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We study new types of geometric query problems defined as follows: given a geometric set P , preprocess it such that given a query point q, the location of the largest circle that does not contain any member of P , but contains q can be reported efficiently. The geometric sets we consider for P are boundaries of convex and simple polygons, and point sets.(More)
Given a polyhedral terrain with n vertices, the shortest monotone descent path problem deals with finding the shortest path between a pair of points, called source (s) and destination (t) such that the path is constrained to lie on the surface of the terrain, and for every pair of points p = (x(p), y(p), z(p)) and q = (x(q), y(q), z(q)) on the path, if(More)
We investigate a special class of map labeling problem. Let P = fp1; p2; : : : ; png be a set of point sites distributed on a 2D map. A label associated with each point is a axis-parallel rectangle of a constant height but of variable width. Here height of a label indicates the font size and width indicates the number of characters in that label. For a(More)
Given a set of n points P = {p 1 , p 2 ,. .. , p n } in the plane, we show how to preprocess P such that for any query line segment L we can report in O(log n) time the smallest enclosing circle whose center is constrained to lie on L. The preprocessing time and space complexity are O(n log n) and O(n) respectively. We then show how to use this data(More)
A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. No efficient algorithm is known to find a shortest descending path (SDP) from s to t in a polyhedral terrain. We give two approximation algorithms (more precisely, FPTASs) that solve the SDP problem on general(More)
In the map labeling problem, we are given a set P = {p 1 , p 2 ,. .. , p n } of point sites distributed on a 2D map. The label of a point p i is an axis-parallel rectangle of specified size. The objective is to label maximum number of points on the map so that the placed labels are mutually non-overlapping. Here, we investigate a special class of map(More)
A path from a point s to a point t on the surface of a polyhedral terrain is said to be descent if for every pair of points p = (x(p), y(p), z(p)) and q = (x(q), y(q), z(q)) on the path, if dist(s, p) < dist(s, q) then z(p) ≥ z(q), where dist(s, p) denotes the distance of p from s along the aforesaid path. Although an efficient algorithm to decide if there(More)
Prune-and-search is an important paradigm for solving many important geometric problems. We show that the general prune-and-search technique can be implemented where the objects are given in read-only memory. As examples we consider convex-hull in 2D, and linear programming in 2D and 3D. For the convex-hull problem, designing sub-quadratic algorithm in a(More)