An Efficient Convex Hull Algorithm for a Planer Set of Points

transcript

Kasun Ranga Wijeweera

(krw198708298@gmail.com)

This Presentation is Based on Following Research Paper

K. R. Wijeweera, U. A. J. Pinidiyaarachchi (2013), An Efficient Convex Hull Algorithm for a Planer Set of Points, Ceylon Journal of Science (Physical Sciences), Volume 17, pp. 9-17.

Definitions(Convex Set)

• A set S is convex if x in S and y in S implies that the segment xy is a subset of S

• Example in 2D:

Definitions(Convex Hull of a Set of Points)

• The convex hull of a set S of points is the smallest convex set containing all the points in S

• Example in 2D:

A 2D Convex Hull Algorithm (Introduction)

• This algorithm is based on gradient in coordinate geometry• Achieves parallelism• Achieves data reduction• Manages coincident points• Deals with collinear points • Outputs hull points in boundary traversal order• Low computational cost

A 2D Convex Hull Algorithm (Inputs and Outputs)

• Input: A non-empty set of points

S = {P1, P2, . . . , Pn}

• Output: A non-empty set that contains all the points of the convex hull of S

H = {Q1, Q2, . . . , Qm}

A 2D Convex Hull Algorithm (Example Data Set)

A 2D Convex Hull Algorithm (Step 1)

Calculate following values from the set S

minx // minimum value of x-coordinates

y_minx // y-coordinate corresponding to minx

maxx // maximum value of x-coordinates

y_maxx // y-coordinate corresponding to maxx

miny // minimum value of y-coordinates

x_miny // x-coordinate corresponding to miny

maxy // maximum value of y-coordinates

x_maxy // x-coordinate corresponding to maxy

A 2D Convex Hull Algorithm (Basic Four Points on the Convex Hull)

(x_maxy, maxy)

(minx, y_minx) (maxx, y_maxx)

(x_miny, miny)

Define following subsets of S

A = { ( x, y) in S | ( x <= x_maxy ) AND ( y >= y_minx )}

B = { ( x, y) in S | ( x >= x_maxy ) AND ( y >= y_maxx )}

C = { ( x, y) in S | ( x >= x_miny ) AND ( y <= y_maxx )}

D = { ( x, y) in S | ( x <= x_miny ) AND ( y <= y_minx )}

A 2D Convex Hull Algorithm (Partitioning the Data Set into Four Regions)

• Find the convex hull parts belong to each of the sets A, B, C, and D in parallel

• Merge those hull parts to derive the set H

A 2D Convex Hull Algorithm (After Processing and Merging Hull Parts)

A 2D Convex Hull Algorithm (Processing Each Set: Gradient)

M (mx, my)

N (nx, ny)

m = ( ny – my ) / ( nx – mx )

A 2D Convex Hull Algorithm (Processing Each Set: Modified Gradient)

Region Diagram Modified Gradient Begin Point End Point

A ( ay[i] – hully ) / ( ax[i] – hullx ) x = minxy = y_minx

x = x_maxyy = maxy

B ( bx[i] – hullx ) / (hully - by[i] ) x = x_maxyy = maxy

x = maxxy = y_maxx

C ( hully - cy[i] ) / ( hullx - cx[i] ) x = maxxy = y_maxx

x = x_minyy = miny

D ( hullx – dx[i] ) / ( dy[i] – hully ) x = x_minyy = miny

x = minxy =y_minx

A 2D Convex Hull Algorithm (Processing Set A: General Case)

A 2D Convex Hull Algorithm (Processing Set A: Collinear Case)

A 2D Convex Hull Algorithm (Processing Set A: Indeterminate Case)

A 2D Convex Hull Algorithm (Processing Sets B, C, and D)

• Using similar method B, C, and D sets can also be processed• Using the same algorithm used for A to process B, C, and D

needs additional computational cost• Therefore four separate algorithms are used to process each of

four sets with modified gradient

A 2D Convex Hull Algorithm (Interior Points Algorithm)

Based on the following Lemma

A point is non-extreme if and only if it is inside some (closed) triangle whose vertices are points of the set and is not itself a corner of that triangle

A 2D Convex Hull Algorithm (Interior Points Algorithm)

Algorithm: INTERIOR POINTS

for each i do

for each j != i do

for each k != i != j do

for each l != k != i != j do

if p(l) in Triangle{ p(i), p(j), p(k) }

then p(l) is non-extreme

A 2D Convex Hull Algorithm (Proposed Algorithm vs. Interior Points Algorithm)

• Both of the algorithms were implemented using C++ programming language

• Following hardware and software resources were used– Computer: Intel(R) Pentium(R) Dual CPU; E2180 @ 2.00 GHz; 2.00

International, Inc;

• First, n number of random points were generated in the range 0 – 399 using randomize() function and written them to a text file

• Such text files were generated for different n• Number of clock cycles taken to process each data file k times

was counted using clock() function

Case Proposed Algorithm (1000)

Interior Points Algorithm (1000)

Average Ratio (IPA : PA)

n = 10; k = 32000; 5.53 37.11 6.71

n = 20; k = 1000; 10.60 693.80 65.45

n = 30; k = 500; 15.20 3881.6 255.36

n = 40; k = 100; 17.00 12769.00 751.11

n = 50; k = 50; 24.00 32554.00 1356.41

Case 1 Case 2 Case 3 Case 4 Case 50

A(1000)B(1000)

A 2D Convex Hull Algorithm (Drawbacks of the Proposed Algorithm)

• It cannot be easily extended to higher dimensions• The notion of gradient causes the problem• Need to solve partial differential equations in order to derive

the gradient• Because of this higher dimensional extension is not cost

effective• The entire data set is needed from the beginning

– The algorithm is static

Any Questions?

Thank You!

An Efficient Convex Hull Algorithm for a Planer Set of Points

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