Communication Cost of Transforming a Nearest Plane Partition to the Voronoi Partition

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by  V. A. Vaishampayan

Communication Cost of Transforming a Nearest
Plane Partition to the Voronoi Partition
Vinay A. Vaishampayan Maiara F. Bollauf
Engineering Science and Physics Dept. Institute of Mathematics, Statistics and Computer Science
City University of New York-College of Staten Island University of Campinas, Sao Paulo, Brazil
Staten Island, NY USA
Abstract—Weconsidertheproblemofdistributedcomputation the vectors which determine the faces of the Voronoi cell,
7 of the nearest lattice point for a two dimensional lattice. An are ±(1,0), ±(ρcosθ,ρsinθ) and ±(ρcosθ−1,ρsinθ). We
1 interactive model of communication is considered. We address
thus consider lattices with generator matrix V as above, with
0 the problem of reconfiguring a specific rectangular partition, a
ρ ≥ 1. From an additional symmetry, and in order to avoid
2 nearest plane, or Babai, partition, into the Voronoi partition.
Expressionsarederivedfortheerrorprobabilityasafunctionof indeterminate solutions we restrict θ such that 0 < ρcosθ <
n
thetotalnumberofcommunicatedbits.Withaninfinitenumber 1/2. Performance at the endpoints 0 and 1/2 can be obtained
a
J of allowed communication rounds, the average cost of achieving by taking limits. More generally, the generator matrix of the
0 zero error probability is shown to be finite. For the interactive lattice is represented by matrix V with ith column vi, i =
model, with a single round of communication, expressions are
3 1,2,...,n. Thus Λ = {Vu, u ∈ Zn}. The (i,j) entry of V
obtained for the error probability as a function of the bits
exchanged. We observe that the error exponent depends on the is vi,j, thus vi = (v1i,v2i,...,vni). The Voronoi cell V(λ)
T] lattice. is defined as the set of all x for which λ ∈ Λ is the closest
I Index terms—Lattices, lattice quantization, Communication lattice point.
s. complexity,distributedfunctioncomputation,Voronoicell,Babai In a companion paper [3] we have developed upper bounds
c cell, rectangular partition. for the communication complexity of constructing a specific
[
rectangular partition for a given lattice along with a closed
1 I. INTRODUCTION form expression for the error probability Pe. The partition is
v Given a lattice 1 Λ⊂Rn, the closest lattice point problem referred to as a Babai partition and is an approximation to the
8 istofindforeachx=(x ,x ,...,x )∈Rn,thepointλ∗(x) Voronoi partition for a given lattice.
5 1 2 n
whichminimizestheEuclideandistance(cid:107)x−λ(cid:107),λ∈Λ.Here, Theremainderofthepaperisorganizedasfollows.Previous
4
8 weassumethatxiisavailableatnodeSiinanetworkofnodes work is presented in Sec. II, assumptions and a preliminary
0 and study the communication cost of this search. We consider analysis are presented in Sec. III, the interactive model is
. an interactive model in which each node S communicates analyzed and quantizer design is presented for a single round
1 i
0 with every other node so that every node such that each of communication (Sec. IV), for unbounded rounds of com-
7 node can determine λ∗(x). Since this may not be possible munication(Sec.V).Numericalresultsandadiscussionarein
1 in general, let λ(x) denote the lattice point determined by Sec. VI. A summary and conclusions is provided in Sec. VII.
:
v concernednodeswhencomputationishalted.Theobjectiveis
i todeterminethetradeoffbetweenthecommunicationrequired II. PREVIOUSWORK
X
and the probability of error P := Pr(λ(X) (cid:54)= λ∗(X)) for a Communication complexity [16], [8] is the minimum
e
r
a known probability distribution on X. amount of communication required to compute a function in
We will assume that generator matrix V of Λ has the upper adistributedsetting.Informationtheoreticcharacterizationsof
triangular form communication complexity are developed for the two node
(cid:18) (cid:19)
1 ρcosθ problem in [14]. Two models are considered: a centralized
V =
0 ρsinθ model, and an interactive model where two messages are
exchanged (one round of communication in our model). Two
where the columns of V are basis vectors for the lattice. The
terminal interactive communication is studied in considerable
associated quadratic form is f(x,y) = x2 + 2ρcosθ xy +
detail in [13], and the benefit of an unbounded number of
ρ2y2. It is known that this form is reduced if and only if
messages is demonstrated. An important and relevant con-
2|ρcosθ| ≤ 1 ≤ ρ2 and the three smallest values taken by f
tribution in [11], [13] is the the strict benefit that inter-
overintegeru=(x,y)(cid:54)=0are1,ρ2,and1−2|ρcosθ|+ρ2see
active communication provides for the computation of the
e.g. Th. II, Ch. II, [4]. Based on a result due to Voronoi,
Boolean AND function. Another stream of related work has
Th. 10, Ch. 21, [5], it follows that the relevant vectors, i.e.
originsinasymptoticquantizationtheory.Theproblemoffine
1AlatticeisadiscreteadditivesubgroupofRn.Thereaderisreferredto quantization for detection problems is addressed in [15], [2]
[5]fordetails. and [6]. More recently, the design of fine scalar quantizers
for distributed function computation with a squared error and X , respectively. Note that since B(λ) is rectangular, X
2 1
distortionmeasureisconsideredin[12]andsucceedingworks. and X are independent.
2
Significant benefits, especially in the interactive setting are Stage-II communication is broken up into rounds, one
obtained. round corresponds to two messages, one from each node in a
predefinedorder.Bothorderings,12and21willbeconsidered.
III. PRELIMINARYANALYSIS
IV. INTERACTIVE,SINGLEROUNDOFCOMMUNICATION
At the conclusion of Stage-I, node S is in possession of
i
X , i = 1,2, and X ∈ B(0). We denote the rectangular cells
i
of the partition at the conclusion of Stage-II by R(i), i =
1,2,...,R.AssociatedwitheachcellR(i)isadecisionλ(i).
Following steps similar to the analysis above, it follows that
P , the error probability at the conclusion of Stage-II is
e,II
given by
R
(cid:88) (cid:88) (cid:92)
P = Area(R(i) V(λ(cid:48)))/Area(B(0)). (1)
e,II
Fig.1. Voronoiregion,Babaipartitionandthreerelevantvectors
i=1λ(cid:48)(cid:54)=λ(i)
The optimum decision rule follows immediately: λ(i) =
We consider a two-stage approach for determining λ(x). (cid:84)
argmax Area(R(i) V(λ)).
In Stage-I, [3], a point λ (x) (defined next) is determined, λ
np
using the nearest plane algorithm [1], and assuming an in-
teractive model. We refer to λ (x) as the Babai point.
np
When the generator matrix for Λ is in upper triangular form,
the nearest plane algorithm determines λ = Vu, with
np
u =[(x −(cid:80)n v u )/v ],i=n,n−1,...,1([x]isthe
i i j=i+1 ij j ii
nearestintegertox).Thenearestplanealgorithmpartitionsthe
plane into congruent Babai cells (rectangles), each of volume
|detV|. The Babai cell associated with lattice vector λ is
denoted B(λ). Once again we denote the nearest lattice point
by λ∗(x).
The analysis of P , the error at the conclusion of
e,I
Stage-I, justifies the modeling assumptions made for the
Stage-II analysis. Specifically, P = (cid:80) Pr(λ∗(X) (cid:54)=
e,I λ∈Λ Fig.2. AtypicalverticalstripcreatedbyS1anditspartitionintothreeparts
λ|X ∈ B(λ))Pr(X ∈ B(λ)). Since an error occurs byS2 (left).ProbabilitydistributionQ(x)whichunderliesthecalculationof
if X is closer to some λ(cid:48) (cid:54)= λ, it follows that H(U2|U1)isontheright.
P = (cid:80) (cid:80) Pr(X ∈ B(λ)(cid:84)V(λ(cid:48))|X ∈
e,I λ∈Λ λ(cid:48)(cid:54)=λ
B(λ))Pr(X ∈ B(λ)). Assuming that p(x) is approximately The scheme for the 12 order is described first. To begin,
constant over each B(λ)2, it follows that Pe,I ≈ node S1 sends U1 = i to S2 indicating an interval of length
(cid:80)λ∈Λ(cid:80)λ(cid:48)(cid:54)=λArea(V(λ(cid:48))(cid:84)B(λ)/Area(B(λ))Pr(X ∈ δi that X1 lies in. This effectively partitions (−1/2,1/2], the
B(λ)). Since the Babai and Voronoi partitions are support of X1 into cells of length δi (and equivalently, parti-
invariant under translations by lattice vectors, it follows tions B(0) into vertical strips of widths δi), i = 1,2,...,N.
that Pe,I ≈(cid:80)λ(cid:48)(cid:54)=0Area(V(λ(cid:48))(cid:84)B(0))/Area(B(0)). Based on this information, S2 makes a decision λ(U1,X2)
From the above analysis for Pe,I, and assuming an inter- andcommunicatesthisdecisionbacktoS1 usingmessageU2.
active model for Stage-I, it follows that at the end of Stage-I, Effectively, S2 partitions each aforementioned vertical strip
λ (X) is known to each node S . Each node thus subtracts into at most three parts using at most two horizontal cuts
np i
off ith coordinate λ from X . The result is also referred or thresholds. The location of each cut is determined by the
np,i i
to as X for notational convenience. We will assume that location of the appropriate boundary wall of V(0). A typical
i
the resulting X = (X ,X ) is uniformly distributed over situation is illustrated in Fig. 2. In this figure, vertical strip i
1 2
B(0). For the lattice that we consider B(0) = (−1/2,1/2]× is partitioned into three rectangles, R0(i), R−1(i) and R1(i),
(−(ρ/2)sinθ,(ρ/2)sinθ]. Since B(0) has length L = 1 where R0(i) corresponds to points decoded to λ = 0, the
and height H = ρsinθ, we have p(x ) = p = 1/L and other two rectangles are decoded to neighboring points. The
1
q(x ) = q = 1/H, where p,q are the marginal pdf’s of X probability of error event E is given by
2 1
(cid:88)(cid:88)
2This is justified under the assumption that the lattice point density is Pe,II = Pr(E|X ∈Rj(i)), (2)
suitablyhigh. i j
whereiindexesthestripsandj indexestherectangleswithina N equal-length intervals, I into N equal-length intervals,
2 −1 1
strip.ForthecutsshowninFig.2,andassumingtheboundary I into 1 interval, I into N equal-length intervals and I
0 1 1 2
lines have slopes s and s and bin size δ we get into N equal-length intervals. The lengths of the intervals
1 2 2
I , I and I are denoted L , L and L , respectively and
δ2 0 1 2 0 1 2
Pr(E|X ∈R (i)) = [(α2+(1−α)2)|s |+ L=L +2L +2L . Let L=(L ,L ,L ,L ,L )/L. Note
j 2|detV| 1 0 1 2 0 1 1 2 2
thatLbehaveslikeaprobabilitydistribution.AlsofromFig.3,
(β2+(1−β)2)|s2|] H = cosθ(1−ρcosθ)/2sinθ, H = ρcos2θ/2sinθ and
1 22
≥ δ2(|s |+|s |)/4|detV|, (3) H =cosθ(1−2ρcosθ)/2sinθ.
1 2 21
From (4), it follows that
and equality holds when α=β =1/2. Thus
P =α /N +α /N , (7)
2 N e,II 1 1 2 2
(cid:88)(cid:88)
P =(1/4) |s |δ2. (4)
e,II l,i i whereα =L (H +H )/2|detV|,α =H L /2|detV|.
1 1 1 22 2 21 2
l=1 i=1 From (5) it follows that
The information rate for Stage-II communication is then
R=H(U1)+H(U2|U1).Theinformationratesarecalculated H(U1)=H(L)+(2L1/L)log2N1+(2L2/L)log2N2. (8)
next. Node S sends H(U ) bits where
1 1 InordertocalculateH(U |U ),wewriteU =(V,W),where
2 1 1
(cid:88)N V identifies the interval IV, (one of I−2,I−1,I0,I1,I2) in
H(U )= (δ /L)log (L/δ ). (5) which X lies and W identifies the bin index, relative to V.
1 i 2 i
i=1 Thus from (6) we obtain
NodeS sendsH(U |U )bitswhichisobtainedbyaveraging
2 2 1 H(U |U )=
2 1
the entropy H(Q(x )) of probability distribution Q(x) =
(Q−1(x),Q0(x),Q1(1x)) over bins of X1 by 2NL22 (cid:80)Nw=21H(U2|V =2,W =w)+
(cid:88)N 2NL11 (cid:80)Nw=11H(U2|V =1,W =w). (9)
H(U |U )= (δ /L)H(Q(x )), (6)
2 1 i i The objective is to minimize P over N and N subject
e,II 1 2
i=1 to the constraint that H(U |U )+H(U ) ≤ R. We observe
2 1 1
where xi is, say, the midpoint of the ith bin. Here that the term H(U2|U1) is weakly dependent on N1 and N2
Q1(x),Q−1(x),Q0(x) are the probabilities that X2 exceeds (see(11)below).ThusweminimizePe,II withaconstrainton
the upper threshold, is smaller than the lower threshold and H(U ). An approximate parametric solution to this optimiza-
1
liesinbetweenthetwothresholds,respectively,givenX1 =x. tionproblemintermsofN2 isgivenbyN1(N2)=(cid:100)αα1L2L2N12(cid:101).
In terms of N we then obtain P = α /N (N )+α /N
2 e 1 1 2 2 2
and R = H(U |U ) + H(L) + (2L /L)log N (N ) +
2 1 1 2 1 2
(2L /L)log N . We note here that α L /α L < 1 for
2 2 2 2 1 1 2
π/3 ≤ θ ≤ π/2. Thus N ≤ N and N > 1 only if
1 2 1
N >α L /α L .
2 2 1 1 2
A. Asymptotic Analysis
We study the behavior of P and R as N →∞. Based
e,II 2
on the information presented, it follows immediately that
α L
P = 2(1+ 1)(1+o(1))
e,II N L
2 2
(cid:18) (cid:19)
α L
R = H(U |U )+H(L)+(2L /L)log 1 2 +
2 1 1 2 α L
2 1
(2(L +L )/L)log N +o(1), (10)
1 2 2 2
where lim o(1)=0. Since
N2→∞
Fig. 3. Babai and Voronoi cells, with key points labeled. x1,x2 are the κ := lim H(U |U )
horizontal,verticalcoordinates,resp. 2 1
N2→∞
(cid:90) t−1
We now specialize the analysis to V(0) and B(0) for the = (2/L) H(Q(x))dx (11)
given lattice. The geometry of the lattice, with all the signif- −1/2
icant boundary points, lengths, heights, and slopes is shown it follows that
in Fig. 3. We identify four thresholds t = (ρcosθ−1)/2,
−2 lim P 2LR/2(L1+L2) =
t = (−ρcosθ)/2, t = −t and t = −t and five e,II
−1 1 −1 2 −2 R→∞
iIn1ter=vals(tI1−,t22]=a(n−d1/I22,t=−2],(tI2−,11/=2](.t−W2e,t−p1a]r,tiIti0on=I(t−−21,itn1t]o, α2(cid:16)1+ LL12(cid:17)(cid:16)αα12LL21(cid:17)(L1L+1L2) 2L2((κL+1H+L(L2))). (12)
The above expression is in terms of geometric parameters V. INTERACTIVE:INFINITEROUNDS
of B(0). An expression in terms of probabilities associated
withB(0)isperhapsmoreintuitiveexpressionforinformation
theoristsandisobtainedbydefiningP=(P ,P ,P ,P ,P ),
0 1 1 2 2
with P =L /L, i=0,1,2. In terms of the P ’s we obtain
i i i
lim P 2R/(1−P0) =
e,II
R→∞
α2(cid:16)1+ LL21(cid:17)(cid:16)αα21LL12(cid:17)(L1L+1L2) 2(κ(+1−HP(0P))). (13)
Observe that P = 1 − ρcosθ. Thus for ρ = 1 and θ ∈
0
(π/3,π/2),therateatwhichP decaystozerodependson
e,II
θ and is maximum when θ →π/3.
Fig.4. Redsolidlinesshowpartitionafterthefirstroundofcommunication.
We note here that identical results are obtained using the
Dashedlinesarecreatedinthesecondroundofcommunication.
heaviermachineryofpointdensityfunctions.Wehavechosen
to present the work using a simpler approach.
We now analyze the interactive model in which an infinite
B. Interactive: Single Round, Reversed Steps
number of communication rounds are allowed. Node S com-
2
Analysis is now presented for 21 order of communication. municatesfirst.InRound-1,NodeS partitionsthesupportof
2
We will summarize the description of the quantizer, and X into three intervals as in Sec. IV-B (see Figs. 4 and 3),
2
present the final results, since the derivation is similar. In J , and J , and J . Let random variable U be the index
−1 0 1 2
fact, the derivation for this case is simpler. The support of the interval in which X lies. In Round-1, upon receiving
2
for X2 is partitioned into 2N + 1 bins. With reference to U2 and if U2 = 1, node S1 partitions the support of X1
Fig. 3, a single large bin spans the interval J0 := (τ−1,τ1]. into three intervals I−1 = (−1/2,t−2], I0 = (t−2,t1] and
The interval J−1 := (−ρsinθ/2,τ−1] is partitioned into N I1 = (t1,1/2] (see Fig. 3). If U2 = −1, the support of
intervals of equal length ∆. The same holds for the interval X is partitioned into intervals −I ,−I ,−I . If U = 0,
2 1 0 −1 2
J1 =(τ1,ρsinθ/2].Equalbinsizesarejustifiedbysymmetry. no partitioning step is taken. Random variable U1 describes
Observe that there is only a single step size parameter here, the interval in which X lies. Let Pr(U = i) =: Q ,
1 2 i
as opposed to the 12 case, where two step sizes were called i = −1,0,1. Let P = Pr(U = i|U = 1), i = −1,0,1.
i 1 2
for. With H = ρsinθ, the vertical (X2) dimension of B(0) Let Q=(Q0,Q1,Q2) and P =(P0,P1,P2).
and with H1 as in Fig. 3, let H0 := (H − 2H1). Let We assume that for every round, upon sending Ui, node Si
Q:=(H0/H,H1/H,H1/H) and Q0 =H0/H. updates Xi by subtracting the lower endpoint of the interval
S2 sends U2, the index of the bin that X2 lies in, and that it lies in.
partitions B(0) into horizontal strips. S1 then partitions each The partition of B(0) into rectangular cells after a single,
horizontal strip into at most three parts using at most two and after two rounds of communication is shown in Fig. 4.
vertical cuts or thresholds, referred to as the left and right Define a rectangular cell to be error-free if its interior does
thresholds, and sends U1 to S2. For a given x2, let P−1(x2) not contain a boundary of V(0). Of the seven rectangles in
be the probability that X1 lies to the left of the left threshold the partition at the conclusion of Round-1, all but four are
(U1 = −1), P1(x2) the probability that X1 lies to the right error-free. If X = (X1,X2) lies in an error-free rectangle,
of the right threshold (U1 = 1) and P0(x2) the probability communication halts after Round-1. Else a second round of
that X1 lies in between the two thresholds (U1 = 0). Let communicationoccurs,duringwhichatotalof2bitsarecom-
P(x)=(P−1(x),P0(x),P1(x)).Withanequivalentdefinition municated. This process of partitioning and communication
of κ, namely, continues until each node determines that X lies in an error
free rectangle of the current partition. When the algorithm
κ := lim H(U |U )
1 2
N→∞ halts, Pe,II = 0. Let N(X), R(X) denote the number of
(cid:90) τ−1 rounds, and number of bits communicated, respectively, when
= (2/H) H(P(x))dx, (14)
the algorithm halts. Let R¯ = E[R(X)] and N¯ = E[N(X)]
−ρsinθ/2
denote averages over X.
it follows that the total number of bits sent is given by
Theorem1. Fortheinteractivemodelwithunlimitedroundsof
R=H(Q)+(1−Q )log N +κ. (15)
0 2
communication, a nearest plane partition can be transformed
and into the Voronoi partition using, on average, a finite number
Pe,II =β/N (16) of bits and rounds of communication. Specifically,
with β = (1/2)((2L2 + L1)/L)(H1/H). Taking limits we R¯ =H(Q)+(1−Q )H(P)+4(1−P )(1−Q ) (18)
0 0 0
obtain
and
lim Pe,II2R/(1−Q0) =β2(H(Q)+κ)/(1−Q0). (17) N¯ =1+2(1−P )(1−Q ). (19)
N→∞ 0 0
Proof: We assume that an optimum entropy code is used performance at the same rate for the 12 and 21 sequences
(thus if U =0, the codeword length is log (1/Q ) bits). The highlights the importance of selecting the sequence of order
2 2 0
termH(Q)+(1−Q )H(P)in(18)isthecostofresolvingthe in which nodes communicate in this case. Under the infinite
0
Round-1partition.AttheconclusionofRound-1,ifX belongs roundinteractivemodel,thehexagonallatticeistheworstcase,
toaregionwhichisnoterror-free,thentheaveragenumberof with R¯ =2.42 bits.
bits transmitted is obtained by the following argument. At the
VII. SUMMARYANDCONCLUSIONS
conclusionofRound-1,therearetwokindsoferrorrectangles,
determinedbythesignoftheslopeoftheboundaryofV(0)in For the nearest lattice point problem, we have considered
the rectangle. Note that error rectangles are designed so that theproblemofrefininganapproximationtothenearestlattice
the boundary of V(0) is a diagonal of the corresponding rect- pointtoobtainthetruenearestlatticepoint,andhaveobtained
angle.LetanerrorrectanglehavelengthLandheightH.Ifthe the communication cost of doing so. More specifically, we
slope is positive, construct the binary expansion 1−x /L = have assumed that the approximate lattice point is obtained
1
(cid:80)∞ b 2−i,elseconstructx /L=(cid:80)∞ b 2−i.Inbothcases using Babai’s nearest plane algorithm. The quality of the
i=1 i 1 i=1 i
construct the binary expansion x /H = (cid:80)∞ c 2−i. From approximation has been measured by the error probability.
2 i=1 i
the independence and uniformity of X and X it follows Aninteractivecommunicationmodelhasbeenconsidered.We
1 2
that the bits B and C are independent unbiased Bernoulli have shown that the rate of decay of the error probability is
i i
random variables. Further, the algorithm halts after n rounds, lattice dependent. Somewhat surprisingly, the communication
with 2n total bits communicated if and only if B (cid:54)= C , cost has been observed to be finite when an infinite number
i i
i < n and B = C . Thus, given X in an error rectangle, of communication rounds are possible.
n n
Pr(R(X) = 2n) = Pr(N(X) = n) = 2−n. The result
REFERENCES
follows immediately by computing the average.
[1] L. Babai. “On Lova´sz lattice reduction and the nearest lattice point
Remark 1. The communication strategy is implicit in the problem”,Combinatorica,6(1),1-13. 1986.
proof. Note that the finite value for R¯ is because of the rapid [2] G.R.Benitz,G.R.andJ.A.Bucklew,“Asymptoticallyoptimalquantizers
fordetectionofiiddata”.IEEETransactionsonInformationTheory,vol.
decrease with n of the probability of halting at n rounds.
35,No.2,pp.316-325,March1989.
[3] M.Bollauf,V.A.Vaishampayan,andS.I.R.Costa,“OntheCommuni-
Remark 2. This result has interesting implications when
cationCostoftheDeterminingAnApproximateNearestLatticePoint”,
viewed in the context of distributed classification problems. submitted,IEEEInt.Symp.Inform.Th.,2017.
Suppose we have an optimum two-dimensional classifier with [4] J.W.S.Cassels,AnIntroductiontotheGeometryofNumbers.Springer-
Verelag,BerlinHeidelberg,1997.
separating boundaries that are not axis aligned and also
[5] J.H.ConwayandN.J.A.Sloane,SpherePackings,LatticesandGroups,
a suboptimal classifier with separating boundaries that are 3rded.,Springer-Verlag,NewYork,1998.
axis aligned, e.g. a k-d tree. We expect the communication [6] R.Gupta,andA.O.Hero,“High-ratevectorquantizationfordetection”,
IEEETransactionsonInformationTheory,vol.49,No.8,pp.1951-1969,
complexity of refining the approximate rectangular classifier
Aug.2003.
to the optimum classifier to be finite. [7] J. Korner and K. Marton. “How to encode the modulo-two sum of
binary sources”, IEEE Transactions on Information Theory 25(2), 219-
VI. NUMERICALRESULTSANDDISCUSSION 221. 1979.
[8] E. Kushilevitz and N. Nisan, Communication Complexity. Cambridge,
U.K.:CambridgeUniv.Press,1997.
#10-3 [9] T.LeeandA.Shraibman,“LowerBoundsinCommunicationComplex-
5
ity”, In Foundations and Trends in Theoretical Computer Science, v.3,
PII,e [10]26L3.-L39o9v.a´sz,2“0C0o9m. munication Complexity: A Survey”. In Paths, Flows,
0 andVLSILayout,SpringerVerlag,Berlin,1990.
04.6#510-4 0.7 0.75 0.8 0.85 0.9 0.95 1 [11] N. Ma and P. Ishwar, “Infinite-message distributed source coding for
two-terminal interactive computing”, 47th Annual Allerton Conf. on
PII,e2 Communication,Control,andComputing,Monticello,IL,Sept.2009.
[12] V. Misra, V. K. Goyal, and L. R. Varshney. “Distributed scalar
00.65 0.7 0.75 0.8 0.85 0.9 0.95 1 quantization for computing: High-resolution analysis and extensions”,
4 IEEETransactionsonInformationTheory,vol.57,No.8,pp.5298-5325,
Aug.2011.
]R[E2 [13] M. Nan, and P. Ishwar. “Some results on distributed source coding
forinteractivefunctioncomputation”.IEEETransactionsonInformation
0
0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Theory,vol57,No.9,pp.6180-6195.Sept.2011.
(2/:)*3 [14] A. Orlitsky and J. R. Roche, “Coding for Computing”, IEEE Transac-
tionsonInformationTheory,vol.47,no.3,pp.903–917,March2001.
Fig. 5. Variation of Pe,II with θ for the single-round interactive model, [15] H. V. Poor, “Fine quantization in signal detection and estimation”,
12 (top), 21 (middle) with R = 4.0 bits. R¯ = E[R] for the infinite-round IEEE Transactions on Information Theory, vol. 34, No. 5, pp.960-972,
interactivemodelisshowninthebottompanelforρ=1. Sept.1988.
[16] A.C.Yao,“SomeComplexityQuestionsRelatedtoDistributiveCom-
puting(PreliminaryReport)”.InProceedingsoftheEleventhAnnualACM
PerformanceresultsforallmodelsaresummarizedinFig.5,
SymposiumonTheoryofComputing,STOC’79,209-213.1979.
for ρ=1 and π/3<θ <π/2. Under the 1-round interactive
model the hexagonal lattice is not the worst case for the
12 sequence, but is for the 21 sequence. The large gap in