A quadratically tight partition bound for classical communication complexity and query complexity

0.13 MB·

by  Rahul Jain

A quadratically tight partition bound for classical communication
complexity and query complexity
4
Rahul Jain∗ Troy Lee† Nisheeth K. Vishnoi‡
1
0
2

n
a
Abstract
J

8 In this workwe introduce, both for classicalcommunicationcomplexity and query complex-
1 ity, a modification of the partition bound introduced by Jain and Klauck [JK10]. We call it

the public-coin partition bound. We show that (the logarithm to the base two of) its commu-
]
C nication complexity and query complexity versions form, for all relations, a quadratically tight
C lower bound on the public-coin randomized communication complexity and randomized query
complexity respectively.
.
s
c
Keywords: Partition bound, communication complexity, lower bounds, linear programs.
[

1
v 1 Introduction
2
1
5 The partition bound introduced by Jain and Klauck [JK10] is known to be one of the strongest
4 lower bound methods in classical communication complexity and query complexity. It is known
.
1 to be stronger than most other lower bound methods, both in communication complexity and
0
query complexity, except its relationship with the information complexity lower bound method in
4
1 communication complexity is unknown. It is an interesting open question, in both these settings,
: as to how tight this lower bound method is. We are not aware, to the best of our knowledge,
v
i of any function or relation where this method is asymptotically weaker either for communication
X
complexity or for query complexity.
r
a In this work we introduce, both for communication complexity and query complexity, a mod-
ification of the partition bound which we call the public-coin partition bound. Analogous to the
partition bound, our new bound is also a linear-programming based lower bound method. We
show that (the logarithm to the base two of) its communication and query complexity versions
continue to form a lower bound on the public-coin communication complexity and randomized
query complexity respectively. In addition we show that the square of (the logarithm to the base
two of) its communication and query complexity versions form an upper bound on the public-coin
communication complexity and randomized query complexity respectively. Also it is easily seen
via their linear programs that our new bound is stronger than the partition bound for all relations,
both in communication complexity and query complexity.
∗CentreforQuantumTechnologiesandDepartmentofComputerScience,NationalUniversityofSingapore. Email:
[email protected]
†NanyangTechnological University and Center for Quantum Technologies. Email: [email protected]
‡Microsoft Research, India. Email: [email protected]
1
A quadratically tight partition bound ... 2 COMMUNICATION COMPLEXITY
2 Communication complexity
In this section we introduce our new bound in the communication complexity setting. Let us first
recall the partition bound of [JK10].
Definition 1 (Partition bound [JK10]) Let f ⊆ X ×Y ×Z be a relation. Let ǫ > 0. The ǫ-
partition bound of f, denoted prt (f), is given by the optimal value of the following linear program.
ǫ
Below R represents a rectangle in X ×Y and (x,y,z) ∈ X ×Y ×Z.
Primal Dual
min: XXwz,R max: (1−ǫ) X µx,y+ X φx,y
z R (x,y) (x,y)
∀(x,y): X X wz,R ≥1−ǫ, ∀(z,R): X µx,y+ X φx,y ≤1,
z:(x,y,z)∈fR:(x,y)∈R
(x,y)∈R:(x,y,z)∈f (x,y)∈R
∀(x,y): X Xwz,R =1, ∀(x,y):µx,y ≥0,φx,y ∈R .
R:(x,y)∈R z
∀(z,R):wz,R ≥0 .
Our new bound is defined as follows.
Definition 2 (Public-coin partition bound) Let f ⊆ X ×Y×Z be a relation. Let ǫ > 0. The
ǫ-public-coin partition bound of f, denoted pprt (f), is given by the optimal value of the following
ǫ
linear program. Below R represents a rectangle in X × Y and P represents a partition along
with outputs in Z; that is P = {(z ,R ),(z ,R ),··· ,(z ,R )}, such that {R ,··· ,R } form a
1 1 2 2 m m 1 m
partition of X ×Y into rectangles and ∀i∈[m],z ∈ Z.
i
Primal Dual
min: XXwz,R max: (1−ǫ) X µx,y+ X φx,y+λ
z R (x,y) (x,y)
∀(x,y): X X wz,R ≥1−ǫ, ∀(z,R): X µx,y+ X φx,y+vz,R ≤1,
z:(x,y,z)∈fR:(x,y)∈R (x,y)∈R:(x,y,z)∈f (x,y)∈R
∀(x,y): X Xwz,R =1, ∀P : X vz,R ≥λ,
R:(x,y)∈R z (z,R)∈P
∀(z,R):wz,R = X ap, ∀(x,y):µx,y ≥0,φx,y ∈R; ∀(z,R):vz,R ∈R,
P:(z,R)∈P λ∈R .
XaP =1,
P
∀(z,R):wz,R ≥0; ∀P :aP ≥0 .
We show that (the logarithm to the base two of) it is a lower bound on public-coin random-
ized communication complexity (please refer to [KN97] for standard definitions in communication
complexity).
Lemma 1 Let f ⊆ X ×Y ×Z be a relation. Let ǫ > 0. Let Rpub(f) represents the public-coin
ǫ
communication complexity of f with worst-case error ǫ. Then,
log pprt (f)≤ Rpub(f).
2 ǫ ǫ
2
A quadratically tight partition bound ... 2 COMMUNICATION COMPLEXITY
Proof This proof goes along similar lines as the proof of [JK10] for analogous result about the
partition bound.
Let P be a public coin randomized protocol for f with communication c d=ef Rpub(f) and worst
ǫ
caseerrorǫ. Forbinarystringr,letP representthedeterministiccommunicationprotocolobtained
r
from P on fixing the public coins to r. Every deterministic communication protocol amounts to
partitioning the inputs in X ×Y into rectangles and outputting an element in Z corresponding to
each rectangle in the partition. Let P = {(zr,Rr),(zr,Rr),··· ,(zr ,Rr )}, be the corresponding
r 1 1 2 2 m m
partition along with the outputs, that is {Rr,··· ,Rr } form a partition of X ×Y into rectangles
1 m
and ∀i ∈ [m],zr ∈ Z. Let q represent the probability of string r in P. For P define a′ d=ef q .
i r r Pr r
For the partitions P that do not correspond to any random string r in P, define a′ = 0. For any
P
(z,R) define,
w′ d=ef a′ .
z,R P
P:(z,R)∈P
X
It is easily seen that for all (x,y,z) ∈ X ×Y ×Z:
Pr[P outputs z on input (x,y)] = w′ .
z,R
R:(x,y)∈R
X
Since the protocol has error at most ǫ on all inputs we get the constraints:
∀(x,y): w′ ≥ 1−ǫ.
z,R
z:(x,y,z)∈fR:(x,y)∈R
X X
Also since the Pr[P outputs some z ∈ Z on input (x,y)] = 1, we get the constraints:
∀(x,y) : w′ = 1 .
z,R
z R:(x,y)∈R
X X
We also have by construction:
a′ = 1; ∀(z,R) :w′ ≥ 0; ∀P : a′ ≥ 0.
P z,R P
P
X
Therefore {w′ }∪{a′ } is feasible for the primal of pprt (f).
z,R P ǫ
We know that for each r, |P | ≤ 2c, since the communication in P is at most c bits. Hence,
r r
pprt (f)≤ w′ = a′ ·|P |≤ 2c a′ = 2c .
ǫ z,R Pr r Pr
z R r r
XX X X
(cid:3)
Next we show that the square of (the logarithm to the base two of) our new bound forms an
upper bound on the public-coin communication complexity.
Theorem 1 Let f ⊆ X ×Y ×Z be a relation. Let ǫ > 0. We have,
2
1
Rpub(f)≤ log pprt (f)+log +1 .
2ǫ 2 ǫ 2 ǫ
(cid:18) (cid:19)
3
A quadratically tight partition bound ... 2 COMMUNICATION COMPLEXITY
Proof Let pprt (f)= 2c. Let {w }∪{a } be an optimal solution for the primal. Let n be the
ǫ z,R P P
number of rectangles in P. We have,
a ·n = w = 2c.
P P z,R
P z,R
X X
Define B d=ef {P | n ≥ 12c}. Then δ d=ef a ≤ ǫ. Define a′ d=ef 1 a for P ∈/ B and a′ d=ef 0
P ǫ P∈B P P 1−δ P P
for P ∈ B. Define w′ d=ef a′ . Then we have,
z,R P:(z,R)∈P PP
P
∀(x,y): w′ ≥ 1−2ǫ,
z,R
z:(x,y,z)∈fR:(x,y)∈R
X X
∀(x,y): w′ = 1,
z,R
R:(x,y)∈R z
X X
∀(z,R) :w′ = a′,
z,R p
P:(z,R)∈P
X
a′ = 1,
P
P
X
∀(z,R) :w′ ≥ 0; ∀P :a′ ≥ 0 .
z,R P
We know that a partition with m rectangles can be realized by a communication protocol with
communication (⌈log m⌉)2 (arguments as in the proof of Theorem 2.11 of [KN97], we reproduce
2
them in Section A for completeness). Consider a public-coin communication protocol Π as follows.
1. Alice and Bob (usingpubliccoins) choose a P = {(z ,R ),(z ,R ),··· ,(z ,R )} with prob-
1 1 2 2 m m
ability a′ .
P
2. They communicate to realize the partition {R ,R ,··· ,R } with communication bounded
1 2 m
by (c+log 1 +1)2.
2 ǫ
3. If they end up with rectangle R , they output z .
i i
It is clear that the worst case communication of the protocol is bounded by (c+log 1 +1)2. The
2 ǫ
condition,
∀(x,y): w′ ≥ 1−2ǫ,
z,R
z:(x,y,z)∈fR:(x,y)∈R
X X
implies that the protocol has worst case error at most 2ǫ. Therefore,
2
1
Rpub(f)≤ log pprt (f)+log +1 .
2ǫ 2 ǫ 2 ǫ
(cid:18) (cid:19)
(cid:3)
4
A quadratically tight partition bound ... 3 QUERY COMPLEXITY
3 Query complexity
In this section we introduce our new bound in the query complexity setting.
Let f ⊆ {0,1}n ×Z be a relation. An assignment A : S → {0,1}l is an assignment of values
to some subset S of n variables (with |S| = l). We say that A is consistent with x ∈ {0,1}n
if x = A(i) for all i ∈ S. We write x ∈ A as shorthand for ‘A is consistent with x’. We
i
write |A| to represent the size of A which is the cardinality of S (not to be confused with the
number of consistent inputs). Furthermore we say that an index i appears in A, iff i ∈ S where
S is the subset of [n] corresponding to A. Let A denote the set of all assignments. Below we
assume x ∈ {0,1}n, A ∈ A and z ∈ Z. Below P represents a partition along with outputs in Z;
that is P = {(z ,A ),(z ,A ),··· ,(z ,A )}, such that {A ,··· ,A } form a partition of {0,1}n
1 1 2 2 m m 1 m
into assignments (that is for each x ∈ {0,1}n, there is a unique i ∈ [m] such that x ∈ A ) and
i
∀i∈ [m],z ∈ Z.
i
Let us first recall the partition bound of [JK10].
Definition 3 (Partition bound [JK10]) Let f ⊆ {0,1}n ×Z be a relation. Let ǫ > 0. The ǫ-
partition bound of f, denoted prt (f), is given by the optimal value of the following linear program.
ǫ
Primal Dual
min: XXwz,A·2|A| max: (1−ǫ)Xµx+Xφx
z A x x
∀x: X X wz,A≥1−ǫ, ∀(z,A): X µx+Xφx ≤2|A|,
z:(x,z)∈fA:x∈A x∈A:(x,z)∈f x∈A
∀x: X Xwz,A=1, ∀x:µx ≥0,φx ∈R .
A:x∈A z
∀(z,A):wz,A≥0 .
Our new bound is defined as follows.
Definition 4 (Public-coin partition bound) Let f ⊆ {0,1}n×Z be a relation. Let ǫ > 0. The
ǫ-public-coin partition bound of f, denoted pprt (f), is given by the optimal value of the following
ǫ
linear program.
Primal Dual
min: XXwz,A·2|A| max: (1−ǫ)Xµx+Xφx+λ
z A x x
∀x: X X wz,A≥1−ǫ, ∀(z,A): X µx+Xφx+vz,A≤2|A|,
z:(x,z)∈fA:x∈A x∈A:(x,z)∈f x∈A
∀x: X Xwz,A=1, ∀P : X vz,A≥λ,
A:x∈A z (z,A)∈P
∀(z,A):wz,A= X ap, ∀x:µx ≥0,φx ∈R; ∀(z,A):vz,A∈R,
P:(z,A)∈P λ∈R .
XaP =1,
P
∀(z,A):wz,A≥0; ∀P :aP ≥0 .
Weshowthat(thelogarithm tothebasetwoof)ournewboundisalower boundonrandomized
query complexity.
5
A quadratically tight partition bound ... 3 QUERY COMPLEXITY
Lemma 2 Let f ⊆ {0,1}n×Z be a relation. Let ǫ > 0. Let R (f) represent the randomized query
ǫ
complexity of f with worst case error ǫ. Then,
1
log pprt (f) ≤ R (f).
2 2 ǫ ǫ
Proof Our proof goes along arguments similar to [JK10] for analogous result for the partition
bound.
def
Let P be a randomized query algorithm which achieves c = R (f). Let P be the deterministic
ǫ r
query algorithm, arising from P, corresponding to random string r. We know that each such
deterministic query algorithm is a binary decision tree of depth at most c (please refer to [BW02]
for standard definitions related to query complexity). We note that the roots of a decision tree
(together) represent a partition of the inputs into assignments along with outputs in Z. Let
P = {(zr,Ar),(zr,Ar),··· ,(zr ,Ar )}, represent the partition along with outputs corresponding
r 1 1 2 2 m m
to random string r, where {Ar,··· ,Ar } form a partition of {0,1}n into assignments and ∀i ∈
1 m
[m],zr ∈ Z. Let q represent the probability of string r in P. For P define a′ d=ef q . For the
i r r Pr r
partitions P that do not correspond to any string r in P, define a′ = 0. For any (z,A) define,
P
w′ d=ef a′ .
z,A P
P:(z,A)∈P
X
As in the proof of Lemma 1, we can argue that {w′ } ∪ {a′ } is feasible for the primal of
z,A P
pprt (f). Note that for each (z,A) with w′ > 0, we have |A| ≤ c. Also |P | ≤ 2c since the depth
ǫ z,A r
of the corresponding binary decision tree is at most c. Now,
pprt (f) = w′ 2|A| ≤ 2c w′
ǫ z,A z,A
!
z A z A
XX XX
≤ 2c a′ ·|P | ≤ 22c a′ =22c .
Pr r Pr
!
r r
X X
Hence our result. (cid:3)
Next we show that the square of (the logarithm to the base two of) our new bound forms an
upper bound on randomized query complexity.
Theorem 2 Let f ⊆ {0,1}n ×Z be a relation. Let ǫ > 0. Then,
2
1
R (f) ≤ logpprt (f)+log .
2ǫ ǫ 2 ǫ
(cid:18) (cid:19)
Proof Let pprt (f)= 2c. Let {w }∪{a } be an optimal solution for the primal. We have,
ǫ z,A P
a ·2|A| = w ·2|A| = 2c.
P z,A
P A:(z,A)∈P z,A
X X X
6
A quadratically tight partition bound ... REFERENCES
Define B d=ef {P | ∃(z,A) ∈ P with |A| > c+log 1}. Then δ d=ef a ≤ ǫ. Define a′ d=ef 1 a
2 ǫ P∈B P P 1−δ P
for P ∈/ B and a′ d=ef 0 for P ∈ B. Define w′ d=ef a′ . Then we have,
P z,A P:(z,A)∈P PP
∀x: wP′ ≥ 1−2ǫ,
z,A
z:(x,z)∈fA:x∈A
X X
∀x: w′ = 1,
z,A
A:x∈A z
X X
∀(z,A) : w′ = a′ ,
z,A P
P:(z,A)∈P
X
a′ = 1,
P
P
X
∀(z,A) : w′ ≥ 0; ∀P :a′ ≥ 0 .
z,A P
We know that a partition with assignments each of length at most m can be realized by a query
protocol with m2 queries (arguments as in the proof of Theorem 11 of [BW02], we reproduce them
in Section B for completeness). Consider a randomized query protocol Π as follows.
1. Alice (randomly) chooses a P = {(z ,A ),(z ,A ),··· ,(z ,A )} with probability a′ .
1 1 2 2 s s P
2. She queries to realize the partition {A ,A ,··· ,A } with (c+log 1)2 queries.
1 2 s 2 ǫ
3. If she ends up with assignment A , she outouts z .
i i
It is clear that the worst case queries of the protocol is (c+log 1)2. The condition,
2 ǫ
∀x: w′ ≥ 1−2ǫ,
z,A
z:(x,z)∈fA:x∈A
X X
implies that the protocol has worst case error at most 2ǫ. Therefore,
2
1
R (f)≤ log pprt (f)+log .
2ǫ 2 ǫ 2 ǫ
(cid:18) (cid:19)
(cid:3)
Acknowledgment
Thework doneis supportedby the internal grants of the Center for Quantum Technologies (CQT),
Singapore. Part of the work done when N.K.V was visiting CQT.
References
[BW02] H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: A
survey. Theoretical Computer Science, 288(1):21–43, 2002.
[JK10] RahulJain,HartmutKlauck. Thepartitionboundforclassicalcommunicationcomplex-
ityandquerycomplexity. InProceeding of the 25th IEEE Conference on Computational
Complexity, (CCC), pp.247-258, 2010.
7
A quadratically tight partition bound ...B QUERY PROTOCOL TO REALIZE A PARTITION
[KN97] E.Kushilevitzand N.Nisan. Communication Complexity. CambridgeUniversity Press,
1997.
A Communication protocol to realize a partition
Let {R ,R ,...,R } be a partition of X ×Y. Let x and y be the inputs to Alice and
1 2 m
Bob respectively.
1. Alice determines if there exists an i ∈ [m] such that the row corresponding to
x intersects with R and the number of rectangles in {R ,R ,...,R } that row
i 1 2 m
intersect with R are at most m/2. If such i exists she communicates it to Bob
i
using ⌈log m⌉ bits. They both now consider {R ∩R ,R ∩R ,...,R ∩R } as a
2 1 i 2 i m i
partition of (X ×Y)∩R and repeat. If Alice cannot find any such i she indicates
i
this to Bob by sending 0.
2. On receiving 0 from Alice, Bob determines if there exists a j ∈ [m] such that
the column corresponding to y intersects with R and the number of rectangles
j
in {R ,R ,...,R } that column intersect with R are at most m/2. If such j
1 2 m j
exists he communicates it to Alice using ⌈log m⌉ bits. They both now consider
2
{R ∩R ,R ∩R ,...,R ∩R } as a partition of (X ×Y)∩R and repeat.
1 j 2 j m j j
We can note that either Alice or Bob must succeed in finding desired i,j respectively
sincethe rectangle that contains (x,y) satisfies therequirements in either 1. or 2. above
(since {R ,R ,...,R } is a partition of X×Y). Also thecommunication in each round
1 2 m
is at most ⌈log m⌉ and the number of (non-empty) rectangles surviving after each
2
round reduce by a factor of 2. Hence the process ends after at most ⌈log m⌉ rounds.
2
The total communication hence is bounded by (⌈log m⌉)2.
2
B Query protocol to realize a partition
Let {A ,A ,...,A } be a partition of {0,1}n such that |A |≤ m for each i ∈[s]. Let x
1 2 s i
be the string in the database.
1. Alice queries the bits of x corresponding to A . If the bits revealed are consistent
1
with A then she considers A as desired assignment and stops.
1 1
2. Ifthebitsrevealed arenotconsistent withA then wenotethatonebitis revealed
1
for A for all i ∈ [s] (since {A ,A ,...,A } is a partition of {0,1}n). Hence the
i 1 2 s
size of each A (consistent with the bits revealed so far) reduces by at least 1.
i
Alice considers now the new set of (modified) A s which are consistent with the
i
bits revealed so far and repeats.
We note that the number of such rounds is at most m and in each round at most m
bits are revealed. Hence the total number of queries is at most m2.
8