We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

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.

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Route Planning in Transportation Networks

The continuously increasing demand for mobility more and more frequently asks for the best driving directions to one's destination saving time and money. Using simple-minded approaches to the computation of suitable routes yields slow response times or suboptimal results. Therefore, there is considerable interest in exact and fast speedup techniques, which typically invest some time into a preprocessing step in order to generate auxiliary data that can be used to accelerate all subsequent route planning queries. We present three highly-efficient and provably accurate point-to-point route planning algorithms and one generic many-to-many approach, which computes for given source and target node sets the optimal distances between all source-target node pairs in a very efficient way. The performance is evaluated in an extensive experimental study using large real-world road networks with up to 33 million junctions. This book addresses students, researchers, and software developers that are interested in car navigation systems, route planning services, logistics, or any other application where route planning in road networks plays a crucial role.

/pdf/route-planning-in-road-networks-2m1t6o6c8a.pdf

Route Planning in Road Networks

A dynamic shortest-path algorithm is called a batch algorithm if it is able to handle graph changes that consist of multiple edge updates at a time. In this paper we focus on fully-dynamic batch algorithms for single-source shortest paths in directed graphs with positive edge weights. We give an extensive experimental study of the existing algorithms for the single-edge and the batch case, including a broad set of test instances. We further present tuned variants of the already existing SWSF-FP -algorithm being up to 15 times faster than SWSF-FP . A surprising outcome of the paper is the astonishing level of data dependency of the algorithms. More detailed descriptions and further experimental results of this work can be found in [1].

/pdf/batch-dynamic-single-source-shortest-path-algorithms-an-1en0nxeng2.pdf

Batch Dynamic Single-Source Shortest-Path Algorithms: An Experimental Study

The online shortest path problem aims at computing the shortest path based on live traffic circumstances. This is very important in modern car navigation systems as it helps drivers to make sensible decisions. To our best knowledge, there is no efficient system/solution that can offer affordable costs at both client and server sides for online shortest path computation. Unfortunately, the conventional client-server architecture scales poorly with the number of clients. A promising approach is to let the server collect live traffic information and then broadcast them over radio or wireless network. This approach has excellent scalability with the number of clients. Thus, we develop a new framework called live traffic index (LTI)which enables drivers to quickly and effectively collect the live traffic information on the broadcasting channel. An impressive result is that the driver can compute/update their shortest path result by receiving only a small fraction of the index. Our experimental study shows that LTI is robust to various parameters and it offers relatively short tune-in cost (at client side), fast query response time (at client side), small broadcast size (at server side), and light maintenance time (at server side)for online shortest path problem.

/pdf/towards-online-shortest-path-computation-5gn1o7f8ug.pdf

Towards Online Shortest Path Computation

Multi-level overlay graphs represent a speed-up technique for shortest paths computation which is based on a hierarchical decomposition of a weighted directed graph G. They have been shown to be experimentally efficient, especially when applied to timetable information. However, no theoretical result on the cost of constructing, maintaining and querying multi-level overlay graphs in a dynamic environment is known. In this paper, we show theoretical properties of multi-level overlay graphs that lead us to the definition of a new data structure for the computation and the maintenance of an overlay graph of G while weight decrease or weight increase operations are performed on G. Our solution is theoretically faster than the recomputation from scratch and allows queries that can be performed more efficiently than running Dijkstra’s shortest paths algorithm on G.

Dynamic Multi-level Overlay Graphs for Shortest Paths

In railways systems the timetable is typically represented as a
weighted digraph on which itinerary queries are answered by
shortest path algorithms, usually running Dijkstra's algorithm
Due to the continuously growing size of real-world graphs, there
is a constant need for faster algorithms and many techniques have
been devised to heuristically speed up Dijkstra's algorithm One
of these techniques is the multi-level overlay graph, that
has been recently introduced and shown to be experimentally
efficient, especially when applied to timetable information

In many practical application major disruptions to the normal
operation cannot be completely avoided because of the complexity
of the underlying systems Timetable information
update after disruptions is considered one of the weakest points
in current railway systems, and this determines the need for an
effective online redesign and update of the shortest paths
information as a consequence of disruptions

In this paper, we make a step forward toward this direction by
showing some theoretical properties of multi-level overlay graphs
that lead us to the definition of a new data structure for the
dynamic maintenance of a multi-level overlay graph of a given
graph G while weight decrease or weight increase operations are
performed on G Our solution is theoretically faster than the
recomputation from scratch and allows fast queries

Francesco Bruera

Papers

Dynamic Multi-level Overlay Graphs for Shortest Paths

15. Maintenance of Multi-level Overlay Graphs for Timetable Queries