We are given a set S of n points in the Euclidean plane. We assume that S is in general position. A simple polygon P is an empty polygon of S if each vertex of P is a point in S and every point in S is either outside P or a vertex of P. In this paper, we consider the problem of enumerating all the empty polygons of a given point set. To design an efficient enumeration algorithm, we use a reverse search by Avis and Fukuda with child lists. We propose an algorithm that enumerates all the empty polygons of S in O(n2|ε(S)|)-time, where ε(S) is the set of empty polygons of S. Moreover, by applying the same idea to the problem of enumerating surrounding polygons of a given point set S, we propose an enumeration algorithm that enumerates them in O(n2)-delay, while the known algorithm enumerates in O(n2 log n)-delay, where a surroundingpolygon of S is a polygon such that each vertex of the polygon is a point in S and every point in S is either inside the polygon or a vertex of the polygon.
Shunta TERUI
Iwate University
Katsuhisa YAMANAKA
Iwate University
Takashi HIRAYAMA
Iwate University
Takashi HORIYAMA
Hokkaido University
Kazuhiro KURITA
Nagoya University
Takeaki UNO
National Institute of Informatics
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Shunta TERUI, Katsuhisa YAMANAKA, Takashi HIRAYAMA, Takashi HORIYAMA, Kazuhiro KURITA, Takeaki UNO, "Enumerating Empty and Surrounding Polygons" in IEICE TRANSACTIONS on Fundamentals,
vol. E106-A, no. 9, pp. 1082-1091, September 2023, doi: 10.1587/transfun.2022DMP0007.
Abstract: We are given a set S of n points in the Euclidean plane. We assume that S is in general position. A simple polygon P is an empty polygon of S if each vertex of P is a point in S and every point in S is either outside P or a vertex of P. In this paper, we consider the problem of enumerating all the empty polygons of a given point set. To design an efficient enumeration algorithm, we use a reverse search by Avis and Fukuda with child lists. We propose an algorithm that enumerates all the empty polygons of S in O(n2|ε(S)|)-time, where ε(S) is the set of empty polygons of S. Moreover, by applying the same idea to the problem of enumerating surrounding polygons of a given point set S, we propose an enumeration algorithm that enumerates them in O(n2)-delay, while the known algorithm enumerates in O(n2 log n)-delay, where a surroundingpolygon of S is a polygon such that each vertex of the polygon is a point in S and every point in S is either inside the polygon or a vertex of the polygon.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.2022DMP0007/_p
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@ARTICLE{e106-a_9_1082,
author={Shunta TERUI, Katsuhisa YAMANAKA, Takashi HIRAYAMA, Takashi HORIYAMA, Kazuhiro KURITA, Takeaki UNO, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Enumerating Empty and Surrounding Polygons},
year={2023},
volume={E106-A},
number={9},
pages={1082-1091},
abstract={We are given a set S of n points in the Euclidean plane. We assume that S is in general position. A simple polygon P is an empty polygon of S if each vertex of P is a point in S and every point in S is either outside P or a vertex of P. In this paper, we consider the problem of enumerating all the empty polygons of a given point set. To design an efficient enumeration algorithm, we use a reverse search by Avis and Fukuda with child lists. We propose an algorithm that enumerates all the empty polygons of S in O(n2|ε(S)|)-time, where ε(S) is the set of empty polygons of S. Moreover, by applying the same idea to the problem of enumerating surrounding polygons of a given point set S, we propose an enumeration algorithm that enumerates them in O(n2)-delay, while the known algorithm enumerates in O(n2 log n)-delay, where a surroundingpolygon of S is a polygon such that each vertex of the polygon is a point in S and every point in S is either inside the polygon or a vertex of the polygon.},
keywords={},
doi={10.1587/transfun.2022DMP0007},
ISSN={1745-1337},
month={September},}
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TY - JOUR
TI - Enumerating Empty and Surrounding Polygons
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1082
EP - 1091
AU - Shunta TERUI
AU - Katsuhisa YAMANAKA
AU - Takashi HIRAYAMA
AU - Takashi HORIYAMA
AU - Kazuhiro KURITA
AU - Takeaki UNO
PY - 2023
DO - 10.1587/transfun.2022DMP0007
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E106-A
IS - 9
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - September 2023
AB - We are given a set S of n points in the Euclidean plane. We assume that S is in general position. A simple polygon P is an empty polygon of S if each vertex of P is a point in S and every point in S is either outside P or a vertex of P. In this paper, we consider the problem of enumerating all the empty polygons of a given point set. To design an efficient enumeration algorithm, we use a reverse search by Avis and Fukuda with child lists. We propose an algorithm that enumerates all the empty polygons of S in O(n2|ε(S)|)-time, where ε(S) is the set of empty polygons of S. Moreover, by applying the same idea to the problem of enumerating surrounding polygons of a given point set S, we propose an enumeration algorithm that enumerates them in O(n2)-delay, while the known algorithm enumerates in O(n2 log n)-delay, where a surroundingpolygon of S is a polygon such that each vertex of the polygon is a point in S and every point in S is either inside the polygon or a vertex of the polygon.
ER -