Jan Vahrenhold

On the Complexity of Computing the Hypervolume Indicator

The goal of multiobjective optimization is to find a set of best compromise solutions for typically conflicting objectives. Due to the complex nature of most real-life problems, only an approximation to such an optimal set can be obtained within reasonable (computing) time. To compare such approximations, and thereby the performance of multiobjective optimizers providing them, unary quality measures are usually applied. Among these, the hypervolume indicator (or S-metric) is of particular relevance due to its favorable properties. Moreover, this indicator has been successfully integrated into stochastic optimizers, such as evolutionary algorithms, where it serves as a guidance criterion for finding good approximations to the Pareto front. Recent results show that computing the hypervolume indicator can be seen as solving a specialized version of Klees measure problem. In general, Klees measure problem can be solved with O(n logn + nd/2logn) comparisons for an input instance of size n in d dimensions; as of this writing, it is unknown whether a lower bound higher than Omega(n log n) can be proven. In this paper, we derive a lower bound of Omega(n log n) for the complexity of computing the hypervolume indicator in any number of dimensions d > 1 by reducing the so-called uniformgap problem to it. For the 3-D case, we also present a matching upper bound of O(n log n) comparisons that is obtained by extending an algorithm for finding the maxima of a point set.

Efficient Bulk Operations on Dynamic R-trees

Abstract In recent years there has been an upsurge of interest in spatial databases. A major issue is how to manipulate efficiently massive amounts of spatial data stored on disk in multidimensional spatial indexes (data structures). Construction of spatial indexes (bulk loading ) has been studied intensively in the database community. The continuous arrival of massive amounts of new data makes it important to update existing indexes (bulk updating ) efficiently.In this paper we present a simple, yet efficient, technique for performing bulk update and query operations on multidimensional indexes. We present our technique in terms of the so-called R-tree and its variants, as they have emerged as practically efficient indexing methods for spatial data. Our method uses ideas from the buffer tree lazy buffering technique and fully utilizes the available internal memory and the page size of the operating system. We give a theoretical analysis of our technique, showing that it is efficient both in terms of I/ O communication, disk storage, and internal computation time. We also present the results of an extensive set of experiments showing that in practice our approach performs better than the previously best known bulk update methods with respect to update time, and that it produces a better quality index in terms of query performance. One important novel feature of our technique is that in most cases it allows us to perform a batch of updates and queries simultaneously. To be able to do so is essential in environments where queries have to be answered even while the index is being updated and reorganized.

A Unified Approach for Indexed and Non-Indexed Spatial Joins

Most spatial join algorithms either assume the existence of a spatial index structure that is traversed during the join process, or solve the problem by sorting, partitioning, or on-the-fly index construction. In this paper, we develop a simple plane-sweeping algorithm that unifies the index-based and non-index based approaches. This algorithm processes indexed as well as non-indexed inputs, extends naturally to multiway joins, and can be built easily from a few standard operations. We present the results of a comparative study of the new algorithm with several index-based and non-index based spatial join algorithms. We consider a number of factors, including the relative performance of CPU and disk, the quality of the spatial indexes, and the sizes of the input relations. An important conclusion from our work is that using an index-based approach whenever indexes are available does not always lead to the best execution time, and hence we propose the use of a simple cost model to decide when to follow an index-based approach.

I/O-efficient dynamic planar point location

We present an I/O-efficient dynamic data structure for point location in a general planar subdivision. Our structure uses O(N/B) disk blocks of size B to store a subdivision of size N. Queries can be answered in O(log B 2N) I/Os in the worst-case, and insertions and deletions can be performed in O(log B 2N) and O(log B N) I/Os amortized, respectively. Part of our data structure is based on an external version of the so-called logarithmic method that allows for efficient dynamization of static external-memory data structures with certain characteristics. Another important part of our structure is an external data structure for vertical ray-shooting among line segments in the plane with endpoints on √B + 1 vertical lines, developed using an external version of dynamic fractional cascading. We believe that these methods could prove helpful in the development of other dynamic external memory data structures.

Detecting and understanding students' misconceptions related to algorithms and data structures

We describe the first results of our work towards a concept inventory for Algorithms and Data Structures. Based on expert interviews and the analysis of 400 exams we were able to identify several core topics which are prone to error. In a pilot study, we verified misconceptions known from the literature and identified previously unknown misconceptions related to Algorithms and Data Structures. In addition to this, we report on methodological issues and point out the importance of a two-pronged approach to data collection.

Space-efficient geometric divide-and-conquer algorithms

We develop a number of space-efficient tools including an approach to simulate divide-and-conquer space-efficiently, stably selecting and unselecting a subset from a sorted set, and computing the kth smallest element in one dimension from a multi-dimensional set that is sorted in another dimension. We then apply these tools to solve several geometric problems that have solutions using some form of divide-and-conquer. Specifically, we present a deterministic algorithm running in O(nlogn) time using O(1) extra memory given inputs of size n for the closest pair problem and a randomized solution running in O(nlogn) expected time and using O(1) extra space for the bichromatic closest pair problem. For the orthogonal line segment intersection problem, we solve the problem in O(nlogn+k) time using O(1) extra space where n is the number of horizontal and vertical line segments and k is the number of intersections.

Algorithms for Performing Polygonal Map Overlay and Spatial Join on Massive Data Sets

We consider the problem of performing polygonal map overlay and the refinement step of spatial overlay joins. We show how to adapt algorithms from computational geometry to solve these problems for massive data sets. A performance study with artificial and real-world data sets helps to identify the algorithm that should be used for given input data.

Of motifs and goals: mining trajectory data

In response to the increasing volume of trajectory data obtained, e.g., from tracking athletes, animals, or meteorological phenomena, we present a new space-efficient algorithm for the analysis of trajectory data. The algorithm combines techniques from computational geometry, data mining, and string processing and offers a modular design that allows for a user-guided exploration of trajectory data incorporating domain-specific constraints and objectives.

Line-segment intersection made in-place

We present a space-efficient algorithm for reporting all k intersections induced by a set of n line segments in the plane. Our algorithm is an in-place variant of Balabans algorithm and, in the worst case, runs in time using extra words of memory in addition to the space used for the input to the algorithm.

Reflections on outreach programs in CS classes: learning objectives for "unplugged" activities

To provide a unified view of any scientific field, outreach programs need to realistically portray the subject in question. Consequently, topics and methods actually taught in Computer Science courses should to be touched upon in Computer Science outreach programs or, conversely, elements from successful Computer Science outreach programs can be used to enrich established courses in Computer Science. We follow up on the latter aspect and investigate how outreach material might be used as a teaching resource in lower secondary Computer Science. In particular, we extract and classify learning objectives from the activities of the well-received Computer Science Unplugged program. Based upon this classification, we comment on where and to which extent these activities can be used to enrich teaching Computer Science in secondary education.