arrays - Is A==0 really better than ~A?

Question

Introduction to problem setup

I was doing some benchmarks involving - ~A and A==0for a double array with no NaNs, both of which convert A to a logical array where all zeros are converted to true values and rest are set as false values.

For the benchmarking, I have used three sets of input data –

• Very small to small sized data - 15:5:100
• Small to medium sized data - 50:40:1000
• Medium to large sized data - 200:400:3800

The input is created with A = round(rand(N)*20), where N is the parameter taken from the size array. Thus, N would vary from 15 to 100 with stepsize of 5 for the first set and similarly for the second and third sets. Please note that I am defining datasize as N, thus the number of elements would be datasize^2 or N^2.

Benchmarking Code

N_arr = 15:5:100; %// for very small to small sized input array
N_arr = 50:40:1000; %// for small to medium sized input array
N_arr = 200:400:3800; %// for medium to large sized input array
timeall = zeros(2,numel(N_arr));
for k1 = 1:numel(N_arr)
    A = round(rand(N_arr(k1))*20);

    f = @() ~A;
    timeall(1,k1) = timeit(f);
    clear f

    f = @() A==0;
    timeall(2,k1) = timeit(f);
    clear f
end

Results

Finally the questions

One can see how A==0 performs better than ~A across all datasizes. So here are some observations and related questions alongside them –

1. A==0 has one relational operator and one operand, whereas ~A has only one relational operator. Both produce logical arrays and both accept double arrays. In fact, A==0 would work with NaNs too, wheras ~A won’t. So, why is still ~A at least not as good as A==0 as it looks like A==0 is doing more work or am I missing something here?

2. There’s a peculiar drop of elapsed time with A==0 and thus increased performance at N = 320, i.e. at 102400 elements for A. I have observed this across many runs with that size on two different systems that I have access to. So what’s going on there?

Answer 1

This is not strictly an answer but rather my contribution to the discussion

I used the profiler to investigate a slightly-modified version of your code:

N_arr = 200:400:3800; %// for medium to large sized input array

for k1 = 1:numel(N_arr)

    A = randi(1,N_arr(k1));
    [~]=eq(A,0);
    clear A

    A = randi(1,N_arr(k1));
    [~]=not(A);
    clear A   

end

I used the following profiler flags (as per UndocumentedMatlab's series of posts about Profiler):

profile('-memory','on');
profile('on','-detail','builtin');

And here's an excerpt from the profiler results (link to the larger image):

It seems that the == variant allocates a tiny bit of additional memory that allows it to work its magic....

Regarding your question 2: Before removing the keeping of timeall, I tried plotting the same charts you did in Excel. I didn't observe the behavior you mentioned for N = 320. I suspect this may have something to do with the additional wrappers (i.e. function handles) you're using in your code.

---

I thought I'd attach the available documentation for the discussed functions for quick reference.

The documentation for ~ (MATLABR20???oolboxmatlabops ot.m):

%~   Logical NOT.
%   ~A performs a logical NOT of input array A, and returns an array
%   containing elements set to either logical 1 (TRUE) or logical 0 (FALSE).
%   An element of the output array is set to 1 if A contains a zero value
%   element at that same array location.  Otherwise, that element is set to
%   0.
%
%   B = NOT(A) is called for the syntax '~A' when A is an object.
%
%   ~ can also be used to ignore input arguments in a function definition,
%   and output arguments in a function call.  See "help punct"

%   Copyright 1984-2005 The MathWorks, Inc.

The documentation for == (MATLABR20???oolboxmatlabopseq.m):

%==  Equal.
%   A == B does element by element comparisons between A and B
%   and returns a matrix of the same size with elements set to logical 1
%   where the relation is true and elements set to logical 0 where it is
%   not.  A and B must have the same dimensions unless one is a
%   scalar. A scalar can be compared with any size array.
%
%   C = EQ(A,B) is called for the syntax 'A == B' when A or B is an
%   object.

%   Copyright 1984-2005 The MathWorks, Inc.