2. Random Search with Complete BST

Transcription

1 Tecnical Repor: TR-BUAA-ACT-06-0 Complexiy Analysis of Random Searc wi AVL Tree Gongwei Fu, Hailong Sun Scool of Compuer Science, Beiang Universiy {FuGW, April 7, 006 TR-BUAA-ACT Inroducion AVL ree is a classical daa srucure in compuer science, wic is commonly used o solve searcing problems. Searc from e roo can acieve bes efficiency averagely. However, wen AVL is implemened in disribued sysems, ree s are disribued across nwor. In a case, if all searces sar from e roo, e roo will becomes a bolenec. A sraigforward approac o address is issue is o allow searc o sar randomly from any. We call is ind of searc random searc. In is paper, we analyze e average searc leng in e case of random searc wi AVL. In secion, we perform e analysis of average searc leng over comple BST (Binary Searc Tree). Secion 3 presens e esimaion of average searc leng of AVL random searc.. Random Searc wi Comple BST Algorim presens a recursive algorim for random searc over a BST, in wic searcing can sar randomly from any. We ave e following ree assumpions in is paper: BST is comple, e dep of wic is and e number of s is n; Te probabiliy of searcing for eac ey is equal; Te probabiliy of searcing a sars from eac is equal. - -

2 Tecnical Repor: TR-BUAA-ACT-06-0 Algorim : Searc wi BST searc (curnode, ey) { if (curnode.conains(ey)) { rurn curnode; else if (curnode.inlefsubree(ey)) { searc (curnode.lefcild(), ey); else if (curnode.inrigsubree(ey)) { searc (curnode.rigcild(), ey); else if (paren! null) { searc (curnode.paren(), ey); else rurn null; Wi e above assumpions, e following wo facs can be go: a) n, log ( n ) b) Given a roo is e enrance of searcing, e TSL (Toal Searc Leng) is: roo, subree( roo)) 3 ( ) i i i ( ) Suppose e enrance of searcing is a cerain ( > ) a lies in e level of e BST, en e arg arg is possible o locae in e 3 saded areas sown in Figure. - -

3 Tecnical Repor: TR-BUAA-ACT-06-0 pa(, roo) subree( siblingnod e) subree( siblingnod e ) subree( ) subree( ) subree( ) Case : Figure. Searcing from a in e level For all arg in e sub-ree rooed a, i.e arg subree( ), e dep of e sub-ree is. So e TSL from o all e s wiin e sub-ree is: TSL ( Case : )) ( ) ( ) For all arg on e pa from o roo, i.e. arg pa(, roo), given a ere are s on e pa, e TSL from o all e s on e pa is:, pa(, roo)) 3 4 ( )( ) ( i ) i Case 3: For all arg in e sub-ree wi a ( ) level on pa(, roo) as is paren, i.e. arg subree( ), e TSL from o all e s wiin e sub-ree is: - 3 -

4 TSL ( Tecnical Repor: TR-BUAA-ACT-06-0 )) ( ) ( ) Similarly, for all arg in e sub-ree wi a ASLAVL ( 7, 5], > 3, < level on pa(, roo) as is paren, i.e. subree ), e arg ( TSL from o all e s wiin e sub-ree is: TSL ( )) 3 ( ) ( ) I can be seen a if arg is in e sub-ree wi a ( i) level on pa(, roo) as is paren, i.e. subree ), e TSL from o all e s wiin e sub-ree is: arg ( i TSL ( )) ( i ) ( ) ( ) ( ) i i Tus, e TSL from o all e s wiin e neigboring sub-rees is:, siblingsubree( )) i subree( i )) i [( ) i ] i [( ) i ] i ( i i ) i ( i ) i ( ) i i i i ( )( ) ( ) ( ) ( ) ( ) [( ) [( ) ( )( ) ] ( )( ] ) Hence, e TSL from o all e s wiin e BST is: roo)) - 4 -

5 Tecnical Repor: TR-BUAA-ACT-06-0 TSL ( )), pa(, roo)), siblingsubree( )) ( ) ( )( ) ( ) ( ) ( )( [( ) ] [ ( ) ( ) ( ) ( 4) 4] [( 4) 4] ( 4) Ten e TSL from eac wiin e BST o all e s wiin e BST is: ) TSL [ NumberOf ( ) roo))] { {[ [ ( 4) ( 4) ] ] [ ( ) ( 4) ] [( ) ] [ ( 4) ] ( ) [( ) ] Wi our ree assumpions menioned above, we ave e average searc leng ASL: ASL TSL SearcTimes [ ( 4) ] ( ) [( ) ( ) ] ( 4) ( ) 4 5 [( ) ( ) ] ( ) Te conclusion can also be drawn roug e following approac, in wic p ) is ( e probabiliy a all e level s exis in e BST. ASL ( roo)) ) ( 4) - 5 -

6 Tecnical Repor: TR-BUAA-ACT-06-0 ASL [ p( ) ASL( )] [ ( 4) ] [ {[ ( ) ( ( 4) ( 4) ] ] ) ( ) ( 4) ] {[ [( ) ] ( 4) 4 5 [( ) ( ) ] ( ) Moreover, i is can be easily calculaed a wen <, e ASL is. 3. Analysis of Random Searc wi AVL In secion, we ave e conclusion a e average searc leng of random searc wi comple BST is rougly equal o -5. Here, we give an esimaion of average searc leng of random searc wi AVL. Te ree assumpions menioned in secion for comple BST also old for AVL ree. Te difference bween a comple BST and an AVL is a e number of leaf s in an AVL can ranges from o -. Terefore, e average searc leng will be less an a of comple BST wi e same dep. Furermore, e average searc leng sould be larger an a of comple BST wi - dep. Ten, we can conclude: ASLAVL ( 7, 5], > 3, < For example, e average searc leng of an AVL wi 00 s is larger an 7 and less an/equal o Conclusion Wen AVL ree is used in disribued compuing, searcing sould no ly sar from e roo in order o avoid sysem bolenec. We analyze e average searc leng wi comple BST, and an esimaion o average searc leng wi AVL ree is given