CE 221 Data Structures and Algorithms

Transcription

1 CE 1 ata Structures and Algorthms Chapter 4: Trees BST Text: Read Wess, 4.3 Izmr Unversty of Economcs 1

2 The Search Tree AT Bnary Search Trees An mportant applcaton of bnary trees s n searchng. Let us assume that each node n the tree stores an tem. Assume for smplcty that these are dstnct ntegers deal wth duplcates later. The property that makes a bnary tree nto a bnary search tree s that for every node, X, n the tree, the values of all the tems n the left subtree are smaller than the tem n X, and the values of tems n the rght subtree are larger than the tem n X. The tree on the left s a bnary search tree, but the tree on the rght s not. The tree on the rght has a node wth key 7 n the left subtree of a node wth key 6 whch happens to be the root. Izmr Unversty of Economcs

3 Bnary Search Trees - Operatons escrptons and mplementatons of the operatons that are usually performed on bnary search trees BST are gven. ote that because of the recursve defnton of trees, t s common to wrte these routnes recursvely. Because the average depth of a bnary search tree s Olog, we generally do not need to worry about runnng out of stack space. Snce all the elements can be ordered, we wll assume that the operators <, >, and = can be appled to them. Izmr Unversty of Economcs 3

4 BST Implementaton - I Izmr Unversty of Economcs 4

5 BST Implementaton - II Prevous slde Izmr Unversty of Economcs 5

6 BST Implementaton - III } Izmr Unversty of Economcs 6

7 BST Implementaton - IV contans returns true f element x s n the BST referenced by t, or false f there s no such node. The structure of the tree makes ths smple. If t s ULL, then we can just return. Otherwse, we make a recursve call on ether the left or the rght subtree of the node referenced by t. Izmr Unversty of Economcs 7

8 BST Implementaton - V To perform a fndmn, start at the root and go left as long as there s a left chld. The stoppng pont s the smallest element. The fndmax routne s the same, except that branchng s to the rght chld. otce that the degenerate case of an empty tree s carefully handled. Also notce that t s safe to change t n fndmax, snce we are only workng wth a copy. Always be extremely careful, however, because a statement such as t.rght=t.rght.rght wll make changes. Izmr Unversty of Economcs 8

9 BST Implementaton Inserton I The nserton routne s conceptually smple. To nsert x nto tree t, proceed down the tree as you would wth a contans. If x s found, do nothng or "update" somethng. Otherwse, nsert x at the last spot on the path traversed. uplcates can be handled by keepng an extra feld n the node ndcatng the frequency of occurrence. If the key s only part of a larger record, then all of the records wth the same key mght be kept n an auxlary data structure, such as a lst or another search tree. Insert node 5 Izmr Unversty of Economcs 9

10 BST Implementaton Inserton II Izmr Unversty of Economcs 10

11 BST Implementaton eleton I Once we have found the node to be deleted, we need to consder 3 possbltes. If the node s a leaf, t can be deleted mmedately. If the node has one chld, the node can be deleted after ts parent adjusts a ponter to bypass the node. otce that the deleted node s now unreferenced and can be dsposed of by automatc garbage collecton. elete node 4 Izmr Unversty of Economcs 11

12 BST Implementaton eleton II 3 The complcated case deals wth a node wth two chldren. The general strategy s to replace the key of ths node wth the smallest key of the rght subtree easy and recursvely delete that node whch s now empty. Because the smallest node n the rght subtree cannot have a left chld, the second delete s an easy one. elete node Izmr Unversty of Economcs 1

13 BST Implementaton eleton III Ineffcent, snce calls hghlghted n yellow result n two passes down the tree to fnd and delete the smallest node n the rght subtree. Izmr Unversty of Economcs 13

14 BST Implementaton eleton IV... We can use stacks to convert an expresson n standart form otherwse known as nfx nto postfx. else f t.left!= null && t.rght!= null{ Bnaryode<AnyType> tmp,prev; /* declare references */ } tmp = t.rght; /* pont to smallest n the rght */ prev = t.rght; /* pont to parent of tmp */ whle tmp.left!= null{ /* fnd smallest of rght */ Example: operators = {+, *,, }, usual prev = tmp; precedence tmp = tmp.left; rules; a + b * c + d * e + f * g } Answer = a b c * + d e * f + g * + t.element = tmp.element; /* replace wth smallest */ f tmp == prev /* t.rght s smallest */ t.rght = tmp.rght;/* skp over tmp */ else /* connect left of prev to rght of tmp */ prev.left = tmp.rght;... Effcent Verson Izmr Unversty of Economcs 14

15 BST Implementaton Lazy eleton If the number of deletons s small, then a popular strategy to use s lazy deleton: When an element s to be deleted, t s left n the tree and merely marked as deleted. Ths s especally popular f duplcates are present, because then the feld that keeps count of the tems can be decremented. If the number of real nodes s the same as the number of "deleted" nodes, then the depth of the tree s only expected to go up by a small constant why?, so there s a very small tme penalty assocated wth lazy deleton. Also, f an tem s renserted, the overhead of allocatng a new cell s avoded. Izmr Unversty of Economcs 15

16 Average-Case Analyss - I All of the operatons of BST, except MakeEmpty, take Od tme where d s the depth of the node contanng the accessed key. As a result, they are O depth of tree. Why? Because n constant tme we descend a level n the tree, thus operatng on a tree that s now roughly half as large. MakeEmpty smply ntalzes root to null; hence O? Observaton: The average depth over all nodes n a BST s Olog assumng all nserton sequences are equally lkely. Proof: The sum of the depths of all nodes n a tree s the nternal path length. Let s calculate the average nternal path length over all possble nserton sequences. Izmr Unversty of Economcs 16

17 Average-Case Analyss - II Let be the nternal path length for some BST T of nodes. = 0. = // Subtree nodes are 1 level deeper All subtree szes are equally lkely for BSTs, snce t depends only on the rank of the frst element nserted nto BST. Ths does not hold for bnary trees though. Let s, then, average: 1/ 1 0 / 1 0 1, 0 If the recurrence s solved, = O log. Thus, the expected depth of any node s Olog. Izmr Unversty of Economcs 17

18 ervaton of /, / subtract from / /...dvde by + / /... *1 /3 / *3 1 / /...sum the equatons sde by sde 1 Izmr Unversty of Economcs 18

19 Izmr Unversty of Economcs 19 ervaton of / 1 / / / 1/ / 1/3 / / 1 / 3/ log / 1/ 1/ 1/ 3/ log / 1/ 1/ 1/ 1/ / 1/3 / e e 4 4 log 4/ 4 log / e e log log log log 4 4 log O e O O e e

20 Average-Case Analyss - III As an example, the randomly generated 500 node BST has nodes at expected depth Izmr Unversty of Economcs 0

21 Average-Case Analyss - IV eleton algorthm descrbed favors makng left subtrees deeper than the rght a deleted node s replaced wth a node from the rght. The exact effect of ths stll unknown, but f nsertons and deletons are alternated Ɵ tmes, expected depth s.in the absence of deletons or when lazy deleton s used; average runnng tmes for BST operatons are Olog. After a quarter-mllon random nsert/remove pars, rght-heavy tree on the prevous slde, looks decdedly unbalanced and average depth becomes Izmr Unversty of Economcs 1

22 Homework Assgnments 4.9.a, 4.9b, 4.16, 4.37, 4.48 You are requested to study and solve the exercses. ote that these are for you to practce only. You are not to delver the results to me. Izmr Unversty of Economcs