I came across this term while reading an academic paper describing a genetic algorithm. I'm not sure what it actually refers to as it is not explained clearly in the paper.
What is clear about C1 operator:
It is one of the crossover operators used in a genetic algorithm. There are other operators such as two-point crossover, uniform crossover, cycle crossover.
I wish to know the procedure that the C1 operator performs and how it affects the crossover results.
Hope someone can explain it or point to some useful references.
C1 operator crossover is another name to the One-Point Crossover.
A single crossover point on both parents is selected. All data beyond
that point in either is swapped between the two parents.
Related
I have a series of points with lengths and rotations like this:
I need to create separate chains from points whose lines overlap but I’m having real trouble doing this efficiently.
I have an array of simple Point objects, in no particular order, and I can loop through them and test them with a simple "intersect" function. I need to end up with an array of chains, each with an ordered list of points. (Or another way of representing the chains).
At the moment every avenue I explore seems to involve a convoluted hack of arrays, nudges and fudges. Having never studied Computer Science I wonder if there is some sort of data structure or technique that would lend itself well to this sort of thing.
Could anyone point me in the right direction to achieving this? Pseudocode is fine (or indeed any language), although I am coding in Processing/Java if that helps.
Many thanks,
Josh
You can use union-find algorithm to find joined sets.
If your sets always are well-defined (there are no multiple intersections and so on), then some modifications seem possible to build chains during join process:
for every set, besides a 'representative', keep two extreme segments, and modify them when joining.
Looking for examples that show me how to quantify over actions (and perhaps fluents?) in situation calculus (Reiter 2001).
I understand the difference between actions, fluents and situations, but why do they need to be represented in 2nd order logic? Why not use first order? Can you please explain?
All but a few formulas used to encode a dynamic world require Second Order Logic (SOL). In particular,
the [properties of the] Initial State
the Action Preconditions
the Action Effects
the Successor State axioms (used to circumvent the "Frame problem")
can all be expressed with First Order Logic (FOL).
With some domains it may be convenient -and more concise- to use SOL for the above items, but AFAIK, it is always possible to convert such SOL to FOL in the context of the items listed above, for a finite domain, and hence SOL is not necessary (again, for the above items).
Typically the need for SOL with Situation Calculus only comes from some of the "Foundational axioms" such as the axiom to perform induction on Situations.
Furthermore, depending on the particular application, one may not need the SOL-based Foundational axiom(s) and hence the whole world can be exclusively described in terms of FOL.
I'm not a specialist in the field, but I think that in many cases, for the purpose of [Logical] Filtering, Planning, Temporal Projection, we can do away with the need for induction and hence only rely on FOL only.
I think I understand the basic concept of simulated annealing. It's basically adding random solutions to cover a better area of the search space at the beginning then slowly reducing the randomness as the algorithm continues running.
I'm a little confused on how I would implement this into my genetic algorithm.
Can anyone give me a simple explanation of what I need to do and clarify that my understand of how simulated annealing works is correct?
When constructing a new generation of individuals in a genetic algorithm, there are three random aspects to it:
Matching parent individuals to parent individuals, with preference according to their proportional fitness,
Choosing the crossover point, and,
Mutating the offspring.
There's not much you can do about the second one, since that's typically a uniform random distribution. You could conceivably try to add some random factor to the roulette wheel as you're selecting your parent individuals, and then slowly decrease that random function. But that goes against the spirit of the genetic algorithm and (more importantly) I don't think it will do much good. I think it would hurt, actually.
That leaves the third factor-- change the mutation rate from high mutation to low mutation as the generations go by.
It's really not any more complicated than that.
What are the relevant differences, in terms of performance and use cases, between simulated annealing (with bean search) and genetic algorithms?
I know that SA can be thought as GA where the population size is only one, but I don't know the key difference between the two.
Also, I am trying to think of a situation where SA will outperform GA or GA will outperform SA. Just one simple example which will help me understand will be enough.
Well strictly speaking, these two things--simulated annealing (SA) and genetic algorithms are neither algorithms nor is their purpose 'data mining'.
Both are meta-heuristics--a couple of levels above 'algorithm' on the abstraction scale. In other words, both terms refer to high-level metaphors--one borrowed from metallurgy and the other from evolutionary biology. In the meta-heuristic taxonomy, SA is a single-state method and GA is a population method (in a sub-class along with PSO, ACO, et al, usually referred to as biologically-inspired meta-heuristics).
These two meta-heuristics are used to solve optimization problems, particularly (though not exclusively) in combinatorial optimization (aka constraint-satisfaction programming). Combinatorial optimization refers to optimization by selecting from among a set of discrete items--in other words, there is no continuous function to minimize. The knapsack problem, traveling salesman problem, cutting stock problem--are all combinatorial optimization problems.
The connection to data mining is that the core of many (most?) supervised Machine Learning (ML) algorithms is the solution of an optimization problem--(Multi-Layer Perceptron and Support Vector Machines for instance).
Any solution technique to solve cap problems, regardless of the algorithm, will consist essentially of these steps (which are typically coded as a single block within a recursive loop):
encode the domain-specific details
in a cost function (it's the
step-wise minimization of the value
returned from this function that
constitutes a 'solution' to the c/o
problem);
evaluate the cost function passing
in an initial 'guess' (to begin
iteration);
based on the value returned from the
cost function, generate a subsequent
candidate solution (or more than
one, depending on the
meta-heuristic) to the cost
function;
evaluate each candidate solution by
passing it in an argument set, to
the cost function;
repeat steps (iii) and (iv) until
either some convergence criterion is
satisfied or a maximum number of
iterations is reached.
Meta-heuristics are directed to step (iii) above; hence, SA and GA differ in how they generate candidate solutions for evaluation by the cost function. In other words, that's the place to look to understand how these two meta-heuristics differ.
Informally, the essence of an algorithm directed to solution of combinatorial optimization is how it handles a candidate solution whose value returned from the cost function is worse than the current best candidate solution (the one that returns the lowest value from the cost function). The simplest way for an optimization algorithm to handle such a candidate solution is to reject it outright--that's what the hill climbing algorithm does. But by doing this, simple hill climbing will always miss a better solution separated from the current solution by a hill. Put another way, a sophisticated optimization algorithm has to include a technique for (temporarily) accepting a candidate solution worse than (i.e., uphill from) the current best solution because an even better solution than the current one might lie along a path through that worse solution.
So how do SA and GA generate candidate solutions?
The essence of SA is usually expressed in terms of the probability that a higher-cost candidate solution will be accepted (the entire expression inside the double parenthesis is an exponent:
p = e((-highCost - lowCost)/temperature)
Or in python:
p = pow(math.e, (-hiCost - loCost) / T)
The 'temperature' term is a variable whose value decays during progress of the optimization--and therefore, the probability that SA will accept a worse solution decreases as iteration number increases.
Put another way, when the algorithm begins iterating, T is very large, which as you can see, causes the algorithm to move to every newly created candidate solution, whether better or worse than the current best solution--i.e., it is doing a random walk in the solution space. As iteration number increases (i.e., as the temperature cools) the algorithm's search of the solution space becomes less permissive, until at T = 0, the behavior is identical to a simple hill-climbing algorithm (i.e., only solutions better than the current best solution are accepted).
Genetic Algorithms are very different. For one thing--and this is a big thing--it generates not a single candidate solution but an entire 'population of them'. It works like this: GA calls the cost function on each member (candidate solution) of the population. It then ranks them, from best to worse, ordered by the value returned from the cost function ('best' has the lowest value). From these ranked values (and their corresponding candidate solutions) the next population is created. New members of the population are created in essentially one of three ways. The first is usually referred to as 'elitism' and in practice usually refers to just taking the highest ranked candidate solutions and passing them straight through--unmodified--to the next generation. The other two ways that new members of the population are usually referred to as 'mutation' and 'crossover'. Mutation usually involves a change in one element in a candidate solution vector from the current population to create a solution vector in the new population, e.g., [4, 5, 1, 0, 2] => [4, 5, 2, 0, 2]. The result of the crossover operation is like what would happen if vectors could have sex--i.e., a new child vector whose elements are comprised of some from each of two parents.
So those are the algorithmic differences between GA and SA. What about the differences in performance?
In practice: (my observations are limited to combinatorial optimization problems) GA nearly always beats SA (returns a lower 'best' return value from the cost function--ie, a value close to the solution space's global minimum), but at a higher computation cost. As far as i am aware, the textbooks and technical publications recite the same conclusion on resolution.
but here's the thing: GA is inherently parallelizable; what's more, it's trivial to do so because the individual "search agents" comprising each population do not need to exchange messages--ie, they work independently of each other. Obviously that means GA computation can be distributed, which means in practice, you can get much better results (closer to the global minimum) and better performance (execution speed).
In what circumstances might SA outperform GA? The general scenario i think would be those optimization problems having a small solution space so that the result from SA and GA are practically the same, yet the execution context (e.g., hundreds of similar problems run in batch mode) favors the faster algorithm (which should always be SA).
It is really difficult to compare the two since they were inspired from different domains..
A Genetic Algorithm maintains a population of possible solutions, and at each step, selects pairs of possible solution, combines them (crossover), and applies some random changes (mutation). The algorithm is based the idea of "survival of the fittest" where the selection process is done according to a fitness criteria (usually in optimization problems it is simply the value of the objective function evaluated using the current solution). The crossover is done in hope that two good solutions, when combined, might give even better solution.
On the other hand, Simulated Annealing only tracks one solution in the space of possible solutions, and at each iteration considers whether to move to a neighboring solution or stay in the current one according to some probabilities (which decays over time). This is different from a heuristic search (say greedy search) in that it doesn't suffer from the problems of local optimum since it can get unstuck from cases where all neighboring solutions are worst the current one.
I'm far from an expert on these algorithms, but I'll try and help out.
I think the biggest difference between the two is the idea of crossover in GA and so any example of a learning task that is better suited to GA than SA is going to hinge on what crossover means in that situation and how it is implemented.
The idea of crossover is that you can meaningfully combine two solutions to produce a better one. I think this only makes sense if the solutions to a problem are structured in some way. I could imagine, for example, in multi-class classification taking two (or many) classifiers that are good at classifying a particular class and combining them by voting to make a much better classifier. Another example might be Genetic Programming, where the solution can be expressed as a tree, but I find it hard to come up with a good example where you could combine two programs to create a better one.
I think it's difficult to come up with a compelling case for one over the other because they really are quite similar algorithms, perhaps having been developed from very different starting points.
I understand they aren't real and they seem to branch computation whenever there are 2 options, instead of picking one. But, for example, if I say this:
"Non deterministically guess a bijection p of vertices from Graph G to Graph H" (context here is Graph Isomorphism)
What is that supposed to mean? I understand the bijection, but it says "non deterministically guess". If it's guessing, how is that an algorithmic approach? How can it guarantee it's going to work?
They don't, they just sort of illustrate a point. Basically what they do is guess an answer, and check if it's right(deterministically). It's not the guessing the answer part that's important though, it's checking that the answer is right. It's just like saying given an arbitrary solution, is it correct? So for example there are problems that take exponential time to compute, and some of their answers can be checked in polynomial time, but some can't. So what the non-deterministic TM does is it divides those two, the ones that can be checked quickly from the ones that can't. And then this brings up the bigger question, if one group of questions solutions can be verified much quicker than another, can their solutions also be generated quicker? This question hasn't been answered, yet.
There's different ways to picture one. One I find useful is the oracle model. Did you ever see the Far Side cartoon where a derivation on the blackboard has "Here a miracle occurs" as one of the intermediate steps? In this version of a NDTM, when you need to choose something, the oracle writes the correct version on the right part of the tape. (This is taken from Garey and Johnson, Computers and Intractability, their classic book on NP-complete problems.) You aren't allowed to assume you've got the right one, though, and there may not be a correct one.
Therefore, when you non-deterministically guess a bijection, you're getting the correct bijection for your purposes, provided one exists.
It isn't a good basis for an algorithm, since the complexity of implementing a non-deterministic Turing machine is basically exponential in the nondeterministic states, and the algorithmic equivalent of the nondeterministic guess is to try every possible bijection.
From a theoretical point of view, I'd translate it as "If there is a bijection such that....". From an algorithmic point of view, find another book, or another chapter of the same book, since that approach is useless for even moderately large graphs.
I believe what is meant is "non deterministically choose a solution" and then test that the solution is true. Since all possible choices (guesses) are tested, the solution is guaranteed.
A physical implementation of the non-deterministic Turing machine is the DNA computer. For example, here's an outline of how to solve the traveling salesman problem in DNA:
Get/make a bunch of DNA sequences, each with length proportional to the cost of an edge in your graph and sticky ends with sequences uniquely identifying one of the vertices that the edge connects.
Mix them together, with DNA ligase in a big beaker. They'll anneal to each other in sequences that represent every possible path through the graph (ok, not the really long ones).
Remove all the sequences that are missing at least one vertex. To do this, sequentially select for each vertex using hybridization. For example, if "ACGTACA" encodes vertex 1, select for sequences that bind to "TGTACGA". Then repeat this selection for every other vertex.
Sort the remaining sequences by size using gel electrophoresis. Then sequence the shortest one. The sequence encodes the shortest path through your graph.