On the enumeration of closures and environments with an application to random generation

Environments and closures are two of the main ingredients of evaluation in lambda-calculus. A closure is a pair consisting of a lambda-term and an environment, whereas an environment is a list of lambda-terms assigned to free variables. In this paper we investigate some dynamic aspects of evaluation in lambda-calculus considering the quantitative, combinatorial properties of environments and closures. Focusing on two classes of environments and closures, namely the so-called plain and closed ones, we consider the problem of their asymptotic counting and effective random generation. We provide an asymptotic approximation of the number of both plain environments and closures of size $n$. Using the associated generating functions, we construct effective samplers for both classes of combinatorial structures. Finally, we discuss the related problem of asymptotic counting and random generation of closed environemnts and closures

On the number of lambda terms with prescribed size of their De Bruijn representation

John Tromp introduced the so-called 'binary lambda calculus' as a way to encode lambda terms in terms of binary words. Later, Grygiel and Lescanne conjectured that the number of binary lambda terms with $m$ free indices and of size $n$ (encoded as binary words of length $n$) is $o(n^{-3/2} \tau^{-n})$ for $\tau \approx 1.963448\ldots$. We generalize the proposed notion of size and show that for several classes of lambda terms, including binary lambda terms with $m$ free indices, the number of terms of size $n$ is $\Theta(n^{-3/2} \rho^{-n})$ with some class dependent constant $\rho$, which in particular disproves the above mentioned conjecture. A way to obtain lower and upper bounds for the constant near the leading term is presented and numerical results for a few previously introduced classes of lambda terms are given

Counting Polygon Triangulations is Hard

We prove that it is #P-complete to count the triangulations of a (non-simple) polygon

Engineering Algorithms for Dynamic and Time-Dependent Route Planning

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

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Multi-scale Pedestrian Navigation and Movement in Urban Areas

Sustainable transport planning highlights the importance of walking to low-carbon and healthy urban transport systems. Studies have identified multiple ways in which vehicle traffic can negatively impact pedestrians and inhibit walking intentions. However, pedestrian-vehicle interactions are underrepresented in models of pedestrian mobility. This omission limits the ability of transport simulations to support pedestrian-centric street design. Pedestrian navigation decisions take place simultaneously at multiple spatial scales. Yet most models of pedestrian behaviour focus either on local physical interactions or optimisation of routes across a road network. This thesis presents a novel hierarchical pedestrian route choice framework that integrates dynamic, perceptual decisions at the street level with abstract, network based decisions at the neighbourhood level. The framework is based on Construal Level Theory which states that decision makers construe decisions based on their psychological distance from the object of the decision. The route choice framework is implemented in a spatial agent-based simulation in which pedestrian and vehicle agents complete trips in an urban environment. Global sensitivity analysis is used to explore the behaviour produced by the multi-scale pedestrian route choice model. Finally, simulation experiments are used to explore the impacts of restrictions to pedestrian movement. The results demonstrate the potential insights that can be gained by linking street scale movement and interactions with neighbourhood level mobility patterns