Why Graham-Scan Needs to Sort Vertices Before Scanning
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1 Why Graham-Scan Needs to Sort Vertices Before Scanning Bill Thies November 12, Recitation Supplement
2 Graham Scan: Sorting Step The first stage of Graham-Scan sorts the points by their polar angle from the bottom-left vertex, p 0 : In recitation, we asked: would the scanning phase of Graham-Scan work on any simple polygon? That is, can you omit this sorting phase if you start from a simple polygon? The answer is NO.
3 Graham Scan: Sorting Step The first stage of Graham-Scan sorts the points by their polar angle from the bottom-left vertex, p 0 : In recitation, we asked: would the scanning phase of Graham-Scan work on any simple polygon? That is, can you omit this sorting phase if you start from a simple polygon? The answer is NO.
4 Graham Scan: Sorting Step The first stage of Graham-Scan sorts the points by their polar angle from the bottom-left vertex, p 0 : In recitation, we asked: would the scanning phase of Graham-Scan work on any simple polygon? That is, can you omit this sorting phase if you start from a simple polygon? The answer is NO.
5 Graham Scan: Sorting Step The first stage of Graham-Scan sorts the points by their polar angle from the bottom-left vertex, p 0 : In recitation, we asked: would the scanning phase of Graham-Scan work on any simple polygon? That is, can you omit this sorting phase if you start from a simple polygon? The answer is NO. Simple polygon (not convex)
6 Graham Scan on Simple Polygons The first stage of Graham-Scan sorts the points by their polar angle from the bottom-left vertex, p 0 : In recitation, we asked: would the scanning phase of Graham-Scan work on any simple polygon? That is, can you omit this sorting phase if you start from a simple polygon? The answer is NO. Simple polygon (not convex)
7
8 Run Graham s Scan starting from
9 Run Graham s Scan starting from
10 Run Graham s Scan starting from
11 Run Graham s Scan starting from
12 Run Graham s Scan starting from
13 Run Graham s Scan starting from
14 Run Graham s Scan starting from
15 Run Graham s Scan starting from
16 Run Graham s Scan starting from
17 Run Graham s Scan starting from
18 Run Graham s Scan starting from
19 Run Graham s Scan starting from
20 Run Graham s Scan starting from
21 Run Graham s Scan starting from
22 Run Graham s Scan starting from At end, get convex hull = <,,,,, > But actually, convex hull = <,,, >
23 original What went wrong? Vertex was not visible from, so the line -> crossed -> when building the convex hull. That is, the polygon became non-simple during course of the algorithm.
24 Graham-Scan Builds Star-Shaped Polygons When vertices are sorted by polar angle from, all other vertices are visible from in resulting polygon: A polygon with a point visible from each vertex is called star-shaped (CLRS p. 957, Ex ). Graham-Scan works for all star-shaped polygons, but not for all simple ones
25 For More Information There do exist linear-time algorithms for building the convex hull of a simple polygon. Many of the first algorithms proposed were actually incorrect! See here for an interesting history: Is the question we considered a million-dollar question? Probably not, but it can be worth up to $200! Click here for a good time:
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