65 research outputs found
Dynamic fractional cascading
The problem of searching for a key in many ordered lists arises frequently in computational geometry. Chazelle and Guibas recently introduced fractional cascading as a general technique for solving this type of problem. In the present paper we show that fractional cascading also supports insertions into and deletions from the lists efficiently. More specifically, we show that a search for a key in n lists takes time O(log N+nloglogN) and an insertion or deletion takes time O(loglogN). Here N is the total size of all lists. If only insertions or deletions have to be supported the O(loglogN) factor reduces to O(1). As an application we show that queries, insertions and deletions into segment trees or range trees can be supported in t ime O(lognloglogn), when n is the number of segments (points)
Dynamic fractional cascading
The problem of searching for a key in many ordered lists arises frequently in computational geometry. Chazelle and Guibas recently introduced fractional cascading as a general technique for solving this type of problem. In the present paper we show that fractional cascading also supports insertions into and deletions from the lists efficiently. More specifically, we show that a search for a key in n lists takes time O(log N+nloglogN) and an insertion or deletion takes time O(loglogN). Here N is the total size of all lists. If only insertions or deletions have to be supported the O(loglogN) factor reduces to O(1). As an application we show that queries, insertions and deletions into segment trees or range trees can be supported in t ime O(lognloglogn), when n is the number of segments (points)
Hidden surface removal for rectangles
AbstractA simple but important special case of the hidden surface removal problem is one in which the scene consists of n rectangles with sides parallel to the x- and y-axes, with viewpoint at z=∞ (that is, an orthographic projection). This special case has application to overlapping windows in computer displays. An algorithm with running time O(n log n + k log n) is given for static scenes, where k is the number of line segments in the output. Algorithms are given for a dynamic setting (that is, rectangles may be inserted and deleted) that take time O(log2n log log n + k log2 n) per insert or delete, where k is now the number of visible line segments that change (appear or disappear). Algorithms for point location in the visible scene are also given
Fully Retroactive Approximate Range and Nearest Neighbor Searching
We describe fully retroactive dynamic data structures for approximate range
reporting and approximate nearest neighbor reporting. We show how to maintain,
for any positive constant , a set of points in indexed by time
such that we can perform insertions or deletions at any point in the timeline
in amortized time. We support, for any small constant ,
-approximate range reporting queries at any point in the timeline
in time, where is the output size. We also show how to
answer -approximate nearest neighbor queries for any point in the
past or present in time.Comment: 24 pages, 4 figures. To appear at the 22nd International Symposium on
Algorithms and Computation (ISAAC 2011
Mixed Map Labeling
Point feature map labeling is a geometric problem, in which a set of input
points must be labeled with a set of disjoint rectangles (the bounding boxes of
the label texts). Typically, labeling models either use internal labels, which
must touch their feature point, or external (boundary) labels, which are placed
on one of the four sides of the input points' bounding box and which are
connected to their feature points by crossing-free leader lines. In this paper
we study polynomial-time algorithms for maximizing the number of internal
labels in a mixed labeling model that combines internal and external labels.
The model requires that all leaders are parallel to a given orientation , whose value influences the geometric properties and hence the
running times of our algorithms.Comment: Full version for the paper accepted at CIAC 201
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