CS100M Lab Exercise 14
Writing efficient code
1. Download the script LargestTriangle from the Section Exercises page. The script (also shown below) is a
first attempt at finding the largest triangle that can be formed from npoints on a unit circle. Add code (tic,
toc) to the script to determine how long it takes to find the answer for n= 100,150,200. Store the results (time)
in vector t1 such that t1(i) corresponds to n(i), i=1,2,3.
for n=100:50:200
theta = rand(n,1)*2*pi; % Angle of random pts on the unit circle
% Determine how long it takes to compute the largest
% possible triangle obtained by selecting vertices from the
% points represented by theta
A=0;
for i=1:n
for j=1:n
for k=1:n
% theta --> Cartesian
c1 = cos(theta(i));
s1 = sin(theta(i));
c2 = cos(theta(j));
s2 = sin(theta(j));
c3 = cos(theta(k));
s3 = sin(theta(k));
% Area using Heron’s Formula
a = sqrt((c1-c2)^2 + (s1-s2)^2);
b = sqrt((c1-c3)^2 + (s1-s3)^2);
c = sqrt((c2-c3)^2 + (s2-s3)^2);
s = (a+b+c)/2;
Aijk = sqrt((s-a)*(s-b)*(s-c)*s);
A = max(A,Aijk);
end
end
end
end
2. We now start to make the computation more efficient. Append the script rather than modifying the code
directly—copy and paste your code from Part 1 to Part 2 of the script and make the modification in Part 2.
Notice that there are several levels of inefficiency. The area for each different triangle is computed 6 times. Modify
the loop ranges to eliminate this redundancy. Also, there are a lot of redundant sine and cosine evaluations.
Address this issue by moving the c1,s1,c2 and s2 assignments. In Part 2, store the time taken to do the
computation in vector t2 such that t2(i) corresponds to n(i). How much speed-up did you get?
Even with the change in where we compute c1,s1,c2 and s2, we are still doing more sine and cosine evaluations
than necessary—given nvalues of theta we should only need to make nsine evaluations and ncosine evaluations.
This suggests that we can reduce the time further by precomputing the sine and cosine of theta. We will combine
this insight with another improvement in Part 3 below.
3. There is additional redundancy associated with the side length computations a,b, and c. In Part 3, eliminate
this redundancy by precomputing an n×narray Dwith the property that D(i,j) is the distance from point
(cos(theta(i)),sin(theta(i))) to point (cos(theta(j)),sin(theta(j))). Note that you only need the “upper half” of D
since D(i,j) = D(j,i). Store the time taken to do the computation in vector t3 such that t3(i) corresponds to
n(i).
4. Draw a plot of the computation time (three graphs of time vs. n). Also show in a table the ratio of t1 to t3
for all n.
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