F. P. Preparata and R. Tamassia. Fully dynamic point location in a monotone subdivision. SIAM J. Computing, 18(4):811--830, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....oliveira cs.uiowa.edu Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA. E mail: dstewart math.uiowa.edu sets f S i ae R j i = 1; 2; N g, rather than a collection of points. Also note that this task is not the point location problem studied previously [9, 10], in that the sets S i are not usually disjoint. Our analysis is also different to that of [9, 10] in that here we are concerned with the expected time for a random query point x, rather than with worst case analysis. x S S S S S S 1 2 3 4 5 7 Figure 1: Example of distribution of ....

....USA. E mail: dstewart math.uiowa.edu sets f S i ae R j i = 1; 2; N g, rather than a collection of points. Also note that this task is not the point location problem studied previously [9, 10] in that the sets S i are not usually disjoint. Our analysis is also different to that of [9, 10] in that here we are concerned with the expected time for a random query point x, rather than with worst case analysis. x S S S S S S 1 2 3 4 5 7 Figure 1: Example of distribution of support sets S i For each x and S i , we assume that we can test if x 2 S i in Theta(1) time. The ....

F. P. Preparata and R. Tamassia. Fully dynamic point location in a monotone subdivision. SIAM J. Computing, 18(4):811--830, 1989.

....of the problem considers only monotone subdivisions, where the subdivision consists of polygons that are monotone (so no horizontal line crosses any polygon more than twice) In the dynamic version of this problem updates manipulate the geometry of the subdivision. Preparata and Tamassia [24] give an algorithm that runs in time O(log n) per operation, this was improved to query time O(log n) by Baumgarten, Jung, and Mehlhorn [3] The lower bound for this problem in [16] applies only to algorithms returning the name of the region containing the queried point. The techniques of the ....

F.P. Preparata and R. Tamassia. Fully dynamic point location in a monotone subdivision. SIAM J. Comp., 18(4):811--830, 1989.

....planar subdivisions. Furthermore in our context we do not need fully dynamic algorithms since we are not interested in deletions. If we demand O(n) storage space then the best relevant algorithms available, from the worst case complexity point of view, are the algorithms of Preparata Tamassia [28] and Cheng Janardan [7] The former algorithm has both a point location query time and an insertion (update) time of O(log n) The latter algorithm is slightly better because it has a point location query time of O(log n) but only an O(log n) insertion time. However, in our context every ....

Preparata, F., and R. Tamassia, Fully dynamic point location in a monotone subdivision, SIAM Journal on Computing, 18, pp. 811-830, 1989.

....planar subdivisions. Furthermore in our context we do not need fully dynamic algorithms since we are not interested in deletions. If we demand O(n) storage space then the best relevant algorithms available, from the worst case complexity point of view, are the algorithms of Preparata Tamassia [31] and Cheng Janardan [7] The former algorithm has both a point location query time and an insertion (update) time of O(log n) The latter algorithm is slightly better because it has a point location query time of O(log n) but only an O(log n) insertion time. However, in our context every ....

Preparata, F., and R. Tamassia, Fully dynamic point location in a monotone subdivision, SIAM Journal on Computing, 18, pp. 811-830, 1989.

....model. With this, we easily derive lower bounds for half a dozen natural problems, including the following: Computational Geometry: Dynamic planar point location in monotone subdivisions cannot be solved faster than Omega Gammaan n= log log n) The important algorithm of Preparata and Tamassia [16] achieves upper bounds of O(log per update and O(log n) per query. Graph algorithms: Dynamic transitive closure in planar acyclic digraphs with one source and one sink that are on the same face cannot be solved faster than Omega Gammaan n= log log n) Tamassia and Preparata [17] achieve ....

....of vertices or (chains of) edges. An important restriction of the problem, for which our bound will apply, considers only monotone subdivisions, where the subdivision consists of polygons that are monotone (so no straight line crosses any polygon more than twice) Preparata and Tamassia [16] give an algorithm that runs in time O(log per operation. Several other dynamic algorithms for this and other types of subdivisions have been found since, see [4] for a survey. 4 To prove a lower bound for this problem we construct a monotone subdivision from the signed prefix sum instance y ....

Franco P. Preparata and Roberto Tamassia. Fully dynamic point location in a monotone subdivision. SIAM Journal of Computing, 18(4):811--830, 1989.

....deletion of vertices or (chains of) edges. An important restriction of the problem, for which our bound will apply, considers only monotone subdivisions, where the subdivision consists of polygons that are monotone (so no horizontal line crosses any polygon more than twice) Preparata and Tamassia [59] give an algorithm that runs in n) per operation, this was improved to query time O(logn) by Baumgarten, Jung, and Mehlhorn [7] The literature does not quite agree on the exact choice of operations, since the representation of the polygons defines what updates are feasible. Our lower bound ....

....return yes if and only if x is in the same polygon as the origin. Our operations are very weak, since we want to prove a useful lower bound. Efficient algorithms are known for more powerful operations that return the name of a queried polygon, and insert and delete (chains of) edges, see [7, 14, 59]. Theorem 6 Every algorithm for dynamic planar point location in monotone subdivisions uses Omega Gammaes n= log log n) steps per operation. Proof. To prove a lower bound for this problem we construct a monotone subdivision from the instance x 2 f0; Sigma1g that is similar to the upward ....

Franco P. Preparata and Roberto Tamassia. Fully dynamic point location in a monotone subdivision. SIAM Journal of Computing, 18(4):811--830, 1989.

....be performed using a point location data structure. Lemma 10. We can maintain a point location data structure in the farthest point Voronoi diagram in expected time O(log 2 n) per update or query. Proof: We can achieve these bounds per change and per query using any of a number of algorithms [8, 12, 13, 24, 35]. By Lemma 1, the expected amount of change per update is O(1) # Thus we are left with the problem of updating the potentially large sets of points in each diagram cell, after each change to the diagram. We no longer use the expected case model for these updates, since our analysis does not ....

F. P. Preparata and R. Tamassia. Fully dynamic point location in a monotone subdivision. SIAM J. Comput., 18:811--830, 1989.

....using the following point location data structure: Lemma 9. We can maintain a point location data structure in the farthest point Voronoi diagram in expected time O(log 2 n) per update or query. Proof: We can achieve these bounds per change and per query using any of a number of algorithms [6, 10, 11, 19, 30]. By Lemma 1, the amount of change per update is O(1) # Thus we are left with the problem of updating the potentially large sets of points in each diagram cell, after each change to the diagram. We no longer use the expected case model for these updates, since our analysis does not indicate when ....

F. P. Preparata and R. Tamassia. Fully dynamic point location in a monotone subdivision. SIAM J. Comput., 18:811--830, 1989.

....using the following point location data structure: Lemma 4.2. We can maintain a point location data structure in the farthest point Voronoi diagram in expected time O(log 2 n) per update or query. Proof. We can achieve these bounds per change and per query using any of a number of algorithms [7, 10, 11, 19, 27]. By Lemma 2.1, the amount of change per update is O(1) # We note that the O(log n) time dynamic Voronoi diagram algorithm maintains a point location data structure in the dual triangulation, however this does not provide the point location in the Voronoi diagram itself that is supplied by Lemma ....

F. P. Preparata and R. Tamassia. Fully dynamic point location in a monotone subdivision. SIAM J. Comput., 18:811--830, 1989.

....lower bounds for well studied problems. 1 For example, we prove optimality or near optimality of a number of dynamic algorithms in the literature from various fields like dynamic string and graph algorithms and computational geometry. These include planar point location in monotone subdivisions [3, 23], reachability in upward planar digraphs [25] and incremental parsing of balanced parentheses [9] We show that all these problems require time Omega Gammame n= log log n) per operation. It is known [8, 12, 15, 20] that this is also a lower bound for reachability in grid graphs. However, grid ....

....of the problem considers only monotone subdivisions, where the subdivision consists of polygons that are monotone (so no horizontal line crosses any polygon more than twice) In the dynamic version of this problem updates manipulate the geometry of the subdivision. Preparata and Tamassia [23] give an algorithm that runs in time O(log 2 n) per operation, this was improved to query time O(log n) by Baumgarten, Jung, and Mehlhorn [3] The lower bound for this problem in [16] applies only to algorithms returning the name of the region containing the queried point. The techniques of the ....

Franco P. Preparata and Roberto Tamassia. Fully dynamic point location in a monotone subdivision. SIAM Journal of Computing, 18(4):811--830, 1989.

....as tries, such as are used for string matching [2] However, the task studied in this paper involves preprocessing a collection of sets S i # R d i = 1, 2, N , rather than a collection of points. Also note that this task is not the point location problem previously studied [11, 12], in that here the sets S i are usually not disjoint. Our analysis is also di erent to that of [11, 12] in that here we are concerned with the expected time for a random query point x, rather than with worst case analysis. While the data structures described here are those of Samet [15] Samet ....

....preprocessing a collection of sets S i # R d i = 1, 2, N , rather than a collection of points. Also note that this task is not the point location problem previously studied [11, 12] in that here the sets S i are usually not disjoint. Our analysis is also di erent to that of [11, 12] in that here we are concerned with the expected time for a random query point x, rather than with worst case analysis. While the data structures described here are those of Samet [15] Samet considers their use only for the intersection problem: nd all pairs (i, j) where S i #S j #= #. Here ....

F. P. Preparata and R. Tamassia. Fully dynamic point location in a monotone subdivision. SIAM J. Computing, 18(4):811830, 1989.

....planar subdivisions. Furthermore in our context we do not need fully dynamic algorithms since we are not interested in deletions. If we demand O(n) storage space then the best relevant algorithms available, from the worst case complexity point of view, are the algorithms of Preparata Tamassia [28] and Cheng Janardan [7] The former algorithm has both a point location query time and an insertion (update) time of O(log 2 n) The latter algorithm is slightly better because it has a point location query time of O(log 2 n) but only an O(log n) insertion time. However, in our context every ....

Preparata, F., and R. Tamassia, Fully dynamic point location in a monotone subdivision, SIAM Journal on Computing, 18, pp. 811-830, 1989.

....Mathematics, The University of Iowa, Iowa City, IA 52242, USA. E mail: dstewart math.uiowa.edu 1 2 X. Han, S. Oliveira and D.E. Stewart sets f S i ae R d j i = 1; 2; N g, rather than a collection of points. Also note that this task is not the point location problem studied previously [9, 10], in that the sets S i are not usually disjoint. Our analysis is also different to that of [9, 10] in that here we are concerned with the expected time for a random query point x, rather than with worst case analysis. x S S S S 6 S S S 1 2 3 4 5 7 Figure 1: Example of distribution of support sets ....

....2 X. Han, S. Oliveira and D.E. Stewart sets f S i ae R d j i = 1; 2; N g, rather than a collection of points. Also note that this task is not the point location problem studied previously [9, 10] in that the sets S i are not usually disjoint. Our analysis is also different to that of [9, 10] in that here we are concerned with the expected time for a random query point x, rather than with worst case analysis. x S S S S 6 S S S 1 2 3 4 5 7 Figure 1: Example of distribution of support sets S i For each x and S i , we assume that we can test if x 2 S i in Theta(1) time. The point about ....

F. P. Preparata and R. Tamassia. Fully dynamic point location in a monotone subdivision. SIAM J. Computing, 18(4):811--830, 1989.

....Danish National Research Foundation Received October 1996. 2 HUSFELDT, RAUHE, SKYUM half a dozen natural problems, including the following: Computational Geometry: Dynamic planar point location in monotone subdivisions cannot be solved faster than ##an# n log log n) The important algorithm of Preparata and Tamassia [1989] achieves upper bounds of O(log 2 n) per operation. Graph algorithms: Dynamic transitive closure in planar acyclic digraphs with one source and one sink that are on the same face cannot be solved faster than ##an# n (log log n) 2 ) Tamassia and Preparata [1990] show a logarithmic upper bound ....

....of insertion and deletion of vertices or (chains of) edges. An important restriction of the problem, for which our bound will apply, considers only monotone subdivisions, where the subdivision consists of polygons that are monotone (so no horizontal line crosses any polygon more than twice) Preparata and Tamassia [1989] give an algorithm that runs in time O(log 2 n) per operation, this was improved to query time O(log n) by Baumgarten et al. 1992] To prove a lower bound for this problem we construct a monotone subdivision from the signed prefix sum instance y # 0, 1 n . This is easier drawn than ....

Preparata, Franco P. and Tamassia, Roberto. 1989. Fully Dynamic Point Location in a Monotone Subdivision. SIAM Journal of Computing 18, 4, 811--830.

....cell probe model. With this, we easily derive lower bounds for half a dozen natural problems, including the following: Computational Geometry: Dynamic planar point location in monotone subdivisions cannot be solved faster than ##an# n log log n) The important algorithm of Preparata and Tamassia [16] achieves upper bounds of O(log 2 n) per operation. # Basic Research in Computer Science, Centre of the Danish National Research Foundation Graph algorithms: Dynamic transitive closure in planar acyclic digraphs with one source and one sink that are on the same face cannot be solved faster ....

....deletion of vertices or (chains of) edges. An important restriction of the problem, for which our bound will apply, considers only monotone subdivisions, where the subdivision consists of polygons that are monotone (so no straight line crosses any polygon more than twice) Preparata and Tamassia [16] give an algorithm that runs in time O(log 2 n) per operation. Several other dynamic algorithms for this and other types of subdivisions have been found since, see [4] for a survey. To prove a lower bound for this problem we construct a monotone subdivision from the signed prefix sum instance y ....

Franco P. Preparata and Roberto Tamassia. Fully dynamic point location in a monotone subdivision. SIAM Journal of Computing, 18(4):811--830, 1989.

....Their second method achieves o(log n log log n k) time for inserting deleting monotone chains, but cost O(log n log log n) time to insert delete a vertex, and the increase the query time to O(log 2 n log log n) 4. 2 Monotone Planar Point Location The data structure of Preparata and Tamassia[32] for monotone subdivision is based on the chain method. The repertory of update operations includes inserting vertices on edges, inserting monotone chains of edges and their inverses. The space requirement is O(n) and the query time is O(log 2 n) Vertices can be inserted and deleted in time ....

F.P.Preparata and R.Tamassia, Fully Dynamic Point Location in Monotone Subdivision. SIAM Journal of Computing 18,1989 811-830.

....planar subdivisions. Furthermore in our context we do not need fully dynamic algorithms since we are not interested in deletions. If we demand O(n) storage space then the best relevant algorithms available, from the worst case complexity point of view, are the algorithms of Preparata Tamassia [PT89] and Cheng Janardan [CJ92] The former algorithm has both a point location query time and an insertion (update) time of O(log 2 n) The latter algorithm is slightly better because it has a point location query time of O(log 2 n) but only an O(log n) insertion time. However, in the present ....

Preparata, F. P. and Tamassia, R., "Fully dynamic point location in a monotone sub- division," SIAM Journal on Computing, vol. 18, 1989, pp. 811-830.

....a lower bound for the query time in a linear space comparison based point location data structure. All of our methods assume that the planar subdivision does not change over time, but environments allowing for dynamic changes to the subdivision over time are well motivated and wellstudied, as well [3, 4, 10, 11, 19, 32, 33, 34, 38]. Thus, another interesting open problem is whether one can, say, achieve the adaptive query bounds of Theorem 3.5 in such dynamic environments, where one allows insertions and deletions of vertices and edges in the subdivision S. ....

F. P. Preparata and R. Tamassia. Fully dynamic point location in a monotone subdivision. SIAM J. Comput., 18:811--830, 1989.

....lower bound for the query time in a linear space comparison based point location data structure. All of our methods assume that the planar subdivision does not change over time, but environments allowing for dynamic changes to the subdivision over time are well motivated and well studied, as well [3, 4, 10, 11, 19, 32, 33, 34, 38]. Thus, another interesting open problem is whether one can, say, achieve the adaptive query bounds of Theorem 3.4 in such dynamic environments, where one allows insertions and deletions of vertices and edges in the subdivision S. ....

F. P. Preparata and R. Tamassia. Fully dynamic point location in a monotone subdivision. SIAM J. Comput., 18:811--830, 1989.

.... first defined in conjunc tion with a planarity testing algorithm [18, 33] and were later used for a variety of topological and geometric graph problems, such as embedding (see, e.g. 6, 15, 42] visibility (see, e.g. 36, 43, 53] drawing (see, e.g. 1, 12, 44] point location (see, e.g. [35, 45]) and floorplanning (see, e.g. 30] One of the notable properties of planar bipolar orientations is that they induce a 2 dimensional lattice [31] on the vertices of the graph. See [10] for a recent comprehensive study of bipolar orientations. Canonical orderings were first defined by de ....

F. P. Preparata and R. Tamassia. Fully dynamic point location in a monotone subdivision. SIAM J. Cornput., 18:811 830, 1989. 34

....practical flavor. Such methods, however, refer to the static case where no alteration of the map is allowed during its use. Due to the obvious importance of the dynamic setting, in recent years considerable attention has been devoted to the development of dynamic point location algorithms [6, 8, 14, 15, 21, 25, 26, 30]. The best results to date for dynamic point location in an n vertex map are due to ChengJanardan [6] with O(log 2 n) query time, O(log n) update time, and O(n) space. Baumgarten, Jung, and Mehlhorn are reported to have recently developed a fractional cascading method with O(log n log log n) ....

....have O( p n polylog(n) query update time [1, 7] and support ray shooting in a set of possibly intersecting segments without taking advantage of the structure of planar maps. A property that appears to greatly facilitate the development of dynamic point location techniques is monotonicity ([8, 15, 25]) Whereas the restriction to monotone maps is quite adequate for many important applications (e.g. convex maps, such as Voronoi diagrams) yet the exclusion of general maps is a severe shortcoming. In the static case, a general map can be reduced to monotone (or, as we say in this paper, ....

F.P. Preparata and R. Tamassia, "Fully Dynamic Point Location in a Monotone Subdivision," SIAM J. Computing, Vol. 18, 811--830, 1989.

....chain method [10] The update operations supported are inserting and deleting edges and isolated vertices. If only insertions are performed, the update time is reduced to O(log 2 n) amortized. Preparata and Tamassia give two dynamic techniques for monotone and convex subdivisions, respectively [18, 19]. The data structure for monotone subdivisions [18] supports inserting vertices on edges, inserting monotone chains of edges, and their inverses. The space requirement is O(n) The query time is O(log 2 n) Vertices can be inserted or deleted in time O(log n) and a monotone chain with k edges ....

....inserting and deleting edges and isolated vertices. If only insertions are performed, the update time is reduced to O(log 2 n) amortized. Preparata and Tamassia give two dynamic techniques for monotone and convex subdivisions, respectively [18, 19] The data structure for monotone subdivisions [18] supports inserting vertices on edges, inserting monotone chains of edges, and their inverses. The space requirement is O(n) The query time is O(log 2 n) Vertices can be inserted or deleted in time O(log n) and a monotone chain with k edges can be inserted or deleted in time O(log 2 n k) ....

[Article contains additional citation context not shown here]

F.P. Preparata and R. Tamassia, "Fully Dynamic Point Location in a Monotone Subdivision," SIAM J. Computing, Vol. 18, 811--830, 1989.

.... first defined in conjunction with a planarity testing algorithm [18, 33] and were later used for a variety of topological and geometric graph problems, such as embedding (see, e.g. 6, 15, 42] visibility (see, e.g. 36, 43, 53] drawing (see, e.g. 1, 12, 44] point location (see, e.g. [35, 45]) and floorplanning (see, e.g. 30] One of the notable properties of planar bipolar orientations is that they induce a 2 dimensional lattice [31] on the vertices of the graph. See [10] for a recent comprehensive study of bipolar orientations. Canonical orderings were first defined by de ....

F. P. Preparata and R. Tamassia. Fully dynamic point location in a monotone subdivision. SIAM J. Comput., 18:811--830, 1989.

....[95] The update operations supported are inserting and deleting edges and isolated vertices. If only insertions are performed, the update time is reduced to O(log 2 n) 109 (pp. 135 143) Preparata and Tamassia give two dynamic techniques for monotone and convex subdivisions, respectively [135,136]. The data structure for monotone subdivisions [135] is based on the chain method [95] The repertory of update operations includes inserting vertices on edges, inserting monotone chains of edges, and their inverses. The space requirement is O(n) The query time is O(log 2 n) Vertices can be ....

....deleting edges and isolated vertices. If only insertions are performed, the update time is reduced to O(log 2 n) 109 (pp. 135 143) Preparata and Tamassia give two dynamic techniques for monotone and convex subdivisions, respectively [135,136] The data structure for monotone subdivisions [135] is based on the chain method [95] The repertory of update operations includes inserting vertices on edges, inserting monotone chains of edges, and their inverses. The space requirement is O(n) The query time is O(log 2 n) Vertices can be inserted or deleted in time O(log n) and a monotone ....

[Article contains additional citation context not shown here]

F.P. Preparata and R. Tamassia, "Fully Dynamic Point Location in a Monotone Subdivision," SIAM J. Computing 18 (1989), 811--830.

....practical flavor. Such methods, however, refer to the static case where no alteration of the map is allowed during its use. Due to the obvious importance of the dynamic setting, in recent years considerable attention has been devoted to the development of dynamic point location algorithms [2, 6, 8, 14, 15, 21, 25, 26, 31]. All the known dynamic point location results are for connected maps, since maintaining region names in a disconnected map would require solving half planar range searching in a dynamic environment, for which no polylog time algorithm is known. The best results to date for dynamic point location ....

....[1, 7] with O( p n polylog(n) query update time; they support ray shooting in a set of possibly intersecting segments without taking advantage of the structure of planar maps. A property that appears to greatly facilitate the development of dynamic pointlocation techniques is monotonicity ([8, 15, 25]) Whereas the restriction to monotone maps is quite adequate for many important applications, yet the exclusion of more general maps is a severe shortcoming. In the static case, a connected map can be reduced to monotone (or, as we say in this paper, normalized) by the straightforward insertion ....

F. P. Preparata and R. Tamassia, Fully dynamic point location in a monotone subdivision, SIAM J. Comput., 18 (1989), pp. 811--830.

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