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EECS 281: Data Structures and Algorithms Dictionary Abstract Data Type Dictionary ADT Store elements so that they can be quickly located using keys Typically, useful additional information in addition to key Examples – bank accounts with SSN as key – student records with UMID or uniqname as key Dictionary ADTs Types Log File Ordered Dictionary Hash Table Skip List Operations Search Insertion Removal 1 Dictionary ADTs Stores items by key – element pairs (k,e) k and e may be of any type k and e may be the same In general, items with the same key may be stored in same Dictionary Unordered vs Ordered Ordered Relative order determined by comparator between keys Total Order relation defined on keys Unordered No order relation is assumed on keys Only equality testing between keys Dictionary ADT: Log File Defn: implementation of Dictionary ADT using a sequence to store items in arbitrary order Obviously, Unordered Dictionary Useful implementation for case with many insertions and few searches Implemented as array (vector) or linked list 2 Dictionary ADT: Log File Search Look through entire data set O( ) Insertion Always insert at end O( ) Removal First must find, then remove O( ) Space, proportional to size of log file O( ) Applications of Log Files Database systems File systems Security audit trails Dict. ADT: Ordered Dictionary Defn: implementation of Dictionary ADT in which usual operations may be used and there exists an order relationship between keys Useful implementation for few insertions / removals, but many searches 3 Binary Search Iterative int search(int a[], int v, int left, int right) while (right >= left) { int mid = (left + right)/2; if (v == a[mid]) return mid; if (v < a[mid]) right = mid - 1; else left = mid + 1; } return -1; Binary Search Recursive int searchR(int a[], int v, int left, int right) if (left > right) return -1; int mid = (left + right)/2; if (v == a[mid]) return mid; if (v < a[mid]) return searchR(a[], v, left, mid - 1); else return searchR(a[], v, mid + 1, right); } Dict. ADT: Ordered Dictionary Search Binary Search (iterative or recursive) O( ) Insertion Always insert in correct location O( ) to find location and O( ) to move subsequent items Removal First must find, then remove O( ) to find and O( ) to move subsequent items 4 Notes Binary search is an example of divide-andconquer technique Rule: An algorithm is O(log n) if each constant time operation (e.g., CMP) can cut the problem size by a fraction (e.g., half). Corollary: If constant time is required to reduce the problem size by a constant amount, the algorithm is O(N). Divide and conquer doesn’t work with linked list, unfortunately Summary: Dictionary ADT Many implementations – Log File (today) – Ordered Dictionary (today) – Hash Table (soon) Different styles lead to different O( ) for – search – insertion – removal 5