I'm faced with a problem where I have to solve puzzles.
E.g. I have an (variable) area of 20x20 (meters for example). There are a number of given set pieces having variable sizes. Such as 4x3, 4x2, 1x5 pieces etc. These pieces can also be turned to add more pain to my problem. The point of the puzzle is to fill the entire area of 20x20 with the given pieces.
What would be a good starting algorithm to achieve such a feat?
I'm thinking of using a heuristic that calculates the open space (for efficiency purposes).
Thanks in advance
That's an Exact Cover problem, with a nice structure too, usually, depending on the pieces. I don't know about any heuristic algorithms, but there are several exact options that should work well.
As usual with Exact Covers, you can use Dancing Links, a way to implement Algorithm X efficiently.
Less generally, you can probably solve this with zero-suppressed decision diagrams. It depends on the tiles though. As a bonus, you can represent all possible solutions and count them or generate one with some properties, all without ever explicitly storing the entire (usually far too large) set of solutions.
BDDs would work about as well, using more nodes to accomplish the same thing (because the solutions are very sparse, as in, using few of the possible tile-placements - ZDDs like that but BDDs like symmetry better than sparseness).
Or you could turn it into a SAT problem, then you get less information (no solution count for example), but faster if there are easy solutions.
Related
When is the right time to use the Elite\Elitist mode in a Genetic Algorithm? I have no idea when to use it. What kind problems can be solved using this?
All I know is an elitist model is where you choose the elite (the solution with highest fitness function) and they have a reserve slot for the next generation, and they are the one up for crossover.
You pretty much always use some form of elitism. What varies is the percentage (p) of best performers that you allow to survive to the next generation. So no elitism is basically saying p=0.
The higher p, the more your algorithm will have a tendency to find local peaks of fitness. i.e. once it finds a chromosome with a good fitness, it'll tend to focus more on optimizing it than trying to find new completely different solutions. On the contrary, if it's smaller, your GA will look for possible solutions all over the place and won't zero in as fast once it finds something close to the optimum solution.
So setting p correctly is going to have a direct impact on your algorithm's performance. But it depends on what you're after and your problem space. Play around with it a bit to adjust properly. I typically use 20% for the problems I work with, to give enough room for innovation. It works ok for me.
I am working on edge detection in images and would like to evaluate the performance of algorithm, if any any one could give me a reference or method on how to proceed it will be really helpful. :)
I do not have ground truth and data set includes color as well as gray images.
Thank you.
Create a synthetic data set with known edges, for example by 3D rendering, by compositing 2D images with precise masks (as may be obtained in royalty free photosets), or by introducing edges directly (thin/faint lines). Remember to add some confounding non-edges that look like edges, of a type appropriate for what you're tuning for.
Use your (non-synthetic) data set. Run the reference algorithms that you want to compare against. Also produce combinations of the reference algorithms, for example by voting (majority, at least K out of N, etc). Calculate stats on your algo vs reference algo performance, in terms of (a) number of points your algo classifies as edge which each reference algo, or the combination, does not classify as edge (false positive), or (b) number of points which the reference algo classifies as edge that your algo does not (false negative). You can also calculate a rank correlation-type number for algos by looking at each point and looking at which algos do (or don't) classify that as an edge.
Create ground truth manually. Use reference edge-finding algos as a starting point, then fix up by hand. Probably valuable to do for a small number of images in any case.
Good luck!
For comparisons, quantitative measures like what #Alex I explained is best. To do so, you need to define what is "correct" with a ground truth set and a way to consistently determine if a given image is correct or on a more granular level, how correct (some number like a percentage) it is. #Alex I gave a way to do that.
Another option that is often used in graphics research where there is no ground truth is user studies. Usually less desirable as they are time consuming and often more costly. However, if it is a qualitative improvement that you are after or if a quantitative measurement is just too hard to do, a user study is an appropriate solution.
When I mean user study I mean to poll people on how well a result is given the input image. You could give them a scale to rate things on and randomly give them samples from both your results and the results of another algorithm
And of course, if you still want more ideas, be sure to check out edge detection papers to see how they measured their results (I'd actually look here first as they've already gone through this same process and determined what was best for them: google scholar).
I saw this game here Flow, it looks quite interesting.
Connect matching colors with pipe to create a flow. Pair all colors,
and cover the entire board to solve each puzzle. But watch out, pipes
will break if they cross or overlap.
Given a set of pairs (x, y), is there an algorithm to solve the puzzle, i.e. fill in the whole grid (assuming there is a solution) that I'm not aware of?
This is a very specific instance of the global routing problem. Global routing is a well studied problem in VLSI CAD (where one needs to route millions of nets in an integrated circuit). The problem is NP-complete and can be solved in many ways depending upon the tradeoff you need between runtime and quality. Following wiki is a good starting point:
https://en.wikipedia.org/wiki/Routing_(electronic_design_automation)
Paper here gives a survey of various techniques:
http://dropzone.tamu.edu/~jhu/publications/HuIntegration01.pdf
Bear in mind that the pointers I had given typically try to solve a far more complex version of the problem you had stated. Never-the-less, the mathematical concepts remain the same.
I'm looking for algorithms to find a "best" set of parameter values. The function in question has a lot of local minima and changes very quickly. To make matters even worse, testing a set of parameters is very slow - on the order of 1 minute - and I can't compute the gradient directly.
Are there any well-known algorithms for this kind of optimization?
I've had moderate success with just trying random values. I'm wondering if I can improve the performance by making the random parameter chooser have a lower chance of picking parameters close to ones that had produced bad results in the past. Is there a name for this approach so that I can search for specific advice?
More info:
Parameters are continuous
There are on the order of 5-10 parameters. Certainly not more than 10.
How many parameters are there -- eg, how many dimensions in the search space? Are they continuous or discrete - eg, real numbers, or integers, or just a few possible values?
Approaches that I've seen used for these kind of problems have a similar overall structure - take a large number of sample points, and adjust them all towards regions that have "good" answers somehow. Since you have a lot of points, their relative differences serve as a makeshift gradient.
Simulated
Annealing: The classic approach. Take a bunch of points, probabalistically move some to a neighbouring point chosen at at random depending on how much better it is.
Particle
Swarm Optimization: Take a "swarm" of particles with velocities in the search space, probabalistically randomly move a particle; if it's an improvement, let the whole swarm know.
Genetic Algorithms: This is a little different. Rather than using the neighbours information like above, you take the best results each time and "cross-breed" them hoping to get the best characteristics of each.
The wikipedia links have pseudocode for the first two; GA methods have so much variety that it's hard to list just one algorithm, but you can follow links from there. Note that there are implementations for all of the above out there that you can use or take as a starting point.
Note that all of these -- and really any approach to this large-dimensional search algorithm - are heuristics, which mean they have parameters which have to be tuned to your particular problem. Which can be tedious.
By the way, the fact that the function evaluation is so expensive can be made to work for you a bit; since all the above methods involve lots of independant function evaluations, that piece of the algorithm can be trivially parallelized with OpenMP or something similar to make use of as many cores as you have on your machine.
Your situation seems to be similar to that of the poster of Software to Tune/Calibrate Properties for Heuristic Algorithms, and I would give you the same advice I gave there: consider a Metropolis-Hastings like approach with multiple walkers and a simulated annealing of the step sizes.
The difficulty in using a Monte Carlo methods in your case is the expensive evaluation of each candidate. How expensive, compared to the time you have at hand? If you need a good answer in a few minutes this isn't going to be fast enough. If you can leave it running over night, it'll work reasonably well.
Given a complicated search space, I'd recommend a random initial distributed. You final answer may simply be the best individual result recorded during the whole run, or the mean position of the walker with the best result.
Don't be put off that I was discussing maximizing there and you want to minimize: the figure of merit can be negated or inverted.
I've tried Simulated Annealing and Particle Swarm Optimization. (As a reminder, I couldn't use gradient descent because the gradient cannot be computed).
I've also tried an algorithm that does the following:
Pick a random point and a random direction
Evaluate the function
Keep moving along the random direction for as long as the result keeps improving, speeding up on every successful iteration.
When the result stops improving, step back and instead attempt to move into an orthogonal direction by the same distance.
This "orthogonal direction" was generated by creating a random orthogonal matrix (adapted this code) with the necessary number of dimensions.
If moving in the orthogonal direction improved the result, the algorithm just continued with that direction. If none of the directions improved the result, the jump distance was halved and a new set of orthogonal directions would be attempted. Eventually the algorithm concluded it must be in a local minimum, remembered it and restarted the whole lot at a new random point.
This approach performed considerably better than Simulated Annealing and Particle Swarm: it required fewer evaluations of the (very slow) function to achieve a result of the same quality.
Of course my implementations of S.A. and P.S.O. could well be flawed - these are tricky algorithms with a lot of room for tweaking parameters. But I just thought I'd mention what ended up working best for me.
I can't really help you with finding an algorithm for your specific problem.
However in regards to the random choosing of parameters I think what you are looking for are genetic algorithms. Genetic algorithms are generally based on choosing some random input, selecting those, which are the best fit (so far) for the problem, and randomly mutating/combining them to generate a next generation for which again the best are selected.
If the function is more or less continous (that is small mutations of good inputs generally won't generate bad inputs (small being a somewhat generic)), this would work reasonably well for your problem.
There is no generalized way to answer your question. There are lots of books/papers on the subject matter, but you'll have to choose your path according to your needs, which are not clearly spoken here.
Some things to know, however - 1min/test is way too much for any algorithm to handle. I guess that in your case, you must really do one of the following:
get 100 computers to cut your parameter testing time to some reasonable time
really try to work out your parameters by hand and mind. There must be some redundancy and at least some sanity check so you can test your case in <1min
for possible result sets, try to figure out some 'operations' that modify it slightly instead of just randomizing it. For example, in TSP some basic operator is lambda, that swaps two nodes and thus creates new route. Your can be shifting some number up/down for some value.
then, find yourself some nice algorithm, your starting point can be somewhere here. The book is invaluable resource for anyone who starts with problem-solving.
Let's imagine I have an unknown function that I want to approximate via Genetic Algorithms. For this case, I'll assume it is y = 2x.
I'd have a DNA composed of 5 elements, one y for each x, from x = 0 to x = 4, in which, after a lot of trials and computation and I'd arrive near something of the form:
best_adn = [ 0, 2, 4, 6, 8 ]
Keep in mind I don't know beforehand if it is a linear function, a polynomial or something way more ugly, Also, my goal is not to infer from the best_adn what is the type of function, I just want those points, so I can use them later.
This was just an example problem. In my case, instead of having only 5 points in the DNA, I have something like 50 or 100. What is the best approach with GA to find the best set of points?
Generating a population of 100,
discard the worse 20%
Recombine the remaining 80%? How?
Cutting them at a random point and
then putting together the first
part of ADN of the father with the
second part of ADN of the mother?
Mutation, how should I define in
this kind of problem mutation?
Is it worth using Elitism?
Any other simple idea worth using
around?
Thanks
Usually you only find these out by experimentation... perhaps writing a GA to tune your GA.
But that aside, I don't understand what you're asking. If you don't know what the function is, and you also don't know the points to being with, how do you determine fitness?
From my current understanding of the problem, this is better fitted by a neural network.
edit:
2.Recombine the remaining 80%? How? Cutting them at a random point and then putting together the first part of ADN of the father with the second part of ADN of the mother?
This is called crossover. If you want to be saucey, do something like pick a random starting point and swapping a random length. For instance, you have 10 elements in an object. randomly choose a spot X between 1 and 10 and swap x..10-rand%10+1.. you get the picture... spice it up a little.
3.Mutation, how should I define in this kind of problem mutation?
usually that depends more on what is defined as a legal solution than anything else. you can do mutation the same way you do crossover, except you fill it with random data (that is legal) rather than swapping with another specimen... and you do it at a MUCH lower rate.
4.Is it worth using Elitism?
experiment and find out.
Gaussian adaptation usually outperforms standard genetic algorithms. If you don't want to write your own package from scratch, the Mathematica Global Optimization package is EXCELLENT -- I used it to fit a really nasty nonlinear function where standard fitters failed miserably.
Edit:
Wikipedia Article
If you hunt down prints of the listed papers on the article, you can find whitepapers and implementations. In general though, you should have some idea what the solution space for your maximizing the fitness function look like. If the number of variables is small, or the number of local maxima is small or they are connected/slope down to a global maxima, simple least squares works fine. If the area around each local maxima is small (IE you have to get a damned good solution to hit the best one, otherwise you hit a bad one), then fancier algorithms are needed.
Choosing variables for a genetic algorithm depends on what the solution space will look like.