Introduction to Algorithms
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1 Itroductio to Algorithms 6.046J/8.40J LECTURE 9 Radomly built biary search trees Epected ode depth Aalyzig height Coveity lemma Jese s iequality Epoetial height Post mortem Pro. Eri Demaie October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.
2 Biary-search-tree sort T Create a empty BST or i to do TREE-INSERT(T, A[i] Perorm a iorder tree wal o T. Eample: A [ ] Tree-wal time O(, but how log does it tae to build the BST? October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.2
3 Aalysis o BST sort BST sort perorms the same comparisos as quicsort, but i a dieret order! The epected time to build the tree is asymptotically the same as the ruig time o quicsort. October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.3
4 Node depth The depth o a ode the umber o comparisos made durig TREE-INSERT. Assumig all iput permutatios are equally liely, we have Average ode depth E # comparisos i O( lg O(lg. to isert ode ( (quicsort aalysis October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.4 i
5 Epected tree height But, average ode depth o a radomly built BST O(lg does ot ecessarily mea that its epected height is also O(lg (although it is. Eample. lg Ave. depth lg + O(lg 2 h October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.5
6 Height o a radomly built biary search tree Outlie o the aalysis: Prove Jese s iequality, which says that (E[X] E[(X] or ay cove uctio ad radom variable X. Aalyze the epoetial height o a radomly built BST o odes, which is the radom variable 2 X, where X is the radom variable deotig the height o the BST. Prove that 2 E[X ] E[2 X ] E[ ] O( 3, ad hece that E[X ] O(lg. October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.6
7 Cove uctios A uctio : R R is cove i or all,β 0 such that + β, we have ( + βy ( + β (y or all,y R. ( + β(y (y ( ( + βy + βy y October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.7
8 Coveity lemma Lemma. Let : R R be a cove uctio, ad let, 2,, be oegative real umbers such that. The, or ay real umbers, 2,,, we have (. Proo. By iductio o. For, we have, ad hece ( ( trivially. October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.8
9 October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.9 Proo (cotiued Iductive step: + ( Algebra.
10 October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.0 Proo (cotiued Iductive step: + + ( ( ( Coveity.
11 October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7. Proo (cotiued Iductive step: ( ( ( ( ( ( Iductio.
12 October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.2 Proo (cotiued Iductive step: ( ( ( ( ( ( ( Algebra..
13 Coveity lemma: iiite case Lemma. Let : R R be a cove uctio, ad let, 2,, be oegative real umbers such that. The, or ay real umbers, 2,, we have (, assumig that these summatios eist. October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.3
14 October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.4 Coveity lemma: iiite case Proo. By the coveity lemma, or ay, ( i i i i.
15 October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.5 Coveity lemma: iiite case Proo. By the coveity lemma, or ay, ( i i i i. Taig the limit o both sides (ad because the iequality is ot strict: ( lim lim i i i i (
16 Jese s iequality Lemma. Let be a cove uctio, ad let X be a radom variable. The, (E[X] E[ (X]. Proo. ( E[ X ] Pr{ X } Deiitio o epectatio. October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.6
17 Jese s iequality Lemma. Let be a cove uctio, ad let X be a radom variable. The, (E[X] E[ (X]. Proo. ( E[ X ] Pr{ X } ( Pr{ X } Coveity lemma (iiite case. October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.7
18 Jese s iequality Lemma. Let be a cove uctio, ad let X be a radom variable. The, (E[X] E[ (X]. Proo. ( E[ X ] Pr{ X } ( Pr{ X } E[ ( X ]. Tricy step, but true thi about it. October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.8
19 Aalysis o BST height Let X be the radom variable deotig the height o a radomly built biary search tree o odes, ad let 2 X be its epoetial height. I the root o the tree has ra, the X + ma{x,x }, sice each o the let ad right subtrees o the root are radomly built. Hece, we have 2 ma{, }. October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.9
20 Aalysis (cotiued Deie the idicator radom variable Z as Z i the root has ra, 0 otherwise. Thus, Pr{Z } E[Z ] /, ad Z 2 ma{., } ( October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.20
21 Epoetial height recurrece [ ] E Z ( 2 ma{, E } Tae epectatio o both sides. October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.2
22 October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.22 Epoetial height recurrece [ ] ( ( [ ] Z E Z E E }, ma{ 2 }, ma{ 2 Liearity o epectatio.
23 October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.23 Epoetial height recurrece [ ] ( ( [ ] E Z E Z E Z E E }], [ma{ ] [ 2 }, ma{ 2 }, ma{ 2 Idepedece o the ra o the root rom the ras o subtree roots.
24 October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.24 Epoetial height recurrece [ ] ( ( [ ] + E E Z E Z E Z E E ] [ 2 }], [ma{ ] [ 2 }, ma{ 2 }, ma{ 2 The ma o two oegative umbers is at most their sum, ad E[Z ] /.
25 October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.25 Epoetial height recurrece [ ] ( ( [ ] + 0 ] [ 4 ] [ 2 }], [ma{ ] [ 2 }, ma{ 2 }, ma{ 2 E E E Z E Z E Z E E Each term appears twice, ad reide.
26 Solvig the recurrece Use substitutio to show that E[ ] c 3 or some positive costat c, which we ca pic suicietly large to hadle the iitial coditios. 4 E[ 0 [ ] E ] October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.26
27 Solvig the recurrece Use substitutio to show that E[ ] c 3 or some positive costat c, which we ca pic suicietly large to hadle the iitial coditios. [ ] E E[ c 3 Substitutio. ] October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.27
28 Solvig the recurrece Use substitutio to show that E[ ] c 3 or some positive costat c, which we ca pic suicietly large to hadle the iitial coditios. [ ] E 4 4 4c E[ c 3 3 d ] Itegral method. October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.28
29 Solvig the recurrece Use substitutio to show that E[ ] c 3 or some positive costat c, which we ca pic suicietly large to hadle the iitial coditios. [ ] E 4 4 4c 4c E[ c d Solve the itegral. ] October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.29
30 Solvig the recurrece Use substitutio to show that E[ ] c 3 or some positive costat c, which we ca pic suicietly large to hadle the iitial coditios. [ ] E 4 4 4c 4c c E[ c d ] 4. Algebra. October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.30
31 The grad iale Puttig it all together, we have 2 E[X ] E[2 X ] Jese s iequality, sice ( 2 is cove. October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.3
32 The grad iale Puttig it all together, we have 2 E[X] E[2 X ] E[ ] Deiitio. October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.32
33 The grad iale Puttig it all together, we have 2 E[X] E[2 X ] E[ ] c 3. What we just showed. October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.33
34 The grad iale Puttig it all together, we have 2 E[X] E[2 X ] E[ ] c 3. Taig the lg o both sides yields E[X ] 3lg +O(. October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.34
35 Post mortem Q. Does the aalysis have to be this hard? Q. Why bother with aalyzig epoetial height? Q. Why ot just develop the recurrece o directly? X + ma{x,x } October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.35
36 Post mortem (cotiued A. The iequality ma{a, b} a + b. provides a poor upper boud, sice the RHS approaches the LHS slowly as a b icreases. The boud ma{2 a,2 b } 2 a + 2 b allows the RHS to approach the LHS ar more quicly as a b icreases. By usig the coveity o ( 2 via Jese s iequality, we ca maipulate the sum o epoetials, resultig i a tight aalysis. October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.36
37 Thought eercises See what happes whe you try to do the aalysis o X directly. Try to uderstad better why the proo uses a epoetial. Will a quadratic do? See i you ca id a simpler argumet. (This argumet is a little simpler tha the oe i the boo I hope it s correct! October 7, 2005 Copyright by Eri D. Demaie ad Charles E. Leiserso L7.37
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