Communication Complexity

Transcription

1 Communication Complexity Jie Ren Adaptive Signal Processing and Information Theory Group Nov 3 rd, 2014 Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

2 1 E. Kushilevitz and N. Nisan, Communication Complexity, Cambridge University Press, L. Lovasz, Communication Complexity: A Survey, in Paths, Flows, and VLSI Layout, B. H. Korte, Ed., Springer Verlag: Berlin T. Lee and A. Shraibman, Lower Bounds in Communication Complexity: A Survey, Now Publishers Inc., A. C. Yao, Some Complexity Questions Related to Distributed Computing, Proc. of 11th ACM Symposium on Theory of Computing, 1981, P. Beame and J. Lawry, Randomized versus Nondeterministic Communication Complexity, Proc. of 24th ACM Symposium on Theory of Computing, 1992, A. K. Chandra, M. L. Furst and R. J. Lipton, Multi-party Protocols, Proc. of 15th ACM Symposium on Theory of Computing, 1983, P. B. Miltersen, N. Nisan, S. Safra and A. Wigderson, On Data Structures and Asymmetric Communication Complexity, Proc. of 27th ACM Symposium on Theory of Computing, 1995, Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

3 Deterministic Communication Complexity Outline 1 Deterministic Communication Complexity Problem Setup Protocol Tree Combinatorial Rectangles Fooling Sets Rectangle Rank 2 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Nondeterministic Communication Complexity: definitions Randomized Communication Complexity Distributional Complexity and Discrepancy 3 Some Analysis Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

4 Deterministic Communication Complexity Problem Setup Outline 1 Deterministic Communication Complexity Problem Setup Protocol Tree Combinatorial Rectangles Fooling Sets Rectangle Rank 2 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Nondeterministic Communication Complexity: definitions Randomized Communication Complexity Distributional Complexity and Discrepancy 3 Some Analysis Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

5 Deterministic Communication Complexity Problem Setup Problem Setup Alice x X 0/1 0/1 f(x, y) Bob y Y f(x, y) {0, 1} Two party communication Each knows an input x X /y Y Let one/both sides compute a function f with no error Only care about communication cost Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

6 Deterministic Communication Complexity Problem Setup Problem Setup Alice x X 0/1 0/1 f(x, y) Bob y Y f(x, y) {0, 1} Sending binary messages f usually binary Deterministic protocol P: who to talk/what to send Communication cost: sum of total bits/rounds CC(P) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

7 Deterministic Communication Complexity Problem Setup Deterministic Communication Complexity Alice x X 0/1 0/1 f(x, y) Bob y Y f(x, y) {0, 1} D(f ) = min P max (x,y) X Y CC(P) (1) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

8 Deterministic Communication Complexity Problem Setup A Naive Upper Bound Proposition (naive upper bound): For every function f : X Y Z D(f ) log 2 X + log 2 Z (2) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

9 Deterministic Communication Complexity Problem Setup A Naive Upper Bound Example: MAX of the union Alice and Bob hold subsets x, y {1,..., n} respectively, and they with to compute MAX (x, y). D(MAX ) 2 log 2 n (3) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

10 Deterministic Communication Complexity Protocol Tree Outline 1 Deterministic Communication Complexity Problem Setup Protocol Tree Combinatorial Rectangles Fooling Sets Rectangle Rank 2 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Nondeterministic Communication Complexity: definitions Randomized Communication Complexity Distributional Complexity and Discrepancy 3 Some Analysis Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

11 Deterministic Communication Complexity Protocol Tree Definition: Protocol Tree Definition : A protocol P over domain X Y with range Z is a binary tree where each internal node v is labeled either by a function a v : X {0, 1} or by a function b v : Y {0, 1}, and each leaf is labeled with an element z Z. The communication cost CC(P) will be the depth of the protocol tree. Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

12 Deterministic Communication Complexity Protocol Tree Example: Protocol Tree a 1 (x = 1, 2) = 0 a 1 (x = 3) = 1 b 2 (y = 1, 2) = 0 b 2 (y = 3) = 1 1 b 3 (y = 1) = 1 b 3 (y = 2, 3) = a 4 (x = 1) = 0 a 4 (x = 2, 3) = 1 f(x, y) = { 1 x y 0 OTH 0 1 x {1, 2, 3} y {1, 2, 3} Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

13 Deterministic Communication Complexity Protocol Tree Why Binary Message? Entropy not involved - simple? No block coding (compute a single function) Worst case - always exists p = 1/2 s.t. h 2 (p) = 1 Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

14 Deterministic Communication Complexity Protocol Tree One Side Compute f Vs. Both Sides Compute f Equivalent setup Need one more bit if f is binary Second last round: one side must know f (x, y) Some Lower Bounds of Communication Complexity D(f ) : unknown for general f Interested in lower bounds Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

15 Deterministic Communication Complexity Combinatorial Rectangles Outline 1 Deterministic Communication Complexity Problem Setup Protocol Tree Combinatorial Rectangles Fooling Sets Rectangle Rank 2 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Nondeterministic Communication Complexity: definitions Randomized Communication Complexity Distributional Complexity and Discrepancy 3 Some Analysis Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

16 Deterministic Communication Complexity Combinatorial Rectangles Combinatorial Rectangles Definition: Let P be a protocol and v be a node of the protocol tree. R v is the set of inputs (x, y) that reach node v. Proposition: If L is the set of leaves of a protocol P, then {R l, l L} is a partition of X Y. Definition: A combinatorial rectangle is a subset R X Y such that R = A B for some A X and B Y. Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

17 Deterministic Communication Complexity Combinatorial Rectangles Proposition: R X Y is a rectangle iff (x 1, y 1 ) R & (x 2, y 2 ) R (x 1, y 2 ) R. (4) Proposition: For every protocol P and leaf l, R l is a rectangle Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

18 Deterministic Communication Complexity Combinatorial Rectangles a 1 (x = 1, 2) = 0 a 1 (x = 3) = 1 b 2 (y = 1, 2) = 0 b 2 (y = 3) = 1 1 b 3 (y = 1) = 1 b 3 (y = 2, 3) = 0 0 y=1 y=2 y=3 1 a 4 (x = 1) = 0 a 4 (x = 2, 3) = 1 x= x= x= Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

19 Deterministic Communication Complexity Combinatorial Rectangles Rectangle lower bound Definition: A subset R X Y is called f -monochromatic if f is fixed on R. Theorem 1.17 (Kushilevitz & Nisan): If any partition of X Y into f -monochromatic rectangles requires at least t rectangles, then log 2 t D(f ) (5) P partitions X Y into monochromatic rectangles Depth of its protocol tree: log 2 t Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

20 Deterministic Communication Complexity Combinatorial Rectangles y=1 y=2 y=3 x= x= x= D(f) log 2 5 D(f) = 3 Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

21 Deterministic Communication Complexity Fooling Sets Outline 1 Deterministic Communication Complexity Problem Setup Protocol Tree Combinatorial Rectangles Fooling Sets Rectangle Rank 2 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Nondeterministic Communication Complexity: definitions Randomized Communication Complexity Distributional Complexity and Discrepancy 3 Some Analysis Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

22 Deterministic Communication Complexity Fooling Sets Motivation: If we exhibit a large set of input pairs such that no two of them can be in a single monochromatic rectangle, then the number of partitions of P must be large y1 y2 x1 z? x2? z Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

23 Deterministic Communication Complexity Fooling Sets Definition : Let f : X Y {0, 1}. A set S X Y is called a fooling set if there exits a value z {0, 1} such that For every (x, y) S, f (x, y) = z For every two distinct pairs (x 1, y 1 ) and (x 2, y 2 ) in S, either f (x 1, y 2 ) z or f (x 2, y 1 ) z y1 y2 x1 z? x2? z Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

24 Deterministic Communication Complexity Fooling Sets Fooling set lower bound Theorem 1.20 (Kushilevitz & Nisan) : If f has a fooling set S of size t, then log 2 t D(f ) (6) Proof : No monochromatic rectangle contains more than one element of S Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

25 Deterministic Communication Complexity Fooling Sets Example: Alice and Bob each hold an n-bit integer 0 x, y < 2 n. The greater than or equal to function, GTE(x, y), is defined to be 1 iff x y. D(GT ) = n + 1 (7) y=0 y=1 y=2 y=3 x= x= x= x= Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

26 Deterministic Communication Complexity Fooling Sets Example: Alice and Bob each hold an n-bit integer 0 x, y < 2 n. The greater than or equal to function, GTE(x, y), is defined to be 1 iff x y. D(GT ) = n + 1 (8) y=0 y=1 y=2 y=3 x= x= x= x= Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

27 Deterministic Communication Complexity Rectangle Rank Outline 1 Deterministic Communication Complexity Problem Setup Protocol Tree Combinatorial Rectangles Fooling Sets Rectangle Rank 2 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Nondeterministic Communication Complexity: definitions Randomized Communication Complexity Distributional Complexity and Discrepancy 3 Some Analysis Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

28 Deterministic Communication Complexity Rectangle Rank Motivation: Give communication complexity lower bound in an algebraic way Definition: Associate with every function f : X Y {0, 1} a matrix M f of dimensions X Y. The rows/columns of M f are indexed by the elements of X /Y. Then rank(f ) is the linear rank of M f over the field of reals. y=0 y=1 y=2 y=3 x= x= x= x= Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

29 Deterministic Communication Complexity Rectangle Rank Rank lower bound Theorem 1.28 (Kushilevitz & Nisan): For any function f : X Y {0, 1} log 2 rank(f ) D(f ) (9) y=0 y=1 y=2 y=3 x= x= x= x= Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

30 Deterministic Communication Complexity Rectangle Rank Proof: Let L 1 be the set of leaves in which the output is 1. For each l L 1, { 1 if (x, y) Rl M l (x, y) = 0 otherwise (10) M f = l L 1 M l (11) and rank(m f ) l L 1 rank(m l ) L 1 L (12) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

31 Deterministic Communication Complexity Rectangle Rank Rank lower bound Theorem 1.28 (Kushilevitz & Nisan): For any function f : X Y {0, 1} log 2 rank(f ) D(f ) (13) y=0 y=1 y=2 y=3 x= x= x= x= Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

32 Deterministic Communication Complexity Rectangle Rank Rank upper bound Proposition 2.3 (Lovasz 1990): For any function f : X Y {0, 1} D(f ) rank(f ) (14) Proof: We know that row rank = column rank = rank(f ), and we can form the row vector space with dimension rank(f ). We then claim that there are at most 2 rank(f ) distinct row vectors, the reason is because, although the coefficients for the polynomials that represent the row vectors can be real, the entries of the matrix M(f ) can only be 0 or 1. Hence we can build a protocol as follows: Alice merge the repeated rows of M(f ) on the table to have M (f ), and then sends the index of row in M (f ) that contains x. Bob compute f (x, y) based on what he received. Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

33 Deterministic Communication Complexity Rectangle Rank Summary Concept: protocol tree, combinatorial rectangles, fooling sets, rank Naive upper bound: log 2 X + 1 Rectangle lower bound: log 2 t r Fooling set lower bound: log 2 t f Rank lower bound: log 2 rank(f ) Rank upper bound: rank(f ) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

34 Nondeterministic CC & Randomized CC Outline 1 Deterministic Communication Complexity Problem Setup Protocol Tree Combinatorial Rectangles Fooling Sets Rectangle Rank 2 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Nondeterministic Communication Complexity: definitions Randomized Communication Complexity Distributional Complexity and Discrepancy 3 Some Analysis Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

35 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Outline 1 Deterministic Communication Complexity Problem Setup Protocol Tree Combinatorial Rectangles Fooling Sets Rectangle Rank 2 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Nondeterministic Communication Complexity: definitions Randomized Communication Complexity Distributional Complexity and Discrepancy 3 Some Analysis Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

36 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Motivation How good are the rectangle lower bounds? Relaxing the need to partition by allowing covering of the same space y=0 y=1 y=2 y=3 y=0 y=1 y=2 y=3 x= x= x= x= x= x= x= x= Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

37 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Motivation: Alice has a n-bit string x {0, 1} n, Bob has a n-bit string y {0, 1} n, either side wants NEQ(x, y). D(NEQ) = n (15) Now assume a third person knows everything: x,y and NEQ(x, y) and want to convince Alice and Bob, Alice and Bob need to check the correctness Sends the index of the first bit differs Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

38 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Setup: A prover, who sees both x and y, is trying to convince Alice and Bob that f (x, y) = 1. If f (x, y) 1, then Alice and Bob must be able to detect the proof is wrong. f (x, y) = 1 z P(x, y, z) = 1 (16) f (x, y) = 0 z P(x, y, z) = 0 (17) Two-stage nondeterministic protocol P N 1 Alice and Bob receive a message z from the third person. 2 Alice and Bob run a deterministic protocol P D,z based on z. The interesting cost in this protocol is the maximum length of z plus the number of bits exchanged over all x, y. Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

39 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Alternative Setup: Let f : X Y {0, 1} be a function. Let L = {(x, y) : f (x, y) = 1}. A successful nondeterministic protocol for f consists of functions f A : X {0, 1} k {0, 1} and f B : Y {0, 1} k {0, 1} such that 1 (x, y) L, z {0, 1} k s.t. f A (x, z) f B (y, z) = 1 2 (x, y) L, z {0, 1} k, f A (x, z) f B (y, z) = 0 One stage nondeterministic protocol 1 Alice and Bob receive a message z and compute f (x, y) successfully. The interesting cost in this protocol is the length of z only. Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

40 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Two-stage One-stage: Given a two-stage nondeterministic protocol with k bits first stage cost and d bits second stage cost, we can always build a one-stage nondeterministic protocol by adding the d bits deterministic communication to the witness z with each party accepting if the message agrees with what Alice and Bob would have said in the protocol. Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

41 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation In the language of Rectangles : A prover, who sees both x and y, is trying to convince Alice and Bob that f (x, y) = 1 by broadcasting a 1-monochromatic rectangle that cover (x, y). By Nondeterministic we mean: this 1-monochromatic rectangle may not be unique y=0 y=1 y=2 y=3 x= x= x= x= Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

42 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: definitions Outline 1 Deterministic Communication Complexity Problem Setup Protocol Tree Combinatorial Rectangles Fooling Sets Rectangle Rank 2 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Nondeterministic Communication Complexity: definitions Randomized Communication Complexity Distributional Complexity and Discrepancy 3 Some Analysis Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

43 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: definitions Definitions: Let f : X Y {0, 1} be a binary function. C P (f ): the smallest number of leaves in a protocol P C D (f ): the smallest number of monochromatic rectangles that partition X Y C(f ): the smallest number of monochromatic rectangles needed to cover X Y C z (f ): the smallest number of monochromatic rectangles needed to cover the z-inputs Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

44 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: definitions Proposition 2.2 (Kushilevitz & Nisan): For all f : X Y {0, 1}, ( log 2 C 0 (f ) + C 1 (f ) ) log 2 C D (f ) log 2 C P (f ) D(f ) (18) Theorem 29 (Lee & Shraibman): Let f : X Y {0, 1} be a function, N 1 (f ) = log 2 C 1 (f ) (19) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

45 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: definitions Proof: N 1 (f ) log 2 C 1 (f ) : Let {R l } be a cover. If f (x, y) = 1, the players receive the index l that (x, y) R l N 1 (f ) log 2 C 1 (f ) : Let k = N 1 (f ), and let f A : X {0, 1} k {0, 1}, f B : Y {0, 1} k {0, 1} be functions in the one-stage nondeterministic protocol. Define R z = {(x, y) : f A (x, z) f B (y, z) = 1}, R z is a rectangle. We claim {R z, z {0, 1} k } is a cover of the 1s. This is because by the definition of nondeterministic protocol: (x, y) pairs that f (x, y) = 1, there must exists some z s.t. (x, y) R z. (x, y) pairs that f (x, y) = 0, (x, y) R z for all z. Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

46 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: definitions Definition (Lee & Shraibman): N 1 (f ) = log 2 C 1 (f ) (20) N 0 (f ) = log 2 C 0 (f ) (21) N(f ) = max(n 1 (f ), N 0 (f )) (22) Definition (Kushilevitz & Nisan): N 1 (f ) = log 2 C 1 (f ) (23) N 0 (f ) = log 2 C 0 (f ) (24) ( N(f ) = log 2 C 0 (f ) + C 1 (f ) ) (25) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

47 Nondeterministic CC & Randomized CC Randomized Communication Complexity Outline 1 Deterministic Communication Complexity Problem Setup Protocol Tree Combinatorial Rectangles Fooling Sets Rectangle Rank 2 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Nondeterministic Communication Complexity: definitions Randomized Communication Complexity Distributional Complexity and Discrepancy 3 Some Analysis Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

48 Nondeterministic CC & Randomized CC Randomized Communication Complexity Motivation: Introduce randomness in the protocol r A and r B : flip coins Allow protocols that may have error Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

49 Nondeterministic CC & Randomized CC Randomized Communication Complexity Randomized Protocol Tree Definition : A randomized protocol P over domain X Y with range Z is a binary tree where each internal node v is labeled either by a function a v : X R A {0, 1} or by a function b v : Y R B {0, 1}, and each leaf is labeled with an element z Z. Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

50 Nondeterministic CC & Randomized CC Randomized Communication Complexity Definition: Let P be a randomized protocol. All the probabilities below are over random choices of r A and r B. P computes a function f with zero error P computes a function f with ɛ error if for all (x, y) P[P(x, y) = f (x, y)] 1 ɛ (26) P computes a function f with one-sided ɛ error if for all (x, y) such that f (x, y) = 0 P[P(x, y) = 0] = 1, (27) and for all (x, y) such that f (x, y) = 1, P[P(x, y) = 1] 1 ɛ (28) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

51 Nondeterministic CC & Randomized CC Randomized Communication Complexity Definition: Let f : X Y {0, 1} be a binary function. We consider the following complexity measure for f R 0 (f ) is the minimum average case cost of a randomized protocol that computes f with zero error R ɛ (f ) is the minimum worst case cost of a randomized protocol that computes f with error ɛ. We typically use ɛ = 1/3 R 1 ɛ (f ) is the minimum worst case cost of a randomized protocol that computes f with one-sided error ɛ. Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

52 Nondeterministic CC & Randomized CC Randomized Communication Complexity Why we care these measures: worst case zero error = deterministic for all average case ɛ error, there exists a worst case problem that can convert to Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

53 Nondeterministic CC & Randomized CC Randomized Communication Complexity Proposition: Given a protocol P that makes an error ɛ/2 and the average number of bits exchanged is t, it can always be modified as follows: execute P as long as at most 2t/ɛ bits are exchanged, if the protocol finishes, use its output, otherwise output 0. This gives a worst case cost 2t/ɛ with error upper bounded by ɛ. Proof: t = cc p(r a, r b, x, y) Hence, r a,r b,x,y = cc 2t/ɛ cc>2t/ɛ cc p(r a, r b, x, y) + cc>2t/ɛ cc p(r a, r b, x, y) cc p(r a, r b, x, y) 2t Pr[cc > 2t/ɛ] ɛ (29) Pr[err] ɛ Pr[P ends] + 1 Pr[P not ends] 2 ɛ 2 + t 2t/ɛ = ɛ (30) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

54 Nondeterministic CC & Randomized CC Randomized Communication Complexity 1 For all 0 < ɛ ɛ < 1/2, R ɛ (f ) O(log ɛ ɛ R ɛ (f )) (31) 2 For all 0 < ɛ 1/2, R ɛ (f ) Rɛ 1 (f ) O(log ɛ 1 R 0 (f )) (32) 3 For all 0 < ɛ 1/2, R 0 (f ) = Θ(max[Rɛ 1 (f ), Rɛ 1 (not(f ))]) (33) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

55 Nondeterministic CC & Randomized CC Randomized Communication Complexity Proof of Property 1: We first prove a similar result for the 1-sided error problem: for all 0 < ɛ ɛ < 1/2, Rɛ 1 (f ) O(log ɛ ɛ R 1 (f )) (34) ɛ Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

56 Nondeterministic CC & Randomized CC Randomized Communication Complexity Proof of Property 1: Given a randomized protocol P with worst case cost T bits and one-sided error no greater than ɛ < 1/2, we can build a new protocol P with worst case cost nt bits by simply running P n times. In the new protocol, Alice and Bob will claim f (x, y) = 1 if and only if there exists at least one time among the n repeating protocols that they will output 1. Now we bound the error for the new protocol P : P[err f (x, y) = 0] = 0 (35) P[err f (x, y) = 1] = P[all n trails output 0 f (x, y) = 1] = (ɛ ) n (36) Therefore, if we repeat P log ɛ ɛ times, we can guarantee to reduce the one-sided error to ɛ. Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

57 Nondeterministic CC & Randomized CC Randomized Communication Complexity Proof of Property 1: Now we prove property 1. We still run P n times, each gives an output X i, i {1,..., n}. In the new protocol, Alice and Bob will claim f (x, y) = 1 if and only if 1 n X i > 1 2 Now we bound the error for the new protocol P : n ( n P[err f (x, y) = 0] i P[err f (x, y) = 1] i ) (ɛ ) i (1 ɛ ) n i (37) i= n/2 (38) (ɛ ) n n ( ) n (ɛ ) i (1 ɛ ) n i i i= n/2 (39) (ɛ ) n Therefore, if we repeat P log ɛ ɛ times, we can also guarantee to reduce the two-sided error to ɛ. Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

58 Nondeterministic CC & Randomized CC Distributional Complexity and Discrepancy Outline 1 Deterministic Communication Complexity Problem Setup Protocol Tree Combinatorial Rectangles Fooling Sets Rectangle Rank 2 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Nondeterministic Communication Complexity: definitions Randomized Communication Complexity Distributional Complexity and Discrepancy 3 Some Analysis Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

59 Nondeterministic CC & Randomized CC Distributional Complexity and Discrepancy Distributional Complexity Motivation: Consider probability distributions over the inputs Definition: Let µ be a probability distribution on X Y. The (µ, ɛ)-distributional communication complexity of f, D µ ɛ (f ), is the cost of the best deterministic protocol that gives the correct answer for f with a probability at least 1 ɛ. Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

60 Nondeterministic CC & Randomized CC Distributional Complexity and Discrepancy Discrepancy Motivation: Allow those rectangles that partition the support to be almost f -monochromatic. Definition: Let f : X Y {0, 1} be a function, R be any rectangle, and µ be a probability distribution on X Y, Denote Disc µ (R, f ) = P µ [f (x, y) = 0 & (x, y) R] P µ [f (x, y) = 1 & (x, y) R] The discrepancy of f according to µ is, (40) Disc µ (f ) = max R Disc µ(r, f ) (41) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

61 Nondeterministic CC & Randomized CC Distributional Complexity and Discrepancy Discrepancy Proposition 3.28 (Kushilevitz & Nisan): For every function f : X Y {0, 1}, every probability distribution µ on X Y, and every ɛ 0, D µ 1/2 ɛ (f ) log 2(2ɛ/Disc µ (f )) (42) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

62 Nondeterministic CC & Randomized CC Distributional Complexity and Discrepancy Discrepancy Proof: Given any P with c bits to compute f, we have 2ɛ P[P(x, y) = f (x, y)] P[P(x, y) f (x, y)] = (P[P(x, y) = f (x, y)&(x, y) R l ] l P[P(x, y) f (x, y)&(x, y) R l ]) P [f (x, y) = 0 & (x, y) R l ] P [f (x, y) = 1 & (x, y) R l ] µ µ l 2 c Disc µ (f ) (43) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

63 Some Analysis Outline 1 Deterministic Communication Complexity Problem Setup Protocol Tree Combinatorial Rectangles Fooling Sets Rectangle Rank 2 Nondeterministic CC & Randomized CC Nondeterministic Communication Complexity: Motivation Nondeterministic Communication Complexity: definitions Randomized Communication Complexity Distributional Complexity and Discrepancy 3 Some Analysis Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

64 Some Analysis Recall Definitions: Let f : X Y {0, 1} be a binary function. C P (f ): the smallest number of leaves in a protocol P C D (f ): the smallest number of monochromatic rectangles that partition X Y C(f ): the smallest number of monochromatic rectangles needed to cover X Y C z (f ): the smallest number of monochromatic rectangles needed to cover the z-inputs Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

65 Some Analysis Recall Proposition: For all f : X Y {0, 1}, log 2 C(f ) log 2 C D (f ) log 2 C P (f ) D(f ) (44) C(f ) = C 0 (f ) + C 1 (f ) (45) Definition: The nondeterministic communication complexity, N 1 (f ) = log 2 C 1 (f ) (46) N 0 (f ) = log 2 C 0 (f ) (47) N(f ) = log 2 C(f ) (48) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

66 Some Analysis Protocol partition number Theorem 2.8 (Kushilevitz and Nisan): The protocol partition number of a function determines the deterministic communication complexity. log 2 C P (f ) D(f ) 2 log 3/2 C P (f ) (49) Proof: Given any protocol P with t number of leaves, it can be converted into a balanced protocol. Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

67 Some Analysis Protocol partition number Proof: Given any protocol P with t number of leaves, there must exist an internal node v such that t/3 < t v 2t/3 (50) We build a new protocol based on this internal node: 1 Alice and Bob determine whether or not (x, y) R v 2 If (x, y) R v, Alice and Bob recursively solve f in the rectangle R v. 3 If (x, y) R v, Alice and Bob recursively solve f on X Y where f (x 1, y 1 ) = { f (x1, y 1 ) if (x 1, y 1 ) R v 0 otherwise (51) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

68 Some Analysis Protocol partition number Analysis: Step 1 Requires 2 bits In Step 3, we take P and replace Tree(v) by a 0-leaf, we get a protocol for f with t t v + 1 leaves, hence D(t) 2 + D(2t/3) (52) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

69 Some Analysis Recall Proposition: For all f : X Y {0, 1}, log 2 C(f ) log 2 C D (f ) log 2 C P (f ) D(f ) 2 log 3/2 C P (f ) (53) Definition: The nondeterministic communication complexity, N 1 (f ) = log 2 C 1 (f ) (54) N 0 (f ) = log 2 C 0 (f ) (55) N(f ) = log 2 C(f ) (56) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

70 Some Analysis Deterministic CC Vs. Nondeterministic CC How good is the rectangle lower bound? D(f )? = O(log C D (f )) (57) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

71 Some Analysis Deterministic CC Vs. Nondeterministic CC Theorem 2.11 (Kushilevitz & Nisan): For every function f : X Y {0, 1}, D(f ) = O(N 0 (f )N 1 (f )) (58) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

72 Some Analysis Deterministic CC Vs. Nondeterministic CC Property: Let R = S T be a 0-monochromatic rectangle, and let R = S T be a 1-monochromatic rectangle, then either S S = or T T =. Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

73 Some Analysis Deterministic CC Vs. Nondeterministic CC Proof of Theorem 2.11 (Kushilevitz & Nisan): We give a protocol P as follows, Alice and Bob search for a 0-rectangle that contains the input (x, y), and they conclude f (x, y) = 1 if they fail. In each round, Alice and Bob exchange log 2 C 1 (f ) + 1 bits and reduce the number of alive 0-rectangles by a factor of 2. There will be no more than log 2 C 0 (f ) rounds, hence D(f ) CC(P) = O(log 2 C 0 (f )(log 2 C 1 (f ) + 1)) = O(N 0 (f )N 1 (f )) (59) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

74 Some Analysis Deterministic CC Vs. Nondeterministic CC Proof of Theorem 2.11 (Kushilevitz & Nisan): In each round, the players do the following: 1 Alice outputs f (x, y) = 0 if no 0-rectangles are alive. Otherwise, Alice looks for a 1-rectangle that contains row x and intersects in rows with at most half of the alive 0-rectangles and send the name of this 1-rectangle. 2 Bob looks for a 1-rectangle that contains column y and intersects in columns with at most half of the alive 0-rectangles and send the name of this 1-rectangle. Otherwise, Bob outputs f (x, y) = 0 Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

75 Some Analysis Deterministic CC Vs. Nondeterministic CC Protocol Analysis: If f (x, y) = 0, it must belong to some 0-rectangle R, then R remains alive during the protocol. Therefore, if no 0-rectangle is alive, f (x, y) must be 1 If neither Alice nor Bob can find a 1-rectangle to announce (which means both of them output f (x, y) = 1), we claim this output must be correct by the property. Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

76 Some Analysis Public coin Theorem (Theo. 3 in Newman 1991, Theo in K & N): Let f : {0, 1} n {0, 1} n {0, 1} be a function. For every δ > 0 and every ɛ > 0, we have R δ+ɛ (f ) R pub ɛ (f ) + O(logn + logδ 1 ) (60) a set of t(δ, n) = O(n/δ 2 ) public coin protocols with error ɛ + δ Alice tells Bob which protocol to use log 2 t = O(log n + log δ 1 ) Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77

77 Some Analysis Randomized CC Vs. Distributional CC Theorem (Theo. 3 in Yao 1979, Theo in K & N): Rɛ pub (f ) = max µ Dµ ɛ (f ) (61) The randomized protocol is correct for every distribution µ with probability 1 ɛ Use min-max theorem of zero-sum game Jie Ren (Drexel ASPITRG) CC Nov 3 rd, / 77