#### DMCA

## Self-Approaching Graphs

Citations: | 4 - 0 self |

### Citations

387 | Geographic Routing without Location Information,"
- Rao, Ratnasamy, et al.
- 2003
(Show Context)
Citation Context ...awing (or “geometric graph”) in the plane has points for vertices, and straight line segments for edges, where the weight of an edge is its Euclidean length. The drawingneed not be planar. Rao et al. =-=[27]-=- introduced the idea of greedy drawings.Agreedy drawing ofagraphis a straight-linedrawing in which, for each origin vertex s and destination vertex t, there is a neighbor of s thatisclosertotthansis,i... |

339 |
An efficient algorithm for determining the convex hull of a finite planar set. Information Processing Letters,
- Graham
- 1972
(Show Context)
Citation Context ..., or we will be able to use the properties of self-approaching paths to construct the convex hull of the traversed vertices incrementally in linear total time by an algorithm similar to Graham’s scan =-=[20]-=-. We now describe a step of the algorithm. Assume that the directed path Pi = {vi, . . . , vn} is self-approaching and assume the convex hull C of vertices {vi, . . . , vn} has already been computed a... |

273 | Optimal search in planar subdivisions.
- Kirkpatrick
- 1981
(Show Context)
Citation Context ...cture combining several known ideas. There is a static data structure for half-space range emptiness in R3 with O(n) space and O(log n) query time, by reduction to planar point location in dual space =-=[26]-=-; the preprocessing time is O(n) if we are given the convex hull. The static data structure can be transformed into a semidynamic data structure with O(b logb n) amortized insertion time and O(logb n ... |

267 | A decomposition of multidimensional point sets with applications to -nearest-neighbors and ¤ -body potential fields
- Callahan, Kosaraju
- 1995
(Show Context)
Citation Context ...the diameter of Bi is at most εd(Ai,Bi), where d(Ai,Bi) is the minimum distance between Ai and Bi. It is known that a well-separated pair decomposition consisting of s = O(n/ε 2 ) pairs always exists =-=[9]-=-. Furthermore, such a decomposition can be constructed by a simple quadtree-based algorithm (for example, see [20] or [10]), where the sets Ai and Bi are in fact squares appearing in the compressed qu... |

220 | Lower Bounds for Algebraic Computation Trees,”
- Ben-Or
- 1983
(Show Context)
Citation Context ...mbedded in R 3 is self-approaching. We do this by reducing from the set intersection problem, for which a solution requires Ω(nlogn) time on an input of size n in the algebraic computation tree model =-=[5]-=-. We can show the following: Theorem 3. Given a straight-line drawing of a path P = (v1,v2,...,vn) in R 3 , at least Ω(nlogn) time is required in the algebraic computation tree model to test whether P... |

145 | Spanning trees and spanners
- Eppstein
- 2000
(Show Context)
Citation Context ... the paper.Self-Approaching Graphs 3 2 Background Aspanner isagraphofboundeddilation.Spannershavebeenverywell-studied— see for example the book by Narasimhan and Smid [25] and the survey by Eppstein =-=[15]-=-. A main goal is to efficiently construct a spanner on a given set of points, with the objective of minimizing dilation while keeping the number or total length of edges small. For recent results, see... |

140 |
Geometric Spanner Networks.
- Narasimhan, Smid
- 2007
(Show Context)
Citation Context ... be found in the long version of the paper.Self-Approaching Graphs 3 2 Background Aspanner isagraphofboundeddilation.Spannershavebeenverywell-studied— see for example the book by Narasimhan and Smid =-=[25]-=- and the survey by Eppstein [15]. A main goal is to efficiently construct a spanner on a given set of points, with the objective of minimizing dilation while keeping the number or total length of edge... |

129 | Online routing in triangulations
- Bose, Morin
(Show Context)
Citation Context ...aph, see the survey by Gudmundsson and Knauer [19] and the paper by Wulff-Nilsen [29]. The Delaunay triangulationhasseveralgood properties:it has dilation factor below 2 [30], and is a greedy drawing =-=[8]-=-, although greedy paths in a Delaunay triangulation do not necessarily have bounded dilation. We show that the Delaunay triangulation is not necessarily self-approaching. 3 Preliminaries We let D(u,v)... |

114 | J.B.: Decomposable searching problems I. Static-to-dynamic transformation.
- Bentley, Saxe
- 1980
(Show Context)
Citation Context ...c data structure with O(b logb n) amortized insertion time and O(logb n log n) query time for a given parameter b, by known techniques—namely, a b-ary version of Bentley and Saxe’s logarithmic method =-=[6]-=-, using Chazelle’s linear-time algorithm for merging two convex hulls [12] as a subroutine. By setting b = log n, both amortized insertion time and query time are bounded by O(log2 n/ log log n), yiel... |

69 | an optimal algorithm for intersecting three-dimensional convex polyhedra,”
- Chazelle
- 1992
(Show Context)
Citation Context ...og n) query time for a given parameter b, by known techniques—namely, a b-ary version of Bentley and Saxe’s logarithmic method [6], using Chazelle’s linear-time algorithm for merging two convex hulls =-=[12]-=- as a subroutine. By setting b = log n, both amortized insertion time and query time are bounded by O(log2 n/ log log n), yielding the desired result. Next, we show that Theorem 5 is tight up to a fac... |

59 | D.: On a conjecture related to geometric routing
- Papadimitriou, Ratajczak
- 2005
(Show Context)
Citation Context ...or of s thatisclosertotthansis,i.e.,thereisagreedy st-pathP = (s = p1,p2,...,pk = t)suchthattheEuclideandistancesD(pi,t)decreaseasiincreases.Thisideahas attracted great interest in recent years (e.g. =-=[3,7,17,21,23,26]-=-) mainly because a greedy drawing of a graph permits greedy local routing. It is a very natural and desirable property that a path should always get closer to its destination, but there is more than o... |

43 | Approximation Algorithm
- Har-Peled
- 2010
(Show Context)
Citation Context ... we can rotate the coordinate axes by π/4 and apply the same construction to obtain another Steiner network G ′ . We can then return the union of G and G ′ . To construct G, we first build a quadtree =-=[20]-=-, defined as follows: The root stores an initial square enclosing P. At each node, we divide its square into four congruent subsquares and create a child for each subsquare that is not empty of points... |

37 | Some results on greedy embeddings in metric spaces.
- Leighton, Moitra
- 2010
(Show Context)
Citation Context ...or of s thatisclosertotthansis,i.e.,thereisagreedy st-pathP = (s = p1,p2,...,pk = t)suchthattheEuclideandistancesD(pi,t)decreaseasiincreases.Thisideahas attracted great interest in recent years (e.g. =-=[3,7,17,21,23,26]-=-) mainly because a greedy drawing of a graph permits greedy local routing. It is a very natural and desirable property that a path should always get closer to its destination, but there is more than o... |

29 |
Lower bounds for algebraic computation trees with integer inputs
- Yao
- 1991
(Show Context)
Citation Context ...f the set intersection problem, where we are asked to check if there is a common element in sets A and B. Using Yao’s improvement to Ben-Or’s lower bound constructions for algebraic computation trees =-=[36]-=-, it suffices to consider the case where A and B are sets of nonnegative integers. Letting M be the maximum element in A and B, we first divide each element of A and B by 2M/π so that both A and B are... |

24 | An algorithm to construct greedy drawings of triangulations
- Angelini, Frati, et al.
(Show Context)
Citation Context ...or of s thatisclosertotthansis,i.e.,thereisagreedy st-pathP = (s = p1,p2,...,pk = t)suchthattheEuclideandistancesD(pi,t)decreaseasiincreases.Thisideahas attracted great interest in recent years (e.g. =-=[3,7,17,21,23,26]-=-) mainly because a greedy drawing of a graph permits greedy local routing. It is a very natural and desirable property that a path should always get closer to its destination, but there is more than o... |

22 | Well-separated pair decomposition in linear time?
- Chan
- 2008
(Show Context)
Citation Context ...l-separated pair decomposition consisting of s = O(n/ε 2 ) pairs always exists [9]. Furthermore, such a decomposition can be constructed by a simple quadtree-based algorithm (for example, see [20] or =-=[10]-=-), where the sets Ai and Bi are in fact squares appearing in the compressed quadtree T. To finish the construction of G, we consider each pair {Ai,Bi} in the decomposition such that Ai and Bi are sepa... |

19 | Curves with increasing chords
- Rote
- 1994
(Show Context)
Citation Context ...e). We also explore the related notion of an increasing-chord graph drawing, which has the stronger property that every pair of vertices is joined by a path thatisself-approachinginbothdirections.Rote=-=[28]-=-provedthatincreasing-chord paths have geometric dilation at most 2.094. Our first result is a linear time algorithm to recognize a self-approaching polygonal path in the plane. This extends to R 3 , w... |

17 | On the geometric dilation of finite point sets
- Ebbers-Baumann, Grüne, et al.
- 2003
(Show Context)
Citation Context ...ite. By contrast, a self-approaching drawing may have crossing edges, for example, any complete geometric graph is self-approaching. Constructing a network to minimize detour has also been considered =-=[13,12]-=-, though not as extensively as spanners. Relevantbackgroundongreedydrawingsisasfollows.Answeringaconjecture of Papadimitriou and Ratajczak [26], Leighton and Moitra [23] showed that any 3-connected pl... |

14 | Sparse geometric graphs with small dilation
- Aronov, Berg, et al.
- 2005
(Show Context)
Citation Context ... goal is to efficiently construct a spanner on a given set of points, with the objective of minimizing dilation while keeping the number or total length of edges small. For recent results, see, e.g., =-=[4,16]-=-. If Steiner vertices are allowed, their number should also be minimized, and different versions of the problem arise if we include the Steiner points in measuring the dilation, see [14]. The detour o... |

13 | Computing geometric minimum-dilation graphs is np-hard
- Klein, Kutz
- 2006
(Show Context)
Citation Context ... goal is to efficiently construct a spanner on a given set of points, with the objective of minimizing dilation while keeping the number or total length of edges small. For recent results, see, e.g., =-=[4,16]-=-. If Steiner vertices are allowed, their number should also be minimized, and different versions of the problem arise if we include the Steiner points in measuring the dilation, see [14]. The detour o... |

12 |
C.: Dilation and Detours in Geometric Networks
- Gudmundsson, Knauer
(Show Context)
Citation Context ...very point set admits a self-approaching network of linear size.4 Self-Approaching Graphs For results on computing the dilation or detour of a path or graph, see the survey by Gudmundsson and Knauer =-=[19]-=- and the paper by Wulff-Nilsen [29]. The Delaunay triangulationhasseveralgood properties:it has dilation factor below 2 [30], and is a greedy drawing [8], although greedy paths in a Delaunay triangula... |

10 | Competitive routing in the half-θ6-graph
- Bose, Fagerberg, et al.
- 2012
(Show Context)
Citation Context |

10 | Succinct greedy geometric routing in the Euclidean plane
- Goodrich, Strash
- 2009
(Show Context)
Citation Context |

10 | E.: Self-approaching curves
- Icking, Klein, et al.
- 1999
(Show Context)
Citation Context ...graph drawing—the maximum, over vertices s and t, of the ratio of their distance in the graph to their Euclidean distance. The dilation factor of greedy graph drawings can be unbounded. Icking et al. =-=[22]-=- introduced a stronger notion of “getting closer” to a destination, that addresses both shortcomings of greedy paths. A curve from s to t is self-approaching if for any three points a,b,c appearing in... |

10 |
Improved upper bound on the stretch factor of Delaunay triangulations
- Xia
- 2011
(Show Context)
Citation Context ...tion or detour of a path or graph, see the survey by Gudmundsson and Knauer [19] and the paper by Wulff-Nilsen [29]. The Delaunay triangulationhasseveralgood properties:it has dilation factor below 2 =-=[30]-=-, and is a greedy drawing [8], although greedy paths in a Delaunay triangulation do not necessarily have bounded dilation. We show that the Delaunay triangulation is not necessarily self-approaching. ... |

9 | Minimum Manhattan network is NP-complete
- Chin, Guo, et al.
(Show Context)
Citation Context ...on finding Manhattan networks of minimum total length (socalled “minimum Manhattan networks”). There are efficient algorithms with approximation factor 2, and the problem has been shown to be NP-hard =-=[11]-=-. More relevant to us is the result of Gudmundsson et al. [18] that every point set admits a Manhattan network of O(nlogn) vertices and edges, and there are point sets for which any Manhattan network ... |

7 | Monotone drawings of graphs.
- Angelini, Colasante, et al.
- 2012
(Show Context)
Citation Context ...aching drawings are related to monotone drawings in which, for every pair of vertices s and t, there is an st-path that is monotone in some direction. This concept was introduced by Angelini, et al., =-=[1]-=- who gave algorithms to construct monotone planar drawings of trees and planar biconnected graphs. A follow-up paper [2] considers the case where a planar embedding is specified. Self-approaching draw... |

7 | Small Manhattan networks and algorithmic applications for the Earth Mover’s Distance. In:
- Gudmundsson, Klein, et al.
- 2007
(Show Context)
Citation Context ...d “minimum Manhattan networks”). There are efficient algorithms with approximation factor 2, and the problem has been shown to be NP-hard [13]. More relevant to us is the result of Gudmundsson et al. =-=[21]-=- that every point set admits a Manhattan network of O(n log n) vertices and edges, and there are point sets for which any Manhattan network has size at least Ω(n log n). This contrasts with our result... |

4 |
On succinct convex greedy drawing of 3-connected plane graphs
- He, Zhang
- 2011
(Show Context)
Citation Context |

3 | Monotone drawings of graphs with fixed embedding.
- Angelini, Didimo, et al.
- 2011
(Show Context)
Citation Context ...is monotone in some direction. This concept was introduced by Angelini, et al., [1] who gave algorithms to construct monotone planar drawings of trees and planar biconnected graphs. A follow-up paper =-=[2]-=- considers the case where a planar embedding is specified. Self-approaching drawings are not necessarily monotone, and monotone drawings are not necessarily self-approaching. The one relationship is t... |

3 | Light orthogonal networks with constant geometric dilation
- Dumitrescu, Tóth
- 2009
(Show Context)
Citation Context ...ite. By contrast, a self-approaching drawing may have crossing edges, for example, any complete geometric graph is self-approaching. Constructing a network to minimize detour has also been considered =-=[13,12]-=-, though not as extensively as spanners. Relevantbackgroundongreedydrawingsisasfollows.Answeringaconjecture of Papadimitriou and Ratajczak [26], Leighton and Moitra [23] showed that any 3-connected pl... |

3 | Embedding point sets into plane graphs of small dilation
- Ebbers-Baumann, Grüne, et al.
- 2005
(Show Context)
Citation Context ...see, e.g., [4, 18]. If Steiner vertices are allowed, their number should also be minimized, and different versions of the problem arise if we include the Steiner points in measuring the dilation, see =-=[16]-=-. The detour of a graph drawing is defined to be the supremum, over all points p, q of the drawing (whether at vertices, or interior to edges) of the ratio of their distance in the graph to their Eucl... |

3 | Euclidean greedy drawings of trees.
- Nollenburg, Prutkin
- 2013
(Show Context)
Citation Context ...l coordinates, but instead using local information of small size. Moitra [28] used combinatorial conditions to classify the trees that have greedy embeddings and very recently Nöllenburg and Prutkin =-=[30]-=- completely characterized greedy drawable trees. Connecting the ideas of greedy drawings and spanners, Bose et al. [8] showed that every triangulation has an embedding in which local routing produces ... |

2 | Near-linear-time deterministic plane Steiner spanners and TSP approximation for well-spaced point sets
- Borradaile, Eppstein
- 2012
(Show Context)
Citation Context ...-separated pair decomposition. The theorem generalizes to any constant dimension. We note that our construction bears some similarity to the construction used independently by Borradaile and Eppstein =-=[6]-=- to create small low-weight plane Steiner spanners in which the paths stay within a bounded range of angles. Whether planar self-approaching Steiner networks of linear size can be constructed or not i... |

1 |
Embeddingpointsets intoplanegraphsofsmall dilation
- Ebbers-Baumann, Grüne, et al.
(Show Context)
Citation Context ... see, e.g., [4,16]. If Steiner vertices are allowed, their number should also be minimized, and different versions of the problem arise if we include the Steiner points in measuring the dilation, see =-=[14]-=-. The detour of a graph drawing is defined to be the supremum, over all points p,q of the drawing (whether at vertices, or interior to edges) of the ratio of their distance in the graph to their Eucli... |

1 |
Small Manhattan networks andalgorithmic applications for theearth mover’s distance
- Gudmundsson, Klein, et al.
- 2007
(Show Context)
Citation Context ...d “minimum Manhattan networks”). There are efficient algorithms with approximation factor 2, and the problem has been shown to be NP-hard [11]. More relevant to us is the result of Gudmundsson et al. =-=[18]-=- that every point set admits a Manhattan network of O(nlogn) vertices and edges, and there are point sets for which any Manhattan network has size at least Ω(nlogn). This contrasts with our result tha... |

1 | A solution to the Papadimitriou-Ratajczak conjecture
- Moitra
- 2009
(Show Context)
Citation Context ...6], Leighton and Moitra [23] showed that any 3-connected planar graph has a greedy drawing, and Goodrich and Strash [17] reduced the number of bits needed for the coordinates in the embedding. Moitra =-=[24]-=- used combinatorial conditions to classify the trees that have greedy embeddings. Connecting the ideas of greedy drawings and spanners, Bose et al. [7] showed that every triangulation has an embedding... |

1 |
Computing the maximum detour of a plane geometric graph in subquadratic time
- Wulff-Nilsen
(Show Context)
Citation Context ...ching network of linear size.4 Self-Approaching Graphs For results on computing the dilation or detour of a path or graph, see the survey by Gudmundsson and Knauer [19] and the paper by Wulff-Nilsen =-=[29]-=-. The Delaunay triangulationhasseveralgood properties:it has dilation factor below 2 [30], and is a greedy drawing [8], although greedy paths in a Delaunay triangulation do not necessarily have bounde... |

1 | Computing geometric minimumdilation graphs is np-hard
- Giannopoulos, Klein, et al.
(Show Context)
Citation Context ... goal is to efficiently construct a spanner on a given set of points, with the objective of minimizing dilation while keeping the number or total length of edges small. For recent results, see, e.g., =-=[4, 18]-=-. If Steiner vertices are allowed, their number should also be minimized, and different versions of the problem arise if we include the Steiner points in measuring the dilation, see [16]. The detour o... |