Computational geometry

TetGen is a Cpp program for generating good quality tetrahedral meshes aimed to support numerical methods and scientific computing. The problem of quality tetrahedral mesh generation is challenged by many theoretical and practical issues. TetGen uses Delaunay-based algorithms which have theoretical guarantee of correctness. It can robustly handle arbitrary complex 3D geometries and is fast in practice. The source code of TetGen is freely available.This article presents the essential algorithms and techniques used to develop TetGen. The intended audience are researchers or developers in mesh generation or other related areas. It describes the key software components of TetGen, including an efficient tetrahedral mesh data structure, a set of enhanced local mesh operations (combination of flips and edge removal), and filtered exact geometric predicates. The essential algorithms include incremental Delaunay algorithms for inserting vertices, constrained Delaunay algorithms for inserting constraints (edges and triangles), a new edge recovery algorithm for recovering constraints, and a new constrained Delaunay refinement algorithm for adaptive quality tetrahedral mesh generation. Experimental examples as well as comparisons with other softwares are presented.

TetGen, a Delaunay-Based Quality Tetrahedral Mesh Generator

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

In this chapter, you will see how the simplex method simplifies when it is applied to a class of optimization problems that are known as “network flow models.” You will also see that if a network flow model has “integer-valued data,” the simplex method finds an optimal solution that is integer-valued.

Flows in Networks

We present a route planning technique solely based on the concept of node contraction. The nodes are first ordered by 'importance'. A hierarchy is then generated by iteratively contracting the least important node. Contracting a node υ means replacing shortest paths going through v by shortcuts. We obtain a hierarchical query algorithm using bidirectional shortest-path search. The forward search uses only edges leading to more important nodes and the backward search uses only edges coming from more important nodes. For fastest routes in road networks, the graph remains very sparse throughout the contraction process using rather simple heuristics for ordering the nodes. We have five times lower query times than the best previous hierarchical Dijkstra-based speedup techniques and a negative space overhead, i.e., the data structure for distance computation needs less space than the input graph. CHs can be combined with many other route planning techniques, leading to improved performance for many-to-many routing, transit-node routing, goal-directed routing or mobile and dynamic scenarios.

/pdf/contraction-hierarchies-faster-and-simpler-hierarchical-44o5lbyhfa.pdf

Contraction hierarchies: faster and simpler hierarchical routing in road networks

When you drive to somewhere far away, you will leave your current location via one of only a few important traffic junctions. Starting from this informal observation, we developed an algorithmic approach, transit node routing, that allows us to reduce quickest path queries in road networks to a small number of table lookups. For road maps of Western Europe and the United States, our best query times improved over the best previously published figures by two orders of magnitude. This is also more than one million times faster than the best known algorithm for general networks.

https://science.sciencemag.org/content/sci/316/5824/566.full.pdf

Fast Routing in Road Networks with Transit Nodes

The identification of holes in a wireless sensor network is of primary interest since the breakdown of sensor nodes in a larger area often indicates one of the special events to be monitored by the network in the first place (e.g. outbreak of a fire, destruction by an earthquakes etc.). This task of identifying holes is especially challenging since typical wireless sensor networks consist of lightweight, low-capability nodes that are unaware of their geographic location.But there is also a secondary interest in detecting holes in a network: recently routing schemes have been proposed that do not assume knowledge of the geographic location of the network nodes but rather perform routing decisions based on the topology of the communication graph. Holes are salient features of the topology of a communication graph.In the first part of this paper we propose a simple distributed procedure to identify nodes near the boundary of the sensor field as well as near hole boundaries. Our hole detection algorithm is based purely on the topology of the communication graph, i.e. the only information available is which nodes can communicate with each other. In the second part of this paper we illustrate the secondary interest of our hole detection procedure using several examples.

Topological hole detection in wireless sensor networks and its applications

When you drive to somewhere 'far away', you will leave your current location via one of only a few 'important' traffic junctions. Starting from this informal observation, we develop an algorithmic approach---transit node routing---that allows us to reduce quickest-path queries in road networks to a small number of table lookups. We present two implementations of this idea, one based on a simple grid data structure and one based on highway hierarchies. For the road map of the United States, our best query times improve over the best previously published figures by two orders of magnitude. Our results exhibit various trade-offs between average query time (5 μs to 63 μs), preprocessing time (59 min to 1200 min), and storage overhead (21 bytes/node to 244 bytes/node).

/pdf/in-transit-to-constant-time-shortest-path-queries-in-road-4zhjvyccak.pdf

In transit to constant time shortest-path queries in road networks

Several routing schemes in ad hoc networks first establish a virtual backbone and then route messages via backbone nodes. One common way of constructing such a backbone is based on the construction of a connected dominating set (CDS). In this article we present a very simple distributed algorithm for computing a small CDS. Our algorithm has an approximation factor of at most 6.91, improving upon the previous best-known approximation factor of 8 due to Wan et al. [2002]. The improvement relies on a refined analysis of the relationship between the size of a maximal independent set and a minimum CDS in a unit disk graph. This subresult also implies improved approximation factors for many existing algorithm.

Stefan Funke

Papers

Fast Routing in Road Networks with Transit Nodes

Topological hole detection in wireless sensor networks and its applications

In transit to constant time shortest-path queries in road networks

In transit to constant time shortest-path queries in road networks

A simple improved distributed algorithm for minimum CDS in unit disk graphs