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Sparse Interpolation of Symmetric Polynomials
ADVANCES IN APPLIED MATHEMATICS ARTICLE NO.
18, 271]285 Ž1997.
AM960508
Sparse Interpolation of Symmetric Polynomials Alexander Barvinok* Department of Mathematics, Uni¨ ersity of Michigan, Ann Arbor, Michigan 48109-1003
and Sergey Fomin† Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Theory of Algorithms Laboratory, St. Petersburg Institute of Informatics, Russian Academy of Sciences, St. Petersburg, Russia Received April 20, 1996
We develop efficient algorithms for computing the expansion of a given symmetric polynomial into Schur functions. This problem frequently arises in applications as the problem of decomposing a given representation of the symmetric Žor general linear. group into irreducible constituents. Our algorithms are probabilistic, and run in time which is polynomial in the sizes of the input and output. They can be used to compute Littlewood]Richardson coefficients, Kostka numbers, and irreducible characters of the symmetric group. Q 1997 Academic Press
I. INTRODUCTION The general sparse interpolation problem can be posed as follows. Let V be a vector space over the rationals or a Z-module of functions with a fixed basis fl4 . Suppose there is a ‘‘black box’’ that computes the value of some function F from V at any given argument x. We want to find the decomposition Fs
Ý
lg L
al fl : al g Q
Ž or al g Z .
Ž 1.1.
* Research partially supported by the NSF ŽDMS-96-01129. and an Alfred P. Sloan research fellowship. † Research partially supported by the NSF ŽDMS-94-00914.. 271 0196-8858r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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as fast as possible. The computational complexity of any such algorithm is of course bounded from below by the actual length of the decomposition Ž1.1., that is, by the number L s < L < of basis functions fl that occur in Ž1.1. with nonzero coefficients Žsince computing the decomposition requires, at the very least, writing the answer.. So we want to perform the decomposition efficiently, preferably in time polynomial in L. The term ‘‘sparse’’ refers to the fact that the complexity of the algorithm must be bounded from above by a function of L. The most important particular case is V s Qw x 1 , . . . , x n x, the ring of polynomials, with fl4 being the basis of monomials. Then there exists an algorithm w2x that finds the decomposition Ž1.1. in time that is polynomial in the number of nonzero coefficients al, even if this number is unknown a priori. Another interesting case is the interpolation with respect to the eigenfunctions of some operator Žsee w5x.. In this paper, we study the sparse interpolation problem for symmetric polynomials. Let us consider the ring Zw x 1 , . . . , x n x of polynomials with integer coefficients in n variables x s Ž x 1 , . . . , x n .. We are interested in the subring L n ; Zw x 1 , . . . , x n x of symmetric polynomials, i.e., polynomials pŽ x . which are invariant under any permutation of the variables x 1 , . . . , x n . As a Z-module, L n possesses quite a few distinguished bases, most prominently the Schur functions sl, which are indexed by vectors l s Ž l1 , . . . , l n . g Z n, where l1 G l2 G ??? G l n G 0. Following the tradition, we call such a vector l a partition Žnote that, contrary to the usual convention, some of the l i are allowed to vanish.. The Schur function slŽx. can be defined in many different ways, in particular, as the following ratio of two n = n determinants: sl Ž x . s det Ž x il jqj y1 . rdet Ž x ijy1 . Žsee w7, Chap. 1, Sect. 3x.. The sparse interpolation problem in L n with respect to the basis sl4 can now be stated as follows. 1.1. Sparse Interpolation of Symmetric Polynomials Input: Ži. A ‘‘black box’’ that computes the value F Žx. of some polynomial F g L n for any x. Žii. The upper bound l on the degree of F. Output: The decomposition of F into a linear combination of Schur functions: Fs
Ý
lg L
al sl .
Ž 1.1.1.
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273
Our study of this problem is motivated by the fact that many important computational problems in the theory of symmetric functions and representation theory of the symmetric and general linear groups can be formulated in the form Ž1.1.. Here are three typical examples. 1.2. Characters of the Symmetric Group Let pm g L n denote the power sum corresponding to a partition m Žsee Section 2.1 or w7, Chap. 1, Sect. 2x.. The coefficients xlŽ m . of the decomposition pm s
Ý
lg L
xl Ž m . sl
Ž 1.2.1.
are known to be integers, namely, the values of irreducible characters xl of the symmetric group Sn on the conjugacy class defined by m. This problem naturally fits our sparse interpolation setup because any value of pmŽx. can be computed very fast Žsee Section 2.4.. Furthermore, many of the coefficients xlŽ m . can be zero. For instance, the classical Murnaghan]Nakayama rule implies that xlŽ m . s 0 unless the number of parts of m is at least the size of the Durfee square of l Žsee, e.g., w6x.. 1.3. Kostka Numbers For m a partition, let hm g L n be the corresponding complete homogeneous symmetric function Žsee Section 2.1 or w7, Chap. 1, Sect. 2x.. The coefficients Kml of the decomposition hm s
Ý
lg L
Klm sl
Ž 1.3.1.
are nonnegative integers called Kostka numbers Žcf. w7x.. These numbers are nothing but the coefficients of the Schur polynomials, sl s
Ý Klm mm ,
Ž 1.3.2.
m
where mmŽ x . denotes the sum of all monomials of type m. Combinatorially, Klm is the number of semistandard Young tableaux of shape l and weight m. The computation of the expansion Ž1.3.1. is equivalent to decomposing the representation of the symmetric group Sn induced from the trivial representation of its Young subgroup Sm , into the sum of irreducibles. This problem can be posed as a sparse interpolation problem because for every x the value of hmŽx. can be computed very fast Žsee Section 2.4.. Further-
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more, in this particular case there is a simple characterization of the set L of partitions l with Klm / 0 Žsee w7, Chap. 1, Sect. 6x and also Section 2.2.. For partitions with few parts, this set is relatively small. 1.4. Littlewood]Richardson Coefficients Let m and n be partitions such that
mi G n i
for all i.
Ž 1.4.1.
Then one can define a skew Schur function sm r n Žsee w7, Chap. 1, Sect. 5x.. m The coefficients gnl of the decomposition sm r n s
Ý
lg L
m gnl sl
Ž 1.4.2.
are nonnegative integers called the Littlewood]Richardson coefficients. The importance of computing decompositions of the form Ž1.4.2. comes from the fact that this is essentially equivalent to expanding the product of two Žordinary. Schur functions in the basis of Schur functions. In other words, the Littlewood]Richardson coefficients are the structure constants for this basis. In representation theory of the symmetric group, the computation of expansions Ž1.4.2. amounts to decomposing a certain product of two irreducible Sn-modules Žor, alternatively, a skew Specht module. into irreducible components. The celebrated Littlewood]Richardson rule provides an explicit Žalbeit m4 complicated. combinatorial description of the numbers gnl as counting certain tableaux Žthe ‘‘Littlewood]Richardson fillings’’.. Most known algorithms computing these numbers make use of this rule; hence their running time has to be at least linear in the number of the Littlewood]Richardson fillings ŽLRF., which is equal to LRFnm s
Ý gnlm . l
ŽThe complexity O ŽLRFnm . was indeed achieved by the algorithm of Remmel and Whitney w8x.. The number LRFnm can be very large; in fact, it m can be much larger than the length L s a l: gnl / 04 of the output expansion Ž1.4.2.. On the other hand, one can observe that for any x the numerical value of sm r n Žx. can be computed very fast Žsee Section 2.4.. Thus the problem of computing the decomposition Ž1.4.2. can be naturally stated as a sparse interpolation problem of a symmetric polynomial. Generally, it may be interesting to find expansions Ž1.1. for some algebraic expressions involving the pm , hm , and sm r n , such as hm 9 y hm 0 or
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275
sm9r n 9 y sm 0 r n 0 , where partitions m9 and m0 Žresp., n 9 and n 0 . are ‘‘close’’ to each other, so that one can expect a short expansion in the Schur functions basis, although it is hard to tell in advance which l’s do come up. In this paper, we construct a probabilistic algorithm, which solves problem Ž1.1. in time that is polynomial in the number of variables n, the upper bound l on the degree of F, the length L s < L < of the resulting expansion, and the value log 2 M, where Ms
Ý
lg L
< al <
Žthis takes care of the binary size of the output.. Note that neither M nor L are known in advance. The paper is organized as follows. In Section 2 we discuss the necessary facts and notions concerning symmetric functions, on one side, and computational complexity, on the other. The purpose of this section is to bridge the gap between these two topics. In Section 3 we discuss the general idea of our algorithms and prove the main technical lemmas. The algorithms are presented in Section 4.
2. PRELIMINARIES 2.1. Power Sums and Complete Homogeneous Symmetric Functions Following w7x, we introduce the standard notation and terminology related to symmetric functions. For a positive integer k, define the power sum pk by n
pk Ž x . s
Ý x ik .
Ž 2.1.1.
is1
It will also be convenient to use the convention p 0 Žx. s 1. For a partition l s Ž l1 , . . . , l n ., we let pl s pl1 ??? pl n .
Ž 2.1.2.
For m s 1, 2, . . . , define the complete homogeneous symmetric polynomial h m as the sum of all monomials of degree m. By convention, h 0 s 1 and h i s 0 for i - 0. The values of polynomials h m can be computed inductively via the formulas Žsee Ž2.11. in w7x.: h1 Ž x . s x 1 q ??? qx n , h m Ž x. s
1 m
m
Ý pr Ž x . h myr Ž x . rs1
for m ) 1.
Ž 2.1.3.
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For a partition l s Ž l1 , . . . , l n ., let hl s hl1 ??? hl n .
Ž 2.1.4.
2.2. Schur Functions and their Newton Polytopes The Schur function slŽx. can be expressed as the Jacobi]Trudi determinant Žsee Ž3.4. of w7x. sl s det Ž hl iyiqj . 1Fi , jFn .
Ž 2.2.1.
For a pair of partitions m and n satisfying Ž1.4.1., the skew Schur function sm r n can be defined as Žsee Ž5.4. of w7x. sm r n s det Ž hm iy n jyiqj . 1F i , jFn .
Ž 2.2.2.
The Schur function sl is a homogeneous symmetric polynomial of degree < l < s l1 q ??? ql n . To write down its expansion into monomials x a s x 1a 1 ??? x na n , let us first generalize the notation for the Kostka numbers, by denoting Kla s Kl b , where b s Ž b 1 , . . . , bn . is the nondecreasing rearrangement of a s Ž a 1 , . . . , a n .. Then sl Ž x . s
Ý
a 1q ??? q a n s < l < a iG0
Kl a x a
wcf. Ž1.3.2.x. We have already mentioned that Kla G 0, that is, the Schur functions are polynomials with nonnegative integer coefficients. The symmetry of sl is reflected in the fact that Kl a does not change if we permute the coordinates of a . Besides, Kll s 1 Žsee w7, Chap. 1, Sect. 6x.. Let us now assume that a is a partition, that is, a 1 G ??? G a n . It is known that Kla ) 0 if and only if k
k
Ý a i F Ý li is1
for k s 1, . . . , n y 1;
is1 n
n
Ý a i s Ý li is1
is1
Žsee w7, Chap. 1, Sect. 6x.. This condition can be expressed geometrically as follows. Let us define the polytope Pl ; R n Ža permutohedron. as the convex hull of all permutations of the vector l s Ž l1 , . . . , l n .. Then Kla ) 0 if and only if a g Pl Žsee, e.g., Rado’s theorem in w4, Sect. 3.1x..
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277
In other words, the Newton polytope of slŽx. is the permutohedron Pl . We will need a very rough estimate of Kla : Kla F '< l < ! ,
Ž 2.2.3.
which follows from the interpretation of Kla as the multiplicity of some representation Žsee w7, Chap. 1, Sect. 6x.. 2.3. Computational Complexity and Algorithms We adopt the usual computational model, namely, random access memory ŽRAM. with the uniform cost criterion Žsee w1x.. Thus our machine operates with integral Žrational. numbers represented by bit strings. The input size of a number M is approximately log 2 M. We assume that our machine can perform arithmetic operations Žaddition, subtraction, multiplication, division, and comparison. at the unit time. We also make sure that the size of all numbers appearing in the course of the algorithm is bounded by a polynomial in the inputroutput size. We will use probabilistic algorithms, meaning that our machine has a built-in device that can ‘‘toss a coin.’’ More precisely, we assume that, at any time, our machine is able to choose, uniformly at random, an integer from the interval w1, N x. By introducing randomness, we allow a certain probability that our algorithm fails to work at a certain step or produces an incorrect answer or does not stop at all. However, we will make sure that, with probability at least 0.99, our algorithm stops after certain time t and produces the correct answer. To ensure an overwhelming probability, we will then run several copies of our algorithm in parallel and pick the most frequent answer. For example, if we run m independent copies of the algorithm, then the probability of failure Žthat is, more than mr2 copies do not stop in time t with the correct answer. does not exceed
Ý kFmr2
m k
ž /Ž
0.01.
my k
k Ž 0.99. F 5ym ,
i.e., is exponentially small. 2.3.1. EXAMPLE. As an illustration, we refer to the randomized polynomial time algorithm for testing polynomial identities w9x. Suppose we are given a ‘‘black box’’ that computes the value of a certain polynomial F Žx. at any point x s Ž x 1 , . . . , , x n . g Z n. Suppose we know that the degree of F does not exceed d. Our goal is to find out whether F identically vanishes
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or not. The algorithm proposed in w9x works as follows. Let N s 3dn and let us choose the coordinates x i independently and uniformly from the set 1, . . . , N 4 . If F Žx. / 0, then we conclude that F is not identically zero. If F Žx. s 0, then the algorithm decides that F ' 0. It is proven in w9x that if F is not identically zero, the probability of choosing an x such that F Žx. s 0 is at most 1r3, so the algorithm makes an error with probability F 1r3. To make the probability of error as small as 3ym , we can generate m vectors x in the above-described fashion. If for some of them F Žx. / 0, then F is not identically 0. Otherwise our algorithm will conclude that F ' 0. 2.4. Complexity of Symmetric Functions Let us discuss the computational complexity of the symmetric polynomials pl, hl , sl , and sm r n . For each of these, we want to construct a ‘‘black box’’ that efficiently computes the value of a function at a given point x. Our presentation of such constructions will be self-contained, except for making use of the fact that the determinant of an n = n matrix can be computed in O Ž n3 . time Žsee, for example, w3x.. Using the definition Ž2.1.1. in a straightforward way, we observe that pk Žx., for any given x s Ž x 1 , . . . , x n ., can be computed in O Ž kn. time. Then Ž2.1.2. shows that the value of plŽx. can be computed in O Ž< l < n. time Žrecall that < l < s l1 q ??? ql n .. Using recursive formulas Ž2.1.3., one can compute the value of h k Žx. in O Ž k 2 n. time; hence the computation of hlŽx. takes O Ž< l < 2 n. time wsee Ž2.1.4.x. The determinantal expressions Ž2.2.1. ] Ž2.2.2. allow us to compute the values of sm r n Žx. and slŽx. in O Ž< m < 2 n q n3 . and O Ž< l < 2 n q n3 . time, respectively. 2.5. Notation We will need some extra notation to be used throughout the paper. A partition l s Ž l1 , . . . , l n . will often be interpreted as a vector in R n, the latter being endowed with the scalar product ² l , m : s l1 m 1 q l 2 m 2 q ??? ql n m n . We have already used the notation x a to denote x 1a 1 ??? x na n , where a s Ž a 1 , . . . , a n . g Z n. The base of exponentiation and logarithms will be 2 by default, so that exp a4 s 2 a and log a s log 2 a. Finally, w x x will denote the integer nearest to x.
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INTERPOLATION OF POLYNOMIALS
3. THE IDEA OF THE ALGORITHMS. BASIC LEMMAS Our main observation is that the coefficients of the decomposition F Ž x. s
Ý
lg L
al sl Ž x .
can be extracted from some asymptotics of F Žx.. Namely, suppose we are able to choose a vector c s Ž c1 , . . . , c n ., where c1 ) c 2 ) ??? ) c n such that the linear function ² c, l: attains its maximum on L on the unique partition n g L . Let us define yt s Žexp tc14 , . . . , exp tc n4.. Then lim ty1 log F Ž yt . s ² c, n : .
tªq`
Knowing scalar products ² c, n : for various vectors c, we are able to reconstruct n . Furthermore, we can extract the coefficient an from the limit lim F Ž yt . rytn s an .
tª`
Modifying F [ F y an sn , we proceed to find the next summand. The following two issues should be taken care of. First, we want to find a vector c that separates some partition n g L from the others. We cannot do it deterministically without knowing the actual set L , but we can choose such a c at random with a sufficiently high probability. Second, we cannot compute asymptotics and we are barred from using very big numbers. This can be fixed because of the discrete structure of the problem: the coefficients an are integers and partitions l g L are integral vectors as well. The results of this section formalize the foregoing considerations. 3.1. LEMMA. Let al: l g L 4 be a set of integers indexed by partitions l s Ž l1 , . . . , l n .. Let F Ž x. s
Ý
lg L
al sl Ž x . ,
x s Ž x1 , . . . , x n . .
Suppose that c s Ž c1 , . . . , c n .: c1 ) c 2 ) ??? ) c n ) 0 is an integral ¨ ector such that there is a unique n g L maximizing ² c, l: s c1 l1 q ??? qc n l n o¨ er l g L . Let T s 3'< l < !
ž
< l< q n y 1 ny1
/Ý
lg L
< al <
and let us choose t ) 3 ln T. Let y s Ž y 1 , . . . , yn ., where yi s exp tc i 4 . Then an s F Ž y . ry n
Ž 3.1.1.
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and if an / 0, then ² c, n : s ty1 log F Ž y . .
Ž 3.1.2.
Proof. Let X be the set of all vectors a s Ž a1 , . . . , a n . such that the coefficient before x a in F Žx. is nonzero. First, we prove that the coefficient before x n in F Žx. is equal to an . Suppose that for some m g L the monomial x n appears in the expansion of the Schur function smŽx. with a nonzero coefficient. Then Žsee Section 2.2. n belongs to the permutohedron Pm and, therefore, the value ² c, n : does not exceed the maximal value of the linear function ² c, ? : on the vertices of the permutohedron Pm , that is, on the permutations of Ž m 1 , . . . , m n .. Whereas c1 ) ??? ) c n , the vertex of Pm with the maximal value of ² c, ? : is the partition m itself. Thus we have ² c, m : G ² c, n : implying that m s n . Whereas the coefficient before x n in the expansion of sn Žx. is 1 Žsee Section 2.2., we conclude that the coefficient before x n in F Žx. is an . Similarly, we prove that n is the only point in X where the maximal value of ² c, l: is attained. Indeed, suppose that ² c, a : G ² c, n : for some a g X. Then there exists m g L such that the monomial x a occurs in the expansion of smŽx. with a nonzero coefficient. Then a g Pm . Hence ² c, m : G ² c, a : G ² e, n : and, therefore, we must have a s n . Hence we can write F Ž y . s an y n q
Ý ba y a , a
where ² c, a : F ² c, n : y 1 for all a and Ý a < ba < F Tr3 wby Ž2.2.3.x. Thus F Ž y . ry n s an q
Ý ba y ary n . a
Now
Ý ba y ary n a
s
Ý ba exp t Ž ² c, a : q ² c, n :. 4 a
F exp yt 4 Ý < ba < F
and Ž3.1.1. follows. Suppose that an / 0. Then 2 3
y n F F Ž y. F T y n
and hence ty1 log y n y 13 F ty1 ln F Ž y . F ty1 log y n q 13 . Now we observe that ² c, n : s ty1 log y n and Ž3.1.2. is proven.
a
1 3
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INTERPOLATION OF POLYNOMIALS
3.2. COROLLARY. Suppose that under the assumptions of Lemma 3.1 we also ha¨ e < l < F l for all l g L . Let us choose z i s exp t Ž 2 lc i . 4 ,
zq i s exp t Ž 2 lc i q 1 . 4 ,
i s 1, . . . , n,
and z 0 s Ž z1 , . . . , z n . ,
z i s Ž z1 , . . . , z iy1 , zq i , z iq1 , . . . , z n . ,
i s 1, . . . , n.
If an / 0, then n s Ž n 1 , . . . , nn . can be computed as
n i s ty1 log F Ž z i . y ty1 log F Ž z 0 .
for i s 1, . . . , n.
Proof. Let C i s Ž2 lc1 , . . . , 2 lc iy1 , 2 lc i q 1, 2 lc iq1 , . . . , 2 lc n . and C s Ž2 lc1 , . . . , 2 lc n .. Because for any l / n from L we have ² c, l: F ² c, n : y 1 provided l / m , we conclude that ² C, l: F ² C, n : y 2 l for every l / n from L . Whereas n i F l we conclude that for every C i the scalar product ² C i, l: attains its unique maximum on L at the same point n . The proof follows by Ž3.1.2.. 3.3. LEMMA. Let L be a family of points in Z n. Let bi : i s 1, . . . , n be the integers drawn independently and uniformly from the set 1, . . . , N 4 , where N G 100 < L < 2 and let c1 s b1 q ??? qbn , . . . , c 2 s b 2 q ??? qbn , . . . , c k s bk q ??? qbn , . . . , c n s bn . Then c1 ) c 2 ) ??? ) c n ) 0 and with probability at least 0.99, ² c, l1 : / ² c, l 2 :
for any two l1 / l 2 from L .
Proof. The first statement is obvious. Let us prove the second one. Let In s 1, . . . , N 4 n be the integral cube consisting of N n points. We choose at random a uniformly distributed vector b g In and then transform it to the vector c s Ab, where A is an invertible linear operator. Let Lys l1 y l2 : l1 , l2 g L , l1 / l2 4 be the set of differences. Our aim is to show that with the probability at least 0.99, ² c, l: s ² b, A*l: / 0
for any l g Ly.
Ž 3.2.1.
Whereas A is invertible, A*l / 0 provided l / 0. Thus the set of those b g In that violate at least one of the conditions Ž3.2.1. is contained in the union of < Ly< hyperplanes. A hyperplane contains at most N ny 1 points in In because the set of some Ž n y 1. coordinates uniquely determines the remaining coordinate. Finally, < Ly< F < L < 2 , so the total number of points b g In that satisfy Ž3.2.1. is at least N n y N ny1 < L < 2 . The proof now follows.
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4. THE ALGORITHMS 4.1. The Basic Algorithm We start with the situation where we are given some a priori bounds on the number of summands in Ž1.1. and the size of the coefficients al. Input: A polynomial F g L n given by the ‘‘black box,’’ numbers M, L, and l such that in the decomposition FŽ x. s
Ý
lg L
al sl Ž x .
one has < L < F L, Ý lg L < al < F M, and < l < F l for any l g L . We assume that al: l g L are nonzero integers. ALGORITHM. Step 0. Let N s 100 L2 . Choose independently at random n numbers b1 , . . . , bn distributed uniformly in the interval 1, . . . , N 4 . Let c1 s b1 q ??? qbn , c 2 s b 2 q ??? qbn , . . . , c k s bk q ??? qbn , . . . , c n s bn . < < Compute T s 3'< l ! < l q n y 1 M and an integer t s 3wlog T x q 3.
ž
ny 1
/
Ž .4 for i s 1, . . . , n. Let z i s exp 2 tlc i 4 , zq i s exp t 2 lc i q 1 Ž . Ž . for i s Let z 0 s z1 , . . . , z n and z i s z1 , . . . , z iy1 , zq i , z iq1 , . . . , z n 1, . . . , n. Compute f i s F Žz i . for i s 0, . . . , n. Set L s B. Set s s 0. Step 1. If f i s 0 for some i s 0, 1, . . . , n, then stop. If s ) L, then report ‘‘failure’’ and stop. Otherwise compute l i s w ty1 log f i x y w ty1 log f 0 x for i s 1, . . . , n. If l s Ž l1 , . . . , l n . is not a partition, report ‘‘failure’’ and stop. Otherwise, let L s L j l4 and al s w f 0rz l x. Let f i [ f i y al slŽz i . for i s 0, . . . , n. Let s [ s q 1 and return to Step 1. 4.1.1. THEOREM. With the probability at least 0.99 the preceding algorithm computes the expansion Fs
Ý
lg L
al sl
in time that is polynomial in log M, n, L, and l. Proof. Lemma 3.3 implies that with the probability at least 0.99 we have ² c, l1 : / ² c, l 2 : for any two l1 / l 2 from L . We claim that if on Step 0 we managed to choose c so that the condition is indeed satisfied then the algorithm works correctly. Using Lemma 3.1 and Corollary 3.2 we
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INTERPOLATION OF POLYNOMIALS
prove by induction that after the kth iteration of Step 1 we have computed the first k partitions n 1, . . . , n k from L in the decreasing order of ² c, l:, that al are the correct coefficients before sl in the expansion of F for l s n 1, . . . , n k , and that f i s Fk Ž z i . ,
where Fk s
Ý
al sl .
l g L_n 1 , . . . , n k 4
So the algorithm works correctly. To estimate the complexity of the algorithm, we note that because of our choice of t, the bit size of z i Žwhich is approximately log z i . is bounded by a polynomial in n, l, log M, and L. The algorithm calls the black box Ž n q 1. times to compute F Žz i . and uses at most LŽ n q 1. computations of slŽz i . that can be accomplished in polynomial time Žsee Section 2.4.. Using the construction of Section 2.3 we can make the error probability smaller than any given e ) 0 by running O Žlog ey1 . independent copies of the algorithm in parallel. 4.2. The General Algorithm We proceed with the situation where no a priori bounds on < L < and M are given. We do need a bound l on < l <: l g L , that is, a bound on the degree of F. Note that l is implicitly encoded in the input: the size of F Žx. for a general x is proportional to deg F. Input: A polynomial F g L n given by the ‘‘black box’’ and a number l such that deg F F l. ALGORITHM. Step 0. Introduce integral variables M1 , L1 , s. Set M1 s L1 s 1, s s 0. Step 1. Let M1 [ 2 M1 , L1 s L1 q 1, s s s q 1. Run the algorithm from Section 4.1 with M s M1 , L s L1 , and l. If the algorithm produces a decomposition Fs
Ý
lg L
al sl ,
test this decomposition by the algorithm in Example 2.3.1 for testing polynomial identities so as to ensure the probability of error to be at most 0.005 = 2ys . If the identity is accepted, report it and stop. Otherwise go to Step 1.
284 4.2.1. THEOREM.
BARVINOK AND FOMIN
Let Fs
Ý
lg L
al sl
be the expansion of F and let Ms
Ý
lg L
< al <
and L s < L < .
Then with the probability at least 0.99 the preceding algorithm stops in time polynomial in log M, n, L, and l, pro¨ iding the correct decomposition. Proof. There are two types of possible errors. First, the algorithm can accept a decomposition that is not correct. Alternatively, the algorithm may not come up with a decomposition at all. The probability of the first type of error does not exceed Ý`ss 10.005 = 2ys s 0.005. Because the algorithm in Section 4.1 always stops after at most L iterations, in time polynomial in log M, L, and l we either report a decomposition and stop or get to the situation with M1 G M and L1 G L. Theorem 4.1.1 implies that with probability at least 1 y Ž0.01. 2 we will report the correct decomposition within two iterations of Step 1 after that. 4.3. COROLLARY. There exist probabilistic polynomial time algorithms for the problems in Sections 1.2]1.4, that is, }for computing an expansion Ž1.2.1. of a gi¨ en power sum pm ; }for computing all non¨ anishing Kostka numbers Klm , for a fixed ¨ alue of m ; and }for computing the Littlewood]Richardson expansion Ž1.4.2. of a gi¨ en skew Schur function sm r n . Proof. As stated in Section 2.4, for any given x we can compute the values of pmŽx., hmŽx., and sm r n Žx. in time that is polynomial in n and < m <. So we can design a polynomial time ‘‘black box’’ for F in the cases when F is one of pm , hm , or sm r n , and then apply the algorithm in Section 4.2. We remark that the computation of the Littlewood]Richardson expansions was the original motivation for our work. If we know in advance the set L s l4 of the l for which al is nonzero Žthis is the case, for example, in Section 1.3, then the algorithm can be greatly simplified. Namely, we can compute the values of F and sl at < L < randomly generated points x and then extract the coefficients al from the resulting system of linear equations. However, even in this case our methods might appear useful. For example, the set L may be so big that it
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is impossible to solve Žor even to store. the corresponding system of equations. In this case, we cannot hope to find the decomposition Ž1.1.. On the other hand, our algorithm enables us to write some m terms of this decomposition in O Ž m2 . time. To do this, find l g L with the maximal norm 5 l 5 and use Lemma 3.1 and Corollary 3.2 with c s l. Then we let F [ F y al sl , L s L _ l4 , and proceed as before. ACKNOWLEDGMENT The authors are thankful to John Stembridge for a number of valuable comments.
REFERENCES 1. A. V. Aho, J. E. Hopcroft, and J. D. Ullman, ‘‘The Design and Analysis of Computer Algorithms,’’ Addison-Wesley, Reading, MA, 1974. 2. M. Ben-Or and P. Tiwari, A deterministic algorithm for sparse multivariate polynomial interpolation, in ‘‘Proceedings of the 20th Annual ACM Symposium on Theory of Computing,’’ ACM, New York, 1988. 3. L. Csanky, Fast parallel matrix inversion algorithms, SIAM J. Comput. 5 Ž1976., 618]823. 4. V. A. Emelichev, M. M. Kovalev, and M. K. Kravtsov, ‘‘Polytopes, Graphs and Optimization,’’ Cambridge University Press, 1984. 5. D. Yu. Grigoriev, M. Karpinski and M. F. Singer, The interpolation problem for k-sparse sums of eigenfunctions of operators, Ad¨ . in Appl. Math. 12 Ž1991., 76]81. 6. G. D. James, ‘‘The Representation Theory of the Symmetric Group,’’ Lecture Notes in Math., Vol. 682, Springer-Verlag, Berlin, 1978. 7. I. G. MacDonald, ‘‘Symmetric Functions and Hall Polynomials,’’ 2nd ed., Oxford University Press, 1995. 8. J. Remmel and R. Whitney, Multiplying Schur functions, J. Algorithms 5 Ž1984., 471]487. 9. J. T. Schwartz, Fast probabilistic algorithms for verification of polynomial identities, J. Assoc. Comput. Mach. 27 Ž1980., 701]717.
18, 271]285 Ž1997.
AM960508
Sparse Interpolation of Symmetric Polynomials Alexander Barvinok* Department of Mathematics, Uni¨ ersity of Michigan, Ann Arbor, Michigan 48109-1003
and Sergey Fomin† Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Theory of Algorithms Laboratory, St. Petersburg Institute of Informatics, Russian Academy of Sciences, St. Petersburg, Russia Received April 20, 1996
We develop efficient algorithms for computing the expansion of a given symmetric polynomial into Schur functions. This problem frequently arises in applications as the problem of decomposing a given representation of the symmetric Žor general linear. group into irreducible constituents. Our algorithms are probabilistic, and run in time which is polynomial in the sizes of the input and output. They can be used to compute Littlewood]Richardson coefficients, Kostka numbers, and irreducible characters of the symmetric group. Q 1997 Academic Press
I. INTRODUCTION The general sparse interpolation problem can be posed as follows. Let V be a vector space over the rationals or a Z-module of functions with a fixed basis fl4 . Suppose there is a ‘‘black box’’ that computes the value of some function F from V at any given argument x. We want to find the decomposition Fs
Ý
lg L
al fl : al g Q
Ž or al g Z .
Ž 1.1.
* Research partially supported by the NSF ŽDMS-96-01129. and an Alfred P. Sloan research fellowship. † Research partially supported by the NSF ŽDMS-94-00914.. 271 0196-8858r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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as fast as possible. The computational complexity of any such algorithm is of course bounded from below by the actual length of the decomposition Ž1.1., that is, by the number L s < L < of basis functions fl that occur in Ž1.1. with nonzero coefficients Žsince computing the decomposition requires, at the very least, writing the answer.. So we want to perform the decomposition efficiently, preferably in time polynomial in L. The term ‘‘sparse’’ refers to the fact that the complexity of the algorithm must be bounded from above by a function of L. The most important particular case is V s Qw x 1 , . . . , x n x, the ring of polynomials, with fl4 being the basis of monomials. Then there exists an algorithm w2x that finds the decomposition Ž1.1. in time that is polynomial in the number of nonzero coefficients al, even if this number is unknown a priori. Another interesting case is the interpolation with respect to the eigenfunctions of some operator Žsee w5x.. In this paper, we study the sparse interpolation problem for symmetric polynomials. Let us consider the ring Zw x 1 , . . . , x n x of polynomials with integer coefficients in n variables x s Ž x 1 , . . . , x n .. We are interested in the subring L n ; Zw x 1 , . . . , x n x of symmetric polynomials, i.e., polynomials pŽ x . which are invariant under any permutation of the variables x 1 , . . . , x n . As a Z-module, L n possesses quite a few distinguished bases, most prominently the Schur functions sl, which are indexed by vectors l s Ž l1 , . . . , l n . g Z n, where l1 G l2 G ??? G l n G 0. Following the tradition, we call such a vector l a partition Žnote that, contrary to the usual convention, some of the l i are allowed to vanish.. The Schur function slŽx. can be defined in many different ways, in particular, as the following ratio of two n = n determinants: sl Ž x . s det Ž x il jqj y1 . rdet Ž x ijy1 . Žsee w7, Chap. 1, Sect. 3x.. The sparse interpolation problem in L n with respect to the basis sl4 can now be stated as follows. 1.1. Sparse Interpolation of Symmetric Polynomials Input: Ži. A ‘‘black box’’ that computes the value F Žx. of some polynomial F g L n for any x. Žii. The upper bound l on the degree of F. Output: The decomposition of F into a linear combination of Schur functions: Fs
Ý
lg L
al sl .
Ž 1.1.1.
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273
Our study of this problem is motivated by the fact that many important computational problems in the theory of symmetric functions and representation theory of the symmetric and general linear groups can be formulated in the form Ž1.1.. Here are three typical examples. 1.2. Characters of the Symmetric Group Let pm g L n denote the power sum corresponding to a partition m Žsee Section 2.1 or w7, Chap. 1, Sect. 2x.. The coefficients xlŽ m . of the decomposition pm s
Ý
lg L
xl Ž m . sl
Ž 1.2.1.
are known to be integers, namely, the values of irreducible characters xl of the symmetric group Sn on the conjugacy class defined by m. This problem naturally fits our sparse interpolation setup because any value of pmŽx. can be computed very fast Žsee Section 2.4.. Furthermore, many of the coefficients xlŽ m . can be zero. For instance, the classical Murnaghan]Nakayama rule implies that xlŽ m . s 0 unless the number of parts of m is at least the size of the Durfee square of l Žsee, e.g., w6x.. 1.3. Kostka Numbers For m a partition, let hm g L n be the corresponding complete homogeneous symmetric function Žsee Section 2.1 or w7, Chap. 1, Sect. 2x.. The coefficients Kml of the decomposition hm s
Ý
lg L
Klm sl
Ž 1.3.1.
are nonnegative integers called Kostka numbers Žcf. w7x.. These numbers are nothing but the coefficients of the Schur polynomials, sl s
Ý Klm mm ,
Ž 1.3.2.
m
where mmŽ x . denotes the sum of all monomials of type m. Combinatorially, Klm is the number of semistandard Young tableaux of shape l and weight m. The computation of the expansion Ž1.3.1. is equivalent to decomposing the representation of the symmetric group Sn induced from the trivial representation of its Young subgroup Sm , into the sum of irreducibles. This problem can be posed as a sparse interpolation problem because for every x the value of hmŽx. can be computed very fast Žsee Section 2.4.. Further-
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more, in this particular case there is a simple characterization of the set L of partitions l with Klm / 0 Žsee w7, Chap. 1, Sect. 6x and also Section 2.2.. For partitions with few parts, this set is relatively small. 1.4. Littlewood]Richardson Coefficients Let m and n be partitions such that
mi G n i
for all i.
Ž 1.4.1.
Then one can define a skew Schur function sm r n Žsee w7, Chap. 1, Sect. 5x.. m The coefficients gnl of the decomposition sm r n s
Ý
lg L
m gnl sl
Ž 1.4.2.
are nonnegative integers called the Littlewood]Richardson coefficients. The importance of computing decompositions of the form Ž1.4.2. comes from the fact that this is essentially equivalent to expanding the product of two Žordinary. Schur functions in the basis of Schur functions. In other words, the Littlewood]Richardson coefficients are the structure constants for this basis. In representation theory of the symmetric group, the computation of expansions Ž1.4.2. amounts to decomposing a certain product of two irreducible Sn-modules Žor, alternatively, a skew Specht module. into irreducible components. The celebrated Littlewood]Richardson rule provides an explicit Žalbeit m4 complicated. combinatorial description of the numbers gnl as counting certain tableaux Žthe ‘‘Littlewood]Richardson fillings’’.. Most known algorithms computing these numbers make use of this rule; hence their running time has to be at least linear in the number of the Littlewood]Richardson fillings ŽLRF., which is equal to LRFnm s
Ý gnlm . l
ŽThe complexity O ŽLRFnm . was indeed achieved by the algorithm of Remmel and Whitney w8x.. The number LRFnm can be very large; in fact, it m can be much larger than the length L s a l: gnl / 04 of the output expansion Ž1.4.2.. On the other hand, one can observe that for any x the numerical value of sm r n Žx. can be computed very fast Žsee Section 2.4.. Thus the problem of computing the decomposition Ž1.4.2. can be naturally stated as a sparse interpolation problem of a symmetric polynomial. Generally, it may be interesting to find expansions Ž1.1. for some algebraic expressions involving the pm , hm , and sm r n , such as hm 9 y hm 0 or
INTERPOLATION OF POLYNOMIALS
275
sm9r n 9 y sm 0 r n 0 , where partitions m9 and m0 Žresp., n 9 and n 0 . are ‘‘close’’ to each other, so that one can expect a short expansion in the Schur functions basis, although it is hard to tell in advance which l’s do come up. In this paper, we construct a probabilistic algorithm, which solves problem Ž1.1. in time that is polynomial in the number of variables n, the upper bound l on the degree of F, the length L s < L < of the resulting expansion, and the value log 2 M, where Ms
Ý
lg L
< al <
Žthis takes care of the binary size of the output.. Note that neither M nor L are known in advance. The paper is organized as follows. In Section 2 we discuss the necessary facts and notions concerning symmetric functions, on one side, and computational complexity, on the other. The purpose of this section is to bridge the gap between these two topics. In Section 3 we discuss the general idea of our algorithms and prove the main technical lemmas. The algorithms are presented in Section 4.
2. PRELIMINARIES 2.1. Power Sums and Complete Homogeneous Symmetric Functions Following w7x, we introduce the standard notation and terminology related to symmetric functions. For a positive integer k, define the power sum pk by n
pk Ž x . s
Ý x ik .
Ž 2.1.1.
is1
It will also be convenient to use the convention p 0 Žx. s 1. For a partition l s Ž l1 , . . . , l n ., we let pl s pl1 ??? pl n .
Ž 2.1.2.
For m s 1, 2, . . . , define the complete homogeneous symmetric polynomial h m as the sum of all monomials of degree m. By convention, h 0 s 1 and h i s 0 for i - 0. The values of polynomials h m can be computed inductively via the formulas Žsee Ž2.11. in w7x.: h1 Ž x . s x 1 q ??? qx n , h m Ž x. s
1 m
m
Ý pr Ž x . h myr Ž x . rs1
for m ) 1.
Ž 2.1.3.
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For a partition l s Ž l1 , . . . , l n ., let hl s hl1 ??? hl n .
Ž 2.1.4.
2.2. Schur Functions and their Newton Polytopes The Schur function slŽx. can be expressed as the Jacobi]Trudi determinant Žsee Ž3.4. of w7x. sl s det Ž hl iyiqj . 1Fi , jFn .
Ž 2.2.1.
For a pair of partitions m and n satisfying Ž1.4.1., the skew Schur function sm r n can be defined as Žsee Ž5.4. of w7x. sm r n s det Ž hm iy n jyiqj . 1F i , jFn .
Ž 2.2.2.
The Schur function sl is a homogeneous symmetric polynomial of degree < l < s l1 q ??? ql n . To write down its expansion into monomials x a s x 1a 1 ??? x na n , let us first generalize the notation for the Kostka numbers, by denoting Kla s Kl b , where b s Ž b 1 , . . . , bn . is the nondecreasing rearrangement of a s Ž a 1 , . . . , a n .. Then sl Ž x . s
Ý
a 1q ??? q a n s < l < a iG0
Kl a x a
wcf. Ž1.3.2.x. We have already mentioned that Kla G 0, that is, the Schur functions are polynomials with nonnegative integer coefficients. The symmetry of sl is reflected in the fact that Kl a does not change if we permute the coordinates of a . Besides, Kll s 1 Žsee w7, Chap. 1, Sect. 6x.. Let us now assume that a is a partition, that is, a 1 G ??? G a n . It is known that Kla ) 0 if and only if k
k
Ý a i F Ý li is1
for k s 1, . . . , n y 1;
is1 n
n
Ý a i s Ý li is1
is1
Žsee w7, Chap. 1, Sect. 6x.. This condition can be expressed geometrically as follows. Let us define the polytope Pl ; R n Ža permutohedron. as the convex hull of all permutations of the vector l s Ž l1 , . . . , l n .. Then Kla ) 0 if and only if a g Pl Žsee, e.g., Rado’s theorem in w4, Sect. 3.1x..
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277
In other words, the Newton polytope of slŽx. is the permutohedron Pl . We will need a very rough estimate of Kla : Kla F '< l < ! ,
Ž 2.2.3.
which follows from the interpretation of Kla as the multiplicity of some representation Žsee w7, Chap. 1, Sect. 6x.. 2.3. Computational Complexity and Algorithms We adopt the usual computational model, namely, random access memory ŽRAM. with the uniform cost criterion Žsee w1x.. Thus our machine operates with integral Žrational. numbers represented by bit strings. The input size of a number M is approximately log 2 M. We assume that our machine can perform arithmetic operations Žaddition, subtraction, multiplication, division, and comparison. at the unit time. We also make sure that the size of all numbers appearing in the course of the algorithm is bounded by a polynomial in the inputroutput size. We will use probabilistic algorithms, meaning that our machine has a built-in device that can ‘‘toss a coin.’’ More precisely, we assume that, at any time, our machine is able to choose, uniformly at random, an integer from the interval w1, N x. By introducing randomness, we allow a certain probability that our algorithm fails to work at a certain step or produces an incorrect answer or does not stop at all. However, we will make sure that, with probability at least 0.99, our algorithm stops after certain time t and produces the correct answer. To ensure an overwhelming probability, we will then run several copies of our algorithm in parallel and pick the most frequent answer. For example, if we run m independent copies of the algorithm, then the probability of failure Žthat is, more than mr2 copies do not stop in time t with the correct answer. does not exceed
Ý kFmr2
m k
ž /Ž
0.01.
my k
k Ž 0.99. F 5ym ,
i.e., is exponentially small. 2.3.1. EXAMPLE. As an illustration, we refer to the randomized polynomial time algorithm for testing polynomial identities w9x. Suppose we are given a ‘‘black box’’ that computes the value of a certain polynomial F Žx. at any point x s Ž x 1 , . . . , , x n . g Z n. Suppose we know that the degree of F does not exceed d. Our goal is to find out whether F identically vanishes
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or not. The algorithm proposed in w9x works as follows. Let N s 3dn and let us choose the coordinates x i independently and uniformly from the set 1, . . . , N 4 . If F Žx. / 0, then we conclude that F is not identically zero. If F Žx. s 0, then the algorithm decides that F ' 0. It is proven in w9x that if F is not identically zero, the probability of choosing an x such that F Žx. s 0 is at most 1r3, so the algorithm makes an error with probability F 1r3. To make the probability of error as small as 3ym , we can generate m vectors x in the above-described fashion. If for some of them F Žx. / 0, then F is not identically 0. Otherwise our algorithm will conclude that F ' 0. 2.4. Complexity of Symmetric Functions Let us discuss the computational complexity of the symmetric polynomials pl, hl , sl , and sm r n . For each of these, we want to construct a ‘‘black box’’ that efficiently computes the value of a function at a given point x. Our presentation of such constructions will be self-contained, except for making use of the fact that the determinant of an n = n matrix can be computed in O Ž n3 . time Žsee, for example, w3x.. Using the definition Ž2.1.1. in a straightforward way, we observe that pk Žx., for any given x s Ž x 1 , . . . , x n ., can be computed in O Ž kn. time. Then Ž2.1.2. shows that the value of plŽx. can be computed in O Ž< l < n. time Žrecall that < l < s l1 q ??? ql n .. Using recursive formulas Ž2.1.3., one can compute the value of h k Žx. in O Ž k 2 n. time; hence the computation of hlŽx. takes O Ž< l < 2 n. time wsee Ž2.1.4.x. The determinantal expressions Ž2.2.1. ] Ž2.2.2. allow us to compute the values of sm r n Žx. and slŽx. in O Ž< m < 2 n q n3 . and O Ž< l < 2 n q n3 . time, respectively. 2.5. Notation We will need some extra notation to be used throughout the paper. A partition l s Ž l1 , . . . , l n . will often be interpreted as a vector in R n, the latter being endowed with the scalar product ² l , m : s l1 m 1 q l 2 m 2 q ??? ql n m n . We have already used the notation x a to denote x 1a 1 ??? x na n , where a s Ž a 1 , . . . , a n . g Z n. The base of exponentiation and logarithms will be 2 by default, so that exp a4 s 2 a and log a s log 2 a. Finally, w x x will denote the integer nearest to x.
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3. THE IDEA OF THE ALGORITHMS. BASIC LEMMAS Our main observation is that the coefficients of the decomposition F Ž x. s
Ý
lg L
al sl Ž x .
can be extracted from some asymptotics of F Žx.. Namely, suppose we are able to choose a vector c s Ž c1 , . . . , c n ., where c1 ) c 2 ) ??? ) c n such that the linear function ² c, l: attains its maximum on L on the unique partition n g L . Let us define yt s Žexp tc14 , . . . , exp tc n4.. Then lim ty1 log F Ž yt . s ² c, n : .
tªq`
Knowing scalar products ² c, n : for various vectors c, we are able to reconstruct n . Furthermore, we can extract the coefficient an from the limit lim F Ž yt . rytn s an .
tª`
Modifying F [ F y an sn , we proceed to find the next summand. The following two issues should be taken care of. First, we want to find a vector c that separates some partition n g L from the others. We cannot do it deterministically without knowing the actual set L , but we can choose such a c at random with a sufficiently high probability. Second, we cannot compute asymptotics and we are barred from using very big numbers. This can be fixed because of the discrete structure of the problem: the coefficients an are integers and partitions l g L are integral vectors as well. The results of this section formalize the foregoing considerations. 3.1. LEMMA. Let al: l g L 4 be a set of integers indexed by partitions l s Ž l1 , . . . , l n .. Let F Ž x. s
Ý
lg L
al sl Ž x . ,
x s Ž x1 , . . . , x n . .
Suppose that c s Ž c1 , . . . , c n .: c1 ) c 2 ) ??? ) c n ) 0 is an integral ¨ ector such that there is a unique n g L maximizing ² c, l: s c1 l1 q ??? qc n l n o¨ er l g L . Let T s 3'< l < !
ž
< l< q n y 1 ny1
/Ý
lg L
< al <
and let us choose t ) 3 ln T. Let y s Ž y 1 , . . . , yn ., where yi s exp tc i 4 . Then an s F Ž y . ry n
Ž 3.1.1.
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and if an / 0, then ² c, n : s ty1 log F Ž y . .
Ž 3.1.2.
Proof. Let X be the set of all vectors a s Ž a1 , . . . , a n . such that the coefficient before x a in F Žx. is nonzero. First, we prove that the coefficient before x n in F Žx. is equal to an . Suppose that for some m g L the monomial x n appears in the expansion of the Schur function smŽx. with a nonzero coefficient. Then Žsee Section 2.2. n belongs to the permutohedron Pm and, therefore, the value ² c, n : does not exceed the maximal value of the linear function ² c, ? : on the vertices of the permutohedron Pm , that is, on the permutations of Ž m 1 , . . . , m n .. Whereas c1 ) ??? ) c n , the vertex of Pm with the maximal value of ² c, ? : is the partition m itself. Thus we have ² c, m : G ² c, n : implying that m s n . Whereas the coefficient before x n in the expansion of sn Žx. is 1 Žsee Section 2.2., we conclude that the coefficient before x n in F Žx. is an . Similarly, we prove that n is the only point in X where the maximal value of ² c, l: is attained. Indeed, suppose that ² c, a : G ² c, n : for some a g X. Then there exists m g L such that the monomial x a occurs in the expansion of smŽx. with a nonzero coefficient. Then a g Pm . Hence ² c, m : G ² c, a : G ² e, n : and, therefore, we must have a s n . Hence we can write F Ž y . s an y n q
Ý ba y a , a
where ² c, a : F ² c, n : y 1 for all a and Ý a < ba < F Tr3 wby Ž2.2.3.x. Thus F Ž y . ry n s an q
Ý ba y ary n . a
Now
Ý ba y ary n a
s
Ý ba exp t Ž ² c, a : q ² c, n :. 4 a
F exp yt 4 Ý < ba < F
and Ž3.1.1. follows. Suppose that an / 0. Then 2 3
y n F F Ž y. F T y n
and hence ty1 log y n y 13 F ty1 ln F Ž y . F ty1 log y n q 13 . Now we observe that ² c, n : s ty1 log y n and Ž3.1.2. is proven.
a
1 3
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INTERPOLATION OF POLYNOMIALS
3.2. COROLLARY. Suppose that under the assumptions of Lemma 3.1 we also ha¨ e < l < F l for all l g L . Let us choose z i s exp t Ž 2 lc i . 4 ,
zq i s exp t Ž 2 lc i q 1 . 4 ,
i s 1, . . . , n,
and z 0 s Ž z1 , . . . , z n . ,
z i s Ž z1 , . . . , z iy1 , zq i , z iq1 , . . . , z n . ,
i s 1, . . . , n.
If an / 0, then n s Ž n 1 , . . . , nn . can be computed as
n i s ty1 log F Ž z i . y ty1 log F Ž z 0 .
for i s 1, . . . , n.
Proof. Let C i s Ž2 lc1 , . . . , 2 lc iy1 , 2 lc i q 1, 2 lc iq1 , . . . , 2 lc n . and C s Ž2 lc1 , . . . , 2 lc n .. Because for any l / n from L we have ² c, l: F ² c, n : y 1 provided l / m , we conclude that ² C, l: F ² C, n : y 2 l for every l / n from L . Whereas n i F l we conclude that for every C i the scalar product ² C i, l: attains its unique maximum on L at the same point n . The proof follows by Ž3.1.2.. 3.3. LEMMA. Let L be a family of points in Z n. Let bi : i s 1, . . . , n be the integers drawn independently and uniformly from the set 1, . . . , N 4 , where N G 100 < L < 2 and let c1 s b1 q ??? qbn , . . . , c 2 s b 2 q ??? qbn , . . . , c k s bk q ??? qbn , . . . , c n s bn . Then c1 ) c 2 ) ??? ) c n ) 0 and with probability at least 0.99, ² c, l1 : / ² c, l 2 :
for any two l1 / l 2 from L .
Proof. The first statement is obvious. Let us prove the second one. Let In s 1, . . . , N 4 n be the integral cube consisting of N n points. We choose at random a uniformly distributed vector b g In and then transform it to the vector c s Ab, where A is an invertible linear operator. Let Lys l1 y l2 : l1 , l2 g L , l1 / l2 4 be the set of differences. Our aim is to show that with the probability at least 0.99, ² c, l: s ² b, A*l: / 0
for any l g Ly.
Ž 3.2.1.
Whereas A is invertible, A*l / 0 provided l / 0. Thus the set of those b g In that violate at least one of the conditions Ž3.2.1. is contained in the union of < Ly< hyperplanes. A hyperplane contains at most N ny 1 points in In because the set of some Ž n y 1. coordinates uniquely determines the remaining coordinate. Finally, < Ly< F < L < 2 , so the total number of points b g In that satisfy Ž3.2.1. is at least N n y N ny1 < L < 2 . The proof now follows.
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4. THE ALGORITHMS 4.1. The Basic Algorithm We start with the situation where we are given some a priori bounds on the number of summands in Ž1.1. and the size of the coefficients al. Input: A polynomial F g L n given by the ‘‘black box,’’ numbers M, L, and l such that in the decomposition FŽ x. s
Ý
lg L
al sl Ž x .
one has < L < F L, Ý lg L < al < F M, and < l < F l for any l g L . We assume that al: l g L are nonzero integers. ALGORITHM. Step 0. Let N s 100 L2 . Choose independently at random n numbers b1 , . . . , bn distributed uniformly in the interval 1, . . . , N 4 . Let c1 s b1 q ??? qbn , c 2 s b 2 q ??? qbn , . . . , c k s bk q ??? qbn , . . . , c n s bn . < < Compute T s 3'< l ! < l q n y 1 M and an integer t s 3wlog T x q 3.
ž
ny 1
/
Ž .4 for i s 1, . . . , n. Let z i s exp 2 tlc i 4 , zq i s exp t 2 lc i q 1 Ž . Ž . for i s Let z 0 s z1 , . . . , z n and z i s z1 , . . . , z iy1 , zq i , z iq1 , . . . , z n 1, . . . , n. Compute f i s F Žz i . for i s 0, . . . , n. Set L s B. Set s s 0. Step 1. If f i s 0 for some i s 0, 1, . . . , n, then stop. If s ) L, then report ‘‘failure’’ and stop. Otherwise compute l i s w ty1 log f i x y w ty1 log f 0 x for i s 1, . . . , n. If l s Ž l1 , . . . , l n . is not a partition, report ‘‘failure’’ and stop. Otherwise, let L s L j l4 and al s w f 0rz l x. Let f i [ f i y al slŽz i . for i s 0, . . . , n. Let s [ s q 1 and return to Step 1. 4.1.1. THEOREM. With the probability at least 0.99 the preceding algorithm computes the expansion Fs
Ý
lg L
al sl
in time that is polynomial in log M, n, L, and l. Proof. Lemma 3.3 implies that with the probability at least 0.99 we have ² c, l1 : / ² c, l 2 : for any two l1 / l 2 from L . We claim that if on Step 0 we managed to choose c so that the condition is indeed satisfied then the algorithm works correctly. Using Lemma 3.1 and Corollary 3.2 we
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INTERPOLATION OF POLYNOMIALS
prove by induction that after the kth iteration of Step 1 we have computed the first k partitions n 1, . . . , n k from L in the decreasing order of ² c, l:, that al are the correct coefficients before sl in the expansion of F for l s n 1, . . . , n k , and that f i s Fk Ž z i . ,
where Fk s
Ý
al sl .
l g L_n 1 , . . . , n k 4
So the algorithm works correctly. To estimate the complexity of the algorithm, we note that because of our choice of t, the bit size of z i Žwhich is approximately log z i . is bounded by a polynomial in n, l, log M, and L. The algorithm calls the black box Ž n q 1. times to compute F Žz i . and uses at most LŽ n q 1. computations of slŽz i . that can be accomplished in polynomial time Žsee Section 2.4.. Using the construction of Section 2.3 we can make the error probability smaller than any given e ) 0 by running O Žlog ey1 . independent copies of the algorithm in parallel. 4.2. The General Algorithm We proceed with the situation where no a priori bounds on < L < and M are given. We do need a bound l on < l <: l g L , that is, a bound on the degree of F. Note that l is implicitly encoded in the input: the size of F Žx. for a general x is proportional to deg F. Input: A polynomial F g L n given by the ‘‘black box’’ and a number l such that deg F F l. ALGORITHM. Step 0. Introduce integral variables M1 , L1 , s. Set M1 s L1 s 1, s s 0. Step 1. Let M1 [ 2 M1 , L1 s L1 q 1, s s s q 1. Run the algorithm from Section 4.1 with M s M1 , L s L1 , and l. If the algorithm produces a decomposition Fs
Ý
lg L
al sl ,
test this decomposition by the algorithm in Example 2.3.1 for testing polynomial identities so as to ensure the probability of error to be at most 0.005 = 2ys . If the identity is accepted, report it and stop. Otherwise go to Step 1.
284 4.2.1. THEOREM.
BARVINOK AND FOMIN
Let Fs
Ý
lg L
al sl
be the expansion of F and let Ms
Ý
lg L
< al <
and L s < L < .
Then with the probability at least 0.99 the preceding algorithm stops in time polynomial in log M, n, L, and l, pro¨ iding the correct decomposition. Proof. There are two types of possible errors. First, the algorithm can accept a decomposition that is not correct. Alternatively, the algorithm may not come up with a decomposition at all. The probability of the first type of error does not exceed Ý`ss 10.005 = 2ys s 0.005. Because the algorithm in Section 4.1 always stops after at most L iterations, in time polynomial in log M, L, and l we either report a decomposition and stop or get to the situation with M1 G M and L1 G L. Theorem 4.1.1 implies that with probability at least 1 y Ž0.01. 2 we will report the correct decomposition within two iterations of Step 1 after that. 4.3. COROLLARY. There exist probabilistic polynomial time algorithms for the problems in Sections 1.2]1.4, that is, }for computing an expansion Ž1.2.1. of a gi¨ en power sum pm ; }for computing all non¨ anishing Kostka numbers Klm , for a fixed ¨ alue of m ; and }for computing the Littlewood]Richardson expansion Ž1.4.2. of a gi¨ en skew Schur function sm r n . Proof. As stated in Section 2.4, for any given x we can compute the values of pmŽx., hmŽx., and sm r n Žx. in time that is polynomial in n and < m <. So we can design a polynomial time ‘‘black box’’ for F in the cases when F is one of pm , hm , or sm r n , and then apply the algorithm in Section 4.2. We remark that the computation of the Littlewood]Richardson expansions was the original motivation for our work. If we know in advance the set L s l4 of the l for which al is nonzero Žthis is the case, for example, in Section 1.3, then the algorithm can be greatly simplified. Namely, we can compute the values of F and sl at < L < randomly generated points x and then extract the coefficients al from the resulting system of linear equations. However, even in this case our methods might appear useful. For example, the set L may be so big that it
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285
is impossible to solve Žor even to store. the corresponding system of equations. In this case, we cannot hope to find the decomposition Ž1.1.. On the other hand, our algorithm enables us to write some m terms of this decomposition in O Ž m2 . time. To do this, find l g L with the maximal norm 5 l 5 and use Lemma 3.1 and Corollary 3.2 with c s l. Then we let F [ F y al sl , L s L _ l4 , and proceed as before. ACKNOWLEDGMENT The authors are thankful to John Stembridge for a number of valuable comments.
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