Degree-3 Treewidth Sparsifiers

We study treewidth sparsifiers. Informally, given a graph $G$ of treewidth $k$, a treewidth sparsifier $H$ is a minor of $G$, whose treewidth is close to $k$, $|V(H)|$ is small, and the maximum vertex degree in $H$ is bounded. Treewidth sparsifiers of degree $3$ are of particular interest, as routing on node-disjoint paths, and computing minors seems easier in sub-cubic graphs than in general graphs. In this paper we describe an algorithm that, given a graph $G$ of treewidth $k$, computes a topological minor $H$ of $G$ such that (i) the treewidth of $H$ is $\Omega(k/\text{polylog}(k))$; (ii) $|V(H)| = O(k^4)$; and (iii) the maximum vertex degree in $H$ is $3$. The running time of the algorithm is polynomial in $|V(G)|$ and $k$. Our result is in contrast to the known fact that unless $NP \subseteq coNP/{\sf poly}$, treewidth does not admit polynomial-size kernels. One of our key technical tools, which is of independent interest, is a construction of a small minor that preserves node-disjoint routability between two pairs of vertex subsets. This is closely related to the open question of computing small good-quality vertex-cut sparsifiers that are also minors of the original graph.Comment: Extended abstract to appear in Proceedings of ACM-SIAM SODA 201

Optimal Networks from Error Correcting Codes

To address growth challenges facing large Data Centers and supercomputing clusters a new construction is presented for scalable, high throughput, low latency networks. The resulting networks require 1.5-5 times fewer switches, 2-6 times fewer cables, have 1.2-2 times lower latency and correspondingly lower congestion and packet losses than the best present or proposed networks providing the same number of ports at the same total bisection. These advantage ratios increase with network size. The key new ingredient is the exact equivalence discovered between the problem of maximizing network bisection for large classes of practically interesting Cayley graphs and the problem of maximizing codeword distance for linear error correcting codes. Resulting translation recipe converts existent optimal error correcting codes into optimal throughput networks.Comment: 14 pages, accepted at ANCS 2013 conferenc

Network protocol scalability via a topological Kadanoff transformation

A natural hierarchical framework for network topology abstraction is presented based on an analogy with the Kadanoff transformation and renormalisation group in theoretical physics. Some properties of the renormalisation group bear similarities to the scalability properties of network routing protocols (interactions). Central to our abstraction are two intimately connected and complementary path diversity units: simple cycles, and cycle adjacencies. A recursive network abstraction procedure is presented, together with an associated generic recursive routing protocol family that offers many desirable features.Comment: 4 pages, 5 figures, PhysComNet 2008 workshop submissio

Precoding-Based Network Alignment For Three Unicast Sessions

We consider the problem of network coding across three unicast sessions over a directed acyclic graph, where each sender and the receiver is connected to the network via a single edge of unit capacity. We consider a network model in which the middle of the network only performs random linear network coding, and restrict our approaches to precoding-based linear schemes, where the senders use precoding matrices to encode source symbols. We adapt a precoding-based interference alignment technique, originally developed for the wireless interference channel, to construct a precoding-based linear scheme, which we refer to as as a {\em precoding-based network alignment scheme (PBNA)}. A primary difference between this setting and the wireless interference channel is that the network topology can introduce dependencies between elements of the transfer matrix, which we refer to as coupling relations, and can potentially affect the achievable rate of PBNA. We identify all possible such coupling relations, and interpret these coupling relations in terms of network topology and present polynomial-time algorithms to check the presence of these coupling relations. Finally, we show that, depending on the coupling relations present in the network, the optimal symmetric rate achieved by precoding-based linear scheme can take only three possible values, all of which can be achieved by PBNA.Comment: arXiv admin note: text overlap with arXiv:1202.340

Hybrid routing technique for a fault-tolerant, integrated information network

The evolutionary growth of the space station and the diverse activities onboard are expected to require a hierarchy of integrated, local area networks capable of supporting data, voice, and video communications. In addition, fault-tolerant network operation is necessary to protect communications between critical systems attached to the net and to relieve the valuable human resources onboard the space station of time-critical data system repair tasks. A key issue for the design of the fault-tolerant, integrated network is the development of a robust routing algorithm which dynamically selects the optimum communication paths through the net. A routing technique is described that adapts to topological changes in the network to support fault-tolerant operation and system evolvability

A practical fpt algorithm for Flow Decomposition and transcript assembly

The Flow Decomposition problem, which asks for the smallest set of weighted paths that "covers" a flow on a DAG, has recently been used as an important computational step in transcript assembly. We prove the problem is in FPT when parameterized by the number of paths by giving a practical linear fpt algorithm. Further, we implement and engineer a Flow Decomposition solver based on this algorithm, and evaluate its performance on RNA-sequence data. Crucially, our solver finds exact solutions while achieving runtimes competitive with a state-of-the-art heuristic. Finally, we contextualize our design choices with two hardness results related to preprocessing and weight recovery. Specifically, $k$-Flow Decomposition does not admit polynomial kernels under standard complexity assumptions, and the related problem of assigning (known) weights to a given set of paths is NP-hard.Comment: Introduces software package Toboggan: Version 1.0. http://dx.doi.org/10.5281/zenodo.82163

A Force-Directed Approach for Offline GPS Trajectory Map Matching

We present a novel algorithm to match GPS trajectories onto maps offline (in batch mode) using techniques borrowed from the field of force-directed graph drawing. We consider a simulated physical system where each GPS trajectory is attracted or repelled by the underlying road network via electrical-like forces. We let the system evolve under the action of these physical forces such that individual trajectories are attracted towards candidate roads to obtain a map matching path. Our approach has several advantages compared to traditional, routing-based, algorithms for map matching, including the ability to account for noise and to avoid large detours due to outliers in the data whilst taking into account the underlying topological restrictions (such as one-way roads). Our empirical evaluation using real GPS traces shows that our method produces better map matching results compared to alternative offline map matching algorithms on average, especially for routes in dense, urban areas.Comment: 10 pages, 12 figures, accepted version of article submitted to ACM SIGSPATIAL 2018, Seattle, US