Sets: Efficient Searching and Insertion

Transcription

1 Sets: Efficient Searching and Insertion PIC 10B May 18, 2016 PIC 10B Sets: Efficient Searching and Insertion May 18, / 9

2 Overview of Lecture 1 Searching and Inserting 2 Sets 3 Questions PIC 10B Sets: Efficient Searching and Insertion May 18, / 9

3 A Problem Imagine you are in a foreign language class... As you read a text, you want to make flashcards for the words you don t recognize. You don t want duplicate flashcards. Before making a new flashcard, you should check that you haven t already made a flashcard for that word. How should you organize your flashcards? Your organizational scheme should allow you to: Efficiently search for a word/flashcard in your dictionary. Efficiently insert a new word/flashcard in your dictionary. PIC 10B Sets: Efficient Searching and Insertion May 18, / 9

4 Solution Using a Vector/Array Searching a vector or array: If the array is unsorted, searching takes time O(n) (linear search). If the array is sorted, searching takes time O(log n) (binary search). So we should keep our vector/array sorted for efficient searching! Insertion into a sorted vector/array How long does it take to insert a new word into a sorted array/vector? It takes time O(n) (which is slow). Insertion at the end, then sorting would take time O(n log n) (which is even slower!). Conclusion: sorted vector/array would allow for fast searching, but slow insertion. PIC 10B Sets: Efficient Searching and Insertion May 18, / 9

5 Solution Using a Linked List What if we tried to solve the problem with a linked list? Insertion into a linked list takes time O(1)......after we know where to insert a new word. But searching a linked list takes time...o(n) (even if the list is sorted!) Conclusion: insertion into a linked list is fast, but searching is slow! Data Structure Search time Insert time vector/array (sorted) O(log n) O(n) linked list O(n) O(1) Is there a way to get the best of both worlds: efficient searching and insertion? The problem with vectors/arrays/lists is that these data structures are one dimensional... PIC 10B Sets: Efficient Searching and Insertion May 18, / 9

6 A New C++ Datatype: std::set C++ supports a data type called std::set which allows searching and insertion both in time O(log n)! Sets support the following basic operations insert Add a new value into the set if the value is already stored in the set, insert has no effect. Every element in a set is unique! erase Remove a value from the set no effect if the value is not contained in the set. find Search for a value in the set. Returns an iterator to the value if it is found, or set::end if the value is not found. PIC 10B Sets: Efficient Searching and Insertion May 18, / 9

7 Syntax for std::set # include <set >... std :: set < int > vals ; // declare a set of ints vals. insert (1); // insert the value 1 into the set vals. insert (2); vals. insert (1); // vals does not change - std :: set <int >:: iterator it; it = vals. find (3); // it has value set :: end it = vals. find (1); // *it has value 1 vals. erase ( it ); // remove the value 1 from the set // cout ( val ) returns the number of times // val appears in the set (0 or 1) int count = vals. count (1); // count has value 0 count = vals. count (2); // count has value 1 // returns num of ( distinct ) elements in set vals. size (); See also std::set on cplusplus.com. PIC 10B Sets: Efficient Searching and Insertion May 18, / 9

8 Notes and Efficiency of set Operations Notes on std::set operations The data type stored in a std::set must have operator< defined. The elements in the std::set must be ordered! Iterators for std::set support incrementation and decrementation. Elements are visited sorted order according to operator<. Runtime comparisons: Data Structure Search time Insert time Deletion time vector/array (sorted) O(log n) O(n) O(n) linked list O(n) O(1) O(1) std::set O(log n) O(log n) O(log n) See insearch.cpp for a runtime comparison. PIC 10B Sets: Efficient Searching and Insertion May 18, / 9

9 Questions 1 Write a program which counts the distinct words in a text using std::set. 2 Implement the following way of generating the first n prime numbers (known as the sieve of Eratosthenes): Create a std::set which contains the numbers 2 through n. Remove all multiples of 2 not equal to 2 from the set. Find the next smallest number in the set, and remove all of its multiples. Repeat the previous step until you ve removed all multiples. When the process above terminates, your std::set should contain precisely the prime numbers that are at most n. 3 Implement the following procedure: Create a set with the numbers 1 through n. For i = 2, 3,..., n, search for each multiple of i in the set. If the set contains the multiple of i, remove it from the set. If the set does not contain the multiple of i, insert the multiple of i to set. What values does the set contain when you finish? PIC 10B Sets: Efficient Searching and Insertion May 18, / 9