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Data Structures

Lecture 3 Dynamic Sets / Dictionaries Binary Search Trees. Data Structures. Haim Kaplan and Uri Zwick November 2013. Dictionaries/Dynamic sets. Maintain a set of items . Each item has key and info fields.

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Data Structures

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  1. Lecture 3 Dynamic Sets / Dictionaries Binary Search Trees Data Structures Haim Kaplan and Uri ZwickNovember 2013

  2. Dictionaries/Dynamic sets • Maintain a set of items. • Each item has key and info fields. • Keys belong to a totally ordered universe, and can be compared with each other • Support the following operations: Insert, Delete, Search, Min, Max, … Extremely useful data structure!

  3. Abstract Data Type: Dictionaries Dic-Item(k,i) – Create a dictionary item containing keyk and infoi, Key(I), Info(I) – The keyand info contained in Dic-ItemI. • Dictionary() – Create an emptydictionary • Insert(D,I) – Insert I into D • Delete(D,I) – Delete I from D (assuming I is in D) • Search(D,k) – Find a dic-item with key k in D, if any. • Min(D) – Return the dic-item with the minimum key in D. • Max(D) – Return the dic-item with the maximum key in D. • Successor(D,I) – Return the successor of I in D. • Predecessor(D,I) – Return the predecessor of I in D. Assume that dic-items have distinct keys

  4. Implementing dictionaries using doubly linked lists (ver. 1) • Store the dic-items in a list, in no particular order. • Insert a new dic-item to an arbitrary position of the list, e.g., the first or last position. • Delete a dic-item using a supplied pointer to it. • Search, and other operations, are implemented by scanning the list. Insert, Delete – O(1) time Other operations – O(n) time

  5. Implementing dictionaries using doubly linked lists (ver. 2) • Store the dic-items in a list, in increasing order of keys. • Insert a new dic-item to the appropriate position • Delete a dic-item by using a supplied pointer to it. • Search is implemented by scanning the list, stopping when the key of the current item is larger than the key sought. Insert,Search – O(n) time (or O(n/2) “on average”) Delete – O(1) time Min, Max, Successor, Predecessor – O(1) time

  6. Implementing dictionaries using (circular) arrays • Store the dic-items in an array, in increasing order of keys. • Insert a new dic-item to the appropriate position • Delete a dic-item by using a supplied pointer to it. • Search implemented using binary search. Insert, Delete– O(n) time (or O(n/2) ) Min, Max, Successor, Predecessor – O(1) time Search – O(log n)

  7. Binary search Successful search:Search(38) high mid low 10 25 38 47 56 67 73 84 95 0 1 2 3 4 5 6 7 8

  8. Binary search Unsuccessful search:Search(39) high mid low 10 25 38 47 56 67 73 84 95 0 1 2 3 4 5 6 7 8

  9. Binary search Key k was foundin position mid Key k should be insertedin position mid or mid+1

  10. Can we implement alloperations in O(log n) time? Yes! Using Binary Search Trees

  11. 7 2 8 1 5 10 Binary search trees A binary tree in which each node contains a dic-item. Satisfies the binary-search-tree property:If y is in the left subtree of x, then y.key < x.key. If y is in the right subtree of x, then y.key > x.key. x parent key info right left

  12. 7 2 8 1 5 10 Binary search trees D.root x Dic-Item ≡ Tree-Node parent key info right left left, right, parent are initially null

  13. 1 7 2 2 8 8 1 5 10 7 10 5 A set can be represented using several different trees height=4 4 height=2 2 3 1 1 2 0 0 0 1 0 Height – length of a longest path to a leaf 0

  14. 7 2 8 1 10 5 Tree-Search(x,k) – Look for k in the subtree of x x x x Tree-Search(x,5) We usually start the search at the root of the tree: Search(D,k)  Tree-Search(D.root,k)

  15. 7 2 8 1 10 5 Tree-Position(x,k) – Look for k in the subtree of xReturn the last node encountered x x x y y y Tree-Position(x,6) Returns the node containing 5 Tree-Position(x,k) is used to find insertion points

  16. 7 2 8 1 5 10 Printing the elements of a BST in sorted order(In-order walk) 3 1 4 0 2 5 Printing, of course, is just an example…

  17. 7 2 8 1 5 10 Finding the minimum“keep going left”

  18. Successor(x) If x has a right child, the successor of xis the minimal element in x.right. x “Go right once, and then left all the way” What if x.right=null ?

  19. Successor(x) If x.right=null, the successor of x is the lowestancestory of x such that x is in its left subtree y “Go up from x untilthe first turn right’’ x

  20. Successor(x) y If x has the largest key, then Successor(x)=null. x Predecessor is symmetric

  21. Insertions and deletions

  22. 7 2 8 1 5 10 Insertions 6 9 Insert(6) Insert(9)

  23. Deletion: easy cases first 7 2 8 1 5 10 6 9 Delete(6) – 6 is a leaf; simply remove it. Delete(8) – 8 has only one child; bypass it. Delete(10) – 10 has only one child; bypass it. Delete(2) – more complicated…

  24. Deletion of a binary node If z has two children,let y be the successor of z z y has no left child Remove y from the tree Replace z by y y Binary-search-tree property preserved! Is it enough to let z.keyy.key? And maybe also z.infoy.info?

  25. Analysis Each operation takes O(h+1)time, where h is the height of the tree In general h may be as large as n Want to keep the tree with small h

  26. Balanced trees A full tree of height h contains n=2h+1 − 1 nodes h = log2(n+1)−1 How do we keep the tree more or less balanced?

  27. Randomly built BSTs Maybe balancing will take care of itself? Not if we insert the elements in sorted order.We get a path of length n Things are usually ok if weinsert the elements in random order Theorem: If n distinct keys are inserted into a BST in random order, then the expected height of the tree is O(log n). We want worst-case results…

  28. Rotations x y Right rotate y x A C A C B B Left rotate

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