## Geometric Shortest Paths and Network Optimization (1998)

Venue: | Handbook of Computational Geometry |

Citations: | 147 - 13 self |

### BibTeX

@INPROCEEDINGS{Mitchell98geometricshortest,

author = {Joseph S.B. Mitchell},

title = {Geometric Shortest Paths and Network Optimization},

booktitle = {Handbook of Computational Geometry},

year = {1998},

pages = {633--701},

publisher = {Elsevier Science Publishers B.V. North-Holland}

}

### Years of Citing Articles

### OpenURL

### Abstract

Introduction A natural and well-studied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal

### Citations

10922 |
Computers and Intractability: A Guide to the Theory of NP-Completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ... of links, while there exists a path joining start and goal that has only 2 links. 17sMulti-criteria optimization problems tend to be difficult. Even the bicriteria path problem in a graph is NP-hard =-=[166]-=-: Does there exist a path from s to t whose length is less than L and whose weight is less than W ? Pseudo-polynomial time algorithms are known, such as the algorithm of Hansen [193], who finds all Pa... |

8530 |
Introduction to Algorithms
- Cormen, Leiserson, et al.
- 1990
(Show Context)
Citation Context ... lists of results on parallel algorithms in geometry. We will freely use the "big-Oh" notation for upper bounds on time and space requirements. We also use "big-Omega" notation for lower bounds. (See =-=[125]-=- for definitions.) We use " ~O(\Deltas\Deltas\Delta )" to indicate an upper bound in which we suppress polylogarithmic factors. Many of the results discussed in this survey are also reported, in a mor... |

1988 | Robot Motion Planning - Latombe - 1991 |

1762 |
Computational Geometry: An Introduction
- Preparata, Shamos
- 1985
(Show Context)
Citation Context ...s either intersect in a common vertex, a common edge, or not at all. A triangulation of a simple polygon P can be computed in O(n) time [93]; a polygonal domain can be triangulated in time O(n log n) =-=[329]-=- or O(n + h log1+ffl h) [54] time. (See the chapter of Bern and Plassman [66] in this handbook, or the survey by Bern [63] for more information on triangulations.) We will use the term obstacle to ref... |

1436 | A note on two problems in connexion with graphs
- Dijkstra
- 1959
(Show Context)
Citation Context ...uja, Magnanti, and Orlin [10]. Here, we mention the case in which all edge weights are non-negative, as this is the most relevant for geometric instances. Then, a standard algorithm given by Dijkstra =-=[140]-=- allows one to compute a tree of shortest paths from any one source node to all other nodes of the graph. Early implementations of Dijkstra's algorithm required time O(v2) or O(e log v), where v denot... |

1407 |
Network Flows: Theory, Algorithms, and Applications
- Ahuja, Magnanti, et al.
- 1993
(Show Context)
Citation Context ...thm is, of course, robustness; see the survey by Schirra [353] in this handbook. Shortest Paths in Graphs Shortest paths in graphs and networks are well studied; see, e.g., Ahuja, Magnanti, and Orlin =-=[10]-=-. Here, we mention the case in which all edge weights are non-negative, as this is the most relevant for geometric instances. Then, a standard algorithm given by Dijkstra [138] allows one to compute a... |

686 |
Heuristics: intelligent search strategies for computer problem solving
- Pearl
- 1984
(Show Context)
Citation Context ...Ghosh [174], and O'Rourke and Streinu [314] for some recent results and some pointers to related work. 1In practice, it may be faster to apply the A\Lambdasheuristic search algorithm (e.g., see Pearl =-=[324]-=-), using the straight-line Euclidean distance as heuristic function, h(\Delta ) (which is a lower bound, so it implies an "admissible" algorithm). 8sFor further information on visibility, visibility g... |

382 |
On the hardness of approximating minimization problems
- Lund, Yannakakis
- 1993
(Show Context)
Citation Context ...ve is to find a minimum-weight tree having at least one vertex from each group. Because of a reduction from set cover, it is NP-hard to approximate the group Steiner tree to a factor of o(log k); see =-=[221, 242, 365, 272]-=-. A (k \Gammas1)-approximation algorithm is given by Reich and Widmayer [337] and Ihler [220]. (See also Ihler, Reich, and Widmayer [223].) Slav'ik [365] gives an O(log k)-approximation algorithm for ... |

318 | Polynomial Time Approximation Schemes for Euclidean TSP and other Geometric Problems
- Arora
- 1996
(Show Context)
Citation Context ...dy" algorithm [393]. We refer the reader Bern and Eppstein [64] and Du and Hwang [143], for excellent surveys on these problems and the recent results. Finally, though, a PTAS was discovered by A=-=rora [35] and Mitch-=-ell [289]. This result serves to separate the geometric versions of the problem from the "metric" version (in an arbitrary graph whose edge lengths satisfy the triangle inequality), since th... |

313 |
The Traveling Salesman Problem
- Lawler, Lenstra, et al.
- 1985
(Show Context)
Citation Context ...lem in which one wants a shortest path that visits S.) The TSP is a classical problem in combinatorial optimization, and has been studied extensively in many forms, including geometric instances; see =-=[59, 252, 345, 232]-=-. The problem is NP-hard, as shown by Papadimitriou [318], even for points in the Euclidean plane. The TSP has a simple approximation algorithm based on "doubling" the minimum spanning tree. Since an ... |

291 |
On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents
- Dubins
- 1957
(Show Context)
Citation Context ...lve exactly, algorithms for restricted versions of the problem, as well as approximation algorithms, have been the topic of recent investigations. Early investigations into the problem were by Dubins =-=[146]-=-, who characterized shortest curvature constrained paths in the absence of obstacles: a shortest path consists of a sequence of at most three segments, each of which is a straight line segment ("S") o... |

289 |
Triangulating a Simple Polygon in Linear Time
- Chazelle
- 1991
(Show Context)
Citation Context ...s a decomposition of P into triangles such that any two triangles either intersect in a common vertex, a common edge, or not at all. A triangulation of a simple polygon P can be computed in O(n) time =-=[91]-=-; a polygonal domain can be triangulated in time O(n log n) [327] or O(n + h log 1+ffl h) [54] time. (See the chapter of Bern and Plassman [66] in this handbook, or the survey by Bern [63] for more in... |

189 | Optimal paths for a car that goes both forwards and backwards
- Reeds, Shepp
- 1990
(Show Context)
Citation Context ...ee segments, each of which is a straight line segment ("S") or an arc of a unit radius circle ("C"), with the allowable sequences being CCC, CSC, or a subsequence of one of these two. Reeds and Shepp =-=[336]-=- extended this result, obtaining a characterization of shortest paths in the case in which the robot is allowed to move in reverse, as well as forward. Boissonnat, C'er'ezo, and Leblond [75] give an a... |

187 |
On constructing minimum spanning trees in k-dimensional spaces and related problems
- Yao
- 1982
(Show Context)
Citation Context ...ince there can be \Omega (n2) Delaunay edges, even in !3. However, geometry can be exploited to avoid examining the full set of \Gamma n2\Deltas= \Omega (n2) weighted edges in the complete graph. Yao =-=[393]-=- was the first to compute an MST in !d in subquadratic time. His general method yields a time bound of O(n2\Gamma ffd(log n)1\Gamma ffd), where ffd is a constant depending on the dimension d. His algo... |

181 |
New lower bound techniques for robot motion planning problems
- Canny, Reif
- 1987
(Show Context)
Citation Context ...ins open, however, whether or not a polynomial-time algorithm exists in two dimensions. For three or more dimensions, the problem is at least NP-hard, as implied by the lower bounds of Canny and Reif =-=[82]-=-. Approximation methods have been developed by Donald et al. [140], who have given a polynomial-time algorithm that produces a trajectory requiring time at most (1 + ffl) times optimal, for the decoup... |

176 |
Scaling and related techniques for geometry problems
- Gabow, Bentley, et al.
- 1984
(Show Context)
Citation Context ... algorithms are known. Agarwal et al. [5] give a deterministic algorithm requiring O(n logd n) time for any metric having a polyhedral unit ball (e.g., L1 and L1); see also Gabow, Bentley, and Tarjan =-=[163]-=-. In three dimensions, there is now an optimal O(n log n) time algorithm for the MST in the L1 or L1 metric, due to Krznaric, Levcopoulos, and Nilsson [246] (improving an earlier O(n log n log log n) ... |

164 | Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems
- Mitchell
- 1999
(Show Context)
Citation Context ...ved in time 2O(k log k)n+O(n log n), which is simply O(n log n) for fixed k. Blum et al. [72] obtained the first O(1)-approximation; this was greatly simplified with the 2p2-approximation of Mitchell =-=[287, 292]-=-. Ultimately, a PTAS was given by Arora [35] and Mitchell [291]. More details will be given below. In general graphs having nonnegative edge weights, the current best approximation algorithm is a 3- a... |

160 | Faster shortest-path algorithms for planar graphs
- Rauch, Klein, et al.
- 1997
(Show Context)
Citation Context ...heaps, Fredman and Tarjan [162] gave an O(e + v log v) time implementation, and argued that this is optimal in a comparison-based model of computation. Exploiting planarity, Henzinger, Klein, and Rao =-=[200]-=- have obtained a linear-time algorithm for computing all shortest paths from a single source in planar graphs having nonnegative edge weights. There has been some recent progress too in devising new a... |

152 |
The discrete geodesic problem
- MITCHELL, MOUNT, et al.
- 1987
(Show Context)
Citation Context ...g the visibility graph (which may have quadratic size), an alternative paradigm for shortest-path problems is to construct the (linear-size) shortest path map directly. The continuous Dijkstra method =-=[280, 281, 282, 284, 285, 293, 294]-=- was developed for this purpose. Building on the success of the method in solving (in nearly linear time) the shortest-path problem for the L1 metric (see Section 4.1), Mitchell [286, 288] developed a... |

150 |
Nonholonomic multi-body mobile robots: Controllability and motion planning in the presence of obstacles
- Barraquand, Latombe
- 1993
(Show Context)
Citation Context ... in both cases, based on optimal control theory. (See also [368].) Approximation algorithms for a shortest "ffl-robust" path were given by Jacobs and Canny [227, 228]. (See also Barraquand a=-=nd Latomb [55].) Here, &-=-quot;ffl-robust" roughly means that small perturbations of certain points along the path do not cause the path to penetrate an obstacle. They place points that discretize the boundaries of the po... |

149 | Path-planning strategies for a point mobile automaton moving amidst unknown obstacles of arbitrary shape
- Lumelsky, Stepanov
- 1987
(Show Context)
Citation Context ...about the obstacles that constitute the holes of P . Some of the first work that obtained worst-case bounds on the length of a path produced by a navigation strategy was that of Lumelsky and Stepanov =-=[270, 271]-=-. They give navigation strategies for a tactile robot moving among a set of arbitrary obstacles. The robot is assumed to know, at any given time, its own position, the position of the goal, and whethe... |

144 | Voronoi diagrams
- Aurenhammer, Klein
- 2000
(Show Context)
Citation Context ...put (e.g., number of edges). A geodesic Voronoi diagram (VD) is a Voronoi diagram for a set of sites, in which the underlying metric is the geodesic distance. See the chapter of Aurenhammer and Klein =-=[45]-=- in this handbook for details about Voronoi diagrams. The geodesic center of P is a point within P that minimizes the maximum of the shortest-path lengths to any other point in P . The geodesic diamet... |

143 |
Dynamic Steiner tree problem
- Imase, Waxman
- 1991
(Show Context)
Citation Context ...iner tree for S. As shown by Imase 6Many authors have defined the Steiner ratio to be the reciprocal of what we call the Steiner ratio. We follow the notation of Bern and Eppstein [64]. 31sand Waxman =-=[225]-=-, a natural greedy strategy results in an O(log n) competitive ratio, for any metric space: at step i, simply join the ith point vi to the connected graph Ti\Gamma 1 built so far, by linking vi to the... |

140 | Approximation Algorithms for Directed Steiner Problems
- Charikar, Cheung, et al.
- 1999
(Show Context)
Citation Context ... special case of an edge-weighted tree. Bateman et al. [58] give the first sublinear approximation factor for general graphs, with an approximation factor of (1 + ln k 2 ) \Delta p k. Charikar et al. =-=[87]-=- give a k ffl -approximation algorithm that runs in polynomial time, as well as an O(log 2 n)-approximation algorithm that runs in quasi-polynomial time. Garg, Konjevod, and Ravi [167] give a randomiz... |

140 |
Steiner minimal trees
- Gilbert, Pollak
- 1968
(Show Context)
Citation Context ...d extensively in the last several years. A simple example (the three corners of an equilateral triangle) shows that the Euclidean Steiner ratio in the plane can be as high as 2=p3. Gilbert and Pollak =-=[177]-=- conjectured that this ratio can in fact never be greater than 2=p3. This conjecture was finally confirmed by a proof due to Du and Hwang [144]. (For the L1 metric, the Steiner ratio in the plane is 3... |

139 | Spanning trees and spanners
- Eppstein
(Show Context)
Citation Context ... are graphs that, for every pair of points, contain a path whose length is at most t times the interpoint distance (Euclidean, geodesic, etc.); see the survey on spanners in this handbook by Eppstein =-=[152]-=-, as well as [79, 108, 353]. Two-Point Queries Two-point queries in a polygonal domain are much more challenging than the case of simple polygons, where optimal algorithms are known. One approach, obs... |

139 |
Motion planner for nonholonomic mobile robots
- Laumond, Jacobs, et al.
- 1994
(Show Context)
Citation Context ...t it is always possible to obtain a curvature-constrained path from s to t if the s and t lie in the same open, pathconnected component of free space. Further, when allowing reversals, Laumond et al. =-=[251]-=- give an algorithm that determines a path (if one exists), producing a path having a local optimality property. Desaulniers [139] shows that, in the presence of reversals, in may be that no shortest p... |

129 |
Fast algorithms for geometric traveling salesman problems
- Bentley
- 1992
(Show Context)
Citation Context ...lem in which one wants a shortest path that visits S.) The TSP is a classical problem in combinatorial optimization, and has been studied extensively in many forms, including geometric instances; see =-=[59, 250, 343, 230]. The prob-=-lem is NP-hard, as shown by Papadimitriou [316], even for points in the Euclidean plane. The TSP has a simple approximation algorithm based on "doubling" the minimum spanning tree. Since an ... |

129 | A polylogarithmic approximation algorithm for the group Steiner tree problem.Journal of Algorithms
- Garg, Konjevod, et al.
(Show Context)
Citation Context ... k. Charikar et al. [89] give a kffl-approximation algorithm that runs in polynomial time, as well as an O(log2 n)-approximation algorithm that runs in quasi-polynomial time. Garg, Konjevod, and Ravi =-=[169]-=- give a randomized O(log3 n log k)-approximation algorithm for general graphs, which improves to O(log2 n log k) for a class of graphs that includes planar graphs, as well as graphs induced by a set o... |

126 |
Some algebraic and geometric computations in pspace
- Canny
- 1988
(Show Context)
Citation Context ...velocity, that do not collide. (Effectively, the fact that the obstacles are moving lifts the dimension of the problem from two to three, making it substantially more difficult; see Section 6.) Canny =-=[80]-=- has given a PSPACE algorithm to solve the asteroid avoidance problem. 5 On-Line Algorithms and Navigation Without Maps In all of the optimal path problems we have discussed so far, we have assumed th... |

121 |
An Analysis of Several Heuristics for the Traveling Salesman Problem
- Rosenkrantz, Sterns, et al.
- 1977
(Show Context)
Citation Context ...is a self-loop of length zero through site v1.) Various insertion methods are possible based on the choice of ordering of the sites for insertion. In a landmark paper, Rosenkrantz, Stearns, and Lewis =-=[351]-=- show that an arbitrary order of insertion of the sites gives a (dlog ne + 1)-approximation of the TSP, in arbitrary metric spaces. Further, they showed that nearest insertion and cheapest insertion l... |

120 | Searching in the plane
- Baeza-Yates, Culberson, et al.
- 1993
(Show Context)
Citation Context ... that shows a matching upper bound of ae(n) = O( p n), both for a vision-based robot and for a tactile robot (utilizing a "doubling" search procedure, suggested by Baeza-Yates, Culberson, an=-=d Rawlins [48]). If the -=-obstacles are aligned rectangles having aspect ratio at most f and longest side at most g (and shortest side at least 1), then Mei and Igarashi [275] give an "adjusted bias heuristic" that a... |

119 |
There are planar graphs almost as good as the complete graph
- Chew
- 1989
(Show Context)
Citation Context ...for every pair of points, contain a path whose length is at most t times the interpoint distance (Euclidean, geodesic, etc.); see the survey on spanners in this handbook by Eppstein [152], as well as =-=[79, 108, 353]-=-. Two-Point Queries Two-point queries in a polygonal domain are much more challenging than the case of simple polygons, where optimal algorithms are known. One approach, observed by Chen, Daescu, and ... |

113 |
The traveling salesman problem
- Junguer, Reinelt, et al.
- 1995
(Show Context)
Citation Context ...lem in which one wants a shortest path that visits S.) The TSP is a classical problem in combinatorial optimization, and has been studied extensively in many forms, including geometric instances; see =-=[59, 252, 345, 232]-=-. The problem is NP-hard, as shown by Papadimitriou [318], even for points in the Euclidean plane. The TSP has a simple approximation algorithm based on "doubling" the minimum spanning tree. Since an ... |

109 | Motion planning in the presence of moving obstacles
- Reif, Sharir
- 1985
(Show Context)
Citation Context ...g a fixed (known) trajectory, and the problem is to find a minimum-time obstacle-avoiding path for a point robot that is subject to a velocity bound. This problem was first studied by Reif and Sharir =-=[339]-=-, who show that the general problem is PSPACE-hard in three dimensions and that the two-dimensional problem can be solved in exponential time in the case of pure translational motion. Canny and Reif [... |

107 |
The visibility complex
- Pocchiola, Vegter
- 1996
(Show Context)
Citation Context ...s to proceed as follows. Using O(n2) space, we can store the shortest path map, SPM(v; P ), rooted at all n vertices. Then, for any s and t, we can use the visibility complex of Pocchiiola and Vegter =-=[325]-=- to compute the set of ks vertices visible to s and kt vertices visible to t, in time O(K log n), where K = minfks; ktg (using a standard "lock step" computation of the visibility from the two points)... |

103 |
On Steiner minimal trees with rectilinear distance
- Hwang
- 1976
(Show Context)
Citation Context ... ratio can in fact never be greater than 2=p3. This conjecture was finally confirmed by a proof due to Du and Hwang [144]. (For the L1 metric, the Steiner ratio in the plane is 3/2, and this is tight =-=[216]-=-.) Approximation algorithms have also been obtained for the Steiner tree problem. First, because of the Steiner ratio, the MST algorithms already give a 2=p3-approximation for the Euclidean Steiner tr... |

103 |
An 11/6-approximation algorithm for the network Steiner problem, Algorithmica 9
- Zelikovsky
- 1993
(Show Context)
Citation Context ...he Steiner ratio, the MST algorithms already give a 2=p3-approximation for the Euclidean Steiner tree problem in the plane. However, in a series of results, starting with important work by Zelikovsky =-=[394]-=-, improved approximation algorithms were obtained, for both graph versions and geometric versions of the problem. In the Euclidean plane, the approximation factor has been improved to just over 1.1 by... |

97 |
The complexity of computing Steiner minimal trees
- Garey, Graham, et al.
(Show Context)
Citation Context ...llowing the flexibility of adding Steiner points in order to obtain a potentially shorter spanning tree makes the problem much more difficult. In fact, the Steiner tree problem is known to be NP-hard =-=[165]-=-, even for points in the Euclidean plane. The Steiner tree problem is in sharp contrast with the MST problem, which can be solved exactly in low-degree polynomial time. It is natural, therefore, to st... |

96 |
Ray shooting in polygons using geodesic triangulations
- Chazelle, Edelsbrunner, et al.
(Show Context)
Citation Context ...algorithm for constructing the visibility graph of a simple polygon ([199]) and can be used for constructing a geodesic triangulation of a simple polygon, which allows for efficient ray-shooting (see =-=[92, 207]-=-). They also form a crucial step in solving link distance problems (Section 4.2). 3 Paths in a Polygonal Domain In contrast to the situation in simple polygons, where there is a unique taut-string pat... |

96 | A nearly best-possible approximation algorithm for node-weighted steiner trees
- Klein, Ravi
- 1995
(Show Context)
Citation Context ...ve is to find a minimum-weight tree having at least one vertex from each group. Because of a reduction from set cover, it is NP-hard to approximate the group Steiner tree to a factor of o(log k); see =-=[221, 242, 365, 272]-=-. A (k \Gammas1)-approximation algorithm is given by Reich and Widmayer [337] and Ihler [220]. (See also Ihler, Reich, and Widmayer [223].) Slav'ik [365] gives an O(log k)-approximation algorithm for ... |

95 |
On shortest paths in polyhedral spaces
- Sharir, Schorr
- 1986
(Show Context)
Citation Context ..., which may be exponential in general, but far smaller in many practical cases. Open Problem 14 Can one compute a shortest path map for a polyhedral domain in output-sensitive time? Sharir and Schorr =-=[364]-=- gave a doubly exponential time (22 O(n) ) exact algorithm, based on reducing to an algebraic decision problem in the theory of real closed fields. This result was improved by Reif and Storer [342], w... |

94 |
Planning smooth paths for mobile robots
- Jacobs, Canny
- 1989
(Show Context)
Citation Context ...native method of obtaining characterizations in both cases, based on optimal control theory. (See also [370].) Approximation algorithms for a shortest "ffl-robust" path were given by Jacobs and Canny =-=[229, 230]-=-. (See also Barraquand and Latomb [55].) Here, "ffl-robust" roughly means that small perturbations of certain points along the path do not cause the path to penetrate an obstacle. They place points th... |

93 |
The prize collecting traveling salesman problem
- BALAS
- 1989
(Show Context)
Citation Context ... to this problem, since each site can be replicated w i times; the running time is then polynomial in n and R. Another related problem is the prize-collecting salesperson problem, as studied by Balas =-=[52] (see also-=- [69]). It differs from the quota-driven salesperson problem, in that, in addition to "values" w i , there are non-negative penalties associated with each site, and the objective function is... |

93 |
An output sensitive algorithm for computing visibility graphs
- Ghosh, Mount
- 1987
(Show Context)
Citation Context ...O(n)-space algorithm to compute the visibility graph of a simple polygon. Overmars and Welzl [315] obtained a relatively simple O(EV G log n)- time method, requiring O(n) space. Then, Ghosh and Mount =-=[175]-=- obtained an algorithm with worst-case optimal running time, O(EV G + n log n), using O(EV G) working storage space. More recently, Pocchiola and Vegter [326, 327] and Rivi`ere [348] have given algori... |

91 | Nearly linear time approximation schemes for Euclidean TSP and other geometric problems
- Arora
- 1998
(Show Context)
Citation Context ... in this handbook. Other well-studied network optimization problems that we do not attempt to survey here include minimum cost matching (which has polynomial-time exact and approximate solutions; see =-=[36, 379, 380, 389]-=-) and minimum weight triangulation (MWT) (whose complexity status is still open, although constant-factor approximation algorithms exist for both the Steiner and non-Steiner versions; see Bern and Epp... |

88 | Navigating in unfamiliar geometric terrain
- Blum, Raghavan, et al.
- 1991
(Show Context)
Citation Context ... is shown that if the goal region t is an infinite vertical line ("wall"), at distance n from s, and the obstacles are aligned rectangles, then ae(n) = \Omega\Gamma p n). Blum, Raghavan, and=-= Schieber [73] provide a "swe-=-ep algorithm" for this wall problem that shows a matching upper bound of ae(n) = O( p n), both for a vision-based robot and for a tactile robot (utilizing a "doubling" search procedure,... |

88 |
C.H.: The weighted region problem: finding shortest paths through a weighted planar subdivision
- Mitchell, Papadimitriou
- 1991
(Show Context)
Citation Context ...g the visibility graph (which may have quadratic size), an alternative paradigm for shortest-path problems is to construct the (linear-size) shortest path map directly. The continuous Dijkstra method =-=[280, 281, 282, 284, 285, 293, 294]-=- was developed for this purpose. Building on the success of the method in solving (in nearly linear time) the shortest-path problem for the L1 metric (see Section 4.1), Mitchell [286, 288] developed a... |

87 | An improved approximation ratio for the minimum latency problem
- Goemans, Kleinberg
- 1998
(Show Context)
Citation Context ...d by the repairman/deliveryman that is traveling along the tour.) For the problem in graphs, Blum et al. [71] have given a 128-approximation algorithm; this has been improved by Goemans and Kleinberg =-=[179]-=-, who obtain a factor of 29. By a direct application of Theorem 2 of [71], which states that a c-approximation for the k-MST implies an 8c-approximation for the MLP, we see that recent PTAS results on... |

86 | An optimal algorithm for Euclidean shortest paths in the plane
- Hershberger, Suri
- 1999
(Show Context)
Citation Context ...continuous Dijkstra method applicable to the Euclidean shortest-path problem, obtaining the first subquadratic (O(n3=2+ffl)) time bound. Subsequently, this result was improved by Hershberger and Suri =-=[207, 208]-=-, who achieve a nearly optimal algorithm based also on the continuous Dijkstra method. They give an O(n log n) time and O(n log n) space algorithm, coming close to the lower bounds of \Omega (n + h lo... |