Relation of BSTs to Quicksort, Analysis of Random BST. Lecture 9

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1 Reltio o BSTs to Quicsort, Alysis o Rdom BST Lecture 9

2 Biry-serch-tree sort T Crete empty BST or i = to do TREE-INSERT(T, A[i]) Perorm iorder tree wl o T. Emple: 3 A = [ ] 8 Tree-wl time = O(), but how log does it te to build the BST? L9.2

3 Alysis o BST sort BST sort perorms the sme comprisos s quicsort, but i dieret order! The epected time to build the tree is symptoticlly the sme s the ruig time o quicsort. L9.3

4 Node depth The depth o ode = the umber o comprisos mde durig TREE-INSERT. Assumig ll iput permuttios re eqully liely, we hve Averge ode depth E # comprisos i O( lg ) O(lg ). to isert ode (quicsort lysis) i L9.4

5 Epected tree height But, verge ode depth o rdomly built BST = O(lg ) does ot ecessrily me tht its epected height is lso O(lg ) (lthough it is). Emple. lg Ave. depth lg O(lg ) 2 h L9.5

6 Height o rdomly built biry serch tree Outlie o the lysis: Prove Jese s iequlity, which sys tht (E[X]) E[(X)] or y cove uctio d rdom vrible X. Alyze the epoetil height o rdomly built BST o odes, which is the rdom vrible = 2 X, where X is the rdom vrible deotig the height o the BST. Prove tht 2 E[X ] E[2 X ] = E[ ] = O( 3 ), d hece tht E[X ] = O(lg ). L9.6

7 Cove uctios A uctio : R R is cove i or ll,b 0 such tht + b =, we hve or ll,y R. ( + by) () + b(y) () + b(y) (y) () ( + by) + by y L9.7

8 Coveity lemm Lemm. Let : R R be cove uctio, d let {, 2,, } be set o oegtive costts such tht =. The, or y set {, 2,, } o rel umbers, we hve ( Proo. By iductio o. For =, we hve =, d hece ( ) ( ) trivilly. ). L9.8

9 L9.9 Proo (cotiued) ) ( Iductive step: Algebr.

10 L9.0 Proo (cotiued) ) ( ) ( ) ( Iductive step: Coveity.

11 L9. Proo (cotiued) ) ( ) ( ) ( ) ( ) ( ) ( Iductive step: Iductio.

12 L9.2 Proo (cotiued) ) ( ) ( ) ( ) ( ) ( ) ( ) ( Iductive step: Algebr..

13 Jese s iequlity Lemm. Let be cove uctio, d let X be rdom vrible. The, (E[X]) E[ (X)]. Proo. ( E[ X ]) Pr{ X } Deiitio o epecttio. L9.3

14 Jese s iequlity Lemm. Let be cove uctio, d let X be rdom vrible. The, (E[X]) E[ (X)]. Proo. ( E[ X ]) Pr{ X } ( ) Pr{ X } Coveity lemm (geerlized). L9.4

15 Jese s iequlity Lemm. Let be cove uctio, d let X be rdom vrible. The, (E[X]) E[ (X)]. Proo. ( E[ X ]) Pr{ X } ( ) Pr{ X } E[ ( X )]. Tricy step, but true thi bout it. L9.5

16 Alysis o BST height Let X be the rdom vrible deotig the height o rdomly built biry serch tree o odes, d let = 2 X be its epoetil height. I the root o the tree hs r, the X = + m{x, X }, sice ech o the let d right subtrees o the root re rdomly built. Hece, we hve = 2 m{, }. L9.6

17 Alysis (cotiued) Deie the idictor rdom vrible Z s Z = i the root hs r, 0 otherwise. Thus, Pr{Z = } = E[Z ] = /, d Z 2 m{,. } L9.7

18 Epoetil height recurrece E Z 2 m{, E } Te epecttio o both sides. L9.8

19 L9.9 Epoetil height recurrece Z E Z E E }, m{ 2 }, m{ 2 Lierity o epecttio.

20 L9.20 Epoetil height recurrece E Z E Z E Z E E }], [m{ ] [ 2 }, m{ 2 }, m{ 2 Idepedece o the r o the root rom the rs o subtree roots.

21 L9.2 Epoetil height recurrece E E Z E Z E Z E E ] [ 2 }], [m{ ] [ 2 }, m{ 2 }, m{ 2 The m o two oegtive umbers is t most their sum, d E[Z ] = /.

22 L9.22 Epoetil height recurrece 0 ] [ 4 ] [ 2 }], [m{ ] [ 2 }, m{ 2 }, m{ 2 E E E Z E Z E Z E E Ech term ppers twice, d reide.

23 Solvig the recurrece Use substitutio to show tht E[ ] c 3 or some positive costt c, which we c pic suicietly lrge to hdle the iitil coditios. 4 E[ 0 E ] L9.23

24 Solvig the recurrece Use substitutio to show tht E[ ] c 3 or some positive costt c, which we c pic suicietly lrge to hdle the iitil coditios. E E[ c 3 Substitutio. ] L9.24

25 Solvig the recurrece Use substitutio to show tht E[ ] c 3 or some positive costt c, which we c pic suicietly lrge to hdle the iitil coditios. E 4 4 4c E[ c 3 3 d Itegrl method. ] L9.25

26 Solvig the recurrece Use substitutio to show tht E[ ] c 3 or some positive costt c, which we c pic suicietly lrge to hdle the iitil coditios. E 4 4 4c 4c E[ c d Solve the itegrl. ] L9.26

27 Solvig the recurrece Use substitutio to show tht E[ ] c 3 or some positive costt c, which we c pic suicietly lrge to hdle the iitil coditios. E 4 4 4c 4c c E[ c d ] 4. Algebr. L9.27

28 The grd ile Puttig it ll together, we hve 2 E[X ] E[2 X ] Jese s iequlity, sice () = 2 is cove. L9.28

29 The grd ile Puttig it ll together, we hve 2 E[X ] E[2 X ] = E[ ] Deiitio. L9.29

30 The grd ile Puttig it ll together, we hve 2 E[X ] E[2 X ] = E[ ] c 3. Wht we just showed. L9.30

31 The grd ile Puttig it ll together, we hve 2 E[X ] E[2 X ] = E[ ] c 3. Tig the lg o both sides yields E[X ] 3 lg +O(). L9.3

32 Post mortem Q. Does the lysis hve to be this hrd? Q. Why bother with lyzig epoetil height? Q. Why ot just develop the recurrece o directly? X = + m{x, X } L9.32

33 Post mortem (cotiued) A. The iequlity m{, b} + b. provides poor upper boud, sice the RHS pproches the LHS slowly s b icreses. The boud m{2, 2 b } b llows the RHS to pproch the LHS r more quicly s b icreses. By usig the coveity o () = 2 vi Jese s iequlity, we c mipulte the sum o epoetils, resultig i tight lysis. L9.33

34 Thought eercises See wht hppes whe you try to do the lysis o X directly. Try to uderstd better why the proo uses epoetil. Will qudrtic do? See i you c id simpler rgumet. (This rgumet is little simpler th the oe i the boo I hope it s correct!) L9.34

Introduction to Algorithms

Introduction to Algorithms Itroductio to Algorithms 6.046J/8.40J/SMA5503 Lecture 9 Pro. Charles E. Leiserso 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 [3 8