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![Algorithms that Fails to Produce the Minimum
• Let’s consider the polygons without holes disallowing Steiner
points.
• Algorithm by Feng and Pavlidis [1975] runs in O (N3n) time.
• Algorithm by Schachter [1978] runs in O (nN) time.
• Algorithm by Chazelle [1982] runs in O (n log n) time and
produces a partition in which the number of convex polygons
is fewer than 13/3 times the minimum.
• Algorithm by Greene [1983] runs in O (n log n) time and
produces a partition in which the number of convex polygons
is less than or equal to 4 times the minimum.
• Algorithm by Hertel & Mehlhorn [1985] runs in O (n log n)
time and produces a partition in which the number of convex
polygons is less than or equal to 4 times the minimum.
4](https://image.slidesharecdn.com/polygonconvexpartitioning-201016101028/85/Algorithms-for-Convex-Partitioning-of-a-Polygon-4-320.jpg)
![Algorithms that Succeeds Produce the Minimum
• Algorithm by Greene [1983] runs in O (N2n2) time
disallowing Steiner points.
• Algorithm by Keil [1985] runs in O (N2 n log n) time
disallowing Steiner points.
• Algorithm by Chazelle & Dobkin [1985] runs in O (n + N3)
time allowing Steiner points.
5](https://image.slidesharecdn.com/polygonconvexpartitioning-201016101028/85/Algorithms-for-Convex-Partitioning-of-a-Polygon-5-320.jpg)
![Algorithms that Fails to Produce the Minimum
• Let’s consider the polygons without holes disallowing Steiner
points.
• Algorithm by Feng and Pavlidis [1975] runs in O (N3n) time.
• Algorithm by Schachter [1978] runs in O (nN) time.
• Algorithm by Chazelle [1982] runs in O (n log n) time and
produces a partition in which the number of convex polygons
is fewer than 13/3 times the minimum.
• Algorithm by Greene [1983] runs in O (n log n) time and
produces a partition in which the number of convex polygons
is less than or equal to 4 times the minimum.
• Algorithm by Hertel & Mehlhorn [1985] runs in O (n log n)
time and produces a partition in which the number of convex
polygons is less than or equal to 4 times the minimum.
4](https://image.slidesharecdn.com/polygonconvexpartitioning-201016101028/75/Algorithms-for-Convex-Partitioning-of-a-Polygon-4-2048.jpg)
![Algorithms that Succeeds Produce the Minimum
• Algorithm by Greene [1983] runs in O (N2n2) time
disallowing Steiner points.
• Algorithm by Keil [1985] runs in O (N2 n log n) time
disallowing Steiner points.
• Algorithm by Chazelle & Dobkin [1985] runs in O (n + N3)
time allowing Steiner points.
5](https://image.slidesharecdn.com/polygonconvexpartitioning-201016101028/75/Algorithms-for-Convex-Partitioning-of-a-Polygon-5-2048.jpg)
Uploaded byKasun Ranga Wijeweera
Algorithms for Convex Partitioning of a Polygon
This document discusses algorithms for partitioning a polygon into the minimum number of convex polygons. It describes that if the polygon contains holes, finding the minimum convex partition is NP-hard. For polygons without holes and disallowing additional points, algorithms by Feng and Pavlidis (1975), Schachter (1978), Chazelle (1982), and Greene (1983) run in O(n log n) time but do not always produce the minimum partition. Algorithms by Greene (1983) and Keil (1985) run in O(N^2n^2) and O(N^2n log n) time respectively and are guaranteed to find the minimum partition without adding points. The algorithm by Chazelle
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Algorithms for Convex Partitioning of a Polygon
- 1. Algorithms for Convex Partitioning ofa Polygon Kasun Ranga Wijeweera (krw19870829@gmail.com) 1
- 2. Convex Partition ofa Polygon • A partition of a polygon P is defined as a set of polygons such that the interiors of the polygons do not intersect and the union of the polygons is equal to the interior of the original polygon P. • It is called a convex partition if each polygon in the set is convex. 2
- 3. Minimum Number ofConvex Polygons • If the polygon may contain holes then the problem of partitioning the polygon into minimum number of convex polygons is NP-hard, either allowing or disallowing Steiner points. 3
- 4. Algorithms that Failsto Produce the Minimum • Let’s consider the polygons without holes disallowing Steiner points. • Algorithm by Feng and Pavlidis [1975] runs in O (N3n) time. • Algorithm by Schachter [1978] runs in O (nN) time. • Algorithm by Chazelle [1982] runs in O (n log n) time and produces a partition in which the number of convex polygons is fewer than 13/3 times the minimum. • Algorithm by Greene [1983] runs in O (n log n) time and produces a partition in which the number of convex polygons is less than or equal to 4 times the minimum. • Algorithm by Hertel & Mehlhorn [1985] runs in O (n log n) time and produces a partition in which the number of convex polygons is less than or equal to 4 times the minimum. 4
- 5. Algorithms that SucceedsProduce the Minimum • Algorithm by Greene [1983] runs in O (N2n2) time disallowing Steiner points. • Algorithm by Keil [1985] runs in O (N2 n log n) time disallowing Steiner points. • Algorithm by Chazelle & Dobkin [1985] runs in O (n + N3) time allowing Steiner points. 5
- 6.