How to Code a Solution To Deal With Large Numbers?

Question

I'm doing some Project Euler problems and most of the time, the computations involve large numbers beyond int, float, double etc.

Firstly, I know that I should be looking for more efficient ways of calculation so as to avoid the large number problem. I've heard of the Bignum libraries.

But, for academics interests, I'd like to know how to code my own solution to this problem.

Can any expert please help me out? (My language is C)

Answer 1

If you want a nice C++ version (I know, you said C, but this is really interesting code), take a look at the internals of CGAL: http://www.cgal.org/

Answer 2

A simple way is to think of the number as its string representation in base b. Suppose b=10, simple arithmetic operation like addition on two such strings can be done using the same method we use when adding numbers by pen and paper. The same goes for other simple operations. For better results, you can take a larger base.

A simple bignum implementation like that should be enough for most Project Euler problems (probably all, but I haven't solved much at Euler so can't be sure), but there are ways of using much faster algorithms for operations such as multiplication and division/mod.

Although I recommend writing your own bignum for practice, if you are really stuck you can take ideas from the code of already implemented bigint libraries. For a serious implementation something like gmp is the obvious choice. But you cana also find small bigints coded by other people when solving similar practice problem online (e.g. Abednego's bigint.cpp).

Answer 3

Here's a nice and simple bignum module for C. You can learn from it for ideas. The C code isn't the highest quality, but the algorithm is well implemented and quite common.

For more advanced stuff, look up GMP.

Answer 4

You need to store the big numbers in a base that your computer can easily handle with its native types, and then store the digits in a variable length array. I'd suggest that for simplicity you start by storing the numbers in base 10 just to get the hang of how to do this. It will make debugging a lot easier.

Once you have a class that can store the numbers in this form, it's just a matter of implementing the operations add, subtract, multiply, etc. on this class. Each operation will have to iterate over digits of its operands and combine them, being careful to carry correctly so that your digits are never larger than the base. Addition and subtraction are simple. Multiplication requires a bit more work as the naive algorithm requires nested loops. Then once you have that working, you can try implementing exponentiation in an efficient manner (e.g. repeated squaring).

If you are planning to write a serious bignum implementation, base 10 won't cut it. It's wasteful of memory and it will be slow. You should choose a base that is natural for the computer, such as 256 or the word size (2**32). However this will make simple operations more difficult as you will get overflows if you naively add two digits, so you will need to handle that very carefully.

Answer 5

I totally agree with Roger Pate. I have seen many times people running into integer limit issue with C/C++/Java, but with Python, it's a non-issue. For most Project Euler problems, coming up with the right algorithm is most important, and the performance you get from C won't matter much. Furthermore with the associative data types, dictionary, set, etc. that are available in Python and some of the built-in libraries, itertools, just to name one, it's much faster to solve a problem with Python. I started seriously learning Python since I jumped on the Project Euler bandwagon and I have been happy with my decision (my first language is C++, second language is Perl, but I wanted to learn Python).

Answer 6

C is not a good choice for Project Euler. The benefits of C are raw speed, machine portability (to an extent, with standard C), language interoperability (if some language communicates with another, C is a popular first choice), sticking close to a specific library or platform's API (because C is common, e.g. OS API), and a stable language & stdlib. None of these benefits apply to solving Project Euler problems. Not even raw speed, because most of the problems aren't about raw computation, but understanding the algorithm required, and you can sit there all day and wait before submission.

If you are attempting Project Euler problems to broaden your experience with C, that's perfectly fine, just realize this experience doesn't necessarily apply to long-lived and real-world C projects you may work on.

For this kind of short, one-off problem those languages commonly described as "scripting languages" will work better, faster (in dev time), and easier. Try Python, it stays close to C in many ways, including a C API, and out of the various popular "scripting languages" is possibly the one for which you will find the most use in conjunction with C projects.

This may become an unpopular answer, but it isn't a rant—plus I really like C and use C/C++ often—and there is an explicit answer here to your problem: "don't use C", with your final large number solution depending on which alternative you choose. Again picking on Python, integers do not have an upper bound (note below), and I use this to naturally code answers to Project Euler problems, where in other languages I have to use a painful-by-comparison alternative number library.

(Python integers: There are two integer types in 2.x, 'int' and 'long' (which have been completely unified in 3.x). The conversion between them is practically seamless, and 'long' allows arbitrarily large values, instead of just being a bigger 'int' type as C's long is.)