Why is squaring a number faster than multiplying t

问题:

Multiplying two binary numbers takes n^2 time, yet squaring a number can be done more efficiently somehow. (with n being the number of bits) How could that be?

Or is it not possible? This is insanity!

回答1:

1. There exist algorithms more efficient than O(N^2) to multiply two numbers (see Karatsuba, Pollard, Schönhage–Strassen, etc.)

2. The two problems "multiply two arbitrary N-bit numbers" and "Square an arbitrary N-bit number" have the same complexity.

We have

4*x*y = (x+y)^2 - (x-y)^2

So if squaring N-bit integers takes O(f(N)) time, then the product of two arbitrary N-bit integers can be obtained in O(f(N)) too. (that is 2x N-bit sums, 2x N-bit squares, 1x 2N-bit sum, and 1x 2N-bit shift)

And obviously we have

x^2 = x * x

So if multiplying two N-bit integers takes O(f(N)), then squaring a N-bit integer can be done in O(f(N)).

Any algorithm computing the product (resp the square) provides an algorithm to compute the square (resp the product) with the same asymptotic cost.

As noted in other answers, the algorithms used for fast multiplication can be simplified in the case of squaring. The gain will be on the constant in front of the f(N), and not on f(N) itself.

回答2:

Squaring an n digit number may be faster than multiplying two random n digit numbers. Googling I found this article. It is about arbitrary precision arithmetic but it may be relevant to what your asking. In it the authors say this:

In squaring a large integer, i.e. X^2 = (xn-1, xn-2, ... , x1, x0)^2 many cross-product terms of the form xi * xj and xj * xi are equivalent. They need to be computed only once and then left shifted in order to be doubled. An n-digit squaring operation is performed using only (n^2 + n)/2 single-precision multiplications.

回答3:

Like others have pointed out, squaring can only be about 1.5X or 2X faster than regular multiplication between arbitrary numbers. Where does the computational advantage come from? It's symmetry. Let's calculate the square of 1011 and try to spot a pattern that we can exploit. u0:u3 represent the bits in the number from the most significant to the least significant.

    1011 //                               u3 * u0 : u3 * u1 : u3 * u2 : u3 * u3
   1011  //                     u2 * u0 : u2 * u1 : u2 * u2 : u2 * u3       
  0000   //           u1 * u0 : u1 * u1 : u1 * u2 : u1 * u3                 
 1011    // u0 * u0 : u0 * u1 : u0 * u2 : u0 * u3                           

If you consider the elements ui * ui for i=0, 1, ..., 4 to form the diagonal and ignore them, you'll see that the elements ui * uj for i ≠ j are repeated twice.

Therefore, all you need to do is calculate the product sum for elements below the diagonal and double it, with a left shift. You'd finally add the diagonal elements. Now you can see where the 2X speed up comes from. In practice, the speed-up is about 1.5X because of the diagonal and extra operations.

回答4:

I believe you may be referring to exponentiation by squaring . This technique isn't used for multiplying, but for raising to a power x^n, where n may be large. Rather than multiply x times itself N times, one performs a series of squaring and adding operations which can be mapped to the binary representation of N. The number of multiplication operations (which are more expensive than additions for large numbers) is reduced from N to log(N) with respect to the naive exponentiation algorithm.

回答5:

Do you mean multiplying a number by a power of 2? This is usually quicker than multiplying any two random numbers since the result can be calculated by simple bit shifting. However, bear in mind that modern microprocessors dedicate lots of brute force silicon to these types of calculations and most arithmetic is performed with blinding speed compared to older microprocessors

回答6:

I have it!

2 * 2

is more expensive than

2 << 1

(The caveat being it only works for one case.)

回答7:

Suppose you want to expand out the multiplication (a+b)×(c+d). It splits up into four individual multiplications: a×c + a×d + b×c + b×d.

But if you want to expand out (a+b)², then it only needs three multiplications (and a doubling): a² + 2ab + b².

(Note also that two of the multiplications are themselves squares.)

Hopefully this just begins to give an insight into some of the speedups that are possible when performing a square over a regular multiplication.

回答8:

First of all great question! I wish there were more questions like this.

So it turns out that the method I came up with is O(n log n) for general multiplication in the arithmetic complexity only. You can represent any number X as

X = x_{n-1} 2^{n-1} + ... + x_1 2^1 + x_0 2^0
Y = y_{m-1} 2^{m-1} + ... + y_1 2^1 + y_0 2^0

where

x_i, y_i \in {0,1}

then

XY = sum _ {k=0} ^ m+n r_k 2^k

where

r_k = sum _ {i=0} ^ k x_i y_{k-i}

which is just a straight forward application of FFT to find the values of r_k for each k in (n +m) log( n + m) time.

Then for each r_k you must determine how big the overflow is and add it up accordingly. For squaring a number this means O(n log n) arithmetic operations.

You can add up the r_k values more efficiently using the Schönhage–Strassen algorithm to obtain a O(n log n log log n) bit operation bound.

The exact answer to your question is already posted by Eric Bainville.

However, you can get a much better bound than O(n^2) for squaring a number simply because there exist much better bounds for multiplying integers!

回答9:

If you assume fixed length to the word size of the machine and that the number to be squared is in memory, a squaring operation requires only one load from memory, so could be faster.

For arbitrary length integers, multiplication is typically O(N²) but there are algorithms which reduce this for large integers.

If you assume the simple O(N²) approach to multiply a by b, then for each bit in a you have to shift b and add it to an accumulator if that bit is one. For each bit in a you need 3N shifts and additions.

Note that

( x - y )² = x² - 2 xy + y²

Hence

x² = ( x - y )² + 2 xy - y²

If each y is the largest power of two not greater than x, this gives a reduction to a lower square, two shifts and two additions. As N is reduced on each iteration, you may get an efficiency gain ( the symmetry means it visits each point in a triangle rather than a rectangle ), but it's still O(N²).

There may be another better symmetry to exploit.

回答10:

a^2 (a+b)*(a+b)+b^2 eg. 66^2 = (66+6)(66-6)+6^2 = 72*60+36= 4356

for a^n just use the power rule

66^4 = 4356^2

回答11:

I would want to solve the problem by N bit multiplication for a number

A the bits be A(n-1)A(n-2)........A(1)A(0).

B the bits be B(n-1)B(n-2)........B(1)B(0).

for the square of number A the unique multiplication bits generated will be for A(0)->A(0)....A(n-1) A(1)->A(1)....A(n-1) and so on so the total operations will be

OP = n + n-1 + n-2 ....... + 1 Therefore OP = n^2+n/2; so the Asymptotic notation will be O(n^2)

and for multiplication of A and B n^2 unique multiplications will be generated so the Asymptotic notation will be O(n^2)

回答12:

The square root of 2ⁿ is 2^{n / 2} or 2^{n >> 1}, so if your number is a power of two everything is totally simple once you know the power. To multiply is even simplier: 2⁴ * 2⁸ is 2⁴⁺⁸. There's no sense in this statements you've done.

回答13:

If you have a binary number A, it can (always, proof left to the eager reader) be expressed as (2^n + B), this can be squared as 2^2n + 2^(n+1)B + B^2. We can then repeat the expansion, until such a point that B equals zero. I haven't looked too hard at it, but intuitively, it feels as if you should be able to make a squaring function take fewer algorithmical steps than a general-purpose multiplication.

回答14:

I think that you are completely wrong in your statements

Multiplying two binary numbers takes n^2 time

Multiplying two 32bit numbers take exactly one clock cycle. On a 64 bit processor, I would assume that multiplying two 64 bit numbers take exactly 1 clock cycle. It wouldn't even surprise my that a 32bit processor can multiply two 64bit numbers in 1 clock cycle.

yet squaring a number can be done more efficiently somehow.

Squaring a number is just multiplying the number with itself, so that is just a simple multiplication. There is no "square" operation in the CPU.

Maybe you are confusing "squaring" with "multiplying by a power of 2". Multiplying by 2 can be implemeted by shifting all the bits one position to the "left". Multiplying by 4 is shifting all the bits two positions to the "left". By 8, 3 positions. But this trick only applies to a power of two.