Bit Vector Linked List (Sorted and Unsorted) Hash Table Search Trees Set Implementations Bit Vector Linked List (Sorted and Unsorted) Hash Table Search Trees 9/21/2018 CS 303 – Implement Set Lecture 9

Bit Vector A = 1 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 U = ai, 0 <= i <= 15 A = {a0,a3,a4,a7,a9,a12,a13,a15} 1 Bit per possible element If |U| <= wordlength, then very efficient. If not...NOT Easy: UNION, INTERSECTION, DIFFERENCE, MEMBER, NULL, EQUAL Hard: MIN, MAX Impossible: nothing – this is a “direct model” 9/21/2018 CS 303 – Implement Set Lecture 9

Linked List A a0 a3 a15 U = ai Unsorted: ai’s kept in any order (making insert easy? NO! why?) Easy: NULL Hard: Everything else! (motivation for linear order, <) Sorted (by <): ai’s kept sorted Hard: MEMBER Easy: Everything else (most operations are 1 pass through data) 9/21/2018 CS 303 – Implement Set Lecture 9

Dictionary ADT MakeNull(A): A = NULL Member(A): (x in A)? Insert(A): A = A UNION {x} Delete(x,A): A = A – {x} Note: no requirement for Ordering, UNION, INTERSECTION, DIFFERENCE, ... Applications: Phone listings, symbol tables, words (duh!) Note that applications may SORT as a matter of implementation convenience 9/21/2018 CS 303 – Implement Set Lecture 9

Dictionary Implementations Bit Vector – almost never useful Arrays or linked list (unsorted) MEMBER – scan entire list! INSERT – if not MEMBER, add DELETE – if MEMBER, remove (for Array, move everything) All are O(n), because of MEMBER Arrays or linked list (sorted) speeds search (O(n) becomes O(log n) for array) slows INSERT (O(1) becomes O(1)) but perhaps the application doesn’t do a lot of INSERTs 9/21/2018 CS 303 – Implement Set Lecture 9

Binary Search Trees An implementation technique aimed at Sets with INSERT, DELETE, MEMBER, MIN MEMBER, MIN – Binary Search INSERT – Search until x or L is found; add if necessary DELETE – Find x. If a leaf, remove it. If not a leaf find largest(smallest) y in left(right) subtree move y to where x is Special cases at root! [exercise!] a b c b < a < c 9/21/2018 CS 303 – Implement Set Lecture 9

Analysis of Binary Search Tree Best Case: BST is balanced. All operations take O(log n) time Worst Case: BST is linear. All operations take O(n) time Average Case: BST is “random”. The BST is “almost balanced” and operations take O(c log n) time. Worse than fully balanced, but only by a constant factor. Next time – how to avoid the worst case...read about AVL, Splay, and B-trees! But...I’ll talk about 2-3 Trees 9/21/2018 CS 303 – Implement Set Lecture 9