6.851: Advanced Data Structures
Spring 2010
Lecture 11 — Mar 11, 2010
Prof. Erik Demaine
Scribe: Jingjing Liu
1
Overview
In the last lecture we covered static fusion tree. A data structure storering
n w
bit integers that
supports predecessor and successor queries in
O
(log
w
n
) time with
O
(
n
) space.
In this lecture we discuss lower bounds on the cellprobe complexity of the static predecessor
problem with constrained space. In particular, we use round elimination technique to prove the
preprocessor lower bound in communication model and that the min of van Emde Boas trees and
fusion trees is an optimal static predecessor data structure up to loglog factors.
2
Predecessor lower bound results
2.1
The problem
Given a set of
n w
bit integers, the goal is to eﬃciently ﬁnd predecessor of element
x
. Observe that
having
O
(2
w
) space one can precompute and store all the results to achieve constant query time,
we assume
O
(
n
O
(1)
) space for our data structures.
The results we are about to discuss are actually for an easier problem: colored predecessor. Each
element is colored red or blue. Given query on element
x
, the goal is to return the color of
x
’s
predecessor. Since we can solve colored predecessor problem using predecessor, gives a stronger
lower bound for our original problem.
2.2
Results
•
AjtaiCombinatorica 1988[1]
–Proved the ﬁrst superconstant bound,
O
(
w
); claimed that
∀
w,
∃
n
that gives Ω(
√
lg
w
) query time.
•
MiltersenSTOC 1994[2]
–Rephrased the same proof ideas in terms of communication com
plexity:
∀
w
,
∃
n
that gives Ω(
√
lg
w
) query time;
∀
n
,
∃
w
that gives Ω(
3
√
lg
n
) query time.
•
Miltersen,Nisan,Safra,WigdersonSTOC 1995[3]
JCSS 1998[4]
–Introduced round elimina
tion technique and used it to give a clean proof of the same lower bound.
•
Beame,FichSTOC 1999[5]
JCSS2002[6]
manusccript 1994
–Proved two strong bounds:
∀
w,
∃
n
that gives Ω(
lg
w
lg lg
w
) query time;
∀
n,
∃
w
that gives Ω(
q
lg
n
lg lg
n
) query time. Also gave a static
data structure achieving
O
(min
{
lg
w
lg lg
w
,
q
lg
n
lg lg
n
)
}
, which shows that these bounds are optimal
if we insist on pure bound in
n
w
.
1
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Xiao  Ph.D. thesis 1992 at U.C. San Diego[7]
–Independently proved the same lower bound
earlier of Beame and Fich.
•
Sen  CCC 2003[8]; Sen,VenkateshJCSS2008[9]
–Gave a stronger version of the round elim
ination lemma that we about to introduce in this lecture, which gives a cleaner proof of the
same bounds.
•
Patrascu, Thorup  STOC 2006[10]; SODA2007[11]
–Gave tight bounds for optimal searching
predecessors among a static set of integers when
a
= lg
space
n
:
Θ(
min
{
log
w
n,
lg(
w

lg
n
a
)
,
lg
w
a
lg(
a
lg
n
lg
w
a
)
,
lg
w
a
lg(lg
w
a
/
lg
lg
n
a
)
}
)
(1)
This tradeoﬀ between
n
w
space
shows that given
n
lg
O
(1)
n
space, the optimal search
time is Θ(
min
{
log
w
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 Fall '10
 ErikDemaine
 Data Structures, Computational complexity theory, LG, Alice and Bob, lg lg

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