# October 26 2005 copyright 2001 5 by erik d demaine

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October 26, 2005Copyright © 2001-5 by Erik D. Demaine and Charles E. LeisersonL11.26Proof of theoremTHEOREM:With high probability, every searchin ann-element skip list costsO(lgn)COOLIDEA:Analyze search backwards—leaf to rootSearch starts [ends] at leaf (node in bottom level)At each node visited:– If node wasn’t promoted higher (gotTAILShere),then we go [came from] left– If node was promoted higher (gotHEADShere),then we go [came from] upSearch stops [starts] at the root (or−∞)
October 26, 2005Copyright © 2001-5 by Erik D. Demaine and Charles E. LeisersonL11.27Proof of theoremTHEOREM:With high probability, every searchin ann-element skip list costsO(lgn)COOLIDEA:Analyze search backwards—leaf to rootPROOF:Search makes “up” and “left” movesuntil it reaches the root (or−∞)Number of “up” moves<number of levelsclgnw.h.p.(Lemma)w.h.p., number of moves is at most the numberof times we need to flip a coin to getclgnHEADs
October 26, 2005Copyright © 2001-5 by Erik D. Demaine and Charles E. LeisersonL11.28Coin flipping analysisCLAIM:Number of coin flips untilclgnHEADs=Θ(lgn)with high probabilityPROOF:ObviouslyΩ(lgn): at leastclgnProveO(lgn)“by example”:Say we make10clgnflipsWhen are there at leastclgnHEADs?(Later generalize to arbitrary values of10)
October 26, 2005Copyright © 2001-5 by Erik D. Demaine and Charles E. LeisersonL11.30Coin flipping analysis (cont’d)Recall bounds on:Pr{at mostclgnHEADs}ncncnclg921lglg10xyxxxyexyxyncncncncelg9lg21lglg10()ncncelg9lg210=ncncelg9lg)10lg(22=ncelg]9)10[lg(2=αn/1=for[]ce=)10lg(9α
October 26, 2005Copyright © 2001-5 by Erik D. Demaine and Charles E. LeisersonL11.31Coin flipping analysis (cont’d)Pr{at mostclgnHEADs}≤ 1/nαforα= [9−lg(10e)]cKEYPROPERTY:α→ ∞as10 → ∞, for anycSo set10, i.e., constant inO(lgn)bound,large enough to meet desiredαThis completes the proof of the coin-flipping claimand the proof of the theorem.
October 31, 2005Copyright © 2001-5 by Erik D. Demaine and Charles E. LeisersonL13.1Prof. Charles E. LeisersonLECTURE13Amortized AnalysisDynamic tablesAggregate methodAccounting methodPotential methodIntroduction to Algorithms6.046J/18.401J
October 31, 2005Copyright © 2001-5 by Erik D. Demaine and Charles E. LeisersonL13.2How large should a hashtable be?Problem:What if we don’t know the proper sizein advance?Goal:Make the table as small as possible, butlarge enough so that it won’t overflow (orotherwise become inefficient).

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Divide and conquer algorithm, Charles E Leiserson, Prof Erik D Demaine