CSCI 104. Rafael Ferreira da Silva. Slides adapted from: Mark Redekopp and David Kempe

Transcription

1 CSCI 0 fel Ferreir d Silv rfsilv@isi.edu Slides dpted from: Mrk edekopp nd Dvid Kempe

2 LOG STUCTUED MEGE TEES

3 Series Summtion eview Let n = k $ = #%& #. Wht is n? n = k+ - Wht is log () + log () + log () + log (8)+ + log ( k ) $ = k = Arithmetic series: i O(k ) #%& So then wht if k = log(n) s in: log () + log () + log () + log (8)+ + log ( log(n) ) O(log n) ( #%) i = (((,)) = θ n. Geometric series ( c # #%). = c(,) c = θ c(

4 Merge Trees Overview Consider list of (pointers to) rrys with the following constrints Ech rry is sorted though no ordering constrints exist etween rrys The rry t list index k is of exctly size k or empty An rry t list loction k cn e of size k or empty 0 NULL 0 Note: These re the keys for set (or key,vlue pirs for mp) Size = 8 Size = if non-empty

5 Merge Trees Size Define n s the # of keys in the entire structure k s the size of the list (i.e. positions in the list) Given k, wht is n? Let n = k = $ #%& #. Wht is n? n= k - An rry t list loction k cn e of size k or empty 0 NULL Size = 8 Size = if non-empty Note: These re the keys for set (or key,vlue pirs for mp)

6 Merge Trees Find Opertion To find n element (or check if it exists) Iterte through the rrys in order (i.e. strt with rry t list position 0, then the rry t list position, etc.) In ech rry perform inry serch If you rech the end of the list of rrys without finding the vlue it does not exist in the set/mp An rry t list loction k cn e of size k or empty 0 NULL 0 Note: These re the keys for set (or key,vlue pirs for mp) Size = 8 Size = if non-empty

7 7 Find untime Wht is the worst cse runtime of find? When the item is not present which requires, inry serch is performed on ech list T(n) = log () + log () + log ( k ) $ = k = #%& i = O(k ) But let's put tht in terms of the numer of elements in the structure (i.e. n) ecll k = log (n)+ So find is O(log (n) ) An rry t list loction k cn e of size k or empty 0 NULL 0 Note: These re the keys for set (or key,vlue pirs for mp) Size = 8 Size = if non-empty

8 8 Improving Find's untime While we might e oky with [log(n)], how might we improve the find runtime in the generl cse? Hint: I would e willing to py O() to know if key is not in prticulr rry without hving to perform find A Bloom filter could e mintined longside ech rry nd llow us to skip performing inry serch in n rry

9 9 Insertion Algorithm Let j e the smllest integer such tht rry j is empty (first empty slot in the list of rrys) An insertion will cuse Loction j's rry to ecome filled Loctions 0 through j- to ecome empty An rry t list loction k cn e of size k or empty insert(9) j= 0 NULL 0 9 Before insertion 8 0 Size = 8 0 After insertion Size = 8

10 0 Insertion Algorithm Strting t rry 0, itertively merge the previously merged rry with the next, stopping when n empty loction is encountered insert(9) 9 0 NULL 0 NULL 0 NULL Merge List 0 is full so merge two rrys of size 9 Merge List is full so merge two rrys of size Size = 8

11 Insert Exmples insert() 0 NULL insert(8) 0 NULL Cost = / 0 insert() 0 NULL Cost = / Cost = / insert() Cost = / 0 0 NULL insert(7) Cost = / 0 NULL insert(9) Cost = / 0 NULL 9 insert() Cost = / 0 0 NULL 7 8 9

12 Insertion untime: First Look 0 Best cse? First list is empty nd llows direct insertion in O() Worst cse? All list entries (rrys) re full so we hve to merge t ech loction In this cse we will end with n rry of size n= k in position k Also recll merging two rrys of size m is Θ(m) So the totl cost of ll the merges is n = *n- = Θ(n) = Θ( k ) But if the worst cse occurs how soon cn it occur gin? It seems the costs vry from one insert to the next This is good plce to use mortized nlysis insert() insert() insert() insert(9) NULL 0 NULL 0 NULL 0 NULL 9

13 Totl Cost for N insertions Totl cost of n= insertions: =*n/ + *n/ + *n/8 + 8*n/ + n =n/ + n/ + n/ + n/ + n =n/*log (n) + n Amortized cost = Totl cost / n opertions log (n)/ + = O(log (n))

14 Amortized Anlysis of Insert We hve sid when you end (plce n rry) in position k you hve to do O( k+ ) work for ll the merges How often do we end in position k The 0 th position will e free with proility ½ (p=0.) We will stop t the st position with proility ¼ (p=0.) We will stop t the nd position with proility /8 (p=0.) We will stop t the k th position with proility / k = -k So we py k+ with proility -(k+) Suppose we hve n items in the structure (i.e. mx k is log n) wht is the expected cost of inserting new element 789 (() $%& $,) ($,)) 789 (() = $%& = log (n) insert() Cost = / 0 insert() Cost = / insert() Cost = / 0 insert(9) Cost = / 0 NULL 0 NULL 0 NULL 0 NULL 9

15 Summry Vrints of log structured merge trees hve found populr usge in industry Strting rry size might e firly lrge (size of memory of single server) Lrge rrys (from merging) re stored on disk Pros: Ese of implementtion Sequentil ccess of rrys helps lower its constnt fctors Opertions: Find = log(n) Insert = Amortized log(n) emove = often not considered/supported

16 SPLAY TEES

17 7 Sources / eding Mteril for these slides ws derived from the following sources /lecture0-sply.pdf Nice Visuliztion Tool html

18 8 Sply Tree Intro Another mp/set implementtion (storing keys or key/vlue pirs) Insert, emove, Find ecll To do m inserts/finds/removes on n BTree w/ n elements would cost? O(m*log(n)) Sply trees hve worst cse 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 ("mortized" uh-oh) time is O(m*log(n)) They hve further enefit tht recently ccessed elements will e ner the top of the tree In fct, the most recently ccessed item is lwys t the top of the tree

19 9 Sply Opertion Sply mens "spred" As you serch for n item or fter you insert n item we will perform series of sply opertions These opertions will cuse the desired node to lwys end up t the top of the tree A desirle side-effect is tht ccessing key multiple times within short time window will yield fst serches ecuse it will e ner the top See next slide on principle of loclity T If we serch for or insert T T T will end up s the root node with the old root in the top level or two

20 dimensions of this principle: spce & time Sptil Loclity Future ccesses will likely cluster ner current ccesses Instructions nd dt rrys re sequentil (they re ll one fter the next) Temporl Loclity Future ccesses will likely e to recently ccessed items Sme code nd dt re repetedly ccessed (loops, suroutines, if(x > y) x++; 90/0 rule: Anlysis shows tht usully 0% of the written instructions ccount for 90% of the executed instructions Sply trees help exploit temporl loclity y gurnteeing recently ccessed items ner the top of the tree 0 Principle of Loclity

21 Sply Cses. Zig-Zig G. oot/zig Cse (Single ottion) P d P c G c c d c Left rotte of, ight rotte of,. Zig-Zg G G P d P G P G P c d d c d c c

22 Find() Zig 7 7 Zig-Zig 7 Zig-Zig esulting Tree 7

23 Find() Zig-Zg Zig-Zg esulting Tree Notice the tree is strting to look t lot more lnced

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

25 Insert() Zig-Zig 8 Zig-Zig esulting Tree

26 Insert() Zig-Zg Zig-Zig esulting Tree 8

27 7 Activity Go to html Try to e n dversry y inserting nd finding elements tht would cuse O(n) ech time

28 8 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 serch 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() [which splys the min/mx to the root] then delete tht node nd set root to e the non-null child of the min/mx emove(x) Find(x) splying it to the top, then overwrite its vlue with is successor/predecessor, deleting the successor/predecessor node

29 9 FindMin() / DeleteMin() FindMin() Zig-Zig Zig esulting Tree DeleteMin() esulting Tree 0

30 0 emove() Zig-Zg Zig-Zg esulting Tree 7 7 Copy successor or predecessor to root Delete successor (emove node or rettch single child)

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

32 Top-Down Splying. Zig (If Trget is in nd level) oot T L T L oot. Finl Step (when rech Trget) T T L L

33 Top-Down Splying. Zig-Zig Y Z c Y Z L Y Z c L c L Z c L Y. Zig-Zg L Z c L Y Z c Y L Y Z c L Y Z c

34 Find() Zig-Zg 7 Steps tken on our journey to find - - L Y Z c L Y Z c L-Tree -Tree L-Tree -Tree 7 L Z c L Y Z c Y -Tree 7 L-Tree

35 Find() L-Tree -Tree esulting tree fter find esulting tree from ottom-up pproch. Finl Step (when rech Trget) T T L L

36 Insert() L-Tree -Tree L-Tree - -Tree L-Tree 0 8 -Tree Originl esulting Tree from Bottomup pproch

37 7 Summry Sply trees don't enforce lnce ut re selfdjusting to ttempt yield lnced tree Sply trees provide efficient 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 lnce Provides fst ccess to recently used keys