## Rectilinear Paths among Rectilinear Obstacles (1996)

Venue: | Discrete Applied Mathematics |

Citations: | 23 - 3 self |

### BibTeX

@ARTICLE{Lee96rectilinearpaths,

author = {D. T. Lee and C. D. Yang and C. K. Wong},

title = {Rectilinear Paths among Rectilinear Obstacles},

journal = {Discrete Applied Mathematics},

year = {1996},

volume = {70},

pages = {185--215}

}

### Years of Citing Articles

### OpenURL

### Abstract

Given a set of obstacles and two distinguished points in the plane the problem of finding a collision free path subject to a certain optimization function is a fundamental problem that arises in many fields, such as motion planning in robotics, wire routing in VLSI and logistics in operations research. In this survey we emphasize its applications to VLSI design and limit ourselves to the rectilinear domain in which the goal path to be computed and the underlying obstacles are all rectilinearly oriented, i.e., the segments are either horizontal or vertical. We consider different routing environments, and various optimization criteria pertaining to VLSI design, and provide a survey of results that have been developed in the past, present current results and give open problems for future research. 1 Introduction Given a set of obstacles and two distinguished points in the plane, the problem of finding a collision free path subject to a certain optimization function is a fundamental probl...

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Citation Context ...ime and minimizing the total cost of these paths could yield better overall results. Since the general problems that call for an overall optimal solution for arbitrary k pairs of paths are intractable=-=[20]-=-, it is practical to settle for routing of a limited number of pairs of paths at a time. We thus consider the problem of finding k non-crossing paths, where k is a small fixed integer. These paths can... |

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Citation Context ...ucted a path preserving graph which is assured to retain desired goal paths, mbsp, smbp, mcp, bbsp and blmbp (see section 4.4). By applying the shortest path searching algorithm of Fredman and Tarjan =-=[18]-=- to such a graph, an O(K + e log e) algorithm is derived for MBSP, SMBP and MCP, where K, bounded by O(em), is the number of intersections between tracks, i.e., vertical or horizontal lines passing th... |

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Citation Context ...ding algorithms [77, 82]. Furthermore the minimum spanning tree is widely used for constructing a Steiner tree which plays an important role in global or detailed routing for multi-terminal nets (see =-=[76, 29]-=- for Steiner tree surveys). More complex Steiner trees are studied where more factors are considered [63]. The more one can control the parameters involved in the single path routing, the more likely ... |

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Citation Context ...izing the other. In addition to these optimization factors of rectilinear paths, the routing models and the types of obstacles also affect the complexity of the problems. Most of the previous results =-=[4, 11, 15, 22, 23, 34, 36, 37, 43, 47, 48, 49, 73, 75, 77]-=- deal with problems under the assumption that paths do not cross any obstacles, i.e., the path is collision free (Figure 1a). Among these results, many aim at finding a collision free shortest path am... |

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Citation Context ...hich can be used to solve other query problems. f17g SMALLP pol . Linear time and space algorithm by McDonald and Peters [45]. It relies on the linear time polygon triangulation algorithm of Chazelle =-=[8]-=-. f18g Manhattan Path. `(n log e) time algorithm by Lipski [41]. f19g SMALLP pol . O(e log e) time and space algorithm by de Berg [14], in which Query 2 - SMALLP pol is solved in O(log e) time. f20g 2... |

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Citation Context ...(Recall that m is the number of extreme edges of the obstacles.) The time complexity can be improved to O(K+ e log e log log e ), if one applies the trans-dichotomous algorithm of Fredman and Willard =-=[19]-=-. 5 For BBSP an O(B(K + e log e)) time algorithm was given in [79] based on the same graph, where B = minfb ; bg, and b and b are, respectively, the bend number of the mbsp and the given bound for the... |

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Citation Context ...ding algorithms [77, 82]. Furthermore the minimum spanning tree is widely used for constructing a Steiner tree which plays an important role in global or detailed routing for multi-terminal nets (see =-=[76, 29]-=- for Steiner tree surveys). More complex Steiner trees are studied where more factors are considered [63]. The more one can control the parameters involved in the single path routing, the more likely ... |

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Citation Context ...t of marked grid points in this grid structure. The Lee algorithm [35] was the first one finding a shortest path between two points on a routing plane with grid. It applies the algorithm due to Moore =-=[54]-=- on a planar grid structure. It simply runs over the grid step by step starting from the source point till the destination is reached (Figure 3a). The Lee's and Moore's algorithm are called the grid e... |

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Citation Context ...tained by Yang, et al. [80] using a horizontal wave front approach. More of it is discussed and compared with 45-degree wave front approach in section 4.5. 3.4 Weighted X and Weighted X 2 Problems In =-=[52]-=-, Mitchell and Papadimitriou first introduced the Euclidean weighted region problem with applications to motion planning in robotics. They proposed an algorithm to find the shortest weighted distance ... |

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Citation Context ...izing the other. In addition to these optimization factors of rectilinear paths, the routing models and the types of obstacles also affect the complexity of the problems. Most of the previous results =-=[4, 11, 15, 22, 23, 34, 36, 37, 43, 47, 48, 49, 73, 75, 77]-=- deal with problems under the assumption that paths do not cross any obstacles, i.e., the path is collision free (Figure 1a). Among these results, many aim at finding a collision free shortest path am... |

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Citation Context ...izing the other. In addition to these optimization factors of rectilinear paths, the routing models and the types of obstacles also affect the complexity of the problems. Most of the previous results =-=[4, 11, 15, 22, 23, 34, 36, 37, 43, 47, 48, 49, 73, 75, 77]-=- deal with problems under the assumption that paths do not cross any obstacles, i.e., the path is collision free (Figure 1a). Among these results, many aim at finding a collision free shortest path am... |

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Citation Context ...complexity is O(E log e), where E is the number of events while dragging wave fronts on the plane. That E = O(e log e) was proved but E = O(e) was conjectured. The conjecture was later proved true in =-=[50]-=-. 2 Mitchell in a recent paper [50] presented an O(e + n log n) time algorithm, which is asymptotically optimal. 3 Yang, et al. [82] recently proposed a method combining the features from these two pa... |

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Citation Context ...e which plays an important role in global or detailed routing for multi-terminal nets (see [76, 29] for Steiner tree surveys). More complex Steiner trees are studied where more factors are considered =-=[63]-=-. The more one can control the parameters involved in the single path routing, the more likely one can construct a routing scheme where all parameters are well balanced. Acknowledgment The authors wou... |

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Citation Context ...bitrary k, is known to be NP-complete [20]. When k equals 2 and the graph, G = (V; E), is planar or chordal, O(jEj) time suffices, due to a result by Perl and Shiloach [60]. Ohtsuki [55] and Shiloach =-=[65]-=- have also presented O(jEj\ThetajV j) algorithms to find two disjoint paths in an undirected graph G = (V; E). The problem of finding two shortest vertex disjoint paths between a pair of vertices in a... |

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Citation Context ...quired for maze running algorithms, a gridless model was introduced, where obstacles are each represented by a set of line segments on the plane. Mikami and Tabuchi [46], and independently, Hightower =-=[27]-=- proposed a line searching approach to finding a path connecting the source and the destination. It first emits vertical and horizontal lines from the source. If these lines have intersections with ot... |

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Citation Context ...ce to represent an m \Theta m grid structure and has a worst case time complexity O(l 2 ) for finding a path crossing l grid points. Many algorithms were derived to improve the maze running algorithm =-=[1, 2, 3, 21, 24, 28, 62, 66]-=-. They achieved certain amount of speed-up but did not improve the worst case time complexity. See section 4.1 for more details about this approach. To reduce the large amount of memory required for m... |

34 | Efficient algorithms for Euclidean shortest path and visibility problems with polygonal obstacles
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Citation Context ...y used path preserving graph, known as visibility graph, preserves the Euclidean shortest path among obstacles. Most of the Euclidean shortest path algorithms are based on the visibility graph (e.g., =-=[4, 32, 59, 73]-=-). After constructing such a graph, one can apply any suitable graph searching algorithm to find the goal path. Therefore, finding a path preserving graph as small in size as possible is the key to th... |

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Citation Context ...educe the large amount of memory required for maze running algorithms, a gridless model was introduced, where obstacles are each represented by a set of line segments on the plane. Mikami and Tabuchi =-=[46]-=-, and independently, Hightower [27] proposed a line searching approach to finding a path connecting the source and the destination. It first emits vertical and horizontal lines from the source. If the... |

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Citation Context ... MBP and SMALLP pol The problem of finding a minimum bend path has been studied and gained more attention recently. In Euclidean case, several minimum bend path 4 algorithms have been proposed (e.g., =-=[33, 53, 67]-=-). These algorithms for Euclidean cases may provide approximations for rectilinear case [13, 33, 42, 45, 53, 57, 67]. Although the Lee algorithm [35] can be modified to find a minimum bend path, (Figu... |

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Citation Context ...hat finds the shortest geodesic (respectively minimum link distance) path in a simple polygon between two query points is solved in O(log e + k) time by Guibas and Hershberger [22] (respectively Suri =-=[68]-=-). s t s s s s s t t t t t 1 1 2 3 4 5 6 2 3 5 6 t 4 Figure 4: Shortest non-crossing paths in a plane region. f1g SP 2 . `(n log n) time and O(n) space algorithm by deRezende, Lee and Wu [15] using pl... |

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Finding two disjoint paths between two pairs of vertices in a graph
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Citation Context ...or k pairs of vertices for an arbitrary k, is known to be NP-complete [20]. When k equals 2 and the graph, G = (V; E), is planar or chordal, O(jEj) time suffices, due to a result by Perl and Shiloach =-=[60]-=-. Ohtsuki [55] and Shiloach [65] have also presented O(jEj\ThetajV j) algorithms to find two disjoint paths in an undirected graph G = (V; E). The problem of finding two shortest vertex disjoint paths... |

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The Lee path connection algorithm
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Citation Context ...ce to represent an m \Theta m grid structure and has a worst case time complexity O(l 2 ) for finding a path crossing l grid points. Many algorithms were derived to improve the maze running algorithm =-=[1, 2, 3, 21, 24, 28, 62, 66]-=-. They achieved certain amount of speed-up but did not improve the worst case time complexity. See section 4.1 for more details about this approach. To reduce the large amount of memory required for m... |

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Citation Context ...interconnection. It is desirable to have these routing measures, i.e., length and the number of bends, minimized. (In the geometry literature it is referred to as bicriteria optimization problem. See =-=[51] for detai-=-ls.) Unfortunately they cannot be both optimized simultaneously in general. Therefore a "best path" can be categorized by either minimizing each of these measures individually, giving them d... |

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A modification of Lee’s path connection algorithm
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Citation Context ...ce to represent an m \Theta m grid structure and has a worst case time complexity O(l 2 ) for finding a path crossing l grid points. Many algorithms were derived to improve the maze running algorithm =-=[1, 2, 3, 21, 24, 28, 62, 66]-=-. They achieved certain amount of speed-up but did not improve the worst case time complexity. See section 4.1 for more details about this approach. To reduce the large amount of memory required for m... |

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Citation Context ...ntion recently. In Euclidean case, several minimum bend path 4 algorithms have been proposed (e.g., [33, 53, 67]). These algorithms for Euclidean cases may provide approximations for rectilinear case =-=[13, 33, 42, 45, 53, 57, 67]-=-. Although the Lee algorithm [35] can be modified to find a minimum bend path, (Figure 3b) it has the drawback that it is neither time nor space efficient. The Mikami-Tabuchi line searching algorithm ... |

16 |
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16 | A new algorithm for shortest paths among obstacles in the plane - Mitchell - 1991 |

15 |
Dynamic orthogonal segment intersection search
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Citation Context ...m for finding an mbp, which is modified from line searching algorithms. The algorithm runs in O(e log 2 e) time using O(e) space. Alternatively, one can use the data structure given by Imai and Asano =-=[30]-=- and apply Ohtsuki's algorithm to solve MBP in `(e log e) time and space. Sato et al. [64] independently presented a line searching algorithm which finds an mbp in O(e log e) time. Their algorithm has... |

14 | Shortest path queries in rectilinear worlds
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Citation Context ...ntion recently. In Euclidean case, several minimum bend path 4 algorithms have been proposed (e.g., [33, 53, 67]). These algorithms for Euclidean cases may provide approximations for rectilinear case =-=[13, 33, 42, 45, 53, 57, 67]-=-. Although the Lee algorithm [35] can be modified to find a minimum bend path, (Figure 3b) it has the drawback that it is neither time nor space efficient. The Mikami-Tabuchi line searching algorithm ... |

14 |
An optimal algorithm for shortest rectilinear paths among obstacles
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13 |
An algorithm for segment-dragging and its implementation
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Citation Context ...rmation (an increasing entity) when it travels. Traveling of wave fronts is done by segment dragging operations, which find the first hit point and drag the wave front to it. An algorithm of Chazelle =-=[9]-=- for dragging vertical or horizontal segments is modified for dragging the 45-degree wave fronts. Wave fronts are subject to different changes; some become longer while dragging (as the wave fronts ju... |

13 |
An efficient algorithm for link-distance problems
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Citation Context ... MBP and SMALLP pol The problem of finding a minimum bend path has been studied and gained more attention recently. In Euclidean case, several minimum bend path 4 algorithms have been proposed (e.g., =-=[33, 53, 67]-=-). These algorithms for Euclidean cases may provide approximations for rectilinear case [13, 33, 42, 45, 53, 57, 67]. Although the Lee algorithm [35] can be modified to find a minimum bend path, (Figu... |

13 |
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13 |
On bends and lengths of rectilinear paths: a graph-theoretic approach
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Citation Context ...end path obtained by Ohtsuki[57] is by no means close to being the shortest. 4 Sometimes it is referred to as minimum link path in the literature. Using a graph-theoretic approach, Yang, Lee and Wong =-=[79]-=- constructed a path preserving graph which is assured to retain desired goal paths, mbsp, smbp, mcp, bbsp and blmbp (see section 4.4). By applying the shortest path searching algorithm of Fredman and ... |

13 | Rectilinear path problems among rectilinear obstacles revisited
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(Show Context)
Citation Context ...t E = O(e) was conjectured. The conjecture was later proved true in [50]. 2 Mitchell in a recent paper [50] presented an O(e + n log n) time algorithm, which is asymptotically optimal. 3 Yang, et al. =-=[82]-=- recently proposed a method combining the features from these two path preserving graphs and obtained a better algorithm which runs in O(e log m+m log 3=2 m) time using O(e +m log 3=2 m) space. 3.2 Pr... |

12 |
Shortest Rectilinear Paths Among Weighted Obstacles
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Citation Context ...mong the weights and the coordinates of vertices) and n is the number of the weighted regions. The version where paths are rectilinear and obstacles are non-adjacent was studied by Lee, Yang and Chen =-=[38]-=-, who extended the O(e log 3=2 e) result of Clarkson, Kapoor and Vaidya [11] for SP to Weighted SP. For Weighted SP 2 problem Yang, Chen and Lee [78] proposed an optimal `(n log n) time algorithm usin... |