70 research outputs found
Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling
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
stores its distance to the so-called hubs , chosen so that for
any there is belonging to some shortest
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 , for which we show a lowerbound of
for the average size of the hubsets.
Additionally, we show a hub-labeling construction for sparse graphs of average
size for some , where 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 would require a
breakthrough in the study of lower bounds on , which have resisted
substantial improvement in the last 70 years. For general distance labeling of
sparse graphs, we show a lowerbound of , where is the communication complexity of the
Sum-Index problem over . Our results suggest that the best achievable
hub-label size and distance-label size in sparse graphs may be
for some
Effective target arrangement in a deterministic scale-free graph
We study the random walk problem on a deterministic scale-free network, in
the presence of a set of static, identical targets; due to the strong
inhomogeneity of the underlying structure the mean first-passage time (MFPT),
meant as a measure of transport efficiency, is expected to depend sensitively
on the position of targets. We consider several spatial arrangements for
targets and we calculate, mainly rigorously, the related MFPT, where the
average is taken over all possible starting points and over all possible paths.
For all the cases studied, the MFPT asymptotically scales like N^{theta}, being
N the volume of the substrate and theta ranging from (1 - log 2/log3), for
central target(s), to 1, for a single peripheral target.Comment: 8 pages, 5 figure
Exploiting Hopsets: Improved Distance Oracles for Graphs of Constant Highway Dimension and Beyond
For fixed h >= 2, we consider the task of adding to a graph G a set of weighted shortcut edges on the same vertex set, such that the length of a shortest h-hop path between any pair of vertices in the augmented graph is exactly the same as the original distance between these vertices in G. A set of shortcut edges with this property is called an exact h-hopset and may be applied in processing distance queries on graph G. In particular, a 2-hopset directly corresponds to a distributed distance oracle known as a hub labeling. In this work, we explore centralized distance oracles based on 3-hopsets and display their advantages in several practical scenarios. In particular, for graphs of constant highway dimension, and more generally for graphs of constant skeleton dimension, we show that 3-hopsets require exponentially fewer shortcuts per node than any previously described distance oracle, and also offer a speedup in query time when compared to simple oracles based on a direct application of 2-hopsets. Finally, we consider the problem of computing minimum-size h-hopset (for any h >= 2) for a given graph G, showing a polylogarithmic-factor approximation for the case of unique shortest path graphs. When h=3, for a given bound on the space used by the distance oracle, we provide a construction of hopset achieving polylog approximation both for space and query time compared to the optimal 3-hopset oracle given the space bound
Route Planning in Transportation Networks
We survey recent advances in algorithms for route planning in transportation
networks. For road networks, we show that one can compute driving directions in
milliseconds or less even at continental scale. A variety of techniques provide
different trade-offs between preprocessing effort, space requirements, and
query time. Some algorithms can answer queries in a fraction of a microsecond,
while others can deal efficiently with real-time traffic. Journey planning on
public transportation systems, although conceptually similar, is a
significantly harder problem due to its inherent time-dependent and
multicriteria nature. Although exact algorithms are fast enough for interactive
queries on metropolitan transit systems, dealing with continent-sized instances
requires simplifications or heavy preprocessing. The multimodal route planning
problem, which seeks journeys combining schedule-based transportation (buses,
trains) with unrestricted modes (walking, driving), is even harder, relying on
approximate solutions even for metropolitan inputs.Comment: This is an updated version of the technical report MSR-TR-2014-4,
previously published by Microsoft Research. This work was mostly done while
the authors Daniel Delling, Andrew Goldberg, and Renato F. Werneck were at
Microsoft Research Silicon Valle
Near-Optimal Distance Emulator for Planar Graphs
Given a graph G and a set of terminals T, a distance emulator of G is another graph H (not necessarily a subgraph of G) containing T, such that all the pairwise distances in G between vertices of T are preserved in H. An important open question is to find the smallest possible distance emulator.
We prove that, given any subset of k terminals in an n-vertex undirected unweighted planar graph, we can construct in O~(n) time a distance emulator of size O~(min(k^2,sqrt{k * n})). This is optimal up to logarithmic factors. The existence of such distance emulator provides a straightforward framework to solve distance-related problems on planar graphs: Replace the input graph with the distance emulator, and apply whatever algorithm available to the resulting emulator. In particular, our result implies that, on any unweighted undirected planar graph, one can compute all-pairs shortest path distances among k terminals in O~(n) time when k=O(n^{1/3})
Pure Message Passing Can Estimate Common Neighbor for Link Prediction
Message Passing Neural Networks (MPNNs) have emerged as the {\em de facto}
standard in graph representation learning. However, when it comes to link
prediction, they often struggle, surpassed by simple heuristics such as Common
Neighbor (CN). This discrepancy stems from a fundamental limitation: while
MPNNs excel in node-level representation, they stumble with encoding the joint
structural features essential to link prediction, like CN. To bridge this gap,
we posit that, by harnessing the orthogonality of input vectors, pure
message-passing can indeed capture joint structural features. Specifically, we
study the proficiency of MPNNs in approximating CN heuristics. Based on our
findings, we introduce the Message Passing Link Predictor (MPLP), a novel link
prediction model. MPLP taps into quasi-orthogonal vectors to estimate
link-level structural features, all while preserving the node-level
complexities. Moreover, our approach demonstrates that leveraging
message-passing to capture structural features could offset MPNNs'
expressiveness limitations at the expense of estimation variance. We conduct
experiments on benchmark datasets from various domains, where our method
consistently outperforms the baseline methods.Comment: preprin
Lower Bounds and Approximation Algorithms for Search Space Sizes in Contraction Hierarchies
Contraction hierarchies (CH) is a prominent preprocessing-based technique that accelerates the computation of shortest paths in road networks by reducing the search space size of a bidirectional Dijkstra run. To explain the practical success of CH, several theoretical upper bounds for the maximum search space size were derived in previous work. For example, it was shown that in minor-closed graph families search space sizes in ?(?n) can be achieved (with n denoting the number of nodes in the graph), and search space sizes in ?(h log D) in graphs of highway dimension h and diameter D. In this paper, we primarily focus on lower bounds. We prove that the average search space size in a so called weak CH is in ?(b_?) for ? ? 2/3 where b_? is the size of a smallest ?-balanced node separator. This discovery allows us to describe the first approximation algorithm for the average search space size. Our new lower bound also shows that the ?(?n) bound for minor-closed graph families is tight. Furthermore, we deeper investigate the relationship of CH and the highway dimension and skeleton dimension of the graph, and prove new lower bound and incomparability results. Finally, we discuss how lower bounds for strong CH can be obtained from solving a HittingSet problem defined on a set of carefully chosen subgraphs of the input network
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