Computational Geometry

1. Algorithms
for
Convex Partitioning of a Polygon
Kasun Ranga Wijeweera
(krw19870829@gmail.com)
1

2. Convex Partition of a 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 of Convex 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 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

5. 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

6. Thank you!
6