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Convex-Hull (2D) incremental construction

Posted on 2006-11-17
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Hi,

Is there an incremental algorithm for convex-hull (in the plane) construction in O(nlogn) ?
That is, we add the points one by one, and each step takes O(logn) time (not amortized, but exactly).

Thanks!!
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Question by:slavikn
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by:ozo
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yes
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by:ozo
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Sorry, that's amortized.
I don't think it can be exactly because you may have to remove O(n) points from the hull when you add a new point
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by:slavikn
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I saw an amortized one, that uses lower and upper binary trees.
Is there an easier algorithm?
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by:JoseParrot
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Hi, slavikn ,

To achieve O(n log n) time, a really good approach is to divide the problem in two parts, upper and lower parts of the convex hull.
By finding the leftmost and the rightmost vertices of the given set, we have the ending vertices of each part.
To find such vertices, we use a O(n log n) sorting algorithm, say, merge-sort.
The finding of each vertice is actually O(n). The final result is O(n log n) + O(n) = O(n log n).

Below is an algorithm for the upper part of the convex hull.

CH(P)
Input: set P of points
Output: a list of convex hull points in clockwise order
{
   UpperList = empty set
   Sort points by x-coord, s.t. p1.x < p2.x < ... <pn.x
   Add point p1 to UpperList
   Add point p2 to UpperList
   for i=3,n
   {
      append pi to UpperList
      {
          while number of points in UpperList > 2
                              and
          last 3 points in UpperList don't make a RIGHT turn
          {
              delete the middle of last 3 points from UpperList
          }
      }
   }
}

Sort points ... is O(n log n)
Add points is linear: O(1)
The "while" loop is executed at least once, each time the "for" loop is executed. So O(n).
Each additional execution of the "while" loop results in exactly one point deletion.
So, again O(n), because we have n points and each one can deleted just once.
Then final time is O(n log n) + O(1) + O(n) + O(n), resulting in O(n log n).

For lower part is similar, except for direction which is from right to left.
This algorith is easy to implement, I don't know about an easer algorithm for convex hull.
Hope it helps.

Jose
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by:slavikn
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Thanks for the detailed explanation JoseParrot.
However, the algorithm I asked for was supposed to be pure incremental, that is without the O(nlogn) sorting in advance.
Eventually I've found a solution (well, nearly, as each step is O(logn) only in amortized...). It's here: http://ww3.algorithmdesign.net/handouts/IncrementalHull.pdf
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by:slavikn
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By the way, I am still looking for a "pure" incremental algorithm with exact complexity of O(logn) in each step (not amortized). A very simple initialization is allowed.
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JoseParrot earned 80 total points
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Well, as far I know, there are several algorithms, with amortized times O(log n log log log n for insertion and  O(log n log log n)  for deletion); the naive O(n^2); the classic O(n), other for O(log^2 n); other with very small gain on O(log log n).

Riko Jacob's PhD thesis, Dynamic Planar Convex Hull of University of Aarhus,  Denmark, claims O(log n) worst case. I didn't accessed this paper.

As you are looking for "exact" O(log n), it seems to be Theta(log n). In such way,  parallel algorithm could be your answer. You may want take a look at
http://www.cse.buffalo.edu/pub/WWW/faculty/miller/Papers/IEEETC88a.html

Jose

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by:JoseParrot
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Hi, slavikn,

The answer to
"Is there an incremental algorithm for convex-hull (in the plane)
 construction in O(nlogn) ?"
is: YES.

Below some of such algorithms:

ALGORITHM 1
Scan, by Graham, 1972
Starting from leftmost point P1, the algorithm has 2 steps:
1) Sorting angles from P1 to P2 until Pn, by Divide and Conquer: O(n log n)
2) Discovering next point, by Greedy approach (point by point): O(n).
Result is: O(n log n) + O(n) --> O(n log n)

ALGORITHM 2
Divide-and-Conquer, by Preparata and Hong, 1977

OTHERS variants, after that:
Monotone Chain, by Andrew, 1979
Incremental, by Kallay, 1984
Marriage-before-Conquest, by Kirkpatrick & Seidel, 1986

Kallay's incremental algorithm description follows. Rather than determine the hull by looking at the whole set of points at once, as in 1 and 2, Kallay's looks at a subset of the points and then increase the subset until it was the entire set. The minimum subset, of course, is a triangle. The points aren't sorted, but the hull construction complexity is O(n log n).

For a pretty complete article, you may want take a look at
http://geometryalgorithms.com/Archive/algorithm_0109/algorithm_0109.htm
With details, pseudo codes and C code.

Hope slavikn is still around and this answer helps...

Jose
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