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Unformatted text preview: Announcements: MP5 available. Due 4/1. 11 :59p.
Exam 2: Tue. 4/5. 79p. locations TBA on Exams page of web site Today:
Balanced BST— Weiss, 4.4 AVL tree analysis — Is it an efﬁcient structure? \SCS ’
Anifty bug; M M SO kill—W
has o». W. AVL tree removals: . H mm Warmups:
MWorst case for Insertion into a Binary Search Tree. 0 (h) '\ MWorst case for removal from a Binary Search Tree. Worst case for an algorithm to return all keys that are greater
an 20 and that are multiples of 3 in a Binary Search Tree. OMWorst case for insertion into an AVL Tree. 0 ( it) a Worst case for an algorithm to return all keys that are greateb(
n 20 and that are multiples of 3 in an AVL Tree. '0
an) Level order traversal of an AVL Tree. 00“ Build an AVL tree with keys that are the numbers between 0 and
. in that order. by repeated insertion into the tree. O(n .5) MBuiid a binary search tree with keys that are the numbers between 0 and n. in that order by repeated insertion into the tree. “ (m 0
0&‘4Za31...1“’ 2 T , om AVLtree operations: M 4 a JAM  1gs_e_r_t_  ‘coA. eﬂmdﬁj
 Remove , V30“ LMJ (if . Eild' 651 A“. What are the running times? d“) 51000310 AVL tree analysis: 1—” Since running times for Insert. Remove and Find are 001). we'll argue
that I1: 0009 n). DefnofblgO: “n3:0(¢8(n\\ ".1 a (”studs OK So
4M Um A4m écsm elp us In our reasoning: " ‘4 ‘4 ink 10.); c505 Putting an new bound the helghtforatreeofnnodesls mess puttingalowerbou n enu n lnatleeofhelght .
My AVL tree analysis: ﬁnk k ‘ 0003f“); Putting an upper bound on the height for a tree of n nodes Is the same as
putting a lower bound on the number of nodes In a tree of helght h. . DeﬂneN(h): I“, %N “M an ““36 “3M "°°'°'" MN 2 NW!
9? [Emu NW: *N(h0N+(N(h: z) wealmu3(:§:er+n:ln(: h+ N‘“ 3))“ ”(h0+5!“ 2); /(2N h 2)
NM > EM 13 NW 2“" ’ fvgtﬁ'zjbﬁfms is correct. N00 )IZNOPIL ”(IA >/
it M» M» 0 \ , MB > NMzZ 1 m. Lara ”9‘ «mm. H" ‘ ‘7 N(h\21"‘: {'1 my
“3'8"; m Msa‘bs $34k“ Nbﬁ’ :3):
M ”w MW 2 1‘? 2 / Classic balanced BST structures:
 RedBlack trees  max ht 2logzn. Constant # of rotations tor Insert. remove. ﬁnd.
 AVL trees  max ht 1.44log,n. O(log n) rotations upon remove. Balanced BSTs, pros and cons:
0 Pros:
 Insert. Remove. and Find are always 0009 n)
 An Improvement over:
 Range flndlng 8. nearest nelghbor
 Cons:
 Possible to search for single keys faster  Ifdaee lsso blgthatltdoesn'tﬂt In memoryltmustbestoredon dlsk
and we requlre a different structure. ...
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 Spring '09
 Heeren

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