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I want to know that is there any difference between NP- hard problems and Computationally hard problems or are these two terms used for the same thing? I have tried to search the solution but cannot get some reasonable answer. Can anybody please help?
As of current knowledge (until that P=NP question is answered):
All NP-hard problems are computationally hard. But not all computationally hard problems are NP-hard (problems in P, with high exponents in the polynome, for example).
Note that "NP-hard" is a well defined class of problems in computer science.
"Computationally hard" on the other hand isn't, as far as I know.
at workplace, the work I do is hardly near to challenging and doing that I think I might be losing the skills to look at a completely new problem and think about different ideas to solve it.
A friend suggested TopCoder.com to me, but looking at the overwhelming number of problems I can not decide how to get started?
what I want is to sharpen my techniques ( not particular language or framework ).
The only way to get started would be to pick problems. Division I is the more difficult division, so you will probably find that the division I medium and hard problems will be somewhat interesting and challenging (unless you are quite clever.)
If you check the event calendar, you can see what algorithm competition rounds are coming up in your time zone. The competitions have the added virtue of forcing you to read and analyze other people's code in the challenge phase, so even if you would just as soon practice without a clock, you may find them interesting.
TopCoder algorithm contests are a way to develop your coding speed. Solving any of the problems in the practice arena is difficult unless you already have knowledge of various algorithms.
The problems on Project Euler suffer from the same flaw. You already have to know the algorithms to solve the problems in a reasonable time frame.
What I would suggest is to pick a project that you're interested in, and pursue it as you have time. As an example, I'm currently learning how to work with the open street map tiles in an Eclipse rich client platform.
Try whit http://projecteuler.net Problems difficulty can be assumed by number of solvers.
I prefer this page, because it is language invariant and problems are really challenging
You need the experience of solving 2 problems in any online judge (like http://www.spoj.com, http://www.lightoj.com, http://www.codeforces.com) in any programming language of your choice. That will give you an idea about how are your programs tested online.
Then you can follow this -> http://localboyfrommadurai.blogspot.in/2011/12/new-to-topcoder.html
I have a decent grasp of NP Complete problems; that's not the issue. What I don't have is a good sense of where they turn up in "real" programming. Some (like knapsack and traveling salesman) are obvious, but others don't seem obviously connected to "real" problems.
I've had the experience several times of struggling with a difficult problem only to realize it is a well known NP Complete problem that has been researched extensively. If I had recognized the connection more quickly I could have saved quite a bit of time researching existing solutions to my specific problem.
Are there any resources (online or print) that specifically connect NP Complete to real world instances?
Edit:
For example, I was working on a program that tried to divide students into groups based on age, grade, and school of origin, which is essentially a graph partitioning problem. It took me a while to realize the connection.
I have found that Computers and Intractability is the definitive reference on this topic.
Usually the connection you are talking about must be extracted with a so-called reduction, for example you reduce 3-SAT to the problem you are working with and then you can conclude that your problem has the same complexity of it.
This passage is not trivial, since you have to prove that you can turn every problem instance l of a known NP-Hard problem L into an instance c of your problem C using a deterministic polinomyal algorithms.
So, except from learning basical correlations of common NP-Hard problems using your memory, there's no way to be sure if a problem is similar to another NP-Hard without first trying to guessing and then proving it, you have to be smart.
here is a wiki link:
http://wapedia.mobi/en/List_of_NP-complete_problems
Notice it says
This list is in no way comprehensive (there are more than 3000 known NP-complete problems)
probably it would be a great task if anyone could compile such list.
A theorist should try to understand/proof an NP-Complete/Hard problem. But, a programmer doesn't have that time to. He needs a list.
Am I correct?
I think you should google it. And, read through all the links. Add any new problem found in the link to your list.
Hope it helps
PS : Don't forget to post the list when you're finished :P
For developing better intuition the book "The Algorithm Design Manual, Second Edition" by Skiena (excerpts on google books) is simply great.
List in the back with problems
(including hard problems), that
include an illustration and a
discussion (often) with a real world
example.
Covers both the theoretical
and practical side of things, often
talking about actual code.
Read excepts online here (see some examples in chapters 14):
http://books.google.dk/books?id=7XUSn0IKQEgC&printsec=frontcover#v=onepage&q&f=false
Chapter 16 (not online) discusses some hard problems, including graph partition.
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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.
I have recently been reading about the general use of prime factors within cryptography. Everywhere i read, it states that there is no 'PUBLISHED' algorithm which operates in polynomial time (as opposed to exponential time), to find the prime factors of a key.
If an algorithm was discovered or published which did operate in polynomial time, then how would this impact in the real world computing environment as opposed to the world of theory and computer science. Considering the extent we depend on cryptography would the would suddenly come to halt.
With this in mind if P = NP is true, what might happen, how much do we depend on the fact that it is yet uproved.
I'm a beginner so please forgive any mistakes in my question, but i think you'll get my general gist.
With this in mind if N = NP is true, would they ever tell us.
Who are “they”? If it were true, we would know. The computer scientists? That’s us. The cryptographers and mathematicians? The professionals? The experts? People like us. Users of the Internet, even of Stack Overflow.
We wouldn’t need being told. We’d tell.
Science and research isn’t done behind closed doors. If someone finds out that P = NP, this couldn’t be kept secret, simply because of the way that research is published. In principle, everyone has access to such research.
It depends on who discovers it.
NSA and other organizations that research cryptography under state sponsorship, contrary to Konrad's assertion, do research and science behind closed doors—and guns. And they have "scooped" published academic researchers on some important discoveries. Finally, they have a history of withholding cryptanalytic advances for years after they are independently discovered by academic researchers.
I'm not big into conspiracy theories. But I'd be very surprised if a lot of "black" money hasn't been spent by governments on the factorization problem. And if any results are obtained, they would be kept secret. A lot of criticism has been leveled at agencies in the U.S. for failing to coordinate with each other to avert terrorism. It might be that notifying the FBI of information gathered by the NSA would reveal "too much" about the NSA's capabilities.
You might find the first question posed to Bruce Schneier in this interview interesting. The upshot is that NSA will always have an edge over academia, but that margin is shrinking.
For what it is worth, the NSA recommends the use of elliptic curve Diffie-Hellman key agreement, not RSA encryption. Do they like the smaller keys? Are they looking ahead to quantum computing? Or … ?
Keep in mind that factoring is not known to be (and is conjectured not to be) NP-complete, thus demonstrating a P algorithm for factoring will not imply P=NP. Presumably we could switch the foundation of our encryption algorithms to some NP-complete problem instead.
Here's an article about P = NP from the ACM: http://cacm.acm.org/magazines/2009/9/38904-the-status-of-the-p-versus-np-problem/fulltext
From the link:
Many focus on the negative, that if P
= NP then public-key cryptography becomes impossible. True, but what we
will gain from P = NP will make the
whole Internet look like a footnote in
history.
Since all the NP-complete optimization
problems become easy, everything will
be much more efficient. Transportation
of all forms will be scheduled
optimally to move people and goods
around quicker and cheaper.
Manufacturers can improve their
production to increase speed and
create less waste. And I'm just
scratching the surface.
Given this quote, I'm sure they would tell the world.
I think there were researchers in Canada(?) that were having good luck factoring large numbers with GPUs (or clusters of GPUs). It doesn't mean they were factored in polynomial time but the chip architecture was more favorable to factorization.
If a truly efficient algorithm for factoring composite numbers was discovered, I think the biggest immediate impact would be on e-commerce. Specifically, it would grind to a halt until a form of encryption was developed that doesn't rely on factoring being a one-way function.
There has been a lot of research into cryptography in the private sector for the past four decades. This was a big switch from the previous era, where crypto was largely in the purview of the military and secret government agencies. Those secret agencies definitely tried to resist this change, but once knowledge is discovered, it's very hard to keep it under wraps. With that in mind, I don't think a solution to the P = NP problem would remain a secret for long, despite any ramifications it might have in this one area. The potential benefits would be in a much wider range of applications.
Incidentally, there has been some research into quantum cryptography, which
relies on the foundations of quantum mechanics, in contrast to traditional public key cryptography which relies on the computational difficulty of certain mathematical functions, and cannot provide any indication of eavesdropping or guarantee of key security.
The first practical network using this technology went online in 2008.
As a side note, if you enter into the realm of quantum computing, you can factor in polynomial time. See Rob Pike's notes from his talk on quantum computing, page 25, also known as Shor's algorithm.