently mentioned heuristic algorithms use parameters derived from experiments to approach the routing solution without carefully considering the effect of each selected wire segment to the final routing solution. In this dissertation, we propose a new channel routing algorithm in the two-layer restricted-Manhattan routing model (2-RM) in detail. There are three phases involved in developing the new routing algorithm. In the first phase, we distinguish one type of wire from the others using some optimality criteria, which makes the selection of a set of best horizontal wire segments for a track more effective so that good performance of the generated routing solutions can be achieved. In the second phase, we develop a theoretical framework related to two major data structures, column density and vertical constraint graph, which effectively improves search efficiency and routing performance. Finally in the third phase, we develop an efficient powerful heuristic channel routing algorithm based on the concepts shown in phase one and the theoretical framework proposed in phase two. We highlight the application of our algorithm to the channel routing problems in the three-layer restricted-Manhattan overlap (3-RM-O) and three-layer Manhattan overlay (3-M-O) routing models. On many tests we have conducted on the examples known in the literature, our algorithm has performed as well or better than the existing algorithms in both 2-RM and 3-M-O routing models. Our experiments show that our approach has the potential to outperform other algorithms in other routing models

Numerical Loop-Tree Duality: contour deformation and subtraction

We introduce a novel construction of a contour deformation within the framework of Loop-Tree Duality for the numerical computation of loop integrals featuring threshold singularities in momentum space. The functional form of our contour deformation automatically satisfies all constraints without the need for fine-tuning. We demonstrate that our construction is systematic and efficient by applying it to more than 100 examples of finite scalar integrals featuring up to six loops. We also showcase a first step towards handling non-integrable singularities by applying our work to one-loop infrared divergent scalar integrals and to the one-loop amplitude for the ordered production of two and three photons. This requires the combination of our contour deformation with local counterterms that regulate soft, collinear and ultraviolet divergences. This work is an important step towards computing higher-order corrections to relevant scattering cross-sections in a fully numerical fashion.Comment: 87 page