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MIT6_851S10_lec11

MIT6_851S10_lec11 - 6.851 Advanced Data Structures Spring...

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6.851: Advanced Data Structures Spring 2010 Lecture 11 Mar 11, 2010 Prof. Erik Demaine 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 n ) time with O ( n ) space. w In this lecture we discuss lower bounds on the cell-probe 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 Ajtai-Combinatorica 1988[1] –Proved the first superconstant bound, O ( w ); claimed that w, n that gives Ω( lg w ) query time. Miltersen-STOC 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,Wigderson-STOC 1995[3] & JCSS 1998[4] –Introduced round elimina- tion technique and used it to give a clean proof of the same lower bound. Beame,Fich-STOC 1999[5] & JCSS2002[6] & manusccript 1994 –Proved two strong bounds: w, n that gives Ω( lg w ) query time; n, w that gives Ω( lg n ) query time. Also gave a static lg lg w lg lg n data structure achieving O (min { lg w , lg n ) } , which shows that these bounds are optimal lg lg w lg lg n 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,Venkatesh-JCSS2008[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 a a Θ( min { log n, lg( w a lg n ) , lg( lg w lg w ) , lg(lg w lg / w lg lg n ) } ) (1) w a lg n a a a This trade-off between n & w & space shows that given n lg O (1) n space, the optimal search time is Θ( min { log n, lg w } ). The result also implies that van Emde Boas tree is
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