Probably faster multiplication ofsparse polynomials

Let

be polynomials that are represented in the usual way as linear combinations of power products. The problem of sparse polynomial multiplication is to compute the product

in a way that is as efficient as possible in terms of the total bitsize of

, and

(and where we use a similar sparse representation for

as for

and

For pedagogical reasons, we mainly restrict our attention to polynomials with integer coefficients. Together with polynomials with rational coefficients, this is indeed the most important case for practical implementations inside computer algebra systems. Nevertheless, it is not hard to adapt our techniques to coefficients in more general rings (some indications to that effect are given in section 5.2). Still for pedagogical reasons, we will carry out our complexity analysis in the RAM model [11]. We expect our algorithms to adapt to the Turing model [13], but more work will be needed to prove this and some of the constant factors might deteriorate.

For polynomials of modest size, naive algorithms are often most efficient. We refer to [3, 6, 10, 19, 28, 32, 33, 37] for implementation techniques that are efficient in practice. Various types of faster algorithms have been proposed for polynomials with special supports [16, 18, 23, 35].

Asymptotically fast methods for polynomials of large sizes usually rely on sparse interpolation. The seminal paper by Ben Or and Tiwari [4] triggered the development of many fast algorithms for the sparse interpolation of polynomial blackbox functions [1, 5, 9, 20, 24–27, 29, 34]. In this framework, the unknown polynomial

is given through a blackbox functions that can be evaluated at points in suitable extensions of the coefficient ring. We refer to [36] for a nice survey on sparse interpolation and other algorithms to compute with sparse polynomials. The present paper grew out of our recent preprint [21] with Grégoire Lecerf on this topic; the idea to “exploit colliding terms” in section 6.6 forms the starting point of our work.

The most efficient algorithms for sparse interpolation are mostly probabilistic. Here we note that it is usually easy to check that the result is correct with high probability: just evaluate both the blackbox function and its supposed interpolation at a random point and verify that both evaluations coincide. In this paper, all algorithms will be probabilistic, which is suitable for the practical purposes that we are interested in. The running times of our algorithms also rely on suitable heuristics that we will make precise.

Although the multiplication problem for sparse polynomials does not directly fit into the usual blackbox model, it does benefit from the techniques that have been developed for sparse interpolation. Practical algorithms along these lines have appeared in [8, 12, 19, 31]. Most algorithms operate in two phases: we first need to determine the exponents of the product

and then its coefficients. The first phase is typically more expensive when the coefficients of

are small, but it becomes cheap for large coefficients, due to the fact that we may first reduce

modulo a suitable prime. It is also customary to distinguish between the supersparse case in which the total degree of

is allowed to become huge and the normally sparse case in which the total degree remains small. In this paper, we mainly focus on the second problem, which is most important for practical applications.

In order to describe the complexity results, let us introduce some notations. Given a polynomial

, we will write

for its total degree,

for its number of terms,

for the number of powers

that occur in its representation, and

for the maximal absolute of a coefficient. For instance, if

, then we have

, and

. For our multiplication problem

, the degree

of the result is easily determined, but we usually only assume a bound

with

for its number of terms.

It is interesting to keep track of the dependency of our running times on logarithmic factors in certain parameters, but it is also convenient to ignore less important logarithmic and sublogarithmic factors. We do this by introducing the notation

We also wish to compare the cost of our algorithms with the cost of multiplying dense univariate polynomials of approximately the same size. Given integers

, we therefore also introduce the following two complexities:

We make the customary assumption that

and

are non-decreasing as functions in

. By [14], one may take

. If

, then one also has

, using Kronecker substitution [11].

One traditional approach for sparse polynomial multiplication is to evaluate

, and

points in a geometric progression

modulo a sufficiently large prime number

, where

stands for the

-th prime number. In combination with the tangent Graeffe method [21, sections 5 and 7.2], this approach allows the exponents of

to be computed in time

The coefficients can be recovered using fast Vandermonde system solving, in time

where

. In our case when

is small, we usually have

, in which case (1) simplifies into

. The dependence of the complexity on

can be reduced using techniques from [26], among others.

The main results of this paper are two faster probabilistic algorithms. The shorter running times rely on two heuristics HE and HC that will be detailed in section 4. For any

, we show (Theorem 6) that the exponents of

can be computed in time

where

and

is prime. This algorithm is probabilistic of Monte Carlo type. Based on numerical evidence in section 3, we conjecture that

. We also show (Theorem 4) that the coefficients may be computed in expected time

using a probabilistic algorithm of Las Vegas type. In practice, when

is small and

not too large with respect to

, the corrective terms in (3) are negligible and the cost reduces to

. Similarly, the cost (4) usually simplifies to

. If we also have

, then this means that the cost of the entire sparse multiplication becomes

. Here we note that

also corresponds to the time needed to multiply two dense polynomials in

, provided that the product

satisfies

and

The proof of these bounds relies on the evaluation of

at three points of the form

in algebras of the form

, where

. If

is sufficiently large (namely

for some critical value) and we already know the exponents of

, then we show how to recover the coefficients with high probability. One interesting feature of our algorithm is that three evaluations are sufficient with high probability. A logarithmic number of evaluations is necessary when using the more obvious iterative approach for which every additional evaluation allows us to compute a constant fraction of the unknown coefficients (with high probability). Our algorithm is first explained with an example in section 2 and then in general in section 4.1. The probabilistic analysis is done in section 3. In section 4.2, we extend our approach to the computation of the exponents, using three additional evaluations. The last section 5 is devoted to variants and extensions of our approach, and further remarks.

The present paper works out an idea that was first mentioned in [21, section 6.6], in the context of general sparse interpolation. The application to polynomial multiplication is particularly suitable because of the low amortized cost of blackbox evaluations. The idea of using evaluations in (small) cyclic algebras has been used before in [2, 9, 31], but with a less optimal complexity. This approach also seems close to binning techniques that have recently been applied to compute sparse Fourier transforms [15, 30]; we plan to investigate this parallel in future work.

2.A game of mystery balls

Assume that the monomials

are known, but not the corresponding coefficients. Our aim is to determine

through its evaluations at “points” of the form

for suitable

and lengths

. These evaluations are obtained by evaluating

and

at the same points and multiplying the results in

. In what follows, we will use three evaluation points.

For each ball, say

, the corresponding “mystery coefficient”

needs to be determined (it might be hidden inside the ball), whereas the corresponding exponents

are known (and stored in a table or painted on the ball). In fact, our game has three identical sets of balls, one for each of the three evaluation points. For each of these evaluation points, we also have a set of

boxes, labeled by

Now consider the evaluation of

at a point as above, say at

in the ring

. Then each term

evaluates to a term

with

and

modulo

. In our game, we throw the corresponding ball into the box that is labeled by

. For instance, our first ball

evaluates to

and goes into the box labeled by

. Our second ball

evaluates to

and goes into the box labeled by

. Continuing this way, we obtain the upper left distribution in Figure 1. Now the complete evaluation of

gives

For each box, this means that we also know the sum of all coefficients hidden in the balls in that box. Indeed, in our example, the first box

contains three balls

, , and

, with coefficients

, and

that sum up to

. In Figure 1, we indicated these sums below the boxes. In round one of our game, we actually took our chances three times, by using the three evaluation points

, and

, and throwing our balls accordingly. This corresponds to the top row in Figure 1.

Now we play our game as follows. If, in a certain round, a ball ends up alone in its box (we will also say that the ball has a private box), then the number below it coincides with the secret coefficient inside. At that point, we may remove the ball, as well as its copies from the two other throws, and update the numbers below accordingly. In round one of our running example, ball

ends up in a box of its own for our first throw. Similarly, the balls

and

both have private boxes for the second throw. Ball

also has a private box for the third throw. Removing the balls

, and

from the game, we obtain the second row in Figure 1. We also updated the numbers below the boxes: for every box, the number below it still coincides with the sum of the mystery coefficients inside the balls inside that box. Now that the balls

, and

have been removed, we observe that balls

and

have private boxes in their turn. We may thus determine their mystery coefficients and remove them from the game as well. This brings us to round three of our game and the third row in Figure 1. Going on like this, we win our game when all balls eventually get removed. We lose whenever there exists a round in which there are still some balls left, but all non-empty boxes contain at least two balls. In our example, we win after five rounds.

Remark 1. When implementing a computer program to play the game, we maintain a table that associates to each ball its exponents and the three boxes where it ended up for the three throws. Conversely, for each box, we maintain a linked list with all balls inside that box. We finally maintain a list with balls inside a private box, and which are about to be removed. In this way, the total amount of work that needs to be done in each round remains proportional to the number of balls that are removed instead of the total number of boxes.

Remark 2. Playing our game in several “rounds” is convenient for the probabilistic analysis in section 3 below. But the order in which balls in private boxes are removed actually does not matter, as long as we remove all balls in private boxes. Assume for instance that we win our game and that we replay it by removing balls in another order. Assume for contradiction that a ball

does not get removed in our modified game and choose

in such a way that the round

in which it gets removed in the original game is minimal. Then we observe that all balls that were removed before round

in the original game also get removed in the modified version, eventually. When this happens,

is in a private box for one of the throws: a contradiction.

3.On our probability of winning the game

Have we been lucky in our example with

throws of

balls in

boxes? For the probabilistic analysis in this section, we will assume that our throws are random and independent. We will do our analysis for three throws, because this is best, although a similar analysis could be carried out for other numbers of throws. From now on, we will assume that we have

balls and

boxes.

The experiment of throwing

balls in

boxes has widely been studied in the literature about hash tables [7, Chapter 9]. For a fixed ball, the probability that all other

balls end up in another box is given by

More generally, for any fixed

, the probability that

other balls end up in the same box and all the others in other boxes is given by

Stated otherwise, we may expect with high probability that approximately

balls end up in a private box, approximately

balls inside a box with one other ball, and so on.

This shows how we can expect our balls to be distributed in the first round of our game and at the limit when

gets large. Assume more generally that we know the distribution

in round

and let us show how to determine the distribution

for the next round. More precisely, assume that

is the expected number of balls in a box with

balls in round

, where we start with

we notice that

, where

stands for the expected number of balls that are removed during round

Now let us focus on the first throw in round

(the two other throws behave similarly, since they follow the same probability distribution). There are

balls that are in a private box for this throw. For each of the remaining balls, the probability

that it ended up in a private box for at least one of the two other throws is

The probability that a box with

balls becomes one with

balls in the next round is therefore given by

tends to zero for large

and

gets large, then we win our game with high probability. If

tends to a limit

, then we will probably lose and end up with approximately

balls that can't be removed (for each of the three throws).

We have not yet been able to fully describe the asymptotic behavior of the distribution

for

, which follows a non-linear dynamics. Nevertheless, it is easy to compute reliable approximations for the coefficients

using interval or ball arithmetic [ 17 ]; for this purpose, it suffices to replace each coefficient

in the tail of the distribution ( i.e. for large

) by the interval

. Tables 1 , 2 , and 3 show some numerical data that we computed in this way (the error in the numbers being at most

). Our numerical experiments indicate that the “phase change” between winning and losing occurs at a critical value

with

Table 1 shows what happens for

: until the seventh round, a bit less than half of the balls get removed at every round. After round eight, the remaining balls are removed at an accelerated rate. For

, the distributions

numerically tend to a non-zero limit distribution

with

. In Table 3 , we show some of the distributions in round ten, for

near the critical point

. We also computed an approximation of the limit distribution at the critical point itself.

For

, we note that reliable numerical computations can be turned into an actual proof that we lose with high probability. Indeed, assume that

gets very small for some

, whereas

remains bounded (for instance, in Table 2, we have

and

for

). Then

also gets very small and

happens to be

times smaller than

for some fixed

(for instance, in Table 2, we actually have

for

), then a standard contracting ball type argument can be used to prove that

decreases to zero with geometric speed for

, while

remains bounded away from zero.

Conversely, given

, it seems harder to prove in a similar way that we win. Nevertheless, for any

, reliable computations easily allow us to determine some round

with

. For instance, for

and

, we may take

. This is good enough for practical purposes: for

, it means that we indeed win with high probability. Here we also note that a large number of rounds are generally necessary to win when

approaches the critical value

. For instance, for

, we need to wait until round

to get

. Nevertheless, after a certain number of rounds, it seems that

always converges to zero with superlinear speed.

In Table 4 , we conclude with the result of a computer simulation in which we played our game with

and

. As one can see, the results are close to the theoretically expected ones from Table 1 .

Remark 3. Our game of mystery balls readily generalizes to different numbers of throws and different numbers of boxes for each throw. However, if our aim is to take the total number of boxes as small as possible, then computer experiments suggest that using three throws with the same number of boxes is optimal. For simplicity, we therefore restricted ourselves to this case, although it should be easy to extend our analysis to a more general setting.

4.Multiplying sparse polynomials

Let us now turn to the general problem of multiplying sparse polynomials. We will focus on the multiplication

of integer polynomials

in at least two variables. We define

to be the total degree of

and assume that we have a bound

for the number of terms of

As explained in the introduction, we proceed in two phases: we first determine the exponents of the unknown product

. This part is probabilistic of Monte Carlo type, where we tolerate a rate of failure

that is fixed in advance. In the second phase, we determine the unknown coefficients, using a probabilistic algorithm of Las Vegas type. In this section, we start with the second phase, which has already been explained on an example in section 2.

4.1.Determination of the coefficients

Assume that our product

has

terms, so that

for certain

and

, where

for

. It is obvious how to generalize the algorithm from section 2 to this case: for some fixed

, we distribute “our

balls” over

boxes, through the evaluation of

at three points

for

and

. The vectors

are essentially chosen at random, but it is convenient to take them pairwise non-collinear modulo

, so as to avoid any “useless throws”. We assume the following heuristic:

We then play our game of mystery balls as usual, which yields the coefficients

if we win and only a subset of these coefficients if we lose. In view of Remark 2 , this leads to Algorithm 1 .

Proof. The correctness of the algorithm has been explained in section 2. As to the running time, we first note that none of the integer coefficients encountered during our computation exceeds

in absolute value. Now steps 1 and 2 have negligible cost and the running time for the remaining steps is as follows:

Let us finally investigate bookkeeping costs that are implicit in our description of the algorithm. Above all, we have to maintain the linked lists with balls inside each box (see Remark 1). This can be done in time

4.2.Determination of the exponents

In [21], we surveyed several strategies for computing the exponents of the product

. Most of the approaches from sections 4, 6, and 7 of that paper can be adapted to the present setting. We will focus on a probabilistic strategy that we expect to be one of the most efficient ones for practical purposes (a few variants will be discussed in section 5).

For

, let

be the

-th prime number and let

. We let

be a fixed prime number with

(for practical implementations, we may also take

to be a product of prime numbers that fit into machine words, and use multi-modular arithmetic to compute modulo

). For some fixed

, we again use

boxes, and evaluate

over the ring

. This time, we use six evaluation points of the form

and

for

, where the

are chosen at random and pairwise non-collinear modulo

Now consider a term

. Its evaluation at

, where

modulo

. Meanwhile, its evaluation at

with

. If there is no other term that evaluates to an expression of the form

, then the same holds for the evaluation at

. Consequently, the unique representative

of the quotient

coincides with

(the quotient is well defined with high probability). From this, we can determine the exponents

by factoring

. As additional safeguards, we also check that

, that all prime factors of

are in

, and that

modulo

Conversely, assume that there are at least two terms of

that evaluate to an expression of the form

. Let

and

now be the coefficients of

and

. Then the quotient

is essentially a random element in

(see the heuristic HC below for more details), so its unique representative in

is higher than

with probability

. This allows our algorithm to detect that we are dealing with colliding terms; for the

quotients that we need to consider the probability of failure becomes

Using the above technique, the three additional evaluations at

, allow us to determine which balls end up in a private box in our game of mystery balls. Moreover, since we can determine the corresponding exponents, we can also find where these balls landed for the two other throws. This allows us to play our game modulo minor modifications. Besides maintaining the numbers below the boxes for the first three throws, we also maintain the corresponding numbers for the evaluations at

. For every round of the game, the same technique then allows us to determine which balls have private boxes, and iterate.

In our probabilistic analysis, it is important that the quotients

are essentially random elements in

in case of collisions. This assumption might fail when the coefficients of

and

are special (e.g. either zero or one). Nevertheless, it becomes plausible after a change of variables

, where

are random invertible elements in

. Let us formulate this as our second heuristic:

Algorithm 2 summarizes our method, while incorporating the random change of variables.

Proof. We first determine the indices

with

using a divide and conquer technique. At the top level, we start with the remainder

of the division of

. We next compute

and

. If

, then we go on with the computation of

and

, and similarly for

. We repeat this dichotomic process until we have found all prime factors of

. For each of the

prime factors

, we next compute

and

. The total running time of this algorithm is bounded by

Proof. We have already explained why our algorithm returns the correct answer with probability

. The complexity analysis for steps 5, 6 and 8 b is similar as the one for Algorithm 1. The running time for the other steps is as follows:

5.Variants and further experiments

5.1.Supersparse polynomials

In section 4.2, we have described an efficient algorithm for the case when the total degree

is small. This indeed holds for most practical applications, but it attractive to also study the case when our polynomials are “truly sparse” with potentially large exponents. This is already interesting for univariate polynomials, a case that we also did not consider so far. Modulo the technique of Kronecker substitution [21, Section 7.1], it actually suffices to consider the univariate case. For simplicity, let us restrict ourselves to this case. When

gets moderately large, it can also be interesting to consider the incremental approach from [20] and [21, Section 7.3].

One first observation in the univariate case is that we can no longer take the same number of boxes

for our three throws. Next best is to take

boxes for our

-th throw, where

. By construction this ensures that

, and

are pairwise coprime. If the exponents of

are large, then it is also reasonable to assume that our heuristic HE still holds.

For the efficient determination of the exponents, let

. As before, we may take

to be prime or a product of prime numbers that fit into machine words. Following Huang [26], we now evaluate both

and

for

. Any term

then gives rise to a term

with

, so we can directly read off

from the quotient

. Modulo the above changes, this allows us to proceed as in Algorithm 2. One may prove in a similar way as before that the bit complexity of determining the exponents in this way is bounded by

5.2.Other rings of coefficients

Rational coefficients.

Complex numbers.

Finite fields.

Instead of taking our three additional evaluation points of the form

, we may for instance take them of the form

, where

is a primitive root of unity of large smooth order (this may force us to work over an extension field; alternatively, one may use the aforementioned incremental technique and roots

of lower orders). Thanks to the smoothness assumption, discrete logarithms of powers of

can be computed efficiently using Pohlig–Hellman's algorithm. This allows us to recover exponents

from quotients of the form

General rings.

5.3.Other operations

The techniques from this paper can also be used for general purpose sparse interpolation. In that case the polynomial

is given through an arbitrary blackbox function that can be evaluated at points in

-algebras over some ring

. This problem was studied in detail in [21]. In section 6, we investigated in particular how to replace expensive evaluations in cyclic

-algebras of the form

by cheaper evaluations at suitable

-th roots of unity in

or a small extension of

. The main new techniques from this paper were also anticipated in section 6.6.

The algorithms from this paper become competitive with the geometric sequence approach if the blackbox function is particularly cheap to evaluate. Typically, the cost

of one evaluation should be

, depending on the specific variant of the geometric sequence approach that we use.

Technically speaking, it is interesting to note that the problem from this paper actually does not fit into this framework. Indeed, if

and

are sparse polynomial that are represented in their standard expanded form, then the evaluation of

at a single point requires

operations in

, and we typically have

. This is due to the fact that the standard blackbox model does not take into account possible speed-ups if we evaluate our function at many points in geometric progression or at

in a cyclic algebra

Both in theory and for practical applications, it would be better to extend the blackbox model by allowing for polynomials

of the form

where

is a blackbox function and

are sparse polynomials in their usual expanded representation. Within this model, the techniques from this paper should become efficient for many other useful operations on sparse polynomials, such as the computation of gcds or determinants of matrices whose entries are large sparse polynomials.

5.4.Special types of support

The heuristic HE plays an important role in our complexity analysis. It is an interesting question whether it is satisfied for polynomials

whose support is highly regular and not random at all. A particularly important case is when

and

are dense polynomials in

variables of total degrees

and

. Such polynomials are often considered to be sparse, due to the fact that

contains about

times less terms than a fully dense polynomial of degree

in each variable.

is bivariate or trivariate and of very high degree, then it has been shown in [16] that

can be computed in approximately the same time as the product of two dense univariate polynomials which has the same number of terms as

. An algorithm for arbitrary dimensions

has been presented in [23], whose cost is approximately

times larger than the cost of a univariate product of the same size. The problem has also been studied in [18, 19] and it is often used as a benchmark [12, 32].

What about the techniques from this paper? We first note that the exponents of

are known, so we can directly focus on the computation of the coefficients. In case that we need to do several product computations for the same

and

, we can also spend some time on computing a small

and vectors

for which we know beforehand that we will win our game of mystery balls. For our simulations, we found it useful to chose each

among a dozen random vectors in a way that minimizes the sum of the squares of the number of balls in the boxes (this sum equals

for the first throw in Figure 1).

For various

, and

, we played our game several times for different triples of vectors

. The critical value for

for this specific type of support seems to be close to

. In Table 5 below, we report several cases for which we managed to win for this value of

. This proves that optimal polynomial multiplication algorithms for supports of this type are almost as efficient as dense univariate polynomial products of the same size.

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1.Introduction

2.A game of mystery balls

3.On our probability of winning the game

4.Multiplying sparse polynomials

4.1.Determination of the coefficients

4.2.Determination of the exponents

5.Variants and further experiments

5.1.Supersparse polynomials

5.2.Other rings of coefficients

Rational coefficients.

Complex numbers.

Finite fields.

General rings.

5.3.Other operations

5.4.Special types of support

Bibliography