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I have been having trouble finding the best solutions to data structures and algorithms questions issued by interviewers. I was wondering how you guys approach these problems. Is it a matter of just practicing solving all kinds of problems to get the experience, or do you guys have systematic ways to recognize certain types of problems? Can you recommend books that could help me? I've reread a lot of Introduction to Algorithms by CLRS, and I'm sure I could refresh on fundamental CS concepts.
I have developed some common sense in recognizing types of problems. E.g. if I am able to recognize that solutions to later iterations of a problem depend on past solutions, and ultimately stem from known base solutions, I know this is a dynamic programming problem. Maybe I need to study more to further develop this common sense.
Thanks for reading.
I'm not sure SO is a best place for this question, but I recommend you "Cracking the Coding Interview" by Gayle Laakmann McDowell.
Classical books about algorithms are ok, but they focused on more fundamental and "academic" stuff. CCI is focused specially on solving interview questions.
As the title says, I am really really really curious about this.
I've been trying to find the cases where you can't use backtracking to solve them but I can't find them.
Share your knowledge please
If there are too many possibilities to check, backtracking search will be impractical - see e.g. http://en.wikipedia.org/wiki/Lighthill_report. There may also be cases where you cannot state the problem simply and clearly enough to recognize the correct answer when you find it.
I was wondering when some one asks you to solve an algorithmic problem, is it a good way to actually start off with Hastable, Hashset or HashMap. Normally i have heard people saying you shouldn't come up with Hashes as your first answer.
So how should we go about in algorithms: In-place should be given importance or make sure time complexity is best
I'm not trying to generalise, but still some suggestions would be helpful.
Thanks
The best you can hope for is a generalized answer for your generalized question.
It depends.
The reason there are many different algorithms is because there is not always 1 algorithm that is the best. And many algorithms aim to solve different problems from each other. Some algorithms it makes no sense to even talk about hash tables.
If someone asks me to solve an algorithmic problem though, I will probably try to use something that is built in to the language I'm using before designing my own algorithm. The reason is because I value my time. If I find later that the code is not efficient enough, then I can look for a better way to do it.
I think it is really situational. If random access is a priority and you need fast access and little constraint on memory utilization and no sequential access, then Hashtable, (et al), is the choice.
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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.
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I'm a beginner C++ programmer, and to stretch my mind I've been trying some of the problems on projecteuler.net. Despite an interest in maths at school, I've found myself automatically going for brute force solutions to the problems, rather than looking for something streamlined or elegant.
Does this sound like a bad mindset to have? I feel a bit guilty doing it like this, but maybe quick and dirty is OK some of the time...
I think you should look at what your end goal is and what your constraints are.
Sometimes a bruteforce method can solve a problem in 50ms trying out every combination of solutions and a "clever" solution can solve it in 10ms. At that point, the less clever but easier to understand solution trumps the clever solution.
However, there are some problems where brute forcing will not only be inelegant but simply won't work. There are many problems where if you attempt to naively brute force them it will take a significant amount of time to solve them. So obviously, these types of problems need a more elegant approach.
So ask yourself, why you are attempting these Project Euler problems? Are you doing it to learn? Then maybe trying a clever solution would be in your best interest but only after you have initially tried a brute force solution to help get a grasp of the problem.
When doing the Python Challenge problems I try to do it the most succinct way I can, pushing the limits of my abilities. After I solve it I then review other peoples answers and take mental notes of people who were more clever than myself and what they did. Some people will make special use of a data structure I hadn't thought of that is more suited to the task or they will have little mathematical tricks they use to make their algorithm more efficient. In the end I try to absorb as much of their cleverness as I can and make it show the next time I'm presented with a problem of a similar nature.
As a beginner programmer, you will be spending more of your mental energy figuring out how to actually implement things in C++, rather than spending energy on finding a clever solution to each problem. This is fine, because it gives you the opportunity to explore different areas of C++ while working on a range of various kinds of problems.
When you become proficient in C++ and you don't have to think about how to do every little thing, then you will be able to spend more time inventing non-brute-force solutions.
No, this isn't a bad thing. I've had solutions that were so elegant they were wrong.
The elegant solutions weren't created spontaneously; they were derived from the brute-force solutions when more speed or less memory consumption were required from the current solution.
So no, it's not. It's how the elegant solutions came into being.
Ken Thompson: "When in doubt, use brute force"
I've sort of gone through this evolution:
Get it to compile
Make it work as expected
Figure out one solution that works
Figure out one good solution
Figure out multiple solutions, and find the best
Figure out multiple solutions, and find the best for this situation
?? haven't gotten there yet
I would say that no, it's not a bad sign. In fact you're doing yourself a favor by trending away from premature optimizations, which is definitely a Good Thing.
learning is a brute force process. I wouldn't say its bad. In trying to do something that way you may notice a pattern. I think as long as you are thinking about something and trying to find solutions you will learn. There are few people who just jump to the most elegant or efficient solutions.
It would be hard to convince me that people that are trying to learn could ever be called bad. Except maybe an evil scientist :P
good luck.
Do you fit inside the 1 minute runtime rule for the problems? If yes, then your "brute force" solution fulfils all the requirements, and that's actually a very good sign that you can quickly come up with something that works!
These kinds of problems encourage micro-optimisation and very clever algorithms, but in general a very readable straightforward implementation will be much easier to maintain, and will be favoured in the business world.
If it happens to be a situation where "brute force" => "simple" and "elegant" => "complex", then brute force wins. And this is very often true.
Not at all. Get the problem solved correctly and completely then make it more performant or elegant as necessary.
That's not to say you should ignore obvious performance improvements... Just don't focus on them until you understand the problem better.
To put this in a different context:
When you use a library that you don't know very well (for creating UI, for instance) you can solve a simple problem in a perfectly performant way, though you know there's a "correct way" to do it. If you are curious and worried that your brute-force code makes you look like a moron, you will soon find the "correct way" to do it (e.g., on weekends, or while you sleep). In the meantime, through brute force, you will have something that works.
I actually forget to use brute force sometimes, and start scanning the API for the "right" solution. This is definitely an error in many cases. If the brute force solution is easy to implement, scales as you need it to (really, if it works), then forget about the correct solution. You'll find it soon enough (and many times you already knew it!), but in the meantime, you solved the problem and were able to go on to the next one.
Roadblocks are terrible when coding, and should definitely be avoided more than brute force solutions.
It's definitely not a bad sign to trend to brute force, especially as a beginner because you may not know any better. Especially with Project Euler, it is a bad sign to implement a brute force method and not review the comments to learn a more efficient method.
I often end up in the same boat you're in and that's actually why I started doing P.E. problems -- I was implementing a lot of brute force approaches and wanted to expose myself to more elegant solutions...
You have weigh your option. If the brute force solution will get the job done and perform ok, it is a good solution.