Convex Hull l

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2011

Cache-Oblivious Output-SensitiveTwo Dimensional Convex Hull

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Peyman Afshani Arash Farzan


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Contents1. Introduction

2. Cache-Oblivious Output-SensitiveTwo-Dimensional Convex Hull

3. The Algorithm

4. Conclusion


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Introduction[Convex Hull] Convex Hull:

Applications:

Pattern Recognition, Image Processing, Statistics, GIS


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Introduction[Convex Hull Algorithms] General Convex Hull Algorithms:

Complexity in terms of:

n (Number of Input Points)

Lower Bound: (nlogn)

Output-Sensitive Algorithms:

Complexity in terms of:

n (Number of Input Points)

h (Number of Points on the Convex Hull)

Lower Bound: (nlogh)


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Introduction[Convex Hull Algorithms] Gift Wrapping Algorithm (Jarvis March - 1973) Complexity: O(nh)

Graham Scan (Ronald Graham - 1972)

Complexity: O(nlogn)

Divide and Conquer (Preparata and Hong - 1977)

Complexity: O(nlogn)

The Ultimate Convex Hull Algorithm (Kirkpatrick and Seidel - 1986)

Complexity: O(nlogh)

Chans Algorithm (Ronald Graham - 1996)

Complexity: O(nlogh)


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Introduction[Cache Algorithms] Main Idea:

Avoid CacheMisses.

Cache Aware Algorithms: Two Levels of Memory.

Depends on Constant Parameters: Cache Size, Cache Line, etc.

Highly Processor Specific.

Cache Oblivious Algorithms: Multiple Levels of Memory.

Does Not Depend on Constant Parameters.

Behaves Similarly in All Levels of the Memory Hierarchy.


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Cache-Oblivious Output-SensitiveTwo-Dimensional Convex Hull[Introduction] Problem

Minimize the Number of Cache Faults Caused WhenComputing the Convex Hull of a Set of NPoints on a

Plane. Output-Sensitive Case

Number of Cache Misses are Analyzed on Both Nand H.

Problems Lower Bound O(N/BlogM/BH/B)

Algorithms Presented Lower Bound O (max{N/B loglogM/B, N/B logM/BH/B})


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Given N Points in Two Dimensions:

Complexity: O(NlogN)

First Optimal Output-Sensitive Convex Hull

Algorithm: Complexity: O(NlogH)

By Kirkpatrick and Seidel

Obtaining a Cache-Aware or Even a Cache-Oblivious Convex Hull Algorithm in TwoDimensions is not Hard: Use of Cache Oblivious/Aware Sorting Algorithm

Use of Grahams Scan

O(N/BlogM/BN/B) Cache Misses

Cache-Oblivious Output-SensitiveTwo-Dimensional Convex Hull[Preliminaries & Background]


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Goodrichs Output-Sensitive Convex Hull Algorithm:

Cache Misses: O(N/BlogM/BH/B)

Arge and Milterson Showed That in the Cache-AwareModel, These are as Best as One Can Do.

Any Lower Bound in the Cache-Aware Model AlsoHolds in the Cache-Oblivious Model.

Here we Present a Cache-Oblivious Algorithm for theOutput-Sensitive Convex Hull Problem that AlmostMatches the Lower Bound.

Cache-Oblivious Output-SensitiveTwo-Dimensional Convex Hull[Preliminaries & Background]


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The Algorithm[1/3] A Modification of Chans Output-Sensitive Algorithm.

Find the Upper Envelope of NLines in Plane

Lemma 1: For a set H of N lines in R2and a parameter H,one can find H upper hull edges in only O(max{n,nlogmh})cache misses.

Proof: Partition H into r = N/H groups of size H each.

For each group of H lines run the optimal algorithm to obtain the upper hull.

Cache Misses: O(hlogmh)

r Chains will be created of maximum size H

It is necessary to show that it is possible to merge these chains into afinal chain causing a small number of cache misses

Sis the set of all the segments constituting the chains

For a segment s E S, denote the x-coordinate of he left end point of swithxr(s)

Partition S into h sets G1,,GH, of size r such that:

For every i < j and every u E Gi , y E Gj we have xr(u)


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The Algorithm[2/3] Lemma 2: Given k numbers in a contiguous list can be distributed into T

buckets B1,,BT of roughly equal size such that an element in a bucketBi is less than any element in a bucket Bj with a larger index ( i.e. i < j )using O(k/B logmT/B)cache misses in the cache oblivious model underthe tall cache assumption.

Hence, the previous step takes O(nlogmh) under the cache oblivious

setting.

For every i the segments of Gi are stored sequentially but in arbitrary order.

xi and Xi are the minimum and maximum left x-coordinate in Gi.

We merge the chains in H iterations and at each iteration we process one set Gi, startingfrom G1.

At iteration i+1 the upper hull of all segments in S i to the left of the vertical line x = X i has

been found.

Let A be theset of all the segments in Siwhich cross the vertical line x = Xi.

We only need to look at A and Gi+1.

Make a linear search at these in order to find the next upper hull segment.

We repeat this search until we discover the upper hull segment at x = Xi+1 and update A.

Each Linear scan will cause O(r/B) cache misses, the discovery of ki upper hull edgesO(rki/B).

At each point if we discover the H-th edge, we break the algorithm and output all the edgesdiscovered by the algorithm.


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The Algorithm[3/3] Since at most Hedges will be discovered by the

convex hull algorithm, the total number of cachemisses sums up to:

For values H < m as we have to scan over theinput set the cache misses are O(n). Thus, theoverall number of cache misses is O(max {n ,nlogmh}).


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Theorem[1/2] For a set P of N points in R2, one can compute the convex hull of P

with at most

cache misses in which H is the size of the final convex hull.

Proof:

We run the algorithm of Lemma 1 with values of h chosen from a doublyexponential sequence.

We set Hi = c2i , starting from i = 0 until the time all the edges of the upper

hull have been discovered.

The total number of cache misses can be expressed as a geometricseries:


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Theorem[2/2] For smaller values of the sequence, since there is a scan

over the input for each such value, there are O(n) cachemisses in each run.

Hence, the total number of cache misses is:


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ConclusionWe considered the problem of output-sensitive

convex hull in the cache-oblivious model.

We presented an algorithm for it.

The algorithm almost matches the lower boundO(nlogmh) except for an additional termO(nloglogm).

O(nloglogm) is a small term and is negligible in anypractical application.


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References Cache-Oblivious Output-Sensitive Two-Dimensional Convex Hull Peyman

Afshani and Arash Farzan. In Proceedings of the 19th Annual CanadianConference on Computational Geometry, pages 153-155, 2007.

T. Chan. Optimal output-sensitive convex hull algorithms in two and threedimensions. Discrete and Computational Geometry, 16:361-368, 1996.

http://en.wikipedia.org/wiki/Convex_hull

http://en.wikipedia.org/wiki/Cache-oblivious_algorithm


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