M. S. Paterson and F. F. Yao. On nearest-neighbor graphs. In Automata, Languages and Programming, volume 623, pages 416-426. Springer, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....are no ties it is asymptotically Bayes optimal. 7 Proximity Graph Methods 7.1 Proximity graphs The most natural proximity graph de ned on a set of points fX; Y g is the nearest neighbor graph or NNG. Here each point in fX; Y g is joined by an edge to its nearest neighbor (Paterson and Yao [81]) Another well known proximity graph is the minimum spannig tree (MST) Zahn [135] In 1980 the relative neighborhood graph (RNG) was proposed as a tool for extracting the shape of a planar 8 pattern (see Toussaint [118] 116] 123] However, such de nitions are readily extended to higher ....

M. S. Paterson and F. F. Yao. On nearest-neighbor graphs. In Automata, Languages and Programming, volume 623, pages 416-426. Springer, 1992.

....algorithm for the minmax edge length triangulation problem. Many other proximity graphs have been studied. For example, Kirkpatrick and Radke [14] defined and studied the skeleton graphs that include both the modified gabriel and the relative neighborhood graph of a point set. Paterson and Yao [21] derived several combinatorial properties of k nearest neighborhood graphs and gave bounds on the size of such graphs. Toussaint [28] introduced the sphere of influence graph and proposed an O(n log n) algorithm for computing the sphere of influence graph of n points in the plane. An extensive ....

M. S. Paterson and F.F. Yao, On Nearest-Neighbor Graphs. Proc. ICALP '92, 1992, pp. 416-426.

.... was studied in [3, 4, 20] The strictly related question of representing trees as minimum spanning trees is considered in [12] Characterizing triangulations that can be drawn as Delaunay triangulations is studied in [11, 10] Graphs that admit a nearest neighbor drawing are characterized in [24, 13]. A survey on the proximity drawability testing problem is given by [8] In this paper we study the rectangle of influence drawability problem, i.e. the problem of characterizing those graphs that admit a rectangle of influence drawing. We focus on classes of graphs 2 that are traditionally ....

M. S. Paterson and F.F. Yao. On Nearest-Neighbor Graphs. Proceedings Annual International Colloqium on Automata, Languages and Programming, ICALP'92, LNCS 623, Springer-Verlag, 1992, pp. 416-426.

....of NN are the points of R i , and q; v 2 R i define an edge of NN if and only if v is the nearest neighbor of q (denoted by v = NN(q) or q is the nearest neighbor of v in R i . NN is well known to be a subgraph of DT R i , the Delaunay triangulation of R i , and to have maximum degree 6 [PY92]. We denote by d ffi DT i Gamma1 (v) the degree of v in DT i Gamma1 , and by E v2R i aefqg the expectation when v is chosen uniformly in R i ae fqg. Then we have E v2R i aefqg i d ffi DT i Gamma1 (v) j = E v2R i Gamma1 aefqg i d ffi DT i Gamma1 (v) j 6 notice that d ffi ....

M. S. Paterson and F. F. Yao. On nearest-neighbor graphs. In Proc. 19th Internat. Colloq. Automata Lang. Program., volume 623 of Lecture Notes Comput. Sci., pages 416--426. Springer-Verlag, 1992.

.... space with 3 dimensional fi regions is studied in [21] Results that are closely related to the proximity drawability testing problem concern the drawability of trees as minimum spanning trees [10] of triangulations as Delaunay triangulations [9, 8] and of graphs as nearest neighbor graphs [25, 11]. A survey on the proximity drawability testing problem is given by [7] 3 1.2 Overview of the Paper In this paper, we study the drawability problem in both the open rectangle of influence model and in the closed rectangle of influence model. We say that a graph is open rectangle of influence ....

M. S. Paterson and F.F. Yao. On Nearest-Neighbor Graphs. Proceedings Annual International Colloqium on Automata, Languages and Programming, ICALP'92, LNCS 623, Springer-Verlag, 1992, pp. 416-426.

.... graph of R i : that is, the vertices of NN are the points of R i , and q; v 2 R i dene an edge of NN if and only if v is the nearest neighbor of q in R i (denoted by v = NN(q) NN is well known to be a subgraph of DT R i , the Delaunay triangulation of R i , and to have maximum degree 6 [PY92]. We denote by d ffi DT i Gamma1 (v) the degree of v in DT i Gamma1 , and by E v2R i the expectation when v is chosen uniformly in R i . Then we have E v2R i i d ffi DT i Gamma1 (v) j = E v2S i Gamma1 i d ffi DT i Gamma1 (v) j 6 INRIA Incremental randomized Delaunay ....

M. S. Paterson and F. F. Yao. On nearest-neighbor graphs. In Proc. 19th Internat. Colloq. Automata Lang. Program., volume 623 of Lecture Notes Comput. Sci., pages 416426. Springer-Verlag, 1992.

....trees, the maximum degree in a nearest neighbor graph is five. Monma and Suri [1] showed that, conversely, any tree with vertex degree at most five is the minimum spanning tree of some point set; thus minimum spanning tree topologies are exactly characterized by their degrees. Paterson and Yao [2] considered the corresponding question for nearest neighbor graphs. They showed that a tree with depth D can have at most O(D 9 ) vertices. Thus unlike minimum spanning trees, nearest neighbor graphs can not be too bushy: a tree with many vertices must contain a long path. Paterson and Yao also ....

....to the annulus containing them, but a point may be assigned to a larger annulus if its nearest neighbor path goes away from the origin before returning. They then count the number of points that can be assigned to any one annulus, and the number of possible annuli. 1 Lemma 1 (Paterson and Yao [2]) At most O(C 2 D 2 ) points can be assigned to any annulus. Proof: Suppose a point is assigned to an annulus with inner radius r. Then the path from that point to the origin has length at least r. There are at most D edges in the path, and the edge lengths decrease as the path nears the ....

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M.S. Paterson and F.F. Yao. On nearest-neighbor graphs. 19th Int. Colloq. Automata, Languages, and Programming (1992) to appear. 5

....methods for solving Problems 2 and 4. Problem 3 is a generalization of the well known nearest neighbors problem. For classification problems, it is more robust than a simple nearest neighbors search. The graph of k nearest neighbors to each point has certain interesting theoretical properties [18, 19]. Eppstein and Erickson [13] showed how a variety of clustering problems such as those of finding k points with minimum diameter, circumradius, or variance, could all be solved efficiently using algorithms for Problem 3 as a subroutine, improving previous techniques based on kth order Voronoi ....

M.S. Paterson and F.F. Yao. On nearest-neighbor graphs. 19th Int. Colloq. Automata, Languages, and Programming (1992) to appear.

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M. S. Paterson and F. F. Yao. On nearest-neighbor graphs. In Automata, Languages and Programming, volume 623, pages 416-426. Springer, 1992.

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M. S. Paterson and F. F. Yao. On nearest-neighbor graphs. In Proc. 19th Internat. Colloq. Automata Lang. Program., volume 623 of Lecture Notes Comput. Sci., pages 416--426. Springer-Verlag, 1992.

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