2. S. Saraf ‘08

3. Z. Dvir, S. Kopparty, S. Saraf ‘09June 13-18, 2011 Mutliplicities @ CSR TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA 1

4. Agenda A technique for combinatorics, via algebra: Polynomial (Interpolation) Method + Multiplicity method List-decoding of Reed-Solomon Codes Bounding size of Kakeya Sets Extractor constructions (won’t cover) Locally decodable codes June 13-18, 2011 2 Mutliplicities @ CSR

5. Part I: Decoding Reed-Solomon Codes Reed-Solomon Codes: Commonly used codes to store information (on CDs, DVDs etc.) Message: C0, C1, …, Cdє F (finite field) Encoding: View message as polynomial: M(x) = i=0dCixi Encoding = evaluations: { M(®) }_{ ®є F } Decoding Problem: Given: (x1,y1) … (xn,yn) є F x F; integers t,d; Find: deg. dpoly through t of the n points. June 13-18, 2011 3 Mutliplicities @ CSR

6. List-decoding? If #errors (n-t) very large, then several polynomials may agree with t of n points. List-decoding problem: Report all such polynomials. Combinatorial obstacle: There may be too many such polynomials. Hope – can’t happen. To analyze: Focus on polynomials P1,…, PLand set of agreements S1 … SL. Combinatorial question: Can S1, … SLbe large, while n = | [jSj | is small? June 13-18, 2011 4 Mutliplicities @ CSR

7. List-decoding of Reed-Solomon codes Given L polynomials P1,…,PLof degree d;and sets S1,…,SL½ F £ F s.t. |Si| = t Si½ {(x,Pi(x)) | x 2 F} How small can n = |S| be, where S = [iSi ? Algebraic analysis from [S. ‘96, GuruswamiS ’98] basis of decoding algorithms. June 13-18, 2011 5 Mutliplicities @ CSR

8. List-decoding analysis [S ‘96] Construct Q(x,y) ≠ 0 s.t. Degy(Q) < L Degx(Q) < n/L Q(x,y) = 0 for every (x,y) 2 S = [iSi Can Show: Such a Q exists (interpolation/counting). Implies: t > n/L + dL) (y – Pi(x)) | Q Conclude: n ¸ L¢ (t – dL). (Can be proved combinatorially also; using inclusion-exclusion) If L > t/(2d), yield n ¸t2/(4d) June 13-18, 2011 6 Mutliplicities @ CSR

9. Focus: The Polynomial Method To analyze size of “algebraically nice” set S: Find polynomial Q vanishing on S; (Can prove existence of Q by counting coefficients … degree Q grows with |S|.) Use “algebraic niceness” of S to prove Q vanishes at other places as well. (In our case whenever y = Pi(x) ). Conclude Q zero too often (unless S large). … (abstraction based on [Dvir]’s work) June 13-18, 2011 7 Mutliplicities @ CSR

10. Improved L-D. Analysis [G.+S. ‘98] Can we improve on the inclusion-exclusion bound? One that works when n > t2/(4d)? Idea: Try fitting a polynomial Q that passesthrough each point with “multiplicity” 2. Can find with Degy < L, Degx < 3n/L. If 2t > 3n/L + dLthen (y-Pi(x)) | Q. Yields n ¸ (L/3).(2t – dL) If L>t/d, then n ¸t2/(3d). Optimizing Q; letting mult. 1, get n ¸t2/d June 13-18, 2011 8 Mutliplicities @ CSR

11. Aside: Is the factor of 2 important? Results in some improvement in [GS] (allowed us to improve list-decoding for codes of high rate) … But crucial to subsequent work [Guruswami-Rudra] construction of rate-optimal codes: Couldn’t afford to lose this factor of 2 (or any constant > 1). June 13-18, 2011 9 Mutliplicities @ CSR

12. Focus: The Multiplicity Method To analyze size of “algebraically nice” set S: Find poly Q zero on S (w. high multiplicity); (Can prove existence of Q by counting coefficients … degree Q grows with |S|.) Use “algebraic niceness” of S to prove Q vanishes at other places as well. (In our case whenever y = Pi(x) ). Conclude Q zero too often (unless S large). June 13-18, 2011 10 Mutliplicities @ CSR

13. Multiplicity = ? Over reals: Q(x,y) has root of multiplicity m+1 at (a,b) if every partial derivative of order up to m vanishes at 0. Over finite fields? Derivatives don’t work; but “Hasse derivatives” do. What are these? Later… There are {m+n choose n} such derivatives, for n-variate polynomials; Each is a linear function of coefficients of f. June 13-18, 2011 11 Mutliplicities @ CSR

14. Part II: Kakeya Sets June 13-18, 2011 12 Mutliplicities @ CSR

15. Kakeya Sets K ½Fn is a Kakeya set if it has a line in every direction. I.e., 8 y 2Fn9 x 2Fns.t. {x + t.y | t 2 F} ½ K F is a field (could be Reals, Rationals, Finite). Our Interest: F = Fq(finite field of cardinality q). Lower bounds. Simple/Obvious: qn/2· K ·qn Do better? Mostly open till [Dvir 2008]. June 13-18, 2011 13 Mutliplicities @ CSR

16. Kakeya Set analysis [Dvir ‘08] Find Q(x1,…,xn) ≠ 0 s.t. Total deg. of Q < q (let deg. = d) Q(x) = 0 for every x 2 K. (exists if |K| < qn/n!) Prove that (homogenous deg. d part of) Q vanishes on y, if there exists a line in direction y that is contained in K. Line L ½ K )Q|L = 0. Highest degree coefficient of Q|L is homogenous part of Q evaluated at y. Conclude: homogenous part of Q = 0. ><. Yields |K| ¸qn/n!. June 13-18, 2011 14 Mutliplicities @ CSR

17. Multiplicities in Kakeya [Saraf, S ’08] Fit Q that vanishes often? Good choice: #multiplicity m = n Can find Q ≠ 0 of individual degree < q,that vanishes at each point in K with multiplicity n, provided |K| 4n < qn Q|Lis of degree < qn. But it vanishes with multiplicity n at q points! So it is identically zero ) its highest degree coeff. is zero. >< Conclude: |K| ¸ (q/4)n June 13-18, 2011 15 Mutliplicities @ CSR

18. Comparing the bounds Simple: |K| ¸qn/2 [Dvir]: |K| ¸qn/n! [SS]: |K| ¸qn/4n [SS] improves Simple even when q (large) constant and n 1 (in particular, allows q < n) [MockenhauptTao, Dvir]: 9 K s.t. |K| ·qn/2n-1 + O(qn-1) Can we do even better? June 13-18, 2011 16 Mutliplicities @ CSR

19. Part III: Randomness Mergers & Extractors June 13-18, 2011 17 Mutliplicities @ CSR

20. Context One of the motivations for Dvir’s work: Build better “randomness extractors” Approach proposed in [Dvir-Shpilka] Following [Dvir] , new “randomness merger” and analysis given by [Dvir-Wigderson] Led to “extractors” matching known constructions, but not improving them … What are Extractors? Mergers? … can we improve them? June 13-18, 2011 18 Mutliplicities @ CSR

21. Randomness in Computation Readily available randomness Support industry: F(X) X Alg Random Distribution A Uniform Randomness Processors Prgs, (seeded) extractors, limited independence generators, epsilon-biased generators, Condensers, mergers, Distribution B June 13-18, 2011 19 Mutliplicities @ CSR

22. Randomness Extractors and Mergers Extractors: Dirty randomness  Pure randomness Mergers: General primitive useful in the context of manipulating randomness. k random variables  1 random variable June 13-18, 2011 20 Mutliplicities @ CSR (Uniform, independent … nearly) (Biased, correlated) + small pure seed (One of them uniform) (high entropy) (Don’t know which, others potentially correlated) + small pure seed

23. Merger Analysis Problem Merger(X1,…,Xk; s) = f(s), where X1, …, Xk2Fqn; s 2Fq andfis deg. k-1function mapping F Fn s.t.f(i) = Xi. (fis the curve through X1,…,Xk) Question: For what choices of q, n, k isMerger’soutput close to uniform? Arises from [DvirShpilka’05, DvirWigderson’08]. “Statistical high-deg. version” of Kakeya problem. June 13-18, 2011 21 Mutliplicities @ CSR

24. Concerns from Merger Analysis [DW] Analysis: Worked only if q > n. So seed length = log2 q > log2 n Not good enough for setting where k = O(1), and n 1. (Would like seed length to be O(log k)). Multiplicity technique: seems bottlenecked at mult= n. June 13-18, 2011 22 Mutliplicities @ CSR

25. General obstacle in multiplicity method Can’t force polynomial Q to vanish with too high a multiplicity. Gives no benefit. E.g. Kakeya problem: Why stop at mult= n? Most we can hope from Q is that it vanishes on all of qn; Once this happens, Q = 0, if its degree is < q in each variable. So Q|Lis of degree at most qn, so multn suffices. Using larger multiplicity can’t help! Or can it? June 13-18, 2011 23 Mutliplicities @ CSR

26. Extended method of multiplicities (In Kakeya context): Perhaps vanishing of Q with high multiplicity at each point shows higher degree polynomials (deg > q in each variable) are identically zero? (Needed: Condition on multiplicity of zeroes of multivariate polynomials .) Perhaps Q can be shown to vanish with high multiplicity at each point in Fn. (Technical question: How?) June 13-18, 2011 24 Mutliplicities @ CSR

27. Vanishing of high-degree polynomials Mult(Q,a) = multiplicity of zeroes of Qata. I(Q,a) = 1 if mult(Q,a) > 0and0 o.w. =min{1, mult(Q,a)} Schwartz-Zippel: for any S ½ F  I(Q,a) · d. |S|n-1 where sum is overa 2Sn Can we replace Iwithmult above? Would strengthen S-Z, and be useful in our case. [DKSS ‘09]: Yes … (simple inductive proof … that I can never remember) June 13-18, 2011 25 Mutliplicities @ CSR

28. Multiplicities? Q(X1,…,Xn) has zero of mult. m at a = (a1,…,an)if all (Hasse) derivatives of order < m vanish. Hasse derivative = ? Formally defined in terms of coefficients of Q, various multinomial coefficients and a. But really … The i = (i1,…, in)thderivative is the coefficient of z1i1…znin in Q(z + a). Even better … coeff. of zi in Q(z+x) (defines ith derivative Qi as a function of x; can evaluate at x = a). June 13-18, 2011 26 Mutliplicities @ CSR

29. Key Properties Each derivative is a linear function of coefficients of Q. [Used in [GS’98], [SS’09] .] (Q+R)i = Qi + Ri Q has zero of multm ata, and S is a curve that passes through a, then Q|Shas zero of multmat a. [Used for lines in prior work.] Qiis a polynomial of degree deg(Q) - jii (not used in prior works) (Qi)j ≠ Qi+j, but Qi+j(a) = 0 ) (Qi)j(a) = 0 Qvanishes with multmata )Qivanishes with multm - j iiat a. June 13-18, 2011 27 Mutliplicities @ CSR

30. Propagating multiplicities (in Kakeya) FindQthat vanishes with multmonK For everyiof order m/2, Q_ivanishes with multm/2on K. Conclude: Q, as well as all derivatives of Q of order m/2 vanish on Fn ) Qvanishes with multiplicity m/2 onFn Next Question: When is a polynomial (of deg >qn, or even qn) that vanishes with high multiplicity onqnidentically zero? June 13-18, 2011 28 Mutliplicities @ CSR

31. Back to Kakeya Find Q of degree d vanishing on K with multm. (can do if (m/n)n |K| < (d/n)n,dn > mn |K| ) Conclude Q vanishes on Fn with mult. m/2. Apply Extended-Schwartz-Zippel to conclude (m/2) qn < d qn-1 , (m/2) q < d , (m/2)nqn < dn = mn |K| Conclude: |K| ¸ (q/2)n Tight to within 2+o(1) factor! June 13-18, 2011 29 Mutliplicities @ CSR

32. Consequences for Mergers Can analyze [DW] merger when q > k very small, n growing; Analysis similar, more calculations. Yields: Seed length log q (independent of n). By combining it with every other ingredient in extractor construction: Extract all but vanishing entropy (k – o(k) bits of randomness from (n,k) sources) using O(log n) seed (for the first time). June 13-18, 2011 30 Mutliplicities @ CSR

33. Conclusions Combinatorics does have many “techniques” … Polynomial method + Multiplicity method adds to the body Supporting evidence: List decoding Kakeya sets Extractors/Mergers Others … June 13-18, 2011 31 Mutliplicities @ CSR

34. Other applications [Woodruff-Yekhanin ‘05]: An elegant construction of novel “LDCs (locally decodable codes)”. [Outclassed by more recent Yekhanin/Efremenko constructions.] [Kopparty-Lev-Saraf-S. ‘09]: Higher dimensional Kakeya problems. [Kopparty-Saraf-Yekhanin‘2011]: Locally decodable codes with Rate  1. June 13-18, 2011 32 Mutliplicities @ CSR

35. Conclusions New (?) technique in combinatorics … Polynomial method + Multiplicity method Supporting evidence: List decoding Kakeya sets Extractors/Mergers Locally decodable codes … More? June 13-18, 2011 33 Mutliplicities @ CSR

36. Thank You June 13-18, 2011 Mutliplicities @ CSR 34

37. Mutliplicity = ? Q vanishes with multiplicity 2 at (a,b): Q(a,b) = 0; el Q/del x(a,b) = 0; el Q/el y(a,b) = 0. el Q/del x? Hasse derivative; Linear function of coefficients of Q; Q zero with multiplicity m at (a,b) and C=(x(t),y(t)) is curve passing through (a,b), then Q|C has zero of multiplity m at (a,b). June 13-18, 2011 35 Mutliplicities @ CSR

38. Concerns from Merger Analysis Recall Merger(X1,…,Xk; s) = f(s), whereX1, …, Xk2Fqn; s 2Fq and f is deg. k-1 curves.t. f(i) = Xi. [DW08] Say X1random; Let Kbe such that ²fraction of choices of X1,…,Xk lead to “bad” curves such that²fraction of s’s such that Merger outputs value in Kwith high probability. Build low-deg. poly Qvanishing on K; Prove for “bad” curves,Q vanishes on curve; and so Q vanishes on ²-fraction of X1’s (and so ²-fraction of domain). Apply Schwartz-Zippel. >< June 13-18, 2011 36 Mutliplicities @ CSR

39. What is common? Given a set in Fqn with nice algebraic properties, want to understand its size. Kakeya Problem: The Kakeya Set. Merger Problem: Any set T ½Fnthat contains ²-fraction of points on ²-fraction of merger curves. If T small, then output is non-uniform; else output is uniform. List-decoding problem: The union of the sets. June 13-18, 2011 37 Mutliplicities @ CSR

40. Randomness Extractors and Mergers Extractors: Dirty randomness  Pure randomness Mergers: General primitive useful in the context of manipulating randomness. Given: k (dependent) random variables X1 … Xk, such that one is uniform. Add: small seed s (Additional randomness) Output: a uniform random variable Y. June 13-18, 2011 38 Mutliplicities @ CSR (Uniform, independent … nearly) (Biased, correlated) + small pure seed