Overmars, M. and van Leeuwen, J. 1981b. Maintenance of configurations in the plane. Journal of Computer and Systems Science 23, 2, 166--204.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....space. Similarly S can be obtained and sorted within the same bounds. Merging the two sets of slopes (rotation events and S) is done in time O(n ) Initially computing the convex hulls for all sets U i takes O(n ) space. Applying the data structure and algorithm of Overmars and van Leeuwen [18], each update of a convex hull takes O(log n) time, totaling in O(n space for all rotation events. Our range searching algorithm takes O(log time per query and per update, after spending O(n log n) preprocessing time and using O(n log n) space (notice that this is the total space ....

M.H. Overmars and J. van Leeuwen, "Maintenance of configurations in the plane", J. Comput. Syst. Sci. 23 (1981), 166--204.

....appears and disappears at most once. Since any combinatorial change of CH (P(t) is caused by some point appearing or disappearing, this proves part (a) To construct the data structure of part (b) we use a divide and conquer approach, and modify the data structure of Overmars and van Leeuwen [OvL81] to our needs. Here is a sketch of the basic idea: Split the points into a left half and a right half, and process the halves recursively. This yields a binary tree. For every node in the tree, the recursive processing yields the two separate hulls associated with the children of this node. We ....

....Its size is linear in u since every node maintains only a constant amount of information that specifies the bridge between the hulls of its two children. The above recursive initialization process takes O(ulogu) time. We omit the technical details, which can be found in Overmars and van Leeuwen [OvL81] In particular, each structural change can be handled in time O(log u) To correctly handle the motion of the points we need to transform the above into a kinetic data structure (see Basch et al. BGH99] and Guibas [Gui98] Any node in our tree contains a bridge (between the convex hulls of ....

M. H. Overmars and J. van Leeuwen. Maintenance of configurations in the plane. J. Comput. Syst. Sci., 23:166--204, 1981.

....in the plane. Lemma 15 In the plane, the worst case complexity of the k envelope is at most O(n ) The idea here is to adapt Dey s proof [6] for the complexity of a k level. 4. 4 Output sensitive construction for maximum depth in non degenerate planar arrangements The Overmars van Leeuwen [18] algorithm for dynamic convex hulls, when applied to the duals of the lines, allows us to maintain a description of the current cell as we walk from cell to cell in the arrangement. With the characterization of the points of maximum depth from Subsection 4.2, this allows us to compute a ....

....in the cell contained in the wedge of a 1X configuration namely, the point with tangent parallel to the tangent of the wedge. Thus, isolated points occur in strings of 1X configurations that end with a 1A configuration, and we can use a binary search in the Overmars van Leeuwen data structure [18] to find the next candidate and enter the next cell. 4.5 Depth of vertices in degenerate arrangements E#ciently finding a deepest vertex in a degenerate arrangement of lines appears to be di#cult. However, we can e#ciently find a vertex whose depth is within a factor of (1 o(1) from the ....

M. H. Overmars and J. van Leeuwen. Maintenance of configurations in the plane. J. Comput. Syst. Sci., 23:166--204, 1981.

....I 2 (H) minima from the I 1 (H) minima, and so on. As the ( k) level has O(k ) local minima [43] the cost of the algorithm is dominated by the cost of O(k ) dynamic LP operations. The general 2 d problem can be lifted to a feasible 3 d problem, and with the appropriate data structures [45], these O(k ) operations can be carried out in O(n log n k n) time. The first term has been lowered to O(n log k) by this author [9, 8] the second term can probably also be lowered using recent dynamic data structures [13] though the particular queries needed were not explicitly ....

M. H. Overmars and J. van Leeuwen. Maintenance of configurations in the plane. J. Comput. Sys. Sci., 23:166--204, 1981.

.... where all red and all blue segments are connected [1] For further applications see [4] Overmars and van Leeuwen in 1981 gave a solution for the fully dynamic convex hull problem supporting point insertions and deletions in O(log n) time, where n is the maximum number of points in the set [12]. The data structure of Overmars and van Leeuwen stores the convex hull in a search tree and typical queries on the convex hull are supported in O(log n) time. Preparata and Vitter gave a simpler approach achieving the same bounds as Overmars and van Leeuwen in [14] Until recently there was made ....

....in the interior of the convex hull; find the two points adjacent to a point on the convex hull; and given an exterior point find the two tangent points on the convex hull from the point. Combining our semi dynamic data structure with the fully dynamic data structure of Overmars and van Leeuwen [12], we immediately get amortized O(log nlog log n) deletion and insertion time. By bootstrapping, we can use the resulting data structure as the fully dynamic data structure in the construction and the insertion time reduces to amortized O(log n log log log n) time, while the deletion time ....

M. H. Overmars and J. van Leeuwen. Maintenance of configurations in the plane. Journal of Computer and System Sciences, 23:166--204, 1981.

....stored in arrays or balanced binary trees. Because a tangent to a polygon P through a given point q can be found in Theta(log n) time by binary search, there is an easy O(log n) time algorithm for finding a tangent common to P and Q that uses nested binary search. Overmars and van Leeuwen [7], as part of a data structure for dynamic convex hulls, gave a logarithmic time algorithm for the special case in which P and Q have a known vertical separating line. Because one can compute a separating line for disjoint polygons in logarithmic time by finding the shortest segment joining them ....

....in situations where no separating line exists. In the next section, we show that tangents for disjoint convex polygons can be computed in logarithmic time by using a tentative prune and search technique [6] C code is given in an appendix. The approach is much like Overmars and van Leeuwen s [7] starting with lists of vertices for P and for Q that are known to contain the tangent vertices, attempt to discard half of some list by doing a constant time local test. Without a separating line, however, some tests do not give sufficient information. One can proceed by making tentative ....

M. Overmars and J. van Leeuwen. Maintenance of configurations in the plane. Journal of Computer and System Sciences, 23:166--204, 1981.

....structure, which would naturally be a leaf linked balanced search tree, storing the vertices of the convex hull in clockwise order. Such a tree representation allows to perform all kinds of queries in optimal O(log n) time. In this setting the data structure presented by Overmars and van Leeuwen [OvL81] achieves updates in worst case time O(log n) This is still the fastest solution providing an explicit representation of the convex hull (in a search tree as stated above) It also achieves the best known worst case time bounds on the single update operations. The problem gets somewhat ....

....the bridge between A and B. This bridge is defined as the segment of Bd(A#B) It is also given as the only common tangent on A and B. The situation is illustrated in Figure 3.2. This bridge finding task is at the core of the dynamic planar convex hull data structure of Overmars and van Leeuwen [OvL81] It is also described in detail in the textbook by Preparata and Shamos [PS85, page 127] A B c B hB l t B Figure 3.2: Classical bridge finding. The vertical line l separates A and B. Given the current candidate c B B, we can argue that c 1 A is too far to the left, whereas c 2 ....

M. H. Overmars and J. van Leeuwen, Maintenance of configurations in the plane, J. Comput. System Sci. 23 (1981), no. 2, 166--204.

....size O(n) Together with our earlier results, this implies that we can compute a convexi fying sequence of fliptums for any polygon in O(n210g4n) time, or for any orthogonal polygon in O(nlog4n) time. Our data structure is a variant of the dynamic convex hull structure of Overmars and van Leeuwen [6], with some important differences. First, instead of insertions and deletions, we must support an operation that reverses an entire subsequence of edges. This requires us to store the vertices in their order of appearance around the polygon, rather than in any coordinate order. Since a linear ....

M.H. Overmars and J. van Leeuwen. Maintenance of configurations in the plane. J. Comput. Syst. Sci. 23:166404, 1981.

....to the left of fl(p ) on (B r (p) B r (p ) if and only if p lies to the right of p . All the sets P for which we want to maintain K (P ) will be subsets of the given set S. This allows us to use the following variant of the dynamic data structure of Overmars and van Leeuwen [12]. We sort the points of S by their x coordinates, and store them in this order at the leaves of a minimum height binary tree T . Each node v of T maintains the intersection K (P v ) where P v is the subset of the current set P whose points are stored at the leaves of the subtree of T rooted at ....

.... curves fl(p) and the fact that the points of S are stored in T in increasing x order, are easily seen to imply that K ) intersect in exactly one point q, and we also store q at v (with pointers to the pair of curves that intersect at q) We construct, search and update this structure as in [12]. We first describe the searching procedure. We are given a query point z and wish to determine whether z lies in K (P ) To do so, we examine the root v of T . If the x range of K (P v ) is empty, we report that z lies outside K (P v ) Similarly, if the x coordinate of z falls outside ....

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M. Overmars and J. van Leeuwen, Maintenance of configurations in the plane, J. Comp. System Sciences 23 (1981), 166--204.

.... an application where such a scenario arises; the problem studied there is to compute all the area bisectors of a simple polygon, and it arises in part orienting in robotics using MEMS) The only work we are aware of that can deal with this problem efficiently is due to Overmars and van Leeuwen [OvL81] and it only applies to the case of lines (and, with some simple modifications, to certain restricted cases involving line segments as well) It can compute such a walk in (deterministic) O( n m) log n) time, inside an arrangement of n lines, where m is the number of intersections of the ....

.... log n) where w is the minimum number of intersections between a spanning tree connecting those query points and the given arcs (the tree is not constructed by the algorithm; it only shows up in its analysis) Both results improve by almost a logarithmic factor over the best previous result of [OvL81] for the case of lines (and for certain cases involving line segments) For the case of general arcs, we are not aware of any similarly efficient previous result. The results of Chapter 9 appeared in [HP99e] Constructing Cuttings in Theory and Practice conquer paradigm. A typical ....

[Article contains additional citation context not shown here]

M. H. Overmars and J. van Leeuwen. Maintenance of configurations in the plane. J. Comput. Syst. Sci., 23:166--204, 1981.

....size O(n) Together with our earlier results, this implies that we can compute a convexi fying sequence of fliptums for any polygon in O(n21og4n) time, or for any orthogonal polygon in O(nlog4n) time. Our data structure is a variant of the dynamic convex hull structure of Overmars and van Leeuwen [6], with some important differences. First, instead of insertions and deletions, we must support an operation that reverses an entire subsequence of edges. This requires us to store the vertices in their order of appearance around the polygon, rather than in any coordinate order. Since a linear ....

M.H. Overmars and J. van Leeuwen. Maintenance of configurations in the plane. J. Comput. Syst. Sci. 23:166404, 1981.

.... is an (O(log n) decomposable set problem if is the ordering on first coordinate (or second coordinate) because the convex hull of a set can be constructed from two convex hulls of any ordered partition in O(log n) time, by computing the supporting lines (also called bridges) see e.g. [4, 13]) For order decomposable set problems there exist dynamic structures, which are studied in detail in [10, 11] From the structure it can easily be seen that (C(n) decomposable set problems can be split and concatenated with respect to in time O(C(n) log n) Structures for order ....

Overmars, M. H., and J. van Leeuwen, Maintenance of configurations in the plane, J. Comp. Syst. Sci. $ (1981), pp. 166-204.

....search structure at every stage of the recursion. The search is guided by the convex hulls of the sets Ps and PE. Clearly our scheme needs a method for maintaining convex hulls in a dynamical way. The best bound currently known for the dynamic convex hull problem is due to Overmars and van Leeuwen [5]. They prove the existence of a dynamic structure for solving the 2 dimensional convex hull searching problem, such that queries can be done in O(logn) time, insertions and deletions can be done in O(log 2 n) time. Building this structure takes O(n log n) time and uses O(n) storage. We use this ....

M. H. Overmars and J. van Leeuwen. Maintenance of configurations in the plane. J. Cornput. Syst. $ci., 23:166-204, 1981.

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Overmars, M. and van Leeuwen, J. 1981b. Maintenance of configurations in the plane. Journal of Computer and Systems Science 23, 2, 166--204.

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M. H. Overmars and J. van Leeuwen. Maintenance of configurations in the plane. J. Comput. Syst. Sci., 23:166--204, 1981.

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Overmars, M. & Leeuwen, J. V. (1981), `Maintenance of configurations in the plane', Journal of Computer and Systems Sciences 23, 166--204.

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Overmars, M. & Leeuwen, J. V. (1981), `Maintenance of configurations in the plane', Journal of Computer and Systems Sciences 23, 166--204.

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M. H. Overmars and J. van Leeuwen. Maintenance of configurations in the plane. J. Comput. Syst. Sci., 23:166--204, 1981.

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M. Overmars and J. van Leeuwen. Maintenance of configurations in the plane. J. Comput. System Sci., 23:166--204, 1981.

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Mark H. Overmars and Jan Van Leeuwen. Maintenance of configurations in the plane. Journal of Computer and System Sciences, 23:166--204, 1981.

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M. H. Overmars, J. van Leeuwen, Maintenance of configurations in the plane, Journal of Computer and System Sciences 23 (1981) 166--204.

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M. Overmars and J. van Leeuwen. Maintenance of configurations in the plane. Journal of Computer and System Sciences, 23:166--204, 1981.

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Mark H. Overmars and Jan van Leeuwen. Maintenance of configurations in the plane. Journal of Computer and System Sciences, 23:166--204, 1981.

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M. H. Overmars, J. van Leeuwen, Maintenance of configurations in the plane, J. Comput. System Sci. 23 (2) (1981) 166--204.

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M.H. Overmars and J. Van Leeuwen, "Maintenance of Config- urations in the Plane", Journal of Computer and System Sciences, 1981, 23, 166-204.

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