I'm implementing the Euclidian algorithm for finding the GCD (Greatest Common Divisor) of two integers.
Two sample implementations are given: Recursive and Iterative.
http://en.wikipedia.org/wiki/Euclidean_algorithm#Implementations
My Question:
In school I remember my professors talking about recursive functions like they were all the rage, but I have one doubt. Compared to an iterative version don't recursive algorithms take up more stack space and therefore much more memory? Also, because calling a function requires uses some overhead for initialization, aren't recursive algorithms more slower than their iterative counterpart?
It depends entirely on the language. If your language has tail-call recursion support(a lot do now days) then they will go at an equal speed. If it does not, then the recursive version will be slower and take more (precious) stack space.
It all depends on the language and compiler. Current computers aren't really geared towards efficient recursion, but some compilers can optimize some cases of recursion to run just as efficiently as a loop (essentially, it becomes a loop in the machine code). Then again, some compilers can't.
Recursion is perhaps more beautiful in a mathematical sense, but if you feel more comfortable with iteration, just use it.
Related
Is the time complexity of dynamic programming tabular approach and recursion with memoization approach the same? For example, in the Knapsack problem the tabular approach takes O(N*W) where N is the number of items and W is the weight. But what is the time complexity for the memoization approach?
Memoization is a method used to solve dynamic programming (DP) problems recursively in an efficient manner. DP abstracts away from the specific implementation, which may be either recursive or iterative (with loops and a table). Therefore, if used appropriately, the time complexity is the same, i.e. O(NW) in the knapsack problem over the integers.
This is what we used in introduction to CS and algorithm design courses in BGU (I was a T.A. in both if matters), but there might be other terminologies which I'm unaware of.
I hope it was helpful, good luck!
Is the time complexity of dynamic programming tabular approach and recursion with memoization approach the same?
Yes the do have same time complexity of O(N*W) where N is the number of items and W is the weight. However, If the original problem requires all subproblems to be solved like in the case of Knapsack problem,
tabulation usually outperformes memoization by a constant factor.
This is because tabulation has no overhead for recursion and can use a preallocated array rather than, say, a hash map.
What is the difference between tabulation and memoization?
When you solve a dynamic programming problem using tabulation (generally iterative) you solve the problem "bottom up", i.e., by solving all related sub-problems first, typically by filling up an n-dimensional table. Based on the results in the table, the solution to the "top" / original problem is then computed.
If you apply memoization (generally recursive) to solve the problem you do it by maintaining a map of already solved sub problems. You do it "top down" in the sense that you solve the "top" problem first (which typically recurses down to solve the sub-problems).
Which is better? Memoizaiton or tabulation?
If we don’t require to solve all the problems and are just looking for the optimal solution, memoization is better.
If we do require to solve all the problems, that means we are going to make numerous recursive calls which may fill the stack space respectively, and there tabulation is better.
The caveat is that memoization is generally more intuitive to implement especially when we don’t know the solution to subproblems, whereas tabulation requires us to know the solutions, or bottom, in advance, in order to build our way up.
Useful Resources:
What is Dynamic Programming? Memoization and Tabulation
Tabulation vs Memoization
To be clear I don't mean, provided the last two numbers in the sequence provide the next one:
(2, 3, -> 5)
But rather given any index provide the Fibonacci number:
(0 -> 1) or (7 -> 21) or (11 -> 144)
Adding two numbers is a very simple task for any machine learning structure, and by extension counting by ones, twos or any fixed number is a simple addition rule. Recursive calculations however...
To my understanding, most learning networks rely on forwards only evaluation, whereas most programming languages have loops, jumps, or circular flow patterns (all of which are usually ASM jumps of some kind), thus allowing recursion.
Sure some networks aren't forwards only; But can processing weights using the hyperbolic tangent or sigmoid function enter any computationally complete state?
i.e. conditional statements, conditional jumps, forced jumps, simple loops, complex loops with multiple conditions, providing sort order, actual reordering of elements, assignments, allocating extra registers, etc?
It would seem that even a non-forwards only network would only find a polynomial of best fit, reducing errors across the expanse of the training set and no further.
Am I missing something obvious, or did most of Machine Learning just look at recursion and pretend like those problems don't exist?
Update
Technically any programming language can be considered the DNA of a genetic algorithm, where the compiler (and possibly console out measurement) would be the fitness function.
The issue is that programming (so far) cannot be expressed in a hill climbing way - literally, the fitness is 0, until the fitness is 1. Things don't half work in programming, and if they do, there is no way of measuring how 'working' a program is for unknown situations. Even an off by one error could appear to be a totally different and chaotic system with no output. This is exactly the reason learning to code in the first place is so difficult, the learning curve is almost vertical.
Some might argue that you just need to provide stronger foundation rules for the system to exploit - but that just leads to attempting to generalize all programming problems, which circles right back to designing a programming language and loses all notion of some learning machine at all. Following this road brings you to a close variant of LISP with mutate-able code and virtually meaningless fitness functions that brute force the 'nice' and 'simple' looking code-space in attempt to follow human coding best practices.
Others might argue that we simply aren't using enough population or momentum to gain footing on the error surface, or make a meaningful step towards a solution. But as your population approaches the number of DNA permutations, you are really just brute forcing (and very inefficiently at that). Brute forcing code permutations is nothing new, and definitely not machine learning - it's actually quite common in regex golf, I think there's even an xkcd about it...
The real problem isn't finding a solution that works for some specific recursive function, but finding a solution space that can encompass the recursive domain in some useful way.
So other than Neural Networks trained using Backpropagation hypothetically finding the closed form of a recursive function (if a closed form even exists, and they don't in most real cases where recursion is useful), or a non-forwards only network acting like a pseudo-programming language with awful fitness prospects in the best case scenario, plus the virtually impossible task of tuning exit constraints to prevent infinite recursion... That's really it so far for machine learning and recursion?
According to Kolmogorov et al's On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition, a three layer neural network can model arbitrary function with the linear and logistic functions, including f(n) = ((1+sqrt(5))^n - (1-sqrt(5))^n) / (2^n * sqrt(5)), which is the close form solution of Fibonacci sequence.
If you would like to treat the problem as a recursive sequence without a closed-form solution, I would view it as a special sliding window approach (I called it special because your window size seems fixed as 2). There are more general studies on the proper window size for your interest. See these two posts:
Time Series Prediction via Neural Networks
Proper way of using recurrent neural network for time series analysis
Ok, where to start...
Firstly, you talk about 'machine learning' and 'perfectly emulate'. This is not generally the purpose of machine learning algorithms. They make informed guesses given some evidence and some general notions about structures that exist in the world. That typically means an approximate answer is better than an 'exact' one that is wrong. So, no, most existing machine learning approaches aren't the right tools to answer your question.
Second, you talk of 'recursive structures' as some sort of magic bullet. Yet they are merely convenient ways to represent functions, somewhat analogous to higher order differential equations. Because of the feedbacks they tend to introduce, the functions tend to be non-linear. Some machine learning approaches will have trouble with this, but many (neural networks for example) should be able to approximate you function quite well, given sufficient evidence.
As an aside, having or not having closed form solutions is somewhat irrelevant here. What matters is how well the function at hand fits with the assumptions embodied in the machine learning algorithm. That relationship may be complex (eg: try approximating fibbonacci with a support vector machine), but that's the essence.
Now, if you want a machine learning algorithm tailored to the search for exact representations of recursive structures, you could set up some assumptions and have your algorithm produce the most likely 'exact' recursive structure that fits your data. There are probably real world problems in which such a thing would be useful. Indeed the field of optimisation approaches similar problems.
The genetic algorithms mentioned in other answers could be an example of this, especially if you provided a 'genome' that matches the sort of recursive function you think you may be dealing with. Closed form primitives could form part of that space too, if you believe they are more likely to be 'exact' than more complex genetically generated algorithms.
Regarding your assertion that programming cannot be expressed in a hill climbing way, that doesn't prevent a learning algorithm from scoring possible solutions by how many much of your evidence it's able to reproduce and how complex they are. In many cases (most? though counting cases here isn't really possible) such an approach will find a correct answer. Sure, you can come up with pathological cases, but with those, there's little hope anyway.
Summing up, machine learning algorithms are not usually designed to tackle finding 'exact' solutions, so aren't the right tools as they stand. But, by embedding some prior assumptions that exact solutions are best, and perhaps the sort of exact solution you're after, you'll probably do pretty well with genetic algorithms, and likely also with algorithms like support vector machines.
I think you also sum things up nicely with this:
The real problem isn't finding a solution that works for some specific recursive function, but finding a solution space that can encompass the recursive domain in some useful way.
The other answers go a long way to telling you where the state of the art is. If you want more, a bright new research path lies ahead!
See this article:
Turing Machines are Recurrent Neural Networks
http://lipas.uwasa.fi/stes/step96/step96/hyotyniemi1/
The paper describes how a recurrent neural network can simulate a register machine, which is known to be a universal computational model equivalent to a Turing machine. The result is "academic" in the sense that the neurons have to be capable of computing with unbounded numbers. This works mathematically, but would have problems pragmatically.
Because the Fibonacci function is just one of many computable functions (in fact, it is primitive recursive), it could be computed by such a network.
Genetic algorithms should do be able to do the trick. The important this is (as always with GAs) the representation.
If you define the search space to be syntax trees representing arithmetic formulas and provide enough training data (as you would with any machine learning algorithm), it probably will converge to the closed-form solution for the Fibonacci numbers, which is:
Fib(n) = ( (1+srqt(5))^n - (1-sqrt(5))^n ) / ( 2^n * sqrt(5) )
[Source]
If you were asking for a machine learning algorithm to come up with the recursive formula to the Fibonacci numbers, then this should also be possible using the same method, but with individuals being syntax trees of a small program representing a function.
Of course, you also have to define good cross-over and mutation operators as well as a good evaluation function. And I have no idea how well it would converge, but it should at some point.
Edit: I'd also like to point out that in certain cases there is always a closed-form solution to a recursive function:
Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form solution.
The Fibonacci sequence, where a specific index of the sequence must be returned, is often used as a benchmark problem in Genetic Programming research. In most cases recursive structures are generated, although my own research focused on imperative programs so used an iterative approach.
There's a brief review of other GP research that uses the Fibonacci problem in Section 3.4.2 of my PhD thesis, available here: http://kar.kent.ac.uk/34799/. The rest of the thesis also describes my own approach, which is covered a bit more succinctly in this paper: http://www.cs.kent.ac.uk/pubs/2012/3202/
Other notable research which used the Fibonacci problem is Simon Harding's work with Self-Modifying Cartesian GP (http://www.cartesiangp.co.uk/papers/eurogp2009-harding.pdf).
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Possible Duplicate:
Recursion and Iteration
What is the difference between a recursive and a non-recursive function? Fibonacci to be exact.
I looking for answers that relate towards the time and memory.
"Recursive" simply means that a function calls itself. This may or may not be intentional (unintentional recursion is responsible for lots of crashes).
Intentional recursion, where a function performs part of an operation, then calls itself to perform the remaining part, is often a useful programming paradigm, but requires some degree of comprehension/experience/skill to "get your head around it".
Basically, recursion can be used to replace "iteration" (loops) and to replace accompanying array allocations (with variables local to the function body). But not every iterative or array-using function can be effectively converted to its recursive equivalent.
If the problem is suitable for recursion, one can often write a recursive version that is about equivalent in execution efficiency to the non-recursive version ... maybe slightly better or worse depending on how efficient the call mechanism is compared to looping and array indexing in the language/compiler. In terms of storage, recursion is rarely more efficient, but it benefits from not having to pre-allocate (and pre-know the size of the allocation) for the particular problem at hand.
Mostly recursion is better (when it actually is) because it makes an implementation much simpler and less error-prone, and errors are by far the biggest cost in computing. (But of course improperly done it can cost you big time as well.)
When recursion is good it's very good. When recursion is bad it's very bad.
Recursive functions are procedures or subroutines implemented in a programming language, whose implementation references itself.
Non Recursive Function are procedures or subroutines implemented in a programming language, whose implementation does not references itself
Below is a link for recursive and non recursive fibonacci series:- Recursive and Non Recursive Fibonacci Series
Or say, do multicore CPUs process recursion faster than iteration?
Or it simply depends on how one language runs on the machine?
like c executes function calls with large cost, comparing to doing simple iterations.
I had this question because one day I told one of my friend that recursion isn't any amazing magic that can speed up programs, and he told me that with multicore CPUs recursion can be faster than iteration.
EDIT:
If we consider the most recursion-loved situation (data structure, function call),
is it even possible for recursion to be faster?
EDIT ont Oct 12th:
So how are the multicore cpus performing for now?
Are the softwares nowadays all programed for multi-core cpus?
There are really two ways to look at this problem:
1. Looking purely at the compiled code, then yes, iteration is faster than recursion. This is because recursion adds a function call (=overhead), and iteration does not. However, a common type of recursion is tail recursion: the recursive call is made at the end of the function. This is always optimized to iteration by compilers. So in that case it does not matter. Ergo: in some cases recursion is slower, but it is never faster.
2. From a functional programming viewpoint, most of the time recursive functions are written to be without side effects. (Having side effects in a recursive function would make it really difficult to get it to produce correct results.) If a function doesn't have side effects, then it is trivial to parallelize (thus easier to run on a multicore system). This isn't a property of recursive functions per se, but that could be the reason why your friend argues that recursion can be faster than iteration.
While the recursion is elegant and mathematically beautiful, it consumes a lot of resources, especially memory. If you have an efficient iterative solution, You should go for that.
I am participating in Al Zimmermann's Programming Contest.
http://www.azspcs.net/Contest/SonOfDarts
I have written a recursive algorithm but it takes a long time to run. I was wondering what are the most important things to consider about speed of recursive algorithms. I have made most of the properties global so they don't get allocated every time the recursions step. Is there anything else I can do that will speed up my program without changing my algorithm?
It depends on the details of your algorithm. If it is tail recursive you could transform it to an iterative algorithm fairly easily.
Recusrsion is always slower than itterative. Due to stack/heap/memory aloocation performs slower than most. It is alwyas easier to implement a recusive function in complex algorithms, nut if possible, itterative will be faster.
What language are you using for writing your program? Some languages like Haskell are tailor-made for recursive algorithms while others like Python are not.
How much time is being spent within each function call vs the number of recursive calls out of the function? Too less code being executed within the function itself would certainly lead to performance loss.
Variables on stack are usually much faster than global variables. Consider passing them around from function to function rather than putting them in global.
Unfortunately there isn't enough context in the question to provide better answer.
Recursive algorithms can also be designed in such a way that they are tail recursive. In such a situation, compilers support tail recursion optimization leading to much faster code.
There are probably a lot of overlapping sub-questions in your algorithm and you didn't save the intermediate results for each sub-questions. If you do, your program should be fast enough.
EDIT:
I just gave the dart question some thought and felt taking the recursion may not be a good approach to the solution. I did some research in SQL server with the sample given by the question:
create table regions (score int)
insert into regions values (0)
insert into regions values (1)
insert into regions values (2)
insert into regions values (4)
insert into regions values (7)
insert into regions values (11)
create table results (score int)
insert into results
select distinct (s1.score+s2.score+s3.score)
from regions s1, regions s2, regions s3
select * from results
The script clearly reveals a possible solution that can be easily implemented in an imperative programming style, without taking any recursive approach.
Don't assume the problem is with the recursion, or anything else a priori. Just do this, where you find out what's biggest, fix it, and move on to the next. I'm not saying it won't turn out that recursion is the big deal at some point. It's just that chances are very good there are bigger problems you can fix first.
If you can submit compiled code for Intel platforms, then:
Collocation of memory content to favor the CPU cache content wins over best classical algorithms in any area. Make sure to use Intel VTune performance analyzer output fed to your linker options to keep bodies of related functions located close in code memory.