Computing Minimum Spanning Trees with Uncertainty

We consider the minimum spanning tree problem in a setting where information about the edge weights of the given graph is uncertain. Initially, for each edge $e$ of the graph only a set $A_e$, called an uncertainty area, that contains the actual edge weight $w_e$ is known. The algorithm can `update' $e$ to obtain the edge weight $w_e \in A_e$. The task is to output the edge set of a minimum spanning tree after a minimum number of updates. An algorithm is $k$-update competitive if it makes at most $k$ times as many updates as the optimum. We present a 2-update competitive algorithm if all areas $A_e$ are open or trivial, which is the best possible among deterministic algorithms. The condition on the areas $A_e$ is to exclude degenerate inputs for which no constant update competitive algorithm can exist. Next, we consider a setting where the vertices of the graph correspond to points in Euclidean space and the weight of an edge is equal to the distance of its endpoints. The location of each point is initially given as an uncertainty area, and an update reveals the exact location of the point. We give a general relation between the edge uncertainty and the vertex uncertainty versions of a problem and use it to derive a 4-update competitive algorithm for the minimum spanning tree problem in the vertex uncertainty model. Again, we show that this is best possible among deterministic algorithms

Evaluating holistic aggregators efficiently for very large datasets

In data warehousing applications, numerous OLAP queries involve the processing of holistic aggregators such as computing the “top n,” median, quantiles, etc. In this paper, we present a novel approach called dynamic bucketing to efficiently evaluate these aggregators. We partition data into equiwidth buckets and further partition dense buckets into sub-buckets as needed by allocating and reclaiming memory space. The bucketing process dynamically adapts to the input order and distribution of input datasets. The histograms of the buckets and subbuckets are stored in our new data structure called structure trees. A recent selection algorithm based on regular sampling is generalized and its analysis extended. We have also compared our new algorithms with this generalized algorithm and several other recent algorithms. Experimental results show that our new algorithms significantly outperform prior ones not only in the runtime but also in accuracy

The update complexity of selection and related problems

We present a framework for computing with input data specified by intervals, representing uncertainty in the values of the input parameters. To compute a solution, the algorithm can query the input parameters that yield more refined estimates in form of sub-intervals and the objective is to minimize the number of queries. The previous approaches address the scenario where every query returns an exact value. Our framework is more general as it can deal with a wider variety of inputs and query responses and we establish interesting relationships between them that have not been investigated previously. Although some of the approaches of the previous restricted models can be adapted to the more general model, we require more sophisticated techniques for the analysis and we also obtain improved algorithms for the previous model. We address selection problems in the generalized model and show that there exist 2-update competitive algorithms that do not depend on the lengths or distribution of the sub-intervals and hold against the worst case adversary. We also obtain similar bounds on the competitive ratio for the MST problem in graphs

Scheduling under Uncertainty: A Query-based Approach

International audienceWe consider a single machine, a set of unit-time jobs, and a set of unit-time errors. We assume that the time-slot at which each error will occur is not known in advance but, for every error, there exists an uncertainty area during which the error will take place. In order to find if the error occurs in a specific time-slot, it is necessary to issue a query to it. In this work, we study two problems: (i) the error-query scheduling problem, whose aim is to reveal enough error-free slots with the minimum number of queries, and (ii) the lexicographic error-query scheduling problem where we seek the earliest error-free slots with the minimum number of queries. We consider both the off-line and the on-line versions of the above problems. In the former, the whole instance and its characteristics are known in advance and we give a polynomial-time algorithm for the error-query scheduling problem. In the latter, the adversary has the power to decide, in an on-line way, the time-slot of appearance for each error. We propose then both lower bounds and algorithms whose competitive ratios asymptotically match these lower bounds

Dynamics in Logistics

This open access book highlights the interdisciplinary aspects of logistics research. Featuring empirical, methodological, and practice-oriented articles, it addresses the modelling, planning, optimization and control of processes. Chiefly focusing on supply chains, logistics networks, production systems, and systems and facilities for material flows, the respective contributions combine research on classical supply chain management, digitalized business processes, production engineering, electrical engineering, computer science and mathematical optimization. To celebrate 25 years of interdisciplinary and collaborative research conducted at the Bremen Research Cluster for Dynamics in Logistics (LogDynamics), in this book hand-picked experts currently or formerly affiliated with the Cluster provide retrospectives, present cutting-edge research, and outline future research directions

Local bounding technique and its applications to uncertain clustering

Ph.DDOCTOR OF PHILOSOPH