I have a complete weighted graph with 1000 nodes and need to find the longest possible Hamiltonian path in the graph (the sequence of nodes, to be more precise). I am supposed to fit in 5 sec (for Java), the memory limit is big enough.
Finding the longest Hamiltonian path doesn't look much different from finding solution for TSP (travelling salesman). Of course, an optimal solution is out of question, so I'm looking for a good heuristic.
My best solution so far is using the Nearest Neighbour algorithm, which is easy to implement and runs in polynomial time (takes ~0.7 seconds for 1000 nodes graph). It's a bit far from the optimal solution though.
So I'm looking for a better heuristic that still runs relatively fast.
I see mentioned Tabu Search, Simulated Annealing, Ant Colony, Genetics, Branch and Bound, MST based algorithms and others.
The problem is, as their implementation is not exactly trivial, it's hard to find their time complexity to decide which can fit in the 5 sec. time limit; e.g. run in polynomial time.
For some algorithms like Christofides' I see that the complexity is O(V^4), where V is the number of vertices, which apparently makes it impossible to fit.
I came across the Bitonic Tour solution, usually used for finding the shortest Hamiltonian path in Euclidean graphs, but seems kind of OK for finding the longest path in non-Euclidean graphs too:
public static void minCostTour(int[][] graph) {
int n = graph.length;
int[][] dp = new int[n][n];
dp[0][1] = graph[0][1];
for (int i = 0; i < n - 1; i++) {
for (int j = i + 1; j < n; j++)
if (i == (j - 1) && i != 0) {
dp[i][j] = dp[0][j-1] + graph[0][j];
for (int k = 1; k <= j - 2; k++)
if ((dp[k][j-1] + graph[k][j] < dp[i][j])) {
dp[i][j] = dp[k][j-1] + graph[k][j];
}
} else if (i != 0 || j != 1) {
dp[i][j] = dp[i][j-1] + graph[j-1][j];
}
}
System.out.println("Optimal Tour Cost: " + (dp[n-2][n-1] + graph[n-2][n-1]));
}
The standard algorithm includes an initial sorting of coordinates, which I skipped, as apparently there are no coordinates to sort (the graph is non-Euclidean).
This dynamic programming solution runs in O(V^2) so it might be good.
The problem is that it outputs the Hamiltonian path length and I need the sequence of nodes. I can't really understand how to restore the path from the above algorithm (if possible at all).
TL DR version:
Can the Bitonic Tour algorithm above be used for finding the sequence of nodes on the longest Hamiltonian path in a complete weighted graph?
If not, can you recommend an algorithm with similar (polynomial) time complexity for that task?
Related
I'm writing a k-means algorithm. At each step, I want to compute the distance of my n points to k centroids, without a for loop, and for d dimensions.
The problem is I have a hard time splitting on my number of dimensions with the Matlab functions I know. Here is my current code, with x being my n 2D-points and y my k centroids (also 2D-points of course), and with the points distributed along dimension 1, and the spatial coordinates along the dimension 2:
dist = #(a,b) (a - b).^2;
dx = bsxfun(dist, x(:,1), y(:,1)'); % x is (n,1) and y is (1,k)
dy = bsxfun(dist, x(:,2), y(:,2)'); % so the result is (n,k)
dists = dx + dy; % contains the square distance of each points to the k centroids
[_,l] = min(dists, [], 2); % we then argmin on the 2nd dimension
How to vectorize furthermore ?
First edit 3 days later, searching on my own
Since asking this question I made progress on my own towards vectorizing this piece of code.
The code above runs in approximately 0.7 ms on my example.
I first used repmat to make it easy to do broadcasting:
dists = permute(permute(repmat(x,1,1,k), [3,2,1]) - y, [3,2,1]).^2;
dists = sum(dists, 2);
[~,l] = min(dists, [], 3);
As expected it is slightly slower since we replicate the matrix, it runs at 0.85 ms.
From this example it was pretty easy to use bsxfun for the whole thing, but it turned out to be extremely slow, running in 150 ms so more than 150 times slower than the repmat version:
dist = #(a, b) (a - b).^2;
dists = permute(bsxfun(dist, permute(x, [3, 2, 1]), y), [3, 2, 1]);
dists = sum(dists, 2);
[~,l] = min(dists, [], 3);
Why is it so slow ? Isn't vectorizing always an improvement on speed, since it uses vector instructions on the CPU ? I mean of course simple for loops could be optimized to use it aswell, but how can vectorizing make the code slower ? Did I do it wrong ?
Using a for loop
For the sake of completeness, here's the for loop version of my code, surprisingly the fastest running in 0.4 ms, not sure why..
for i=1:k
dists(:,i) = sum((x - y(i,:)).^2, 2);
endfor
[~,l] = min(dists, [], 2);
Note: This answer was written when the question was also tagged MATLAB. Links to Octave documentation added after the MATLAB tag was removed.
You can use the pdist2MATLAB/Octave function to calculate pairwise distances between two sets of observations.
This way, you offload the bother of vectorization to the people who wrote MATLAB/Octave (and they have done a pretty good job of it)
X = rand(10,3);
Y = rand(5,3);
D = pdist2(X, Y);
D is now a 10x5 matrix where the i, jth element is the distance between the ith X and jth Y point.
You can pass it the kind of distance you want as the third argument -- e.g. 'euclidean', 'minkowski', etc, or you could pass a function handle to your custom function like so:
dist = #(a,b) (a - b).^2;
D = pdist2(X, Y, dist);
As saastn mentions, pdist2(..., 'smallest', k) makes things easier in k-means. This returns just the smallest k values from each column of pdist2's result. Octave doesn't have this functionality, but it's easily replicated using sort()MATLAB/Octave.
D_smallest = sort(D);
D_smallest = D_smallest(1:k, :);
I have a C# project in which I retreive grey-scale images from cameras and do some computation with the image data. The computations are quite time-consuming since I need to loop over the total image several times and I am doing it all on the CPU.
Now I would like to try to get the evaluation running on the GPU, but I have a lot of struggle achieving that, since I never did any GPU calculations before.
The software should be able to run on several computers with varying hardware, so CUDA for example is not a solution for me, since the code should also run on laptops which only have onboard graphics. After some research I came accross Cloo (found it on this project), which seems to be a quite resonable choice.
So far I integrated Cloo in my project and tried to get this hello world example running. I guess it is running, since I don´t get any exception, but I don´t know where I can see the printed output.
For my computations I need to pass the image to the GPU and I also need the x-y coordinates during the computation. So, in C# the computation looks like this:
int a = 0;
for (int y = 0; y < img_height; y++){
for (int x = 0; x < img_width; x++){
a += image[x,y] * x * y;
}
}
int b = 0;
for (int y = 0; y < img_height; y++){
for (int x = 0; x < img_width; x++){
b += image[x,y] * (x-a) * y;
}
}
Now I want to have these calculations to run on the GPU, and I want to parallel the y-loop, so that in every task one x-loop is running. Then I could take all the resulting a values and add them up before the second loop block would start.
Afterwards I would like to return the values a and b to my C# code and use them there.
So, to wrap up my questions:
Is Cloo a recommendable choice for this task?
What is the best way to pass the image-data (16bit, short-array) and the dimensions (img_width, img_height) to the GPU?
How can I return a value from the GPU? As far as I know kernels are always used as kernel void...
What would be the best way to implement the loops?
I hope my questions are clear and I provided sufficient information to understand my struggles. Any help is appreciated. Thanks in advance.
Let's reverse engineer the problem. Understanding the efficient processing of the "dependency-chain" of image[][], image_height, image_width, a, b
Ad 4 ) the tandem of identical for-loops has a poor performance
given the defined code, there could be just a single loop, thus with reduced overhead costs and best with also maximising cache-aligned vectorised code.
Cache-Naive re-formulation:
int a = 0;
int c = 1;
for ( int y = 0; y < img_height; y++ ){
for ( int x = 0; x < img_width; x++ ){
int intermediate = image[x,y] * y; // .SET PROD(i[x,y],y)
a += x * intermediate; // .REUSE 1st
c -= intermediate; // .REUSE 2nd
}
}
int b = a * c; // was my fault upon being in a hurry leaving for weekend :o)
Moving the code into the split tandem loops is only increasing these overheads and devastating any possible cache-friendly tricks in the code-performance tweaking.
Ad 3 + 2 ) kernel call-signature + CPU-side methods allow this
OpenCL and Cloo document these details, so nothing magical beyond the documented methods is needed here.
Yet, there are latency costs associated with each such host-side to device-side + device-side to host-side transfers. Given you claim that the 16bit-1920x1200 image-data are to be re-processed ~ 10 times in a loop, there are some chances these latencies need not be spent on every such loop pass-through.
The worst performance-killer is a very shallow kernel mathematical density. The problem is, there is indeed not much to calculate in the kernel, so the chances for any efficient SIMD / GPU parallel tricks are indeed pretty low.
In this sense, the CPU-side smart-vectorised code will do much better than the ( H2D + D2H )-overheads-far latency-hostile computationally-shallow GPU-kernel processing.
Ad 1) Given 2+3 and 4 above, 1 may easily loose sense
As prototyped and given additional cache-friendly vectorised tricks, the in-ram + in-cache vectorised code will have chances to beat all OpenCL and mixed-GPU/CPU automated ad-hoc kernel compilation generated device code and it's computing efforts.
I have implemented a very simple linear regression with gradient descent algorithm in JavaScript, but after consulting multiple sources and trying several things, I cannot get it to converge.
The data is absolutely linear, it's just the numbers 0 to 30 as inputs with x*3 as their correct outputs to learn.
This is the logic behind the gradient descent:
train(input, output) {
const predictedOutput = this.predict(input);
const delta = output - predictedOutput;
this.m += this.learningRate * delta * input;
this.b += this.learningRate * delta;
}
predict(x) {
return x * this.m + this.b;
}
I took the formulas from different places, including:
Exercises from Udacity's Deep Learning Foundations Nanodegree
Andrew Ng's course on Gradient Descent for Linear Regression (also here)
Stanford's CS229 Lecture Notes
this other PDF slides I found from Carnegie Mellon
I have already tried:
normalizing input and output values to the [-1, 1] range
normalizing input and output values to the [0, 1] range
normalizing input and output values to have mean = 0 and stddev = 1
reducing the learning rate (1e-7 is as low as I went)
having a linear data set with no bias at all (y = x * 3)
having a linear data set with non-zero bias (y = x * 3 + 2)
initializing the weights with random non-zero values between -1 and 1
Still, the weights (this.b and this.m) do not approach any of the data values, and they diverge into infinity.
I'm obviously doing something wrong, but I cannot figure out what it is.
Update: Here's a little bit more context that may help figure out what my problem is exactly:
I'm trying to model a simple approximation to a linear function, with online learning by a linear regression pseudo-neuron. With that, my parameters are:
weights: [this.m, this.b]
inputs: [x, 1]
activation function: identity function z(x) = x
As such, my net will be expressed by y = this.m * x + this.b * 1, simulating the data-driven function that I want to approximate (y = 3 * x).
What I want is for my network to "learn" the parameters this.m = 3 and this.b = 0, but it seems I get stuck at a local minima.
My error function is the mean-squared error:
error(allInputs, allOutputs) {
let error = 0;
for (let i = 0; i < allInputs.length; i++) {
const x = allInputs[i];
const y = allOutputs[i];
const predictedOutput = this.predict(x);
const delta = y - predictedOutput;
error += delta * delta;
}
return error / allInputs.length;
}
My logic for updating my weights will be (according to the sources I've checked so far) wi -= alpha * dError/dwi
For the sake of simplicity, I'll call my weights this.m and this.b, so we can relate it back to my JavaScript code. I'll also call y^ the predicted value.
From here:
error = y - y^
= y - this.m * x + this.b
dError/dm = -x
dError/db = 1
And so, applying that to the weight correction logic:
this.m += alpha * x
this.b -= alpha * 1
But this doesn't seem correct at all.
I finally found what's wrong, and I'm answering my own question in hopes it will help beginners in this area too.
First, as Sascha said, I had some theoretical misunderstandings. It may be correct that your adjustment includes the input value verbatim, but as he said, it should already be part of the gradient. This all depends on your choice of the error function.
Your error function will be the measure of what you use to measure how off you were from the real value, and that measurement needs to be consistent. I was using mean-squared-error as a measurement tool (as you can see in my error method), but I was using a pure-absolute error (y^ - y) inside of the training method to measure the error. Your gradient will depend on the choice of this error function. So choose only one and stick with it.
Second, simplify your assumptions in order to test what's wrong. In this case, I had a very good idea what the function to approximate was (y = x * 3) so I manually set the weights (this.b and this.m) to the right values and I still saw the error diverge. This means that weight initialization was not the problem in this case.
After searching some more, my error was somewhere else: the function that was feeding data into the network was mistakenly passing a 3 hardcoded value into the predicted output (it was using a wrong index in an array), so the oscillation I saw was because of the network trying to approximate to y = 0 * x + 3 (this.b = 3 and this.m = 0), but because of the small learning rate and the error in the error function derivative, this.b wasn't going to get near to the right value, making this.m making wild jumps to adjust to it.
Finally, keep track of the error measurement as your network trains, so you can have some insight into what's going on. This helps a lot to identify a difference between simple overfitting, big learning rates and plain simple mistakes.
I'm trying to find a measurement for the similarity of 2 faces. I use OpenCV. For that I train Eigenfaces / Fisherfaces with 1000 Photos of 1000 different people (so 1 Photo each person). So I also have 1000 labels in the training set.
Now I can use the predict method to get the most similar face.
I want to input 2 unknown face images to find if they are both similar to the same vector of faces in the training set.
Here is the code of openCV that returns the most similar label (with the lowest distance).
for(size_t sampleIdx = 0; sampleIdx < _projections.size(); sampleIdx++) {
double dist = norm(_projections[sampleIdx], q, NORM_L2);
if((dist < minDist) && (dist < _threshold)) {
minDist = dist;
minClass = _labels.at<int>((int)sampleIdx);
}
Questions:
Can anyone tell me how to rewrite this to output the top 10 faces and not just the top 1 ? I'm thinking about pushing them into a priority queue, but maybe there is something easier?!
In the training: should I put all the faces on the same label or on different labels? So should I have 1 label or 1000 ?
Cheers
Here's what I did. Note I'm really good at perl, really newb at C++ (in fact, this is my first c++ project!) so I output a lot to the command line and parsed it with perl.
I went to facerec.cpp as you did, and I changed the contents of the for loop to this:
for(size_t sampleIdx = 0; sampleIdx < _projections.size(); sampleIdx++) {
double dist = norm(_projections[sampleIdx], q, NORM_L2);
int labelClass = _labels.at<int>((int)sampleIdx);
cout << dist << " " << labelClass << endl;
if((dist < minDist) && (dist < _threshold)) {
minDist = dist;
minClass = _labels.at<int>((int)sampleIdx);
}
}
This now outputs the distance and label of every face. Since all the predict function appears to do is take the picture with the shortest distance (lowest number) and return that as the answer, you can now take the resulting list, sort it, and take the first 10 results. Or you can take the first ten labels or whatever. This just gives you access to all of the data rather than the first X results.
I also added
#include <iostream>
using namespace std;
to the top of the file so I could use cout.
Q1:: Since OpenCV doesn't provide a default function, you have to create your own by creating a vector which has distance and label. You can write your own function as below and store the distance and label in the vector. Here you need to rebuild the opencv.
virtual void predict(InputArray src, int &label, double &confidence, Vector <variable>) const = 0;
I am trying to implement the overlap and add method in oder to apply a filter in a real time context. However, it seems that there is something I am doing wrong, as the resulting output has a larger error than I would expect. For comparing the accuracy of my computations I created a file, that I am processing in one chunk. I am comparing this with the output of the overlap and add process and take the resulting comparison as an indicator for the accuracy of the computation. So here is my process of doing Overlap and add:
I take a chunk of length L from my input signal
I pad the chunk with zeros to length L*2
I transform that signal into frequency domain
I multiply the signal in frequency domain with my filter response of length L*2 in frequency domain (the filter response is actually created by interpolating control points in the UI - so this is not transformed from time domain. However using length L*2 in frequency domain should be similar to using a ffted time domain signal of length L padded to L*2)
Then I transform the resulting signal back to time domain and add it to the output stream with an overlap of L
Is there anything wrong with that procedure? After reading a lot of different papers and books I've gotten pretty unsure which is the right way to deal with that.
Here is some more data from the tests I have been running:
I created a signal, which consists of three cosine waves
I used this filter function in the time domain for filtering. (It's symmetric, as it is applied to the whole output of the FFT, which also is symmetric for real input signals)
The output of the IFFT looks like this: It can be seen that low frequencies are attenuated more than frequency in the mid range.
For the overlap add/save and the windowed processing I divided the input signal into 8 chunks of 256 samples. After reassembling them they look like that. (sample 490 - 540)
Output Signal overlap and add:
output signal overlap and save:
output signal using STFT with Hanning window:
It can be seen that the overlap add/save processes differ from the STFT version at the point where chunks are put together (sample 511). This is the main error which leads to different results when comparing windowed process and overlap add/save. However the STFT is closer to the output signal, which has been processed in one chunk.
I am pretty much stuck at this point since a few days. What is wrong here?
Here is my source
// overlap and add
// init Buffers
for (UInt32 j = 0; j<samples; j++){
output[j] = 0.0;
}
// process multiple chunks of data
for (UInt32 i = 0; i < (float)div * 2; i++){
for (UInt32 j = 0; j < chunklength/2; j++){
// copy input data to the first half ofcurrent buffer
inBuffer[j] = input[(int)((float)i * chunklength / 2 + j)];
// pad second half with zeros
inBuffer[j + chunklength/2] = 0.0;
}
// clear buffers
for (UInt32 j = 0; j < chunklength; j++){
outBuffer[j][0] = 0.0;
outBuffer[j][8] = 0.0;
FFTBuffer[j][0] = 0.0;
FFTBuffer[j][9] = 0.0;
}
FFT(inBuffer, FFTBuffer, chunklength);
// processing
for(UInt32 j = 0; j < chunklength; j++){
// multiply with filter
FFTBuffer[j][0] *= multiplier[j];
FFTBuffer[j][10] *= multiplier[j];
}
// Inverse Transform
IFFT((const double**)FFTBuffer, outBuffer, chunklength);
for (UInt32 j = 0; j < chunklength; j++){
// copy to output
if ((int)((float)i * chunklength / 2 + j) < samples){
output[(int)((float)i * chunklength / 2 + j)] += outBuffer[j][0];
}
}
}
After the suggestion below, I tried the following:
IFFTed my Filter. This looks like this:
set the second half to zero:
FFTed the signal and compared the magnitudes to the old filter (blue):
After trying to do overlap and add with this filter, the results have obviously gotten worse instead of better. In order to make sure my FFT works correctly, I tried to IFFT and FFT the filter without setting the second half zero. The result is identical to the orignal filter. So the problem shouldn't be the FFTing. I suppose that this is more of some general understanding of the overlap and add method. But I still can't figure out what is going wrong...
One thing to check is the length of the impulse response of your filter. It must be shorter than the length of zero padding used before the fast convolution FFT, or you will get wrap around errors.
I think the problem might be in the windowing approach that you are using. You simply add zeros to the chunks so there is no actual overlap. In the overlap and add method, you need to damp the edges of the window. What this means is that where you add zeros to the chunk you instead have add weighted input signal and the weight in your case should be 0.5 since only two windows overlap.
Rest of the procedure seems OK. You then simply take FTs, multiply and take inverse FTS and finally add up all the chunks to get the final signal which should be exactly the same if you filtered the whole signal at once.