What data-structure should I use to create my own

问题:

As an optional assignment, I'm thinking about writing my own implementation of the BigInteger class, where I will provide my own methods for addition, subtraction, multiplication, etc.

This will be for arbitrarily long integer numbers, even hundreds of digits long.

While doing the math on these numbers, digit by digit isn't hard, what do you think the best datastructure would be to represent my "BigInteger"?

At first I was considering using an Array but then I was thinking I could still potentially overflow (run out of array slots) after a large add or multiplication. Would this be a good case to use a linked list, since I can tack on digits with O(1) time complexity?

Is there some other data-structure that would be even better suited than a linked list? Should the type that my data-structure holds be the smallest possible integer type I have available to me?

Also, should I be careful about how I store my "carry" variable? Should it, itself, be of my "BigInteger" type?

回答1:

Check out the book C Interfaces and Implementations by David R. Hanson. It has 2 chapters on the subject, covering the vector structure, word size and many other issues you are likely to encounter.

It's written for C, but most of it is applicable to C++ and/or Java. And if you use C++ it will be a bit simpler because you can use something like std::vector to manage the array allocation for you.

回答2:

Always use the smallest int type that will do the job you need (bytes). A linked list should work well, since you won't have to worry about overflowing.

回答3:

If you use binary trees (whose leaves are ints), you get all the advantages of the linked list (unbounded number of digits, etc) with simpler divide-and-conquer algorithms. You do not have in this case a single base but many depending the level at which you're working.

If you do this, you need to use a BigInteger for the carry. You may consider it an advantage of the "linked list of ints" approach that the carry can always be represented as an int (and this is true for any base, not just for base 10 as most answers seem to assume that you should use... In any base, the carry is always a single digit)

I might as well say it: it would be a terrible waste to use base 10 when you can use 2^30 or 2^31.

回答4:

Accessing elements of linked lists is slow. I think arrays are the way to go, with lots of bound checking and run time array resizing as needed.

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Clarification: Traversing a linked list and traversing an array are both O(n) operations. But traversing a linked list requires deferencing a pointer at each step. Just because two algorithms both have the same complexity it doesn't mean that they both take the same time to run. The overhead of allocating and deallocating n nodes in a linked list will also be much heavier than memory management of a single array of size n, even if the array has to be resized a few times.

回答5:

Wow, there are some… interesting answers here. I'd recommend reading a book rather than try to sort through all this contradictory advice.

That said, C/C++ is also ill-suited to this task. Big-integer is a kind of extended-precision math. Most CPUs provide instructions to handle extended-precision math at comparable or same speed (bits per instruction) as normal math. When you add 2^32+2^32, the answer is 0… but there is also a special carry output from the processor's ALU which a program can read and use.

C++ cannot access that flag, and there's no way in C either. You have to use assembler.

Just to satisfy curiosity, you can use the standard Boolean arithmetic to recover carry bits etc. But you will be much better off downloading an existing library.

回答6:

I would say an array of ints.

回答7:

An Array is indeed a natural fit. I think it is acceptable to throw OverflowException, when you run out of place in your memory. The teacher will see attention to detail.

A multiplication roughly doubles digit numbers, addition increases it by at most 1. It is easy to create a sufficiently big Array to store the result of your operation.

Carry is at most a one-digit long number in multiplication (9*9 = 1, carry 8). A single int will do.

回答8:

std::vector<bool> or std::vector<unsigned int> is probably what you want. You will have to push_back() or resize() on them as you need more space for multiplies, etc. Also, remember to push_back the correct sign bits if you're using two-compliment.

回答9:

i would say a std::vector of char (since it has to hold only 0-9) (if you plan to work in BCD)

If not BCD then use vector of int (you didnt make it clear)

Much less space overhead that link list

And all advice says 'use vector unless you have a good reason not too'

回答10:

As a rule of thumb, use std::vector instead of std::list, unless you need to insert elements in the middle of the sequence very often. Vectors tend to be faster, since they are stored contiguously and thus benefit from better spatial locality (a major performance factor on modern platforms).

Make sure you use elements that are natural for the platform. If you want to be platform independent, use long. Remember that unless you have some special compiler intrinsics available, you'll need a type at least twice as large to perform multiplication.

I don't understand why you'd want carry to be a big integer. Carry is a single bit for addition and element-sized for multiplication.

Make sure you read Knuth's Art of Computer Programming, algorithms pertaining to arbitrary precision arithmetic are described there to a great extent.