Some problems that require recursion have always put me in a fix. I am not always able to come up with a recursive algorithm, but I know that there is a recursive solution to the problem.
I find problems like factorial and fibonacci easy to implement using recursive approach. But when I face more complex problems such as generating the partition of a number http://en.wikipedia.org/wiki/Partition_%28number_theory%29, I know that there is a possible recursive approach but I get stuck right there. I can't devise a recursive algorithm. Suppose I want to print all combinations of a string or if I want to bruteforce a Coin Change problem using recursion, I can't devise a recursive approach.
Is there any particular way to think so as to come up with a recursive approach? Is there any extensive recursive algorithms tutorial out there which can help me solve more advanced problems?
Look through the Structure and Interpretation of Computer Programs book, which is highly recommended here on Stack Overflow and is freely available online. It uses the Scheme programming language to teach fundamental concepts about programming. Since Scheme is a functional programming language, recursion is widely used everywhere - not only where you would use it even in imperative programming languages such as C or PHP, but also where you would typically use a looping construct. The examples and problems in the book present recursion in its natural habitat, if you will, and not through convoluted made-up scenarios.
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As the question describe itself "What is the core difference between algorithm and pseudocode?".
algorithm
An algorithm is a procedure for solving a problem in terms of the actions to be executed and the order in which those actions are to be executed. An algorithm is merely the sequence of steps taken to solve a problem. The steps are normally "sequence," "selection, " "iteration," and a case-type statement.
Pseudocode
Pseudocode is an artificial and informal language that helps programmers develop algorithms. Pseudocode is a "text-based" detail (algorithmic) design tool.
The rules of Pseudocode are reasonably straightforward. All statements showing "dependency" are to be indented. These include while, do, for, if, switch. Examples below will illustrate this notion.
I think all the other answers give useful explanations and definitions, but I'm going to give mine.
An algorithm is the idea of how to obtain some result from some input. It is an abstract concept; an algorithm is not something material by itself, but more something like an imagination or a deduction, a thing that only exists in the mind. In the broad sense, any sequence of steps that give you some thing(s) from other thing(s) could be called an algorithm. For example, if the screen of your computer is dirty, "spraying some glass cleaner on it and wipe it with a cloth" could be said to be an algorithm to solve the problem of how to obtain a clean screen form a dirty screen. It is important to note the difference between the problem itself (getting a clean screen) and the algorithm (wiping it with a cloth and cleaner); generally, several different algorithms are possible to solve the same problem. The idea of complexity is inherent to the algorithms itself, not the problem or the particular implementation or execution of the algorithm.
Pseudocode is a language to express algorithms. Since, as said before, algorithms are only concepts, we need to use something to express them and explain them to other people. Pseudocode is a convenient way for many computer science algorithms, because it is usually unambiguous, easy to read and somewhat similar to many programming languages. However, a specific programming language like C or Java can also be used to express and algorithm (it's just less convenient to those not familiarized with that language). In other cases, pseudocode may not be the best way to express an algorithm; for example, many graph and tree algorithms can be explained more easily with drawings or diagrams. In the previous example, the algorithm to get your screen cleaned is probably better expressed in a natural language like English, because it is simple and specific enough for that case.
Obviously, terms are frequently used loosely and exchanged depending on the context, and there's no need to be nitpicky about it, but I think it is important to have the difference clear. An algorithm doesn't stop being an algorithm just because it is written in Python instead of pseudocode. Pseudocode is just a convenient and widespread communication tool to express them.
An algorithm is something (a sequence of steps) you can do. Pseudocode is a notation to describe an algorithm.
Algorithm is something which is represented in mathematical terms. It includes, analysis, complexity considerations(best, average and worstcase analysis etc.).Pseudo code is a human readable representation of a program.
From Wikipedia :
Starting from an initial state and initial input (perhaps empty), the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states, eventually producing "output" and terminating at a final ending state.
With a pseudo language one can implement an algorithm without using a programming language such as C.
An example of pseudo language is Flow Charts.
I understand that doing minimization in integer programming is a very complex problem. But what makes this problem so difficult?
If I were to (attempt) to write an algorithm to solve it, what would I need to take into account? I'm only familiar with the branch-and-bound technique for solving it and I'm wondering what sort of roadblocks I will face when attempting to apply this technique programatically.
I'm wondering what sort of roadblocks I will face when attempting to apply this technique programatically.
None in particular (assuming a fairly straightforward implementation without a lot of tricks). The algorithms aren’t complicated – they are complex, that’s a fundamental difference.
Techniques such as branch and bound or branch and cut try to prune the search tree and thus speed up the running time. But the whole problem tree is nevertheless exponentially large, hence the problem.
Like the other said, those problem are very hard and there are no simple solution nor simple algorithm that apply to all classes of problems.
The "classic" way of solving those problem is to do a branch-and-bound and apply the simplex algorithm at each node, as you say in your question. However, I would not recommand implementing this yourself if you are not an expert.
As for a lot of numerical methods, it is very hard to get it right (good parameter values, good optimisations), and a lot have been done (see CPLEX, COIN_OR, etc).
It's not that you can't do it: the branch-and-bound part is pretty straigtforward, but without all the tricks your program will be really slow.
Also, you will need a simplex implementation and this is not something you want to do yourself: you will have to use a third-part lib anyway.
Most likely, wether
if your data set is not that big (try it !), and you are not interested in solving it really fast: use something like COIN-OR or lp_solve with the default method, it will work;
if your data set is really big (and/or you need to find a solution quickly each time), you need to work with an expert in this field.
My main point is that only experienced people will know which algorithm will perform better on your problem, wich form of the model will be the easiest to solve, which method to apply and what kind of optimisations you can try.
If you are interested in those problems, I would recommend this book for an introduction to the math behind all this (with a lot of examples). It is incredibly expansive, so you may want to go to a library instead of buying it: Nemhauser and Wolsey.
Integer programming is NP-hard. That's why it is so difficult.
There is a tutorial that you might be interested.
The first thing you do before you solve any mathematical optimization problem is you categorize it. Except special cases, most of the time, integer programming problems will be np-hard. So instead of using an "algorithm", you will use a "heuristic". The final solution you will find will not be a guaranteed optimum, but it will be a pretty good solution for real life problems.
Your main roadblock will your programming skills. Heuristic programming requires a good level of programming understanding. So instead of programming your own heuristic you are better of using well known package (eg, COIN-OR, free). This way you can focus on your problem instead of the heuristic.
Up until now I've mostly concentrated on how to properly design code, make it as readable as possible and as maintainable as possible. So I alway chose to learn about the higher level details of programming, such as class interactions, API design, etc.
Algorithms I never really found particularly interesting. As a result, even though I can come up with a good design for my programs, and even if I can come up with a solution to a given problem it rarely is the most efficient.
Is there a particular way of thinking about problems that helps you come up with an as efficient solution as possible, or is it simple a matter of practice and/or memorizing?
Also, what online resources can you recommend that teach you various efficient algorithms for different problems?
Data dominates. If you design your program around the right abstract data structures (ADTs), you often get a clean design, the algorithms follow quite naturally and when performance is lacking, you should be able to "plug in" more efficient ones.
A strong background in maths and logic helps here, as it allows you to visualize your program at a high level as the interaction between functions, sets, graphs, sequences, etc. You then decide whether the sets need to be ordered (balanced BST, O(lg n) operations) or not (hash tables, O(1) operations), what operations need to supported on sequences (vector-like or list-like), etc.
If you want to learn some algorithms, get a good book such as Cormen et al. and try to implement the main data structures:
binary search trees
generic binary search trees (that work on more than just int or strings)
hash tables
priority queues/heaps
dynamic arrays
Introduction To Algorithms is a great book to get you thinking about efficiency of different algorithms/data structures.
The authors of the book also teach an algorithms course on MIT . You can find most lectures here
I would say that in coming up with good algorithms (which is actually part of good design IMHO), you have to develop a way of thinking. This is best done by studying algorithm design. By study I don't mean just knowing all the common algorithms covered in a textbook, but actually understanding how and why they work, and being able to apply the basic idea contained in them to actual problems you are trying to solve.
I would suggest reading a good book on algorithms (my favourite is CLRS). For an online resource I would recommend the series of articles in the TopCoder Algorithm Tutorials.
I do not understand why you would mention practice and memorization in the same breath. Memorization won't help you at all (you probably already know this), but practice is essential. If you cannot apply what you learned, its not really learning. You can practice at various online programming contest/puzzle sites like SPOJ, Project Euler and PythonChallenge.
Recommendations:
First of all i recommend the book "Intro to Algorithms, Second Edition By corman", great book has most(if not all) of the algorithms you will need. (Some of the more important topics are sorting-algorithms, shortest paths, dynamic programing, many data structures like bst, hash maps, heaps).
another great way to learn algorithms is http://ace.delos.com/usacogate, great practice after the begining.
To your questions you will just get used to write good fast running code, after a little practice you just wouldnt want to write un-efficient code.
While I think #larsmans is correct inasmuch that understanding logic and maths is a fast way to understanding how to choose useful ADTs for solving a given problem, studying existing solutions may be more instructive for those of us who struggle with those topics. In particular, reviewing code of established software (OSS) that solves some similar problem as the one you're interested in.
I find a particularly good method for this method of study is reviewing unit tests of such a project. Apache Lucene, for example, has a source control repository containing numerous examples. While it doesn't reveal the underlying algorithms, it helps trace to particular functionality that solves a certain problem. This leads to an opportunity for studying its innards - i.e. an interesting algorithm. In Lucene's case inverted indices come to mind.
While this does not guarantee the algorithm you discover is the best, it's likely one that's received a lot scrutiny and probably comes from project with an active mailing that may answer your questions. So it's a good resource for finding a solution that is probably better than what most of us would come up with on our own.
I've been contemplating programming language designs, and from the definition of Declarative Programming on Wikipedia:
This is in contrast from imperative programming, which requires a detailed description of the algorithm to be run.
and further down:
... Any style of programming that is not imperative. ...
It then goes on to express that functional languages, because they are not imperative, are declarative by their very nature.
However, this makes me wonder, are purely functional programming languages able to solve any algorithmic problem, or are the constraints based upon what functions are available in that language?
I'm mostly interested in general thoughts on the subject, although if specific examples can illustrate the point, I certainly welcome them.
According to the Church-Turing Thesis ,
the three computational processes (recursion, λ-calculus, and Turing machine) were shown to be equivalent"
where Turing machine can be read as "procedural" and lambda calculus as "functional".
Yes, Haskell, Erlang, etc. are Turing complete languages. In principle, you don't need mutable state to solve a problem, since you can always create a new object instead of mutating the old one. Of course, Brainfuck is also Turing complete. In other words, just because an algorithm can be expressed in a functional language doesn't mean it's not horribly awkward.
OK, so Church and Turing provied it is possible, but how do we actually do something?
Rewriting imperative code in pure functional style is an exercise I frequently assign to undergraduate students:
Each mutable variable becomes a function parameter
Loops are rewritten using recursion
Each goto is expressed as a function call with arguments
Sometimes what comes out is a mess, but often the results are surprisingly elegant. The only real trick is not to pass arguments that never change, but instead to let-bind them in the outer environment.
The big difference with functional style programming is that it avoids mutable state. Where imperative programming will typically update variables, functional programming will define new, read-only values.
The main place where this will hit performance is with algorithms that use updatable arrays. An imperative implementation can update an array element in O(1) time, while the best a purely functional style of implementation can achieve is O(log N) (using a sorted tree).
Note that functional languages generally have some way to use updateable arrays with O(1) access time (e.g., Haskell provides this with its state transformer monad). However, this is arguably an imperative programming method... nothing wrong with that; you want to use the best tools for a particular job, after all.
The functional style of O(log N) incremental array update is not all bad, though, as functional style algorithms seem to lend themselves well to parallellization.
Too long to be posted as a comment on #SteveB's answer.
Functional programming and imperative programming have equal capability: whatever one can do, the other can do. They are said to be Turing complete. The functions that a Turing machine can compute are exactly the ones that recursive function theory and λ-calculus express.
But the Church-Turing Thesis, as such, is irrelevant. It asserts that any computation can be carried out by a Turing machine. This relates an informal idea - computation - to a formal one - the Turing machine. Nobody has yet found anything we would recognise as computation that a Turing machine can't do. Will someone find such a thing in future? Who can tell.
Using state monads you can program in an imperative style in Haskell.
So the assertion that Haskell is declarative by its very nature needs to be taken with a grain of salt. On the positive side it then is equivalent to imperative programming languages, also in a practical sense which doesn't completely ignore efficiency.
While I completely agree with the answer that invokes Church-Turing thesis, this begs an interesting question actually. If I have a parallel computation problem (which is not algorithmic in a strict mathematical sense), such as multiple producer/consumer queue or some network protocol between several machines, can this be adequately modeled by Turing machine? It can be simulated, but if we simulate it, we lose the purpose why we have the parallelism in the problem (because then we can find simpler algorithm on the Turing machine). So what if we were not to lose parallelism inherent to the problem (and thus the reason why are we interested in it), we couldn't remove the notion of state?
I remember reading somewhere that there are problems which are provably harder when solved in a purely functional manner, but I can't seem to find the reference.
As noted above, the primary problem is array updates. While the compiler may use a mutable array under the hood in some conditions, it must be guaranteed that only one reference to the array exists in the entire program.
Not only is this a hard mathematical fact, it is also a problem in practice, if you don't use impure constructs.
On a more subjective note, stating that all Turing complete languages are equivalent is only true in a narrow mathematical sense. Paul Graham explores the issue in Beating the Averages in the section "The Blub Paradox."
Formal results such as Turing-completeness may be provably correct, but they are not necessarily useful. The travelling salesman problem may be NP-complete, and yet salesman travel all the time. It seems they don't feel the need to follow an "optimal" path, so the theorem is irrelevant.
NOTE: I am not trying to bash functional programming, since I really like it. It is just important to remember that it is not a panacea.
The other day I thought I'd attempt creating the Fibonacci algorithm in my code, but I've never been good at maths.
I ended up writing my own method with a loop but it seemed inefficient or not 'the proper way'.
Does anyone have any recommendations/reading material on implementing algorithms in code?
I find Project Euler useful for this kind of thing. It forces you to think about an algorithm and then implement it. Many of the questions then have extensive discussions on how to solve the problem (from the naive solutions to some pretty ingenious ones) that you can use to see what you did right and wrong.
In the discussion threads you'll find various implementations from other people in many different languages too. Coming up with a solution yourself and then comparing it to that from other people is (imho) a good way to learn.
Both of these introductory books have good information about this sort of thing:
How To Design Programs and moreso Structure and Interpretation of Computer Programs
Both are somewhat funcitonal (and scheme) oriented, but that's a natural fit for these sorts of problems.
On top of that, you might get quite a bit out of Project Euler
Derive your algorithm test-driven. I've been able to write much more complex algorithms correctly by using TDD than I was before.
Go on youtube and look at some of the lectures on Introduction to Algorithms. There are some really, really good lectures that break down some of the most common algorithms such as the Fibonacci series and how to optimize them.
Start reading about O notation so you can understand how your algorithm grows with variable size input and how to classifiy the run-time of the algorithm you have.
Start with this video series which I found excellent material on the subject:
Algorithms Lecture
If you can't translate pseudo code for a fibonacci function to your language, then you should go and find a basic tutorial for your language, since it seems that you have not yet grasped its basic idioms.
If you have a working solution, but feel insecure about it, show it to others for review.