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"Designing the right algorithm for a given application is a difficult job. It requires a major creative act, taking a problem and pulling a solution out of the ether. This is much more difficult than taking someone else's idea and modifying it or tweaking it to make it a little better. The space of choices you can make in algorithm design is enormous, enough to leave you plenty of freedom to hang yourself".
I have studied several basic design techniques of algorithms like Divide and Conquer, Dynamic Programming, greedy, backtracking etc.
But i always fail to recognize what principles to apply when i come across certain programming problems. I want to master the designing of algorithms.
So can any one suggest a best place to understand the principles of algorithm design in depth.....
I suggest Programming Pearls, 2nd edition, by Jon Bentley. He talks a lot about algorithm design techniques and provides examples of real world problems, how they were solved, and how different algorithms affected the runtime.
Throughout the book, you learn algorithm design techniques, program verification methods to ensure your algorithms are correct, and you also learn a little bit about data structures. It's a very good book and I recommend it to anyone who wants to master algorithms. Go read the reviews in amazon: http://www.amazon.com/Programming-Pearls-2nd-Edition-Bentley/dp/0201657880
You can have a look at some of the book's contents here: http://netlib.bell-labs.com/cm/cs/pearls/
Enjoy!
You can't learn algorithm design just from reading books. Certainly, books can help. Books like Programming Pearls as suggested in another answer are great because they give you problems to work. Each problem forces you to think about how to solve a particular type of problem.
The idea is that you expose yourself to many different types of problems and their solutions. In doing so, you learn how to examine a problem and see if it shares anything in common with problems you've already seen. In that regard, it's not a whole lot different than the way you learned how to solve "word problems" in math class. Granted, most algorithm problems are more complex than having to figure out where on the tracks the two trains will collide, but the way you learn how to solve the problems is the same. You learn common techniques used to solve simple problems, then combine those techniques to solve more complex problems, etc.
Read, practice, lather, rinse, repeat.
In addition to books like Programming Pearls, there are sites online that post different programming challenges that you can test yourself on. It helps if you have friends or co-workers who also are interested in algorithms, because you can bounce ideas off each other and pose interesting challenges, or work together to come up with solutions to problems.
Did I mention that it takes practice?
"Mastering" anything takes time. A long time. A popular theory is that it takes 10,000 hours of practice to become an expert at anything. There's some dispute about that for particular endeavors, but in general it's true. You don't master anything overnight. You have to study. And practice. And read what others have done. Study some more and practice some more.
A good book about algorithm design is Kleinbeg Tardos. Every design technique depends on the problem that you are going to tackle. It is very important to do the exercises in the algorithm books and have feedback from teachers about that.
If there exist a locally optimal choice taht brings the globally optimal solution you can use a greedy algorithm.
If the problem has optimal substructure, you can use dynamic programming.
As everyone knows, real life problems when it comes to programming are numerous and often unexpected. Sometimes, those problems even are hard to solve, and without being trained to recognize them, you can quickly get stuck. I like challenge, because the more you get confronted to a recurrent situation, the less time you need to come up with an efficient answer -in time complexity, for instance- when you encounter a similar issue.
This leads to my question :
Does anybody know a good book, or any kind of support, that remains language independent enough, providing problems that tends to be hard at some point, probably crescent difficulty, to practice coding. I mean, these problems that are addictive and interesting, and you feel a real achievement when you solve them. Something like, if you don't find a trick to get your algorithm time-linear, while there also is an expensive brute-force version, it will lead you to failure.
Thanks already for your suggestions.
Code Kata
High School Programming Language
France IOI (in French, but there's tons of exercises)
Have fun :)
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Every time I start a hard problem and if can not figure out the exact solution or can not get started, I get into this never ending discussion with myself, as below:
That problem
solving/mathematics/algorithms skills
are gifted (not that you can learn
by practicing, by practice, you only
master the kind of problems that you
already have solved before)
only those who went to good schools can do it, as they learned it early.
What are your thoughts, can one achieve awesomeness in problem solving/algorithms just by hard work or you need to have that extra-gene in you?
I spent a big part of my life wondering whether talent was something you developed or something you were born with. Then it occurred to me that the answer was irrelevant, at least if you want to achieve things yourself. Even if you have talent, it will only help you if you act as if talent only comes from practice, because you will work that much harder.
With regards to algorithms, as well as any other really difficult skill, it takes practice to get good. Whether or not you have to have some amount of talent too, I don't know. I do know for a fact, however, that people have made huge improvements in competitions like TopCoder by practicing. I myself have learned a lot from that.
If you set up a systematic training program, you will be way ahead of the pack, even if it is not perfect. I have written a few hundred programs on TopCoder by now and it has affected my thinking in a profound way. I have learned a lot of things that could only ever be learned by doing them wrong and then fixing my mistake. A friend of mine has written several thousand programs on TopCoder and he is way better than I am, even though his stats were worse when he started out than mine were. That is no coincidence.
EDIT:
I just came across this answer at math.stackexchange. I think it is one of the best explanations of how to learn algorithms I have read, even though he writes about chess and math.
1) Don't try to solve the problem in its most general abstraction.
2) Choose the right time when your mind is working at maximum.
I got the first point as an advice from a math instructor. It works! try to do different examples and scenarios of the problem. This helps greatly in identifying the edge cases which are the hardest to understand in most problems.
My favorite time for solving this kind of problems is the dawn(4-6 AM). Have a good sleep the night before, and wakeup ready to solve the problem. Silence is your friend.
I do believe that some people have extra intelligence than others, but it is not the most important factor. It is how you utilize this intelligence to solve the problem.
I took magic lessons in a group setting when I was twelve years old. The magician's name was Joe Carota. He did a magic trick one time and I blurted out, "How did you do that?" He said something that day that has stuck with me ever since.
Joe's response, "Michael, if you really want to know how that trick is done you must figure out how you would do it yourself."
Well of course that's not what I wanted to hear but it did get my mind focused on problem solving. This was problem solving from my perspective. If my first attempt at solving the problem took seventeen steps and was really klunky, the good news was I solved the problem.
Then by looking at the solution I had developed and further looking for ways to refine that solution I would learn how to streamline the end result. Later on in my computer programming life I found out that this process was called "Stepwise Refinement".
It worked back in 1971 and it still works today.
For me, i think it's a bit talent, but much more important is experience and practice. If you know many problems and the best solutions to them, you can come up more easily with a solution to a new problem.
Example from my own past: There was some programming contest (good for training, btw) and I did not find a good solution. The winner solved the problem mainly by using a KD-Tree. To come up with this, you first of all need to know what, in this case, a KD-Tree is, and where it's useful. Today, this is clear to me and if i'd encounter a similar problem again, i'd be able to solve it really quickly.
Hardwork beats talent if talent doesn't work hard.
This above statement defines what the true potential of persistence is.Any skill in this world can be developed by practice.This process is analogous to nailing a nail in the wall.It not only requires correct magnitude but also appropriate direction.
To answer the question, first we need to find the ingredients for the capability to solve an issue.
There is a so-called natural talent. This is the talent you are born with. This predetermines your potential. People born with more gray matter will tend to perform better than people with whom nature was less generous with. This means that a person having better talent has a higher probability to perform better than a person not as talented if they had the same parameters (education, personality, resistance to stress, willpower). If one observes that he or she tends to consume a great time to absorb new information until he or she is able to apply it, then the wisest decision for the person is to leave programming and prevent a life full of frustration. Naturally, one cannot expect as a beginner to be able to instantly understand the most complex phenomenon, but if a beginner is too slow to understand beginner concepts, then programming is not his or her cup of tea.
Developed talent. One has a natural talent, but that is, in itself not enough to solve problems. I have never seen newborns writing code. One has to get some education. The earlier, the better. Also, the quality of school is of high importance. We should never deny the fact that a person who did not have the chance to learn programming at a good school early, then he or she has a handicap in the race for success. However, if someone misses good schools early, then the handicap can be covered with hard work. For instance, my wife had an education in another field, but after finishing the university, she did not find proper jobs. So I started to educate her. After a month she learned how to learn and was able to solve almost any problems presented to her, but she was not yet effective. She gradually became to start learning in auto-didacting manner. After a year she was already a professional coder. She does not have a paper from a school that she can code, but she is doing a fantastic job. So, she missed early education, but was later able to neutralize the handicap. Developed talent can be described as the set of information learned and known, along with the right attitude, the scientific approach to new types of challenges.
Practice: Practice is good to increase the level of developed talent, yet, it SHOULD not be the sole source of developing talent. Along with practice, the theoretical horizons must be regularly expanded.
Working strategy: One can be extremely talented, can have a lot of knowledge. If he or she does not have a right working strategy, then he or she has a handicap. Whenever a new task is given, the right questions should be asked:
what was the closest task to this one? Can I reuse my solution to an extent?
what should I learn to be able to solve this problem?
how can I write clear and efficient code to solve the problem?
So the answer is: while it is good to have excellent education as early as possible, it is not necessary. Do not forget, that life is the best school and you can recuperate the lost opportunity later if you have talent, willpower and source of information. Practice is not only showing you the right steps to solve a problem, it also widens your horizons. For instance, if one understands number systems, then he or she will be able to understand a variety of things later, like colors in CSS, PSD, or number overflows. If one learns how to code in Java, then he or she will understand C# very quickly. So, practice is giving you knowledge about the solution to a given problem type, but also, gives new theoretical knowledge which will be useful in various areas. The core skill one has to develop is the ability to learn quickly.
There have been many examples of people having extraordinary talent with minimum success. You see such examples in sports,politics,business and also in general around you. So, I feel after a certain limit, talent is a meaningless virtue. Its mostly the hard word that rewards you with greater success. If you follow cricket, here is a link with good example.
I feel same principle applies to algorithm and problem solving. An year back I use to pick up algorithmic problems to solve and used to find myself completely lost. An year invested in reading algorithmic books, solving its exercises and also practicing some more programming problems, I am confident that now I can solve most problems ( I still have a long way to go in making myself efficient in it). But the point is smart work is enough to develop this knack of solving problems.
Talent is cheap and useless without hardwork. Talent can only take you to some extent, but with hardwork and practice anybody can reach great heights
- Josh Waitzkin, 8-time National Chess Champion, a 13-Time National and 2-time World Champion
He himself says this in his voice over in Chessmaster Grandmaster Edition
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I was thinking about ways to improve my ability to find algorithmic solutions to a problem.I have thought of solving math problems from various math sectors such as discrete mathematics or linear algebra.After "googling" a bit I have read an article that claimed the need of learning game programming in order to achieve this and it seems logical to me.
Do you have/had the same concerns as me or do you have any ideas on this?I am looking forward to hear them.
Thank you all, in advance.
P.S.1:I want to say that I already know about programming and how to program(although I am at amateur level:-) ) and I just want to improve at the specific issue, NOT to start learning it
P.S.2:I think that its a useful topic for future reference so I checked the community wiki box.
Solve problems on a daily basis. Watch traffic lights and ask yourself, "How can these be synced to optimize the flow of traffic? Or to optimize the flow of pedestrians? What is the best solution for both?". Look at elevators and ask yourself "Why should these elevators use different rules than the elevators in that other building I visited yesterday? How is it actually implemented? How can it be improved?".
Try to see a problem everywhere, even if it is solved already. Reflect on the solution. Ask yourself why your own superior solution probably isn't as good as the one you can see - what are you missing?
And so on. Every day. All of the time.
The idea is that almost everything can be viewed as an algorithm (a goal that has some kind of meaning to somebody, and a method with which to achieve it). Try to have that in mind next time you watch a gameshow on TV, or when you read the news coverage of the latest bank robbery. Ask yourself "What is the goal?", "Whose goal is it?" and "What is the method?".
It can easily be mistaken for critical thinking but is more about questioning your own solutions, rather than the solutions you try to understand and improve.
First of all, and most important: practice. Think of solutions to everything everytime. It doesn't have to be on your computer, programming. All algorithms will do great. Like this: when you used to trade cards, how did you compare your deck and your friend's to determine the best way for both of you to trade? How can you define how many trades can you do to do the maximum and yet don't get any repeated card?
Use problem databases and online judges like this site, http://uva.onlinejudge.org/index.php, that has hundreds of problems concerning general algorithms. And you don't need to be an expert programmer at all to solve any of them. What you need is a good ability with logic and math. There, you can find problems from the simplest ones to the most challenging. Most of them come from Programming Marathons.
You can, then, implement them in C, C++, Java or Pascal and submit them to the online judge. If you have a good algorithm, it will be accepted. Else, the judge will say your algorithm gave the wrong answer to the problem, or it took too long to solve.
Reading about algorithms helps, but don't waste too much time on it... Reading won't help as much as trying to solve the problems by yourself. Maybe you can read the problem, try to figure out a solution for yourself, compare with the solution proposed by the source and see what you missed. Don't try to memorize them. If you have the concept well learned, you can implement it anywhere. Understanding is the hardest part for most of them.
Polya's "How To Solve It" is a great book for thinking about how to solve mathematical problems and do proofs, and I'd recommend it for anyone who does problem solving.
But! It doesn't really address the excitement that happens when the real world provides input to your system, via channel noise, user wackiness, other programs grabbing resources, etc. For that it is worth looking at algorithms that get applied to real-world input (obligatory and deserved nod to Knuth's collection), and systems which are fairly robust in the face of same (TCP, kernel internals). Part of coming up with good algorithmic solutions is to know what already exists.
And alongside reading all that, of course practice practice practice.
You should check out Mathematics and Plausible Reasoning by G. Polya. It is a rare math book, which actually deals with the thought process involved in making mathematical discoveries. I think it is the same thought process that is involved in coming up with algorithms.
The saying "practice makes perfect" definitely applies. I'm tutoring a friend of mine in programming, and I remind him that "if you don't know how to ride a bike, you could read every book about it but it doesn't mean you'll be better than Lance Armstrong tomorrow - you have to practice".
In your case, how about trying the problems in Project Euler? http://projecteuler.net
There are a ton of problems there, and for each one you could practice at developing an algorithm. Once you get a good-enough implementation, you can access other people's solutions (for a particular problem) and see how others have done it. Don't think of it as math problems, but rather as problems in creating algorithms for solving math problems.
In university, I actually took a class in algorithm design and analysis, and there is definitely a lot of theory behind it. You may hear people talking about "big-O" complexity and stuff like that - there are quite a lot of different properties about algorithms themselves which can lead to greater understanding of what constitutes a "good" algorithm. You can study quite a bit in this regard as well for the long-term.
Check some online judges, TopCoder (algorithm tutorials). Take some algorithms book (CLRS, Skiena) and do harder exercises. Practice much.
I would suggest this path for you :
1.First learn elementary parts of a language.
2.Then learn about some basic maths.
3.Move to topcoder div2 easy problems.Usually if you cannot score 250 pts. in any given day,then it means you need a lot of practise,keep practising.
4.Now's the time to learn some tools of a programmer,take a good book like Algorithm Design Manual by Steven Skienna and learn about dynamic programming and greedy approach.
5.Now move to marathons,don't be discouraged if you cannot solve it quickly.Improvement will not happen overnight,you will have to patiently keep on working hard.
6.Continue step 5 from now on and you will be a better programmer.
Learning about game programming will probably lead you to good algorithms for game programming, but not necessarily to better algorithms in general.
It's a good start, but I think that the best way to learn and apply algorithmic knowledge is
Learn about good algorithms that currently exist for your area of interest
Expand your knowledge by viewing other areas; for example, what kinds of algorithms are
required when working on genetic analysis? What's the best approach for determining
run-off potential as it relates to flooding?
Read about problems in other domains and attempt to use the algorithms that you're
familiar with to see if they fit. If they don't try to break the problem down and see if
you can come up with your own algorithm.
A few more books worth reading (in no particular order):
Aha! Insight (Martin Gardner)
Any of the Programming Pearls books (Jon Bentley)
Concrete Mathematics (Graham, Knuth, and Patashnik)
A Mathematical Theory of Communication (Claude Shannon)
Of course, most of those are just samples -- other books and papers by the same authors are usually quite good as well (e.g. Shannon wrote a lot that's well worth reading, and far too few people give it the attention it deserves).
Read SICP / Structure and Interpretation of Computer Programs and work all the problems; then read The Art of Computer Programming (all volumes), working all the exercises as you go; then work through all the problems at Project Euler.
If you aren't damned good at algorithms after that, there is probably no hope for you. LOL!
P.S. SICP is available freely online, but you have to buy AoCP (get the international, not-for-release-in-north-america edition used for 30 USD). And I haven't done this yet myself (I'm trying when I have free time).
I can recommend the book "Introductory Logic and Sets for Computer Scientists" by Nimal Nissanke (Addison Wesley). The focus is on set theory, predicate logic etc. Basically the maths of solving problems in code if you will. Good stuff and not too difficult to work through.
Good luck...Kevin
Great
how about trying the problems in Project Euler? http://projecteuler.net
There are a ton of problems there, and for each one you could practice at developing an algorithm. Once you get a good-enough implementation, you can access other people's solutions (for a particular problem) and see how others have done it. Don't think of it as math problems, but rather as problems in creating algorithms for solving math problems
Ok, so to sum up the suggestions:
The most effective way to improve this ability is to solve problem as frequently as possible.Either real world problems(such as the elevators "algorithm" which is already suggested) or exercises from books like CLRS(great, I already own it :-)).But I didn't see comments about maths and I don't know what to suppose(if you agree or not).:-s
The links were great.I will definitely use them.I also think that it will be a good exercise to solve problems from national/international informatics contests or to read the way a mathematician proves a theorem.
Thank you all again.Feel free to suggest more, although I am already satisfied with the solutions mentioned.
When faced with a problem in software I usually see a solution right away. Of course, what I see is usually somewhat off, and I always need to sit down and design (admittedly, I usually don't design enough), but I get a certain intuition right away.
My problem is I don't get that same intuition when it comes to advanced algorithms. I feel much more up to the task of building another Facebook then building another Google search, or a Music Genom project. It's probably because I've been building software for quite some time, but I have little experience with composing algorithms.
I would like the community's advice on what to read and what projects to undertake to be better at composing algorithms.
(This question has nothing to do with Algorithmic composition. Well, almost nothing)
+1 To whoever said experience is the best teacher.
There are several online portals which have a lot of programming problems, that you can submit your own solutions to, and get an automated pass/fail indication.
http://www.spoj.pl/
http://uva.onlinejudge.org/
http://www.topcoder.com/tc
http://code.google.com/codejam/contests.html
http://projecteuler.net/
https://codeforces.com
https://leetcode.com
The USACO training site is the training program that all USA computing olympiad participants go through. It goes step by step, introducing more and more complex algorithms as you go.
You might find it helpful to perform algorithms physically. For example, when you're studying sorting algorithms, practice doing each one with a deck of cards. That will activate different parts of your brain than reading or programming alone will.
Steve Yegge referred to "The Algorithm Design Manual" in one of his rants. I haven't seen it myself, but it sounds like it's just the ticket from his description.
My absolute favorite for this kind of interview preparation is Steven Skiena's The Algorithm Design Manual. More than any other book it helped me understand just how astonishingly commonplace (and important) graph problems are – they should be part of every working programmer's toolkit. The book also covers basic data structures and sorting algorithms, which is a nice bonus. But the gold mine is the second half of the book, which is a sort of encyclopedia of 1-pagers on zillions of useful problems and various ways to solve them, without too much detail. Almost every 1-pager has a simple picture, making it easy to remember. This is a great way to learn how to identify hundreds of problem types.
problem domain
First you must understand the problem domain. An elegant solution to the wrong problem is no good, nor is an inefficient solution to the right problem in most cases. Solution quality, in other words, is often relative. A simple scheduling problem that has a deterministic solution that takes ten minutes to run may be fine if schedules are realculated once per week, but if schedules change several times a day then a genetic algorithm solution that converges in a few seconds may be required.
decomposition and mapping
Second, decompose the problem into sub-problems and known/unknown elements that correspond to elements of the solution. Sometimes this is obvious, e.g. to count widgets you need a way of identifying widgets, an incrementable counter, and a way of storing the count. Sometimes it is not so obvious. Sometimes you have to decompose the problem, the domain, and possible solutions at the same time and try several different mappings between them to find one that leads to the correct results [this is the general method].
model
Model the solution, in your head at least, and walk through it to see if it works correctly. Adjust as necessary (See decomposition and mapping, above).
composition/interfaces
Many times you can find elements of the problem and elements of the solution that map to each other and produce partial results that are useful. This composition and interface construction provides the kernal of the solution, and also serves to reduce the scope of the problem remaining. So then you just loop back to the top with a smaller initial problem, and go through it again.
experience
Experience is the best teacher, of course, but reading about different kinds of problems and solutions will also be helpful. Studying some of the well-known algorithms and their applications is likewise very helpful, e.g. Dijkstra, Bresenham, Unification, and of course, graph theory.
I am not sure intuition can be cultivated, but I think I know what you are asking. The more problems you solve, the more information and experience you have at your disposal for future problems. So, I say just practice. Practice programming real world applications and you run into plenty of problems. Sometimes, solving puzzles can be very educational as well.
I try to find physical analogues when I'm looking at a complex problem.