It seems that it possible to make reduction only for odd number of elements. For example, it needs to sum up numbers. When I have even number of elements, it will be like this:
1 2 3 4
1+2
3+3
6+4
But what to do when I have, for instance 1 2 3 4 5? The last iteration is the sum of three elements 6+4+5 or what? I saw the same question here, but couldn't find the answer.
A parallel reduction will add pairs of elements first:
1 1+3 4+6
2 2+4
3
4
Your example with an odd number of elements would typically be realized as:
1 1+4 5+3 8+7
2 2+5 7+0
3 3+0
4 0+0
5
0
0
0
That is to say, typically a parallel reduction will work with a power-of-2 set of threads, and at most one threadblock (the last one) will have less than a full complement of data to work with. The usual method to handle this is to zero-pad the data out to the threadblock size. If you study the cuda parallel reduction sample code, you'll find examples of this.
Related
I am transforming a signal using the ZeroFFT library. The results I get from it, are not what I would intuitively expect.
As a test, I feed the FFT algorithm with a buffer that contains two full cycles of cosine:
Which is sampled over 512 samples.
Which are fed as int16_t values. What I expected to get back, is 256 amplitudes, with the values [ 0, 4095, 0, 0, ..., 0 ].
Instead, this is the result:
2 2052 4086 2053 0 2 2 2 1 2 2 2 4 4 3 4...
And it gets weirder! If I feed it the same signal, but shifted (so sinf() over 0 .. 4*pi instead of cosf() function) I get a completely different result: 4 10 2 16 2 4 4 4 2 2 2 3 2 4 3 4
This throws up the questions:
1. Doesn't a sine signal and cosine signal with same period, contain exactly the same frequencies?
2. If I feed it a buffer with exactly 2 cycles of cosine, wouldn't the Fourier transform result in all zeros, except for 1 frequency?
I generate my test signal as:
static void setup_reference(void)
{
for (int i=0; i<CAPTURESZ; ++i)
{
const float phase = 2 * 3.14159f * i / (256.0f);
reference_wave[i] = (int16_t) (cosf(phase) * 0x7fff);
}
}
And call the ZeroFFT function as:
ZeroFFT(reference_Wave, CAPTURESZ);
Note: the ZeroFFT docs state that a Hanning window is applied.
Windowing causes some spectral leakage. Including the window function, the wave now looks like this:
If I feed it a buffer with exactly 2 cycles of cosine, wouldn't the Fourier transform result in all zeros, except for 1 frequency?
Yes, if you do it without windowing. Actually two frequencies: both the positive frequency that you expect, and the equivalent negative frequency, though not all FFT functions will include the negative frequencies in their output (for Real input, the result is Hermitian-symmetric, there is no extra information in the negative frequencies). For practical reasons, since neither the input signal nor the FFT calculation are exact, you may not get exactly zero everywhere else either, but it should be close - that's mainly a concern for floating point output.
By the way by this I don't mean that windowing is bad, but in this special case (perfectly periodic input) it didn't work out in your favour.
As for the sine wave, the magnitudes of the result should be similar (within reason - exactness shouldn't be expected), but the comments on the FFT function you used mention
The complex portion is discarded, and the real values are returned.
While phase shifts would not change the magnitudes much, they change the phases of the results, and therefore also their Real component.
I am not understanding how to compute the output bits of a 6-to-4-SBOX with bitslice technique in DES. Matthew Kwan made a brief recap in his paper "Reducing the Gate Count of Bitslice DES" of Biham original paper. He wrote:
Basically, for each S-box, the technique is to take two of the input
bits, expand them to all 16 possible functions of two variables, and
use the remaining four S-box inputs to select from those 16 functions.
However, the details are slightly more complicated
I believe that I understand how to expand 2 variables to 16 functions (from f0 till f15)... But how do I select now with my remaining 4 Input Bits all 4 Outputs?
The paper of Matthew Kwan can be found here: http://fgrieu.free.fr/Mattew%20Kwan%20-%20Reducing%20the%20Gate%20Count%20of%20Bitslice%20DES.pdf
If I am correct, memoization is associated with READ operations. What is the term which refers to the same technique used for WRITE operations?
Example:
Let's say an app receives the following inputs,
0 1 2 2 2 2 2 3 4 3 3 3 3 3 4 4 4 2 1 2 5 5 5 5 3
Instead of saving everything, we tune the app to save only the transitions. (i.e ignore consecutive duplicates)
0 1 2 3 4 3 4 2 1 2 5 3
What is the (standard) term that can be used to describe the above technique?
I feel bad about using the same term since the final outcome is quite different. In READ operations, if memoization is used, the final outcome will remain the same. But in above example for WRITE operations, the final output is different from the original input.
"Deduplication of adjacent/most-recent entries". Your example looks like what the uniq tool does.
If you preserved the count of duplicates, it would be a form of RLE (run-length encoding).
As an aside, I guess you mean memoization as a way to speed up reads, and this as a method to speed up writes, but I wouldn't say that's the opposite of memoization, since it's opposite of the general goal, but not related to the particular method.
As far as I know, there is no applicable terminology for what you are asking.
And it is NOT memoization ... or (to my mind) the reverse of memorization.
(Likewise, there is no word in English for a cat with three legs.)
I am trying to solve this problem: Design a method for representing the state of a tic-tac-toe board in computer memory. Can you fit your representation into three bytes?
This is from a textbook without solutions, thank you!
Any help is appreciated!
The state of a Tic-Tac-Toe board can be encoded using 3 bytes as follows.
To represent the state of each cell, 3 states are necessary, namely X, O and undefined. 3 states can be represented by 2 bits (2 bits can in fact represent 4 states, but only 3 are needed here - on the other hand, 1 bit is insufficient).
There are 9 cells in total, so in total
2 * 9 = 18
bits are necessary to represent the board. 18 bits can be encoded in 3 bytes (which in total have 24 bits, which means that 6 bits are not needed).
A Tic-Tac-Toe board consist of 9 fields. Each field can take 3 states: Empty, Circle, Cross. To represent each state you need 2 bits: 00, 01, 10.
With two bits for each field, you can easily represent whole board in 3 bytes, by using two bits as each field, and each byte as row of board.
(I wish my mathematical vocabulary was more developed)
I have a website. On that website is a video. As a user watches the video, a bit of javascript stores how far they have gotten so far in the video. When they stop watching the video, that number of seconds is stored. There's no pattern to when the js will do this, unfortunately.
So if one person is watching the video, we might see this set:
3
6
8
10
12
16
And another person might get bored immediately:
1
3
This data is all stored in the same place, anonymously. So the sorted table with all this info would look like this:
1
3
3
6
8
10
12
16
Finally, the amount of times the video is started at all is stored. In this case it would be 2.
So. How do I get the average 'high-time' (the farthest reached point in the video) for all of the times the video was played?
I know that if we had a value for every second:
1
2
3
4
5
6
7
...
14
15
16
1
2
3
Then we could count up the values and divide by the number of plays:
(19) / 2 = 9.5
Or if the data was otherwise uniform, say in increments of 5, then we could count that up and multiply it by 5 (in the example, we would have some loss of precision, but that's ok):
5
10
15
5
(4) * 5 / 2 = 10
So it seems like I have a general function which would work:
count * 1/d = avg
where d is the density of the numbers (in the example above with 5 second increments, 1/5).
Is there a way to derive the density, d, from a set of effectively random numbers?
Why not just keep the last time that has been provided, and average across those? If you either throw away, or only pay attention to, the last number, it seems like you could just average over these.
You might also want to check out the term standard deviation as the raw average of this might not be the most useful measurement. If you have the standard deviation as well, it could help you realize that you have an average of 7, but it is composed of mostly 1's and 15's.
If you HAVE to have all the data, like you suggested, I will try and think about this a little bit more. I'm not totally certain how you can associate a value with all the previous values that came with it. Do you ALWAYS know the sequence by which numbers are output? If so, I think I know of a way you could derive the 'last' one, which might be slightly computationally expensive.
If you only have a sequence of integers, I think you may be able to increase each value (exponentially?) to 'compensate' for the fact that a later value 'contains' earlier values. I'm still working through this idea, but maybe it will give someone else a seed. What if you average over the sum of these, and then take the base2 logarithm of this average? Does that provide any kind of useful metric? That should 'weight' the later values to the point where they compensate for the sum of earlier values. I think.
In python-esk:
sum = 0
numberOf = 0
for node in nodes:
sum = sum + node.value ^ 2
numberOf = numberOf + 1
weightedAverage = log(sum/numberOf, 2)
print weightedAverage
print "Thanks Brian"
I think that #brian-stiner is on the right track in one of his comments.
Start with something like:
1
3
3
6
8
10
12
16
Turn that into numbers and counts.
1, 1
3, 2
6, 1
8, 1
10, 1
12, 1
16, 1
And then reading from the end down, find all of the points that happened more often than any remaining ones.
3, 2
16, 1
Take differences in counts.
3, 1
16, 1
And you have an estimate of stopping places.
This will not be an unbiased estimate. But if the JavaScript is independently inconsistent and the number of people is large, the biases should be fairly small.
It won't be right, but it will be close enough for government work.
Assuming increments are always around 5, some missing, some a bit longer or shorter. Then it won't be easy (possible?) to do this exactly. My suggestion: compute something like a 'moving count'. Similar to moving average.
So, for second 7: count how many numbers are 5,6,7,8 or 9 and divide by 5. That will give you a pretty good guess of how many people watched the 7th second. Do the same for second 10. The difference would be close to the number of the people who left between second 7 and 10.
To get the total time watched for each user, you'll have parse the list smallest to largest. If you have 4 views, you'll go through your list until you find that you no longer have 4 identical numbers, the last number where you had 4 identical numbers is the maximum of the first view. Then you'll look for when the 3 identical numbers stop, and so on. For example:
4 views data:
1111222233334445566778
4 views side by side:
1 1 1 1
2 2 2 2
3 3 3 3 <- first view max is 3 seconds
4 4 4 <- second view max is 4 seconds
5 5
6 6
7 7 <- third view max is 7 seconds
8 <- fourth view max is 8 seconds
EDIT- Oh, I just noticed that they are not uniform. In that case, the moving average would probably be your best bet.
The number of values roughly corresponds to the number of time periods in which your javascript sends the values (minus 1/2 if the video stop is accompanied with a obligatory time posting, since its moment is random within the interval).
If all clients have similar intervals and you know them, you may just use:
SELECT (COUNT(*) - 0.5) * 5.0 / (SELECT counter FROM countertable)
FROM ticktable
5.0 is the interval between the posts here.
Note that it does not even look at the values: you could as well just store "ticks".
For the max time, you could use MAX() on your field. Perhaps something like...
SELECT MAX(play_time) AS maxTime FROM video
Which would give you the longest time someone has played the video for.
If you want other things, like AVG() then you'll need more complex queries, for collecting on a per-user basis etc etc.
MySQL also contains a Standard Deviation function called STDDEV() and STD() which could help you too.