43 research outputs found

    Pruning based Distance Sketches with Provable Guarantees on Random Graphs

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    Measuring the distances between vertices on graphs is one of the most fundamental components in network analysis. Since finding shortest paths requires traversing the graph, it is challenging to obtain distance information on large graphs very quickly. In this work, we present a preprocessing algorithm that is able to create landmark based distance sketches efficiently, with strong theoretical guarantees. When evaluated on a diverse set of social and information networks, our algorithm significantly improves over existing approaches by reducing the number of landmarks stored, preprocessing time, or stretch of the estimated distances. On Erd\"{o}s-R\'{e}nyi graphs and random power law graphs with degree distribution exponent 2<β<32 < \beta < 3, our algorithm outputs an exact distance data structure with space between Θ(n5/4)\Theta(n^{5/4}) and Θ(n3/2)\Theta(n^{3/2}) depending on the value of β\beta, where nn is the number of vertices. We complement the algorithm with tight lower bounds for Erdos-Renyi graphs and the case when β\beta is close to two.Comment: Full version for the conference paper to appear in The Web Conference'1

    Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling

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    A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. An important class of distance labeling schemes is that of hub labelings, where a node vGv \in G stores its distance to the so-called hubs SvVS_v \subseteq V, chosen so that for any u,vVu,v \in V there is wSuSvw \in S_u \cap S_v belonging to some shortest uvuv path. Notice that for most existing graph classes, the best distance labelling constructions existing use at some point a hub labeling scheme at least as a key building block. Our interest lies in hub labelings of sparse graphs, i.e., those with E(G)=O(n)|E(G)| = O(n), for which we show a lowerbound of n2O(logn)\frac{n}{2^{O(\sqrt{\log n})}} for the average size of the hubsets. Additionally, we show a hub-labeling construction for sparse graphs of average size O(nRS(n)c)O(\frac{n}{RS(n)^{c}}) for some 0<c<10 < c < 1, where RS(n)RS(n) is the so-called Ruzsa-Szemer{\'e}di function, linked to structure of induced matchings in dense graphs. This implies that further improving the lower bound on hub labeling size to n2(logn)o(1)\frac{n}{2^{(\log n)^{o(1)}}} would require a breakthrough in the study of lower bounds on RS(n)RS(n), which have resisted substantial improvement in the last 70 years. For general distance labeling of sparse graphs, we show a lowerbound of 12O(logn)SumIndex(n)\frac{1}{2^{O(\sqrt{\log n})}} SumIndex(n), where SumIndex(n)SumIndex(n) is the communication complexity of the Sum-Index problem over ZnZ_n. Our results suggest that the best achievable hub-label size and distance-label size in sparse graphs may be Θ(n2(logn)c)\Theta(\frac{n}{2^{(\log n)^c}}) for some 0<c<10<c < 1

    Measuring Effectiveness of Address Schemes for AS-level Graphs

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    This dissertation presents measures of efficiency and locality for Internet addressing schemes. Historically speaking, many issues, faced by the Internet, have been solved just in time, to make the Internet just work~\cite{justWork}. Consensus, however, has been reached that today\u27s Internet routing and addressing system is facing serious scaling problems: multi-homing which causes finer granularity of routing policies and finer control to realize various traffic engineering requirements, an increased demand for provider-independent prefix allocations which injects unaggregatable prefixes into the Default Free Zone (DFZ) routing table, and ever-increasing Internet user population and mobile edge devices. As a result, the DFZ routing table is again growing at an exponential rate. Hierarchical, topology-based addressing has long been considered crucial to routing and forwarding scalability. Recently, however, a number of research efforts are considering alternatives to this traditional approach. With the goal of informing such research, we investigated the efficiency of address assignment in the existing (IPv4) Internet. In particular, we ask the question: ``how can we measure the locality of an address scheme given an input AS-level graph?\u27\u27 To do so, we first define a notion of efficiency or locality based on the average number of bit-hops required to advertize all prefixes in the Internet. In order to quantify how far from ``optimal the current Internet is, we assign prefixes to ASes ``from scratch in a manner that preserves observed semantics, using three increasingly strict definitions of equivalence. Next we propose another metric that in some sense quantifies the ``efficiency of the labeling and is independent of forwarding/routing mechanisms. We validate the effectiveness of the metric by applying it to a series of address schemes with increasing randomness given an input AS-level graph. After that we apply the metric to the current Internet address scheme across years and compare the results with those of compact routing schemes

    Beyond Highway Dimension: Small Distance Labels Using Tree Skeletons

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    International audienceThe goal of a hub-based distance labeling scheme for a network G = (V, E) is to assign a small subset S(u) ⊆ V to each node u ∈ V, in such a way that for any pair of nodes u, v, the intersection of hub sets S(u) ∩ S(v) contains a node on the shortest uv-path. The existence of small hub sets, and consequently efficient shortest path processing algorithms, for road networks is an empirical observation. A theoretical explanation for this phenomenon was proposed by Abraham et al. (SODA 2010) through a network parameter they called highway dimension, which captures the size of a hitting set for a collection of shortest paths of length at least r intersecting a given ball of radius 2r. In this work, we revisit this explanation, introducing a more tractable (and directly comparable) parameter based solely on the structure of shortest-path spanning trees, which we call skeleton dimension. We show that skeleton dimension admits an intuitive definition for both directed and undirected graphs, provides a way of computing labels more efficiently than by using highway dimension, and leads to comparable or stronger theoretical bounds on hub set size

    Engineering Algorithms for Dynamic and Time-Dependent Route Planning

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    Efficiently computing shortest paths is an essential building block of many mobility applications, most prominently route planning/navigation devices and applications. In this thesis, we apply the algorithm engineering methodology to design algorithms for route planning in dynamic (for example, considering real-time traffic) and time-dependent (for example, considering traffic predictions) problem settings. We build on and extend the popular Contraction Hierarchies (CH) speedup technique. With a few minutes of preprocessing, CH can optimally answer shortest path queries on continental-sized road networks with tens of millions of vertices and edges in less than a millisecond, i.e. around four orders of magnitude faster than Dijkstra’s algorithm. CH already has been extended to dynamic and time-dependent problem settings. However, these adaptations suffer from limitations. For example, the time-dependent variant of CH exhibits prohibitive memory consumption on large road networks with detailed traffic predictions. This thesis contains the following key contributions: First, we introduce CH-Potentials, an A*-based routing framework. CH-Potentials computes optimal distance estimates for A* using CH with a lower bound weight function derived at preprocessing time. The framework can be applied to any routing problem where appropriate lower bounds can be obtained. The achieved speedups range between one and three orders of magnitude over Dijkstra’s algorithm, depending on how tight the lower bounds are. Second, we propose several improvements to Customizable Contraction Hierarchies (CCH), the CH adaptation for dynamic route planning. Our improvements yield speedups of up to an order of magnitude. Further, we augment CCH to efficiently support essential extensions such as turn costs, alternative route computation and point-of-interest queries. Third, we present the first space-efficient, fast and exact speedup technique for time-dependent routing. Compared to the previous time-dependent variant of CH, our technique requires up to 40 times less memory, needs at most a third of the preprocessing time, and achieves only marginally slower query running times. Fourth, we generalize A* and introduce time-dependent A* potentials. This allows us to design the first approach for routing with combined live and predicted traffic, which achieves interactive running times for exact queries while allowing live traffic updates in a fraction of a minute. Fifth, we study extended problem models for routing with imperfect data and routing for truck drivers and present efficient algorithms for these variants. Sixth and finally, we present various complexity results for non-FIFO time-dependent routing and the extended problem models

    Labeled Nearest Neighbor Search and Metric Spanners via Locality Sensitive Orderings

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    Chan, Har-Peled, and Jones [SICOMP 2020] developed locality-sensitive orderings (LSO) for Euclidean space. A (τ,ρ)(\tau,\rho)-LSO is a collection Σ\Sigma of orderings such that for every x,yRdx,y\in\mathbb{R}^d there is an ordering σΣ\sigma\in\Sigma, where all the points between xx and yy w.r.t. σ\sigma are in the ρ\rho-neighborhood of either xx or yy. In essence, LSO allow one to reduce problems to the 11-dimensional line. Later, Filtser and Le [STOC 2022] developed LSO's for doubling metrics, general metric spaces, and minor free graphs. For Euclidean and doubling spaces, the number of orderings in the LSO is exponential in the dimension, which made them mainly useful for the low dimensional regime. In this paper, we develop new LSO's for Euclidean, p\ell_p, and doubling spaces that allow us to trade larger stretch for a much smaller number of orderings. We then use our new LSO's (as well as the previous ones) to construct path reporting low hop spanners, fault tolerant spanners, reliable spanners, and light spanners for different metric spaces. While many nearest neighbor search (NNS) data structures were constructed for metric spaces with implicit distance representations (where the distance between two metric points can be computed using their names, e.g. Euclidean space), for other spaces almost nothing is known. In this paper we initiate the study of the labeled NNS problem, where one is allowed to artificially assign labels (short names) to metric points. We use LSO's to construct efficient labeled NNS data structures in this model

    Doctor of Philosophy

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    dissertationNetwork emulation has become an indispensable tool for the conduct of research in networking and distributed systems. It offers more realism than simulation and more control and repeatability than experimentation on a live network. However, emulation testbeds face a number of challenges, most prominently realism and scale. Because emulation allows the creation of arbitrary networks exhibiting a wide range of conditions, there is no guarantee that emulated topologies reflect real networks; the burden of selecting parameters to create a realistic environment is on the experimenter. While there are a number of techniques for measuring the end-to-end properties of real networks, directly importing such properties into an emulation has been a challenge. Similarly, while there exist numerous models for creating realistic network topologies, the lack of addresses on these generated topologies has been a barrier to using them in emulators. Once an experimenter obtains a suitable topology, that topology must be mapped onto the physical resources of the testbed so that it can be instantiated. A number of restrictions make this an interesting problem: testbeds typically have heterogeneous hardware, scarce resources which must be conserved, and bottlenecks that must not be overused. User requests for particular types of nodes or links must also be met. In light of these constraints, the network testbed mapping problem is NP-hard. Though the complexity of the problem increases rapidly with the size of the experimenter's topology and the size of the physical network, the runtime of the mapper must not; long mapping times can hinder the usability of the testbed. This dissertation makes three contributions towards improving realism and scale in emulation testbeds. First, it meets the need for realistic network conditions by creating Flexlab, a hybrid environment that couples an emulation testbed with a live-network testbed, inheriting strengths from each. Second, it attends to the need for realistic topologies by presenting a set of algorithms for automatically annotating generated topologies with realistic IP addresses. Third, it presents a mapper, assign, that is capable of assigning experimenters' requested topologies to testbeds' physical resources in a manner that scales well enough to handle large environments
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