### Abstract

We present efficient algorithms for shortest-path and minimum-link-path queries between two convex polygons inside a simple polygon P, which acts as an obstacle to be avoided. Let n be the number of vertices of P, and h the total number of vertices of the query polygons. We show that shortest-path queries can be performed optimally in time O(logh+log n) (plus O(κ) time for reporting the κ edges of the path) using a data structure with O(κ) space and preprocessing time, and that minimum-link-path queries can be performed in optimal time O(logh+f log n) (plus O(κ) to report the κ links), with O(n
^{3}) space and preprocessing time. We also extend our results to the dynamic case, and give a unified data structure that supports both queries for convex polygons in the same region of a connected planar subdivision S. The update operations consist of insertions and deletions of edges and vertices. Let n be the current number of vertices in S. The data structure uses O(n) space, supports updates in O(log
^{2} n) time, and performs shortest-path and minimum-link-path queries in times O(log h+log
^{2}n) (plus O (κ) to report the κ edges of the path) and O(log h + κlog
^{2} n), respectively. Performing shortest-path queries is a variation of the well-studied separation problem, which has not been efficiently solved before in the presence of obstacles. Also, it was not previously known how to perform minimum-link-path queries in a dynamic environment, even for two-point queries.

Original language | English (US) |
---|---|

Pages (from-to) | 85-121 |

Number of pages | 37 |

Journal | International Journal of Computational Geometry and Applications |

Volume | 7 |

Issue number | 1-2 |

State | Published - 1997 |

### Fingerprint

### Keywords

- Analysis of algorithms
- Computational geometry
- Minimum-link path
- Shortest path
- Static and dynamic data structures

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Applied Mathematics
- Computational Mathematics
- Geometry and Topology
- Theoretical Computer Science

### Cite this

*International Journal of Computational Geometry and Applications*,

*7*(1-2), 85-121.

**Optimal shortest path and minimum-link path queries between two convex polygons inside a simple polygonal obstacle.** / Chiang, Yi Jen; Tamassia, Roberto.

Research output: Contribution to journal › Article

*International Journal of Computational Geometry and Applications*, vol. 7, no. 1-2, pp. 85-121.

}

TY - JOUR

T1 - Optimal shortest path and minimum-link path queries between two convex polygons inside a simple polygonal obstacle

AU - Chiang, Yi Jen

AU - Tamassia, Roberto

PY - 1997

Y1 - 1997

N2 - We present efficient algorithms for shortest-path and minimum-link-path queries between two convex polygons inside a simple polygon P, which acts as an obstacle to be avoided. Let n be the number of vertices of P, and h the total number of vertices of the query polygons. We show that shortest-path queries can be performed optimally in time O(logh+log n) (plus O(κ) time for reporting the κ edges of the path) using a data structure with O(κ) space and preprocessing time, and that minimum-link-path queries can be performed in optimal time O(logh+f log n) (plus O(κ) to report the κ links), with O(n 3) space and preprocessing time. We also extend our results to the dynamic case, and give a unified data structure that supports both queries for convex polygons in the same region of a connected planar subdivision S. The update operations consist of insertions and deletions of edges and vertices. Let n be the current number of vertices in S. The data structure uses O(n) space, supports updates in O(log 2 n) time, and performs shortest-path and minimum-link-path queries in times O(log h+log 2n) (plus O (κ) to report the κ edges of the path) and O(log h + κlog 2 n), respectively. Performing shortest-path queries is a variation of the well-studied separation problem, which has not been efficiently solved before in the presence of obstacles. Also, it was not previously known how to perform minimum-link-path queries in a dynamic environment, even for two-point queries.

AB - We present efficient algorithms for shortest-path and minimum-link-path queries between two convex polygons inside a simple polygon P, which acts as an obstacle to be avoided. Let n be the number of vertices of P, and h the total number of vertices of the query polygons. We show that shortest-path queries can be performed optimally in time O(logh+log n) (plus O(κ) time for reporting the κ edges of the path) using a data structure with O(κ) space and preprocessing time, and that minimum-link-path queries can be performed in optimal time O(logh+f log n) (plus O(κ) to report the κ links), with O(n 3) space and preprocessing time. We also extend our results to the dynamic case, and give a unified data structure that supports both queries for convex polygons in the same region of a connected planar subdivision S. The update operations consist of insertions and deletions of edges and vertices. Let n be the current number of vertices in S. The data structure uses O(n) space, supports updates in O(log 2 n) time, and performs shortest-path and minimum-link-path queries in times O(log h+log 2n) (plus O (κ) to report the κ edges of the path) and O(log h + κlog 2 n), respectively. Performing shortest-path queries is a variation of the well-studied separation problem, which has not been efficiently solved before in the presence of obstacles. Also, it was not previously known how to perform minimum-link-path queries in a dynamic environment, even for two-point queries.

KW - Analysis of algorithms

KW - Computational geometry

KW - Minimum-link path

KW - Shortest path

KW - Static and dynamic data structures

UR - http://www.scopus.com/inward/record.url?scp=0031504214&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031504214&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031504214

VL - 7

SP - 85

EP - 121

JO - International Journal of Computational Geometry and Applications

JF - International Journal of Computational Geometry and Applications

SN - 0218-1959

IS - 1-2

ER -