I started reading Intro to Algorithms by Cormen et al like 3 weeks ago on my free time. I finished the second chapter and have been trying out the exercises for quite a while. I find them a bit difficult.
Is this normal? Should I finish all the exercises before moving on? Or is it alright if I solve the ones I can and move on to the next chapters, possibly coming back to the exercises I can't figure out right now?
If anyone out there has had experience with this book, can you tell me how it was for you? I'm a bit discouraged on not being able to solve quite a few of the exercises here.
That book was hard for me too. We used it at the university I attended and I often had to refer to other sources to get simpler explanations when I found CLRS a little over my head. Once I got the Wikipedia explanation straight in my head, and a code sample working (which CLRS often lacks), I found that I was able to go back to the text and make sense of it.
Don't worry about doing all of the exercises. Even the super-elite MIT students don't have to do them all. Do what you can do and move on. If you need a concept in the next chapter that you had glossed over, it will still be there for you to backtrack to.
MIT OpenCourseWare has also made available the old lectures for Introduction to Algorithms (SMA 5503).
Good for you for diving into CLRS by yourself. You're a braver man than me. I used the book for a grad algorithms course I took last semester, and I had a hard time just finishing the problem sets assigned for the course. Completing all of the exercises would be a truly Herculean effort.
I'd recommend tackling the chapters that interest you most and those that you don't find to difficult. The beginning of the book, if I remember correctly, is one of the harder parts, diving into the mathematical background of a lot of different areas of algorithms. Chapter 5 is especially difficult unless you know a fair bit of probability theory. Also, starred sections and problems are significantly more challenging than the surrounding material (like 21.4, which contains material our professor confessed to being unable to prove in class). Finally at the end of the book, there is just a survey of miscellaneous topics; you can just look at those that interest you, since there are entire books written about each of those topics if you want to learn more about them.
I hope this helps, and most importantly, don't get too discouraged! This is the seminal book on algorithms for a reason.
It's a difficult book, used by one of the pre-eminent technical universities in the world. It's no surprise that it's challenging. There are a LOT of exercises of varying difficulty. It's a noble goal to attempt all of them.
Aren't the course materials on-line? It'd be interesting to see if students taking the course for credit do all the exercises.
I wouldn't be discouraged. Keep plugging, even if you have to pass on some of the exercises. There's nothing saying that you have to master it in one pass, either. Go through, take in what you can, and re-do if necessary. You might find that the extra context helps.
The lectures are available on iTunes if you find that helps.
The important thing is to set a deadline and make steady progress. Good luck.
The problem with not doing all the problems is that when you are self-studying, you really don't have a good gage for how much you should be able to answer.
You can look at the course assignments online, I would recommend that for figuring out problem sets to get done.
I am learning Algorithms on my own from the CLRS book in 2020. Regardless of what people tell you about solutions manuals in general, it is advisable to get "good" solutions manuals if you are self studying with the book.
The two sets of solutions I recommend are (1) The official instructor manual and (2) solutions by Rutger's university students Michelle Bodnar and Andrew Lohr. When one of those solutions is unclear, I simply refer to the other one. If you get stuck at a problem, then give yourself a few minutes to solve it. If you don't get the answer, then use the solutions manuals. You can always test yourself on the problems from other text books or leetcode to see how much you can do on your own vs just following a solutions manual.
I won't post the solutions manuals here. I suggest that you search for them online. The Rutgers one is easily available and is legal. The official one is restricted to instructors only and is hard to get. You might be able to pay obscure online sellers/hackers to get the official one for you. Use a preloaded visa or master card gift card to make that purchase. Make sure that the card is accepted in the sellers country.
Chapter 2 was doable because I used Youtube to understand algorithms and time complexity when it was not clearly explained in the CLRS book, which is quite often. The solutions manuals also helped a little bit.
Chapter 3 is hard and I don't know if I will be able to get past this one. I might have to switch to another book, perhaps the one by Tamassia. I had studied elementary algebra, set theory, functions, probability, mathematical series and calculus a few years ago. But, I remember only a few of those things. So, it is difficult to understand Chapter 3 and move ahead.
In general, it is a comprehensive and rigorous book. However, it is bad because of these:
One-based array indexing (instead of the usual 0-based) - so everytime you translate the algorithm present in the book into code you have to either +1 or -1 and / or use < instead of <= or the other way around and so on.
Bad variable naming in pseudocode - instead of lo, hi or left, right you get p and q.
The fact that is is very rigorous may get you confused in the little details and usually you can miss the overall idea of an algorithm.
It is a famous book because many scientific papers refer to this book in their references.
Otherwise, it is ok.
Related
"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.
I've been tinkering around with code (Basic, Python, C++, PHP, JavaScript) on and off for almost two decades, but have only recently begun to get more "serious" about it (using Java). I can write code to do what I want, but now I want to learn to optimize my programs to run faster (looping through an array for every element in an array can get slow very quickly, etc). What I don't want is to be popping onto this site every 5 minutes for every little question I have. I want to learn to answer my own questions.
That said, what are some good resources for learning algorithm analysis and optimization?
I have a copy of Data Structures and Algorithms in Java (3rd edition) but I feel it's written to mostly be incorporated into a college curriculum and isn't very easy to use sans-professor. The book also has a tendency to over-use abbreviations, making it hard to flip to a particular chapter without having to skim back through the book to understand what each abbreviation stands for.
I do have some knowledge of Calculus, but it's extremely rusty, so I would prefer resources that give more explanation and fewer formulas.
Thank you in advance for all the help you can give!
I can't recommend enough Michael Abrash's "The Zen of Code Optimization". It's easyto read and full of insights. The parts that focus on pre-pentium x86 are dated, but it's real value is the focus on how to think about making code faster.
I believe it's out of print, but you may find a used copy online.
You might start with Skiena's Algorithm Design Manual. The same author also has a book on puzzle-solving called Programming Challenges, which gives you a more entertaining way to get practice with algorithms than slogging through a textbook.
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I've taken everything up to pre-calculus in college, but when trying to get through things like the Donald Knuth books, or even things like this link: http://en.wikipedia.org/wiki/Self-balancing_binary_search_tree I wind up looking at math that means absolutely nothing to me. I'm not looking for magic, I don't expect to make sense of this in a week, I'm just looking for a good graduated plan of things to read / explore to get me there. Any pointers are welcome, after 20+ years as a professional programmer, I feel it would be nice to have this under my belt. Thanks in advance to everyone! :-)
I actually recommend taking a discrete mathematics course at your local university. This helped me out tremendously. Until I had this, I did not understand recursion (which is based on mathematical induction.) There are a number of other concepts which you will learn in a good discrete mathematics course which are extremely, extremely helpful (graph theory, asymptotic notation, combinatorics...)
I also recommend taking the class for a grade. I have always noticed that this makes people take the course more seriously, even if it is not in line with a degree path or anything past the grade.
If your local university is good, they will likely have tutoring sessions and office hours available that you can go to in order to ask questions and get clarification. These are really, really valuable and helped me learn things in a deeper manner, and more quickly, than I ever could have on my own.
You may need to take calculus in order to meet the prerequisites, but that is something I would also recommend if you'd like to increase your mathematical literacy. This 'answer' will take at least a semester, and more like two, but I think this is the way to go. It's not an immediate solution, but you will become better at math if you perform well in these two classes (and you have a good university close by.)
Your profile says you are in Dallas. I found this course (with no prerequisites!) for you. The syllabus looks like it covered a lot of good material, and the course met at 5:30 p.m. (good for working people!). If they are offering anything similar next semester, I'd consider it. If you call up the instructor, I'm sure he'd be happy to talk with you about what he knows for summer and fall scheduling.
This path has worked well for me.
Good luck!
You can try this: http://www.amazon.com/Concrete-Mathematics-Foundation-Computer-Science/dp/0201558025
There's a pdf version of this available online, you can easily google it out.
Many of my friends who are great programmers recommended it.
A lot of talented programmers understand algorithms before understanding the maths behind them. Maths are only there to help, they are not here to make you understand everything. You will need to spend more time reading about algorithms and complexity, then you might get a sense of how to evaluate them.
I recommend you to read more books about algorithm complexities.
In your long experience as a professional programmer, there surely are topics and sub-domains that you are most curious about. My advice is: identify those areas and go after them. It might be code-breaking, number theory, recursion, functional programming, computational origami, logical puzzles, crystal structures, graphs, genetic algorithms, splines...
Take your own remark to heart:
but when trying to get through things like the Donald Knuth books, or even
things like this link:...I wind up looking at math that means
absolutely nothing to me
What sort of math fascinates you?
I could say there are lots of intriguing puzzles at Project Euler. After you solve a programming challenge, you have access to a forum in which other folks share their solutions and occasionally refer to some body of knowledge they were drawing on. I love it. But what matters is what you like. Your own interests are the key to your learning.
If math and programming no longer have any appeal--you don't like doing them in your spare time--find something else to get into: acting, foreign languages, travel, French cooking, biking. Who knows, maybe you're burned out.
I'd say get a good book in discrete math and one in combinatorics as well. Here are a few I've liked. The Rosen book is good place to start.
http://www.amazon.com/Course-Combinatorics-J-van-Lint/dp/0521006015
http://www.amazon.com/Discrete-Mathematics-Applications-Kenneth-Rosen/dp/0073229725/ref=sr_1_2?s=books&ie=UTF8&qid=1305304408&sr=1-2
http://www.amazon.com/Introductory-Combinatorics-5th-Richard-Brualdi/dp/0136020402/ref=sr_1_7?s=books&ie=UTF8&qid=1305304434&sr=1-7
In line with what Vincent said, I recommend Algorithms in a Nutshell from O'Reilly (here).
There is a plenty of good video-lectures on Discrete Math, Calculus and Applied Math. Just watch them every evening, make notes and try to solve simple problems. To prepare yourself for Knuth, try "Discrete Mathematics". To understand deeply what is math and how all things in the universe are interconnected (including algorithms), try "Joy of Mathematics".
I was looking for just the same thing. I couldn't afford any of the material suggested here so far so here's a link to a YouTube lecture series on Discrete Mathematics. I wish there was a playlist but unfortunately there is not.
The videos are taken uploaded from http://www.aduni.org who ask for a donation of 25c per video to cover operation costs.
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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.