Fast algorithms for multivariate interpolation problems

Transcription

1 Fast algorithms for multivariate interpolation problems Vincent Neiger,, Claude-Pierre Jeannerod Éric Schost Gilles Villard AriC, LIP, École Normale Supérieure de Lyon, France ORCCA, Computer Science Department, Western University, London, ON, Canada Supported by the international mobility grant Explo ra doc from Région Rhône-Alpes PolSys Seminar LIP6, Paris, France, July 2, 2015 Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

2 Outline 1 List-decoding and interpolation problem 2 Fast algorithms using structured linear algebra 3 Fast algorithms using linear algebra over K[X] Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

3 List-decoding and interpolation problem Outline 1 List-decoding and interpolation problem 2 Fast algorithms using structured linear algebra 3 Fast algorithms using linear algebra over K[X] Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

4 List-decoding and interpolation problem Reed-Solomon codes Reliable delivery of data over an unreliable communication channel w = w 0 + +w k X k encoding (w(x 1 ),..., w(x n )) At most e errors during transmission: noise y = (y 1,..., y n ) #{i w(x i ) y i } e Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

5 List-decoding and interpolation problem Unique decoding problem w = w 0 + +w k X k encoding (w(x 1 ),..., w(x n )) noise y = (y 1,..., y n ) where #{i w(x i ) y i } e the receiver gets y and looks for w Unique decoding of Reed-Solomon codes Input: x 1,..., x n the n distinct evaluation points in K, k the degree bound, e the error-correction radius, (y 1,..., y n ) the received word in K n Unique decoding assumption: e < n k 2 Output: The polynomial w in K[X] such that deg w k and #{i w(x i ) y i } e. Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

6 List-decoding and interpolation problem Key equations (unique decoding) Define the interpolation polynomial R(X) such that R(x i ) = y i, and the error-locator polynomial Λ(X) = i error (X x i). Λ(X) is an unknown polynomial with deg Λ e Key equations for every i, Λ(x i )R(x i ) = Λ(x i )w(x i ) Quadratic equations in the unknown coefficients of w and Λ... Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

7 List-decoding and interpolation problem Modular key equation (unique decoding) Recall the interpolation and error-locator polynomials R(x i ) = y i, Λ(X) = i error (X x i) Key equations for every i, Λ(x i )R(x i ) = Λ(x i )w(x i ) i.e. for every i, Λ(X)R(X) = Λ(X)w(X) mod (X x i ) Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

8 List-decoding and interpolation problem Modular key equation (unique decoding) Recall the interpolation and error-locator polynomials Key equations R(x i ) = y i, Λ(X) = i error (X x i) for every i, Λ(x i )R(x i ) = Λ(x i )w(x i ) i.e. for every i, Λ(X)R(X) = Λ(X)w(X) mod (X x i ) Define the master polynomial Modular key equation G(X) = 1 i n (X x i) Λ(X)R(X) = Λ(X)w(X) mod G(X) Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

9 List-decoding and interpolation problem Fast algorithm via rational function reconstruction Modular key equation: ΛR = Λw mod G = λ = Λ, ω = Λw solution of the rational reconstruction problem { λr = ω mod G, deg(λ) e, deg(ω) e + k, λ monic. Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

10 List-decoding and interpolation problem Fast algorithm via rational function reconstruction Modular key equation: ΛR = Λw mod G = λ = Λ, ω = Λw solution of the rational reconstruction problem { λr = ω mod G, deg(λ) e, deg(ω) e + k, λ monic. Unique decoding assumption: e + k < n e = unique rational solution ω λ = Λw Λ = w cost bound O (n) using extended Euclidean algorithm [Modern Computer Algebra, von zur Gathen - Gerhard, 2013] Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

11 List-decoding and interpolation problem Non-unique decoding How to decode when more errors? transmission with e errors where e d min /2 possibly two (or more) code words at the same distance... the closest code word is not necessarily the one which was sent... Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

12 List-decoding and interpolation problem Non-unique decoding How to decode when more errors? transmission with e errors where e d min /2 possibly two (or more) code words at the same distance... the closest code word is not necessarily the one which was sent... Return a list of all code words at distance e (called list-decoding) Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

13 List-decoding and interpolation problem List-decoding problem List-decoding Reed-Solomon codes Input: x 1,..., x n the n distinct evaluation points in K, k the degree bound, e the error-correction radius, (y 1,..., y n ) the received word in K n List-decoding assumption: e < n kn [Guruswami - Sudan 1999] Output: list of all polynomials w in K[X] such that deg w k and #{i w(x i ) y i } e. Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

14 List-decoding and interpolation problem Towards the interpolation problem (1/3) For one solution w 1, modular key equation Λ 1 R = Λ 1 w 1 mod G where R(x i ) = y i, G(X) = 1 i n (X x i), Λ 1 (X) = i error 1 (X x i ). Possibly deg(λ 1 ) + deg(λ 1 w 1 ) n = deg G = no uniqueness of a rational solution to λ 1 R = ω 1 mod G Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

15 List-decoding and interpolation problem Towards the interpolation problem (2/3) Key equation Λ 1 (R w 1 ) = 0 For two solutions w 1 and w 2, key equation mod G Λ(R w 1 )(R w 2 ) = 0 mod G where Λ = i error 1 2 (X x i ) = gcd(λ 1, Λ 2 ). = w 1, w 2 are Y -roots of the bivariate polynomial Q(X, Y ) = Λ(Y w 1 )(Y w 2 ) Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

16 List-decoding and interpolation problem Towards the interpolation problem (3/3) Λ(R w 1 )(R w 2 ) = 0 mod G w 1, w 2 are Y -roots of Q(X, Y ) = Λ(Y w 1 )(Y w 2 ) = Λw 1 w 2 Λ(w 1 + w 2 )Y + ΛY 2 Similar properties for all solutions w 1,..., w l Properties of Q(X, Y ): deg Y Q is the number of solutions l coefficients in X of Q have small degree key equation Q(X, R) = 0 mod G that is, Q(x i, y i ) = 0 for every i Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

17 List-decoding and interpolation problem List-decoding: Guruswami-Sudan algorithm w solution: deg w k and #{i w(x i ) y i } e [Guruswami - Sudan, 1999] Interpolation step compute a polynomial Q(X, Y ) such that: Q(X, w) has small degree Q(X, w) has many roots w solution Q(X, w) = 0 Root-finding step find all Y -roots of Q(X, Y ), keep those that are solutions Here we focus on the Interpolation step. Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

18 List-decoding and interpolation problem The interpolation problem Bivariate interpolation Input: points {(x i, y i )} 1 i n, number of points n, degree weight k, weighted-degree bound b, list size l Output: a nonzero polynomial Q in K[X, Y ] such that (i) deg Y Q l, (list-size condition) (ii) deg X Q(X, X k Y ) < b, (weighted-degree condition) (iii) for 1 i n, Q(x i, y i ) = 0 (vanishing condition) [Guruswami - Sudan, 1999]: e < n kn solution exists for some well-chosen l Using Gaussian elimination, cost O(n ω ) (ω = exponent of mat. mult.) Note: in general e < n kn requires vanishing with some multiplicity µ. Here, for simplicity, µ = 1. Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

19 Fast algorithms using structured linear algebra Outline 1 List-decoding and interpolation problem 2 Fast algorithms using structured linear algebra 3 Fast algorithms using linear algebra over K[X] Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

20 Fast algorithms using structured linear algebra Overview [Olshevsky - Shokrollahi, 1999] linearize the vanishing condition on each point Vandermonde-like system, cost O(ln 2 ) [Roth - Ruckenstein, 2000] [Zeh - Gentner - Augot, 2011] linearize the reversed extended key equation Mosaic-Hankel system, cost O(ln 2 ) using an adapted version of [Feng - Tzeng, 1991] [Chowdhury - Jeannerod - Neiger - Schost - Villard, 2015] linearize the extended key equation Toeplitz-like system, cost O (l ω 1 n) using [Bostan - Jeannerod - Schost, 2007] Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

21 Fast algorithms using structured linear algebra Extended Key Equation [Roth - Ruckenstein, 2000] [Zeh - Gentner - Augot, 2011] Assuming x i pairwise distinct, vanishing condition: Q(x i, y i ) = 0 for i {1,..., n} extended key equation: Q(X, R) = 0 mod G where G = 1 i n (X x i) and i, R(x i ) = y i. List-size condition: linearize over K[X], Q(X, Y ) = 0 j l Q j(x)y j Weighted-degree condition: deg Q j (X) < b jk. Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

22 Fast algorithms using structured linear algebra Polynomial approximation problem Vanishing condition + list-size condition + weighted-degree condition Q j (X)R(X) j = 0 mod G(X) 0 j l deg Q j (X) < b jk for j l In [Roth - Ruckenstein, 2000] and [Zeh - Gentner - Augot, 2011] reverse all polynomials Q j (X)S j (X) = B(X) mod X n+b 1 0 j l deg Q j (X) < b jk for j l linearize as a mosaic-hankel system over K, find a solution in O(ln 2 ) using an adapted [Feng - Tzeng, 1991] Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

23 Fast algorithms using structured linear algebra Direct linearization with a companion matrix (1/2) [Chowdhury - Jeannerod - Neiger - Schost - Villard, 2015] Write Q j (X) = r<b jk Q(r) j X r, then the equation becomes 0 j l r<b jk Define the companion matrix C(G) = Q (r) j X r R(X) j }{{} F j (X) G G G G n 1 = 0 mod G(X) K n n Key property: multiplication by C(G) on the left is multiplication by X modulo G(X) Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

24 Fast algorithms using structured linear algebra Direct linearization (2/2) Solution nonzero vector in the nullspace of A = [A 0 A l ] where the block A j K n b jk is defined by its first column F (0) c (0) = j. and the subsequent columns c (r+1) = C(G) c (r) F (n 1) j Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

25 Fast algorithms using structured linear algebra Displacement rank A is n u where u = b + (b k) + (b 2k) + + (b lk) n = number of linear equations Q(x i, y i ) = 0 u = number of unknown coefficients of Q(X, Y ) Z n = K n n displacement operator A A Z n AZ T u Toeplitz structure Fact: A Z n AZ T u has rank l + 2 Conclusion: Toeplitz-like system with displacement rank l + 2 Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

26 Fast algorithms using structured linear algebra Complexity bound for this approach Solving the structured linear system [Bitmead - Anderson, 1980] [Morf, 1980] [Kaltofen, 1994] [Pan, 2001] [Bostan - Jeannerod - Schost, 2007] Two main operations: computing generators (compact representation) computing the first column, last column, first row of each block cost O (ln) solving the system number of equations n, displacement rank l + 2 cost O (l ω 1 n), probabilistic algorithm Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

27 Fast algorithms using structured linear algebra Structured system approach: O (l ω 1 n), probabilistic algorithm Goal: with same cost bound, deterministic algorithm? accepting repeated x i? Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

28 Fast algorithms using linear algebra over K[X] Outline 1 List-decoding and interpolation problem 2 Fast algorithms using structured linear algebra 3 Fast algorithms using linear algebra over K[X] Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

29 Fast algorithms using linear algebra over K[X] Bases of interpolants Recall we have points {(x i, y i )} 1 i n, weight k, degree bounds l and b the set of interpolants {(Q 0,..., Q l ) K[X] l+1 /Q(x i, y i ) = 0 for 1 i n} is a free K[X]-module of rank l + 1 interpolation basis = basis of this module matrix P in K[X] l+1 l+1 with rows P 0,..., P l interpolants P is a k-minimal interpolation basis if (deg X P 0 (X, X k Y ),..., deg X P l (X, X k Y )) is lexicographically minimal = an interpolant Q such that deg X Q(X, X k Y ) < b can be found in a k-minimal interpolation basis (unless no such Q exists) Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

30 Fast algorithms using linear algebra over K[X] Using polynomial lattice reduction [Alekhnovich, 2002] [Reinhard, 2003] [Beelen - Brander, 2010] [Bernstein, 2011] [Cohn - Heninger, 2011] [Cohn - Heninger, 2012] Compute a known basis of interpolants G G Y R R Y (Y R) 0 R Y l (Y R) 0 0 R 1 K[X] l+1 l+1 Lattice basis reduction k-minimal interpolation basis Return row with smallest weighted-degree Fastest known cost: O (l ω n) [Bernstein, 2011] [Cohn - Heninger, 2011] using deterministic reduction [Gupta - Sarkar - Storjohann - Valeriote, 2012] Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

31 Fast algorithms using linear algebra over K[X] Using order basis computation Derive extended key equation as in the structured approach Vanishing condition + list-size condition + weighted-degree condition Q j (X)F j (X) = 0 mod G(X) 0 j l deg(q 0, X k Q 1,..., X lk Q l ) < b Reverse it Hermite-Padé approximation Q j (X)S j (X) = B(X) mod X n+b 1 0 j l deg(q 0, X k Q 1,..., X lk Q l, XB) < b Cost O (l ω 1 n) using [Zhou - Labahn, 2012], deterministic Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

32 Fast algorithms using linear algebra over K[X] Structured system approach: O (l ω 1 n), probabilistic algorithm Order basis approach: O (l ω 1 n), deterministic algorithm Goal: with same cost bound, deterministic algorithm accepting repeated x i? Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

33 Fast algorithms using linear algebra over K[X] Iterative algorithm Sometimes called Kötter algorithm in coding theory. Related literature: [Feng - Tzeng, 1991] [Beckermann - Labahn, 1994 & 2000] [Kötter, 1996] [Nielsen - Høholdt, 1998] Algorithm (Interpolation) 1. P = identity 2. weights w = (0, k, 2k,..., lk) 3. For i from 1 to n do a. Evaluate: compute P 0 (x i, y i ),..., P l (x i, y i ) b. Choose pivot: π with smallest w π such that P π (x i, y i ) 0; w π = w π + 1 c. Eliminate: Pj (xi,yi ) For j π do P j = P j P π(x i,y i ) P π /* j π, P j (x i, y i ) = 0 */ P π = (X x i )P π /* P π (x i, y i ) = 0 */ After i iterations P is a w-minimal interpolation basis for points 1,..., i Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

34 Fast algorithms using linear algebra over K[X] Weights and size of the basis Iterative algorithm: Cost bound: O(ln 2 + l 2 n) O(ln 2 ) when l n accepts repeated x i Goal: divide-and-conquer version in O (l ω 1 n) In general, the size of a w-minimal basis is O(l 2 n) leaves no hope for a O (l ω 1 n) algorithm... Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

35 Fast algorithms using linear algebra over K[X] Weights and size of the basis Iterative algorithm: Cost bound: O(ln 2 + l 2 n) O(ln 2 ) when l n accepts repeated x i Goal: divide-and-conquer version in O (l ω 1 n) In general, the size of a w-minimal basis is O(l 2 n) leaves no hope for a O (l ω 1 n) algorithm... Assumption: weight w satisfies w w l O(n) (satisfied in coding theory applications) Consequence: a w-minimal basis has sum of row degrees O(n) in particular, size O(ln) Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

36 Fast algorithms using linear algebra over K[X] Ingredient 1: recursion base case Base case: l = n, goal: O (l ω ) Complete linearization over K (similar to linearization in [Beckermann - Labahn, 2000]) minimal interpolation basis minimal linear relations between rows fast computation of those relations, in O(l ω log l) Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

37 Fast algorithms using linear algebra over K[X] Ingredient 2: evaluation matrix Divide-and-conquer: P (1) for first n/2 points P (2) for last n/2 points with P (1) -dependent evaluation return P (2) P (1) Store an evaluation matrix E = evaluations of the global basis at points currently processed Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

38 Fast algorithms using linear algebra over K[X] Ingredient 2: evaluation matrix Divide-and-conquer: P (1) for first n/2 points P (2) for last n/2 points with P (1) -dependent evaluation return P (2) P (1) Store an evaluation matrix E = evaluations of the global basis at points currently processed Example: after first recursive call, E =. 0 0 P (1) 0 (x n/2+1, y n/2+1 ) P (1) 0 (x n, y n ) 0 0 P (1) 1 (x n/2+1, y n/2+1 ) P (1) 1 (x n, y n ) P (1) l (x n/2+1, y n/2+1 ) P (1) l (x n, y n ) Using assumption on the weight, Kl+1 n update of the evaluation matrix is computed in O (l ω 1 n) Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

39 Fast algorithms using linear algebra over K[X] Divide-and-conquer algorithm Central result: [Beckermann - Labahn, 1994 & 2000] transitivity of interpolation bases Algorithm (Fast bivariate interpolation) P (1) for first n/2 points, evaluations E, and weights w E updated evaluations, w updated weight P (2) for last n/2 points, evaluations E, weights w return P = P (2) P (1) P is a w-minimal interpolation basis Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

40 Fast algorithms using linear algebra over K[X] Divide-and-conquer algorithm Central result: [Beckermann - Labahn, 1994 & 2000] transitivity of interpolation bases Algorithm (Fast bivariate interpolation) P (1) for first n/2 points, evaluations E, and weights w E updated evaluations, w updated weight P (2) for last n/2 points, evaluations E, weights w return P = P (2) P (1) P is a w-minimal interpolation basis Using assumption on the weight, P = P (2) P (1) is computed in O (l ω 1 n) Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

41 Fast algorithms using linear algebra over K[X] Ingredient 3: assumption on the weights When w w l O(n), complexity bound equation: C(l, n) = 2C(l, n/2) + O (l ω 1 n)? Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

42 Fast algorithms using linear algebra over K[X] Ingredient 3: assumption on the weights When w w l O(n), complexity bound equation: C(l, n) = 2C(l, n/2) + O (l ω 1 n)? Problem: In the recursive call, n becomes n/2 but w is unchanged Preserve the assumption on w throughout recursive calls Compute a minimal interpolation basis for w = 0 Recover a w-minimal basis by change of weights Done in O (l ω 1 n) by computing a minimal nullspace basis [Zhou Labahn Storjohann, 2012] Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36

43 Fast algorithms using linear algebra over K[X] Conclusion Bivariate interpolation algorithm: Cost bound O (l ω 1 n) Deterministic No assumption on x i Returns a minimal interpolation basis With the same algorithm: Multiplicity µ i on each point (x i, y i ) (e.g. for soft-decoding of Reed-Solomon codes) Several variables Q(X, Y 1,..., Y s ) (e.g. for folded Reed-Solomon codes, or Private Information Retrieval) Remark: Does fast order basis as a special case Vincent Neiger (ENS de Lyon) Fast algorithms for multivariate interpolation problems PolSys, July / 36