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169
The BrunnMinkowski inequality
 Bull. Amer. Math. Soc. (N.S
, 2002
"... Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The BrunnMinkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n, and deserves to be better known. This guide explains ..."
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Cited by 184 (9 self)
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Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The BrunnMinkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n, and deserves to be better known. This guide explains the relationship between the BrunnMinkowski inequality and other inequalities in geometry and analysis, and some applications. 1.
Compressive Sensing and Structured Random Matrices
 RADON SERIES COMP. APPL. MATH XX, 1–95 © DE GRUYTER 20YY
"... These notes give a mathematical introduction to compressive sensing focusing on recovery using ℓ1minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to ..."
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Cited by 162 (19 self)
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These notes give a mathematical introduction to compressive sensing focusing on recovery using ℓ1minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to providing conditions that ensure exact or approximate recovery of sparse vectors using ℓ1minimization.
ANALYTIC SOLUTION TO THE BUSEMANNPETTY PROBLEM ON SECTIONS OF CONVEX BODIES
, 1999
"... We derive a formula connecting the derivatives of parallel section functions of an originsymmetric star body in Rn with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n − 1)dimensional Xray) gives the ((n − 1)dimensional) volumes of all hyp ..."
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Cited by 61 (12 self)
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We derive a formula connecting the derivatives of parallel section functions of an originsymmetric star body in Rn with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n − 1)dimensional Xray) gives the ((n − 1)dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in Rn and leads to a unified analytic solution to the BusemannPetty problem: Suppose that K and L are two originsymmetric convex bodies in Rn such that the ((n − 1)dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L; is the (ndimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the BusemannPetty problem, intersection bodies, and positive definite distributions, our formula shows that the answer to the problem depends on the behavior of the (n − 2)nd derivative of the parallel section functions. The affirmative answer to the BusemannPetty problem for n ≤ 4 and negative answer for n ≥ 5 now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.
unknown title
"... We study the problem of computing best connections in public transit networks. A common approach models the network as a graph (Pyrga et al. 2008) on which it runs a shortest path algorithm (Dijkstra 1959). To enable interactive queries, a variety of speedup techniques exist that use a preproces ..."
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We study the problem of computing best connections in public transit networks. A common approach models the network as a graph (Pyrga et al. 2008) on which it runs a shortest path algorithm (Dijkstra 1959). To enable interactive queries, a variety of speedup techniques exist that use a preprocessing stage to accelerate queries (Delling et al. 2009; Bast et al. 2010). Unfortunately, developed with road networks in mind, they fall short on public transit networks due to their different combinatorial structure (Bast 2009). Also, unlike in road networks, one is usually interested in reporting Pareto sets of journeys for several criteria, such as travel time and the number of transfers. While Dijkstra’s algorithm can be augmented (Pyrga et al. 2008), this increases running time even further (Berger et al. 2009; Disser,
TimeDependent SHARCRouting
 In Proceedings of the 16th Annual European Symposium on Algorithms (ESA’08
, 2008
"... In recent years, many speedup techniques for Dijkstra’s algorithm have been developed that make the computation of shortest paths in static road networks a matter of microseconds. However, only few of those techniques work in timedependent networks which, unfortunately, appear quite frequently in ..."
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Cited by 17 (9 self)
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In recent years, many speedup techniques for Dijkstra’s algorithm have been developed that make the computation of shortest paths in static road networks a matter of microseconds. However, only few of those techniques work in timedependent networks which, unfortunately, appear quite frequently in reality: Roads are predictably congested by traffic jams, and efficient timetable information systems rely on timedependent networks. Hence, a fast technique for routing in such networks is needed. In this work, we present an efficient timedependent route planning algorithm. It is based on our recently introduced SHARC algorithm, which we adapt by augmenting its basic ingredients such that correctness can still be guaranteed in a timedependent scenario. As a result, we are able to efficiently compute exact shortest paths in timedependent continentalsized transportation networks, both of roads and of railways. It should be noted that timedependent SHARC was the first efficient algorithm for timedependent route planning. 1
Accelerating MultiModal Route Planning by AccessNodes
 PROCEEDINGS OF THE 17TH ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHMS (ESA’09), LECTURE NOTES IN COMPUTER SCIENCE
, 2009
"... Recent research on fast route planning algorithms focused either on road networks or on public transportation. However, on the long run, we are interested in planning routes in a multimodal scenario: we start by car to reach the nearest train station, ride the train to the airport, fly to an airp ..."
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Cited by 12 (5 self)
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Recent research on fast route planning algorithms focused either on road networks or on public transportation. However, on the long run, we are interested in planning routes in a multimodal scenario: we start by car to reach the nearest train station, ride the train to the airport, fly to an airport near our destination and finally take a taxi. In other words, we need to incorporate public transportation into road networks. However, we do not want to switch the type of transportation too often. We end up in a label constrained variant of the shortest path problem. In this work, we present a first efficient solution to a restricted variant of this problem including experimental results for transportation networks with up to 125 Mio. edges.
Recent Production Projects
"... Designed and helped implement three generations of the routing engine used by Bing Maps to compute driving directions. Developed the algorithm used by Bing Maps for finding journeys in public transportation systems. Developed the algorithm used by Bing Local to quickly rank points of interest based ..."
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Designed and helped implement three generations of the routing engine used by Bing Maps to compute driving directions. Developed the algorithm used by Bing Maps for finding journeys in public transportation systems. Developed the algorithm used by Bing Local to quickly rank points of interest based on driving times from one or multiple locations. Contributed to the algorithm used by Microsoft’s Business Intelligence Unit to transform naturallanguage expressions into formal queries into a database.
Intriguingly Simple and Fast Transit Routing
 In SEA, volume 7933 of LNCS
, 2013
"... Abstract. This paper studies the problem of computing optimal journeys in dynamic public transit networks. We introduce a novel algorithmic framework, called Connection Scan Algorithm (CSA), to compute journeys. It organizes data as a single array of connections, which it scans once per query. Des ..."
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Cited by 5 (0 self)
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Abstract. This paper studies the problem of computing optimal journeys in dynamic public transit networks. We introduce a novel algorithmic framework, called Connection Scan Algorithm (CSA), to compute journeys. It organizes data as a single array of connections, which it scans once per query. Despite its simplicity, our algorithm is very versatile. We use it to solve earliest arrival and multicriteria profile queries. Moreover, we extend it to handle the minimum expected arrival time (MEAT) problem, which incorporates stochastic delays on the vehicles and asks for a set of (alternative) journeys that in its entirety minimizes the user’s expected arrival time at the destination. Our experiments on the dense metropolitan network of London show that CSA computes MEAT queries, our most complex scenario, in 272ms on average. 1
UserConstrained MultiModal Route Planning
"... In the multimodal route planning problem we are given multiple transportation networks (e. g., pedestrian, road, public transit) and ask for a best integrated journey between two points. The main challenge is that a seemingly optimal journey may have changes between networks that do not reflect the ..."
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Cited by 8 (1 self)
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In the multimodal route planning problem we are given multiple transportation networks (e. g., pedestrian, road, public transit) and ask for a best integrated journey between two points. The main challenge is that a seemingly optimal journey may have changes between networks that do not reflect the user’s modal preferences. In fact, quickly computing reasonable multimodal routes remains a challenging problem: Previous approaches either suffer from poor query performance or their available choices of modal preferences during query time is limited. In this work we focus on computing exact multimodal journeys that can be restricted by specifying arbitrary modal sequences at query time. For example, a user can say whether he wants to only use public transit, or also prefers to use a taxi or walking at the beginning or end of the journey; or if he has no restrictions at all. By carefully adapting node contraction, a common ingredient to many speedup techniques on road networks, we are able to compute pointtopoint queries on a continental network combined of cars, railroads and flights several orders of magnitude faster than Dijkstra’s algorithm. Thereby, we require little space overhead and obtain fast preprocessing times.
Results 1  10
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