2-3 search trees red-black BSTs B-trees

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1 2-3 serch trees red-lck BTs B-trees 3

2 2-3 tree llow 1 or 2 keys per node. 2-node: one key, two children. 3-node: two keys, three children. ymmetric order. Inorder trversl yields keys in scending order. erfect lnce. very pth from root to null link hs sme length. 3-node 2-node smller thn J lrger thn J L X nd J null link 4

3 Locl trnsformtions in 2-3 tree plitting 4-node is locl trnsformtion: constnt numer of opertions. e c d less thn nd nd c c nd d d nd e greter thn e c e d less thn nd nd c c nd d d nd e greter thn e 6

4 2-3 tree: performnce erfect lnce. very pth from root to null link hs sme length. Tree height. Worst cse: Best cse: 8

5 2-3 tree: performnce erfect lnce. very pth from root to null link hs sme length. Tree height. Worst cse: lg N. [ll 2-nodes] Best cse: log3 N.631 lg N. [ll 3-nodes] Between 12 nd 20 for million nodes. Between 18 nd 30 for illion nodes. Gurnteed logrithmic performnce for serch nd insert. 9

6 2-3 tree: implementtion? Direct implementtion is complicted, ecuse: intining multiple node types is cumersome. Need multiple compres to move down tree. Need to move ck up the tree to split 4-nodes. Lrge numer of cses for splitting. Bottom line. ould do it, ut there's etter wy. 11

7 2-3 serch trees red-lck BTs B-trees 12

8 Left-lening red-lck BTs 3-node 1. epresent 2 3 tree s BT. lck tree 2. less for Use "internl"left-lening links s "glue" 3 nodes.greter J L less thn thn 3-node nd J less thn greter thn less thn L X J J L lck links connect 2-nodes nd 3-nodes red lck tree nd red links "glue" nodes within 3-node ncoding 3-node with two 2-nodes connected y left-lening red link tree greter thn ncoding 3-node with two 2-nodes connected y left-lening red link greter thn nd lrger key is root zontl red links thn nd X X L X 2-3 tree corresponding red-lck BT 1 correspondence red-lck nd 2-3 trees horizontl red links J 13

9 n equivlent definition BT such tht: No node hs two red links connected to it. very pth from root to null link hs the sme numer of lck links. ed links len left. "perfect lck lnce" ck tree J L X 14

10 Left-lening red-lck BTs: 1-1 correspondence with 2-3 trees Key property. 1 1 correspondence 2 3 nd LLB. red lck tree J L X horizontl red links J L X 2-3 tree J L X 15

erch implementtion for red-lck BTs Oservtion. erch is the sme s for elementry BT (ignore color). ut runs fster ecuse of etter lnce pulic Vl get(key key) { Node x = root; while (x!

11 erch implementtion for red-lck BTs Oservtion. erch is the sme s for elementry BT (ignore color). ut runs fster ecuse of etter lnce pulic Vl get(key key) { Node x = root; while (x!= null) { int cmp = key.compreto(x.key); if (cmp < 0) x = x.left; else if (cmp > 0) x = x.right; else if (cmp == 0) return x.vl; } return null; } ck tree J L X emrk. ost other ops (e.g., ceiling, selection, itertion) re lso identicl. 16

12 ed-lck BT representtion ch node is pointed to y precisely one link (from its prent) cn encode color of links in nodes. privte sttic finl oolen D = true; privte sttic finl oolen BLK = flse; privte clss Node { Key key; Vlue vl; Node left, right; oolen color; // color of prent link } h.left.color is D D h G J h.right.color is BLK privte oolen ised(node x) { if (x == null) return flse; return x.color == D; } null links re lck 17

13 lementry red-lck BT opertions Left rottion. Orient (temporrily) right-lening red link to len left. rotte left (efore) less thn h nd x greter thn privte Node rotteleft(node h) { ssert ised(h.right); Node x = h.right; h.right = x.left; x.left = h; x.color = h.color; h.color = D; return x; } Invrints. intins symmetric order nd perfect lck lnce. 18

14 lementry red-lck BT opertions Left rottion. Orient (temporrily) right-lening red link to len left. rotte left (fter) less thn h nd x greter thn privte Node rotteleft(node h) { ssert ised(h.right); Node x = h.right; h.right = x.left; x.left = h; x.color = h.color; h.color = D; return x; } Invrints. intins symmetric order nd perfect lck lnce. 19

15 lementry red-lck BT opertions ight rottion. Orient left-lening red link to (temporrily) len right. rotte right (efore) less thn x nd h greter thn privte Node rotteight(node h) { ssert ised(h.left); Node x = h.left; h.left = x.right; x.right = h; x.color = h.color; h.color = D; return x; } Invrints. intins symmetric order nd perfect lck lnce. 20

16 lementry red-lck BT opertions ight rottion. Orient left-lening red link to (temporrily) len right. rotte right (fter) less thn x nd h greter thn privte Node rotteight(node h) { ssert ised(h.left); Node x = h.left; h.left = x.right; x.right = h; x.color = h.color; h.color = D; return x; } Invrints. intins symmetric order nd perfect lck lnce. 21

17 lementry red-lck BT opertions olor flip. ecolor to split (temporry) 4-node. flip colors less thn (efore) nd h nd greter thn privte void flipolors(node h) { ssert!ised(h); ssert ised(h.left); sset ised(h.right); h.color = D; h.left.color = BLK; h.right.color = BLK; } Invrints. intins symmetric order nd perfect lck lnce. 22

18 lementry red-lck BT opertions olor flip. ecolor to split (temporry) 4-node. flip colors less thn (fter) nd h nd greter thn privte void flipolors(node h) { ssert!ised(h); ssert ised(h.left); sset ised(h.right); h.color = D; h.left.color = BLK; h.right.color = BLK; } Invrints. intins symmetric order nd perfect lck lnce. 23

19 Insertion in LLB tree: overview Bsic strtegy. intin 1-1 correspondence with 2-3 trees y pplying elementry red-lck BT opertions. insert dd new node here right link red so rotte left Insert into 2-node 24

20 Insertion in LLB tree Wrmup 1. Insert into tree with exctly 1 node. left root serch ends t this null link root red link to new node contining converts 2-node to 3-node right root serch ends t this null link ttched new node with red link root rotted left to mke legl 3-node 25

21 Insertion in LLB tree se 1. Insert into 2-node t the ottom. Do stndrd BT insert; color new link red. If new red link is right link, rotte left. insert dd new node here right link red so rotte left Insert into 2-node 26

22 Insertion in LLB tree Wrmup 2. Insert into tree with exctly 2 nodes. lrger c serch ends t this null link ttched new node with red link smller c serch ends t this null link c ttched new node with red link c serch ends t this null link c c ttched new node with red link colors flipped to lck c c rotted right colors flipped to lck c ed rotted left c rotted right colors flipped to lck c 27

23 Insertion in LLB tree inserting se 2. Insert into 3-node t the ottom. Do stndrd BT insert; color new link red. otte to lnce the 4-node (if needed). Flip colors to pss red link up one level. otte to mke len left (if needed). inserting dd new node here two lefts oth children red so flip colors node here two lefts in row so rotte right right link red so rotte left Insert into 3-node right link red 28

24 Insertion in LLB tree: pssing red links up the tree dd new node here se 2. Insert into 3-node t the ottom. Do stndrd BT insert; color new link red. inserting otte to lnce the 4-node (if needed). Flip colors to pss red link up one level. inserting otte to mke len left (if needed). epet cse inserting 1 or cse 2 up the tree (if needed). inserting dd new node here right link red so rotte left dd new node here two lefts in row so rotte right oth children red so flip colors two lefts in row so rotte right right link red oth so rotte children left red so flip colors two lefts in row so rotte right dd new node here oth children red so flip colors oth children red so flip colors oth children red so flip colors right link red so rotte left two lefts in row so rotte right oth children red so flip colors ssing red link up the tree oth children red so flip colors 29

25 Insertion in LLB tree: Jv implementtion me code for oth cses. ight child red, left child lck: rotte left. Left child, left-left grndchild red: rotte right. Both children red: flip colors. h right rotte h left rotte h flip colors ssing red link up red-lck tree privte Node put(node h, Key key, Vlue vl) { if (h == null) return new Node(key, vl, D); int cmp = key.compreto(h.key); if (cmp < 0) h.left = put(h.left, key, vl); else if (cmp > 0) h.right = put(h.right, key, vl); else if (cmp == 0) h.vl = vl; if (ised(h.right) &&!ised(h.left)) h = rotteleft(h); if (ised(h.left) && ised(h.left.left)) h = rotteight(h); if (ised(h.left) && ised(h.right)) flipolors(h); insert t ottom (nd color red) len left lnce 4-node split 4-node } return h; only few extr lines of code provides ner-perfect lnce 31

26 2-3 serch trees red-lck BTs B-trees 39

Time required for proe is much lrger thn time to ccess dt within pge.

27 File system model ge. ontiguous lock of dt (e.g., file or 4,096-yte chunk). roe. First ccess to pge (e.g., from disk to memory). slow fst roperty. Time required for proe is much lrger thn time to ccess dt within pge. ost model. Numer of proes. Gol. ccess dt using minimum numer of proes. 40

28 B-trees (Byer-creight, 1972) B-tree. Generlize 2-3 trees y llowing up to - 1 key-link pirs per node. t lest 2 key-link pirs t root. t lest / 2 key-link pirs in other nodes. xternl nodes contin client keys. Internl nodes contin copies of keys to guide serch. choose s lrge s possile so tht links fit in pge, e.g., = 1024 * K 2-node sentinel key internl 3-node externl 3-node * D ech red key is copy of min key in sutree externl 5-node (full) K Q U externl 4-node * B D F I J K N O Q T U W X Y client keys (lck) re in externl nodes ntomy of B-tree set ( = 6) ll nodes except the root re 3-, 4- or 5-nodes 41

29 erching in B-tree trt t root. Find intervl for serch key nd tke corresponding link. erch termintes in externl node. serching for follow this link ecuse is * nd K * K * D K Q U follow this link ecuse is D nd * B serch for in this externl node D F I J K N O Q T erching in B-tree set ( = 6) U W X 42

30 Insertion in B-tree erch for new key. Insert t ottom. plit nodes with key-link pirs on the wy up the tree. inserting * K Q U * B F I J K N O Q T U W X * K Q U * B F I J K N O Q T U W X new key () cuses overflow nd split * K Q U new key () cuses overflow nd split * B F I J K N O Q T U W X * K * root split cuses new root to e creted K Q U * B F I J K N O Q T U W X Inserting new key into B-tree set 43

31 Blnce in B-tree roposition. serch or n insertion in B-tree of order with N keys requires log -1 N nd log /2 N proes. f. ll internl nodes (esides root) hve / 2 nd - 1 links. In prctice. Numer of proes is t most 4. = 1024; N = 62 illion log /2 N 4 Optimiztion. lwys keep root pge in memory. 44

32 Building lrge B tree full pge, out to split externl nodes (line segment of length proportionl to numer of keys in tht node) 45 Building lrge B-tree

33 Blnced trees in the wild ed-lck trees re widely used s system symol tles. Jv: jv.util.treep, jv.util.treeet. ++ TL: mp, multimp, multiset. Linux kernel: completely fir scheduler, linux/rtree.h. B-tree vrints. B+ tree, B*tree, B# tree, B-trees (nd vrints) re widely used for file systems nd dtses. Windows: F. c: F, F+. Linux: eiserf, XF, xt3f, JF. Dtses: OL, DB2, ING, QL, ostgreql. 46

Balanced Trees. 2-3 trees red-black trees B-trees. 2-3 trees red-black trees B-trees smaller than. 2-node. 3-node E J S X A C.

Balanced Trees. 2-3 trees red-black trees B-trees. 2-3 trees red-black trees B-trees smaller than. 2-node. 3-node E J S X A C. ymol tle review Blned Trees implementtion gurntee verge se serh insert delete serh hit insert delete ordered itertion? opertions on keys sequentil serh (linked list) N N N N/2 N N/2 no equls() 2-3 trees