04 - Treaps. Dr. Alexander Souza

Transcription

1 Algorths Theory 04 - Treaps Dr. Alexander Souza

2 The dctonary proble Gven: Unverse (U,<) of keys wth a total order Goal: Mantan set S U under the followng operatons Search(x,S): Is x S? Insert(x,S): Insert x nto S f not already n S. Delete(x,S): Delete x fro S. 2

3 Extended set of operatons Mnu(S): Return sallest key. Maxu(S): Return largest key. Lst(S): Output eleents of S n ncreasng order of key. Unon(S,S 2 ): Merge S and S 2. Condton: x S, x 2 S 2 : x < x 2 Splt(S,x,S,S 2 ): Splt S nto S and S 2. x S, x 2 S 2 : x x and x < x 2 3

4 Known solutons Bnary search trees d b g a c h Drawback: Sequence of nsertons ay lead to a lnear lst a, b, c, d, e, f Heght balanced trees: AVL trees, (a,b)-trees Drawback: Coplex algorths or hgh eory requreents. 4

5 Approach for randozed search trees If n eleents are nserted n rando order nto a bnary search tree, the expected depth s.39 log n. Idea: Each eleent x s assgned a prorty chosen unforly at rando pro(x) R The goal s to establsh the followng property. p (*) The search tree has the structure that would result f eleents were nserted n the order of ther prortes. 5

6 Treaps (Tree Heap) Defnton: A treap s a bnary tree. Each node contans one eleent x wth key(x) U and pro(x) R. The followng propertes hold. Search tree property For each eleent x: - eleents y n the left subtree of x satsfy: key(y) < key(x) - eleents y n the rght subtree of x satsfy : key(y) > key(x) Heap property For all eleents x,y: If y s a chld of x, then pro(y) > pro(x). All prortes are parwse dstnct. 6

7 Exaple key a b c d e f g prorty d a 3 f 2 c e g b 7 7

8 Treap unqueness Lea: For eleents x,..., x n wth key(x ) and pro(x ), there exsts a unque treap. It satsfes property (*). Proof: n=: ok n>: 8

9 Search for an eleent d a 3 f 2 c e g b 7 9

10 Search for eleent wth key k v := root; 2 whle v nl do 3 case key(v) = k : stop; eleent found (successful search) 4 key(v) < k : v:= RghtChld(v); 5 key(v) > k : v:= LeftChld(v); 6 endcase; 7 endwhle; 8 eleent not found (unsuccessful search) Runnng te: O(# eleents on the search path) 0

11 Analyss of the search path Eleents x,, x n x has -th sallest key Let M be a subset of the eleents. P n (M) = eleent n M wth lowest prorty Lea: a) Let <. x s ancestor of x ff P n ({x,,x }) = x b) Let <. x s ancestor of x ff P n ({x,,x }) = x

12 Analyss of the search path Proof: a) Use (*). Eleents are nserted n order of ncreasng prortes. P n ({x,,x }) = x x s nserted frst aong {x,,x }. When x s nserted, the tree contans only keys k wth k < key(x ) or k > key(x ) k key(x ) > k k key(x ) < k 2

13 Analyss of the search path Proof: a) (Let <. x s ancestor of x ff P n ({x,,x }) = x ) Let x j = P n ({x,,x }). Show: x = x j Suppose: x x j Case : x j = x Case 2: x j x Part b) follows analogously. 3

14 Analyss of the Search operaton Let T be a treap wth eleents x,, x n x has -th sallest key H n = k n-th Haronc nuber: k = Lea: n / k. Successful search: The expected nuber of nodes on the path to x s H H n-. 2. Unsuccessful search : Let be the nuber of keys that are saller than the search key k. The expected nuber of nodes on the search path s H H n-. 4

15 Analyss of the Search operaton Proof: Part X, = 0 x s ancestor of otherwse x X = # nodes on the path fro the root to x (ncl. x ) X = X, X X, < > E E = X, E [ X ] X, < > 5

16 Analyss of the Search operaton < : E[ X, ] = Prob[ x s ancestor of x ] = /( ) All eleents n {x,, x } have the sae probablty of beng the one wth the sallest prorty. Prob[P n ({x,,x }) = x ] = /(- ) > : E[ X, ] = /( ) 6

17 Analyss of the Search operaton ] [ X E ] [ = > < X E = = n = H H n 7 Part 2 follows analogously Part 2 follows analogously

18 Insertng a new eleent x. Choose pro(x). 2. Search for the poston of x n the tree. 3. Insert x as a leaf. 4. Restore the heap property. whle pro(parent(x)) > pro(x) do f x s left chld then RotateRght(parent(x)) endf endwhle; else RotateLeft(parent(x)); 8

19 Rotatons y RotateRght t R x x RotateLeft y C A B A B C The rotatons antan the search tree property p and restore the heap property. 9