CSCI 104 Splay Trees. Mark Redekopp

Transcription

1 CSCI 0 Sply Trees Mrk edekopp

2 Soures / eding Mteril for these slides ws derived from the following soures /leture0-sply.pdf Nie Visuliztion Tool html

3 Sply Tree Intro Another mp/set implementtion (storing keys or key/vlue pirs) Insert, emove, Find ell To do m inserts/finds/removes on n BTree w/ n elements would ost? O(m*log(n)) Sply trees hve worst se find, insert, delete time of O(n) However, they gurntee tht if you do m opertions on sply tree with n elements tht the totl time is O(m*log(n)) [i.e. mortized time is O(log(n)] They hve further enefit tht reently essed elements will e ner the top of the tree In ft, the most reently essed item is lwys t the top of the tree

4 Sply Opertion Sply mens "spred" As you serh for n item or fter you insert n item we will perform series of sply opertions These opertions will use the desired node to lwys end up t the top of the tree A desirle side-effet is tht essing key multiple times within short time window will yield fst serhes euse it will e ner the top See next slide on priniple of lolity T If we serh for or insert T T T will end up s the root node with the old root in the top level or two

5 dimensions of this priniple: spe & time Sptil olity Future esses will likely luster ner urrent esses Instrutions nd dt rrys re sequentil (they re ll one fter the next) Temporl olity Future esses will likely e to reently essed items Sme ode nd dt re repetedly essed (loops, suroutines, if(x > y) x++; 90/0 rule: Anlysis shows tht usully 0% of the written instrutions ount for 90% of the exeuted instrutions Sply trees help exploit temporl lolity y gurnteeing reently essed items ner the top of the tree Priniple of olity

6 Sply Cses G P G P G P... ig-ig d P G d ight rotte of, d d P G d G P d ig-g eft rotte of, oot/ig Cse (Single ottion)

7 Find() ig ig-ig ig-ig esulting Tree

8 Find() ig-g ig-g esulting Tree Notie the tree is strting to look t lot more lned

9 9 Worst Cse Suppose you wnt to mke the mortized time (verged time over multiple lls to find/insert/remove) look d, you might try to lwys ess the node in the tree Deepest But sply trees hve property tht s we keep essing deep nodes the tree strts to lne nd thus ess to deep nodes strt y osting O(n) ut soon strt osting O(log n)

10 0 Insert() ig-ig ig-ig esulting Tree

11 Insert() ig-g ig-ig esulting Tree

12 Ativity Go to html Try to e n dversry y inserting nd finding elements tht would use O(n) eh time

13 Sply Tree Supported Opertions Insert(x) Norml BST insert, then sply x Find(x) Attempt norml BST find(x) nd sply lst node visited If x is in the tree, then we sply x If x is not in the tree we sply the lef node where our serh ended FindMin(), FindMx() Wlk to fr left or right of tree, return tht node's vlue nd then sply tht node DeleteMin(), DeleteMx() Perform FindMin(), FindMx() [whih splys the min/mx to the root] then delete tht node nd set root to e the non-nu hild of the min/mx emove(x) Find(x) splying it to the top, then overwrite its vlue with is suessor/predeessor, deleting the suessor/predeessor node

14 FindMin() / DeleteMin() FindMin() ig-ig ig esulting Tree DeleteMin() esulting Tree 0

15 emove() ig-g ig-g esulting Tree Copy suessor or predeessor to root Delete suessor (emove node or retth single hild)

16 Top Down Splying ther thn wlking down the tree to first find the vlue then splying k up, we n sply on the wy down We will e "pruning" the ig tree into two smller trees s we wlk, utting off the unused pthwys

17 Top-Down Splying. ig (If Trget is in nd level) oot T T oot. Finl Step (when reh Trget) T T

18 Top-Down Splying. ig-ig. ig-g

19 9 Find() ig-g Steps tken on our journey to find - - -Tree -Tree -Tree -Tree -Tree -Tree

20 0 Find() -Tree -Tree esulting tree fter find esulting tree from ottom-up pproh. Finl Step (when reh Trget) T T

21 Insert() -Tree -Tree Tree - -Tree Tree 0 -Tree Originl esulting Tree from Bottomup pproh

22 Summry Sply trees don't enfore lne ut re selfdjusting to yield lned tree Sply trees provide effiient mortized time opertions A single opertion my tke O(n) m opertions on tree with n elements => O(m(log n)) Uses rottions to ttempt lne Provides fst ess to reently used keys