This paper describes several steps in the derivation of boundaries of imprecise regions using the Web as the information source. We discuss how to obtain locations that are part of and locations that are not part of the region to be delineated, and then we propose methods to compute the region algorithmically. The methods introduced are evaluated to judge the potential of the approach.

Geographic Information Retrieval is concerned with retrieving documents in response to a spatially related query. This paper addresses the ranking of documents by both textual and spatial relevance. To this end, we introduce multi-dimensional scattered ranking, where textually and spatially similar documents are ranked spread in the list, instead of consecutively. The effect of this is that documents close together in the ranked list have less redundant information. We present various ranking methods of this type, efficient algorithms to implement them, and experiments to show the outcome of the methods.

In geographic information retrieval, queries often name geographic regions that do not have a well-defined boundary, such as "Southern France." We provide two algorithmic approaches to the problem of computing reasonable boundaries of such regions based on data points that have evidence indicating that they lie either inside or outside the region. Our problem formulation leads to a number of subproblems related to red-blue point separation and minimum-perimeter polygons, many of which we solve algorithmically. We give experimental results from our implementation and a comparison of the two approaches.

We study the problem of maximizing the overlap of two convex polytopes under translation in R d for some constant d ≥ 3. Let n be the number of bounding hyperplanes of the polytopes. We present an algorithm that, for any ε > 0, finds an overlap at least the optimum minus ε and reports the translation realizing it. The running time is O(n d/2 +1 log d n) with probability at least 1 − n −O(1) , which can be improved to O(n log 3.5 n) in R 3. The time complexity analysis depends on a bounded incidence condition that we enforce with probability one by randomly perturbing the input polytopes. The perturbation causes an additive error ε, which can be made arbitrarily small by decreasing the perturbation magnitude. Our algorithm in fact computes the maximum overlap of the perturbed polytopes. The running time bounds, the probability bound, and the big-O constants in these bounds are independent of ε.

Motivated by geographic information retrieval, we study the problem of partitioning a simple polygon into four parts that can be considered as the North, East, West, and South. We list criteria for such partitionings, propose formalizations into geometric problems, and give efficient algorithms. An implementation and tests on country outlines show the results for three different partitionings.

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Geographic Information Retrieval is concerned with retrieving documents in response to a spatially related query. This paper addresses the ranking of documents by both textual and spatial relevance. To this end, we introduce distributed ranking, where similar documents are ranked spread in the list instead of consecutively. The effect of this is that documents close together in the ranked list have less redundant information. We present various ranking methods, efficient algorithms to implement them, and experiments to show the outcome of the methods.

For a set of n points in the plane, we consider the axis-aligned (p, k)-Box Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together contain at least n − k points. In this paper, we consider the boxes to be either squares or rectangles, and we want to minimize the area of the largest box. For general p we show that the problem is NP-hard for both squares and rectangles. For a small, fixed number p, we give algorithms that find the solution in the following running times: For squares we have O(n + k log k) time for p = 1, and O(n log n + k p log p k) time for p = 2, 3. For rectangles we get O(n + k 3 ) for p = 1 and O(n log n + k 2+p log p−1 k) time for p = 2, 3. In all cases, our algorithms use O(n) space.