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Research Papers

# On Minimum Link Monotone Path Problems

[+] Author and Article Information
Xiangzhi Wei

Ajay Joneja

Department of IELM, HKUST, Clear Water Bay, Kowloon, Hong Kongjoneja@ust.hk

J. Comput. Inf. Sci. Eng 11(3), 031002 (Aug 01, 2011) (10 pages) doi:10.1115/1.3615687 History: Received December 23, 2009; Accepted June 03, 2011; Published August 01, 2011; Online August 01, 2011

## Abstract

The problem of finding monotone paths between two given points has useful applications in path planning, and in particular, it is useful to look for minimum link paths. We are given a simple polygon P or a polygonal domain D with n vertices and a triplet of input parameters: (s, t, d), where s and t are two points in the plane and d is any direction. We show how to answer a query for the existence of a d-monotone path between s and t inside P (or D) in logarithmic time after preprocessing P in O(En) time, or D in O(En + ERlogR) time, where E is the size of the visibility graph of P (or D), and R is the number of reflex vertices in D. Our approach is based on the novel idea utilizing the dual graph of the trapezoidal map of P (or D). For polygonal domains, our approach uses a trapezoidal map associated with each visibility edge of D, and we show how to compute this large set of trapezoidal maps efficiently. Furthermore, we show how to output a minimum linkd-monotone path between points s and t, for an arbitrary input triplet (s, t, d).

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Topics: Algorithms , Chain , Bullets

## Figures

Figure 1

A polygon with its trapezoidal map T(d) and its dual graph H(d)

Figure 2

Relation between visibility edges and canonical monotone directions

Figure 3

Illustration of query process (a) If the query direction is a canonical direction, then s′ is the LCA of s′, t′ in H(d); (b) A d-monotone path from s to t inside Pd ; (c) Node s″ is LCA of s″ and t″ in H(d ″); s′ is not the LCA of s′ and t′ in H(d ′); there is a d-monotone path from s to points to the right (with respect to d) of reflex vertex v, e.g. t1 , and there is no such path for points to the left of v, e.g., t2

Figure 4

The proof of Theorem 2.3

Figure 5

Illustration of T(d′) and H(d′) in D

Figure 6

(a) Illustration for computing the entries in array VE (b) Entries corresponding to directions e and e′ in (a)

Figure 7

Illustration of computing an MLMP between s and t

Figure 8

The interior angle in the unilluminated region at any vertex, v, of an illumination chain created due to the monotonicity constraint is larger than 180 deg

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