1 Programming for Engineers in Python Autumn Lecture 9: Sorting, Searching and Time Complexity Analysis

Design a recursive algorithm by 1. Solving big instances using the solution to smaller instances 2. Solving directly the base cases Recursive algorithms have 1. Stopping criteria 2. Recursive case(s) 3. Construction of a solution using solution to smaller instances 2 Lecture 8: Highlights

3 Today Information Importance of quick access to information How can it be done? Preprocessing the data enables fast access to what we are interested in Example: dictionary The most basic data structure in Python: list Sorting  preprocessing Searching  fast access Time complexity

Information 4 There are about 20,000,000,000 web pages in the internet

Sorting A sorted array is an array whose values are in ascending/descending order Very useful Sorted array example: Super easy in Python – the sorted function

Why is it Important to Sort Information? To find a value, and fast! Finding M values in a list of size N Naive solution: given a query, traverse the list and find the value Not efficient, average of N/2 operations per query Better: sort the array once and than perform each query much faster 6

Why not Use Dictionaries? Good idea! Not appropriate for all applications: Find the 5 most similar results to the query Query's percentile We will refer to array elements of the form (key, value) 7

Naïve Search in a General Array Find location of a value in a given array 8

Binary Search (requires a sorted array) 9 Input: sorted array A, query k Output: corresponding value / not found Algorithm: Check the middle element in A If the corresponding key equals k return corresponding value If k < middle find k in A[0:middle-1] If k > middle find k in A[middle+1:end]

10 Example value index Searching for 56

11 Example value index Searching for 4

12 Code –Binary Search

13 Binary Search – 2 nd Try

Time Complexity 14 Worst case: Array size decreases with every recursive call Every step is extremely fast (constant number of operations - c) There are at most log 2 (n) steps Total of approximately c*log 2 (n) For n = 1,000,000 binary search will take 20 steps - much faster than the naive search

Time Complexity על רגל אחת 15 Algorithms complexity is measured by run time and space (memory) Ignoring quick operations that execute constant number of times (independent of input size) Approximate time complexity in order of magnitude, denoted with O ( Example: n = 1,000,000 O(n 2 ) = constant * trillion (Tera) O(n) = constant * million (Mega) O(log 2 (n)) = constant * 20

16 Order of Magnitude nlog 2 nn log 2 nn2n ,04865,536 4, ,15216,777,216 65, ,048,5654,294,967,296 1,048, ,971,5201,099,511,627,776 16,777, ,653,183281,474,976,710,656

Graphical Comparison 17

18 Code – Iterative Binary Search

19 Testing Efficiency Preparations (Tutorial for timeit:

20 Testing Efficiency

21 Results

Until now we assumed that the array is sorted… 22 How to sort an array efficiently?

Bubble Sort מיון בועות 23 נסרוק את המערך ונשווה כל זוג ערכים שכנים נחליף ביניהם אם הם בסדר הפוך נחזור על התהליך עד שלא צריך לבצע יותר החלפות (המערך ממויין) למה בועות? האלגוריתם "מבעבע" בכל סריקה את האיבר הגדול ביותר למקומו הנכון בסוף המערך.

(done) Bubble Sort Example

25 Code – Bubble Sort n iterations i iterations constant (n-1 + n-2 + n-3 + …. + 1) * const ~ ½ * n 2

26 דוגמאות לחישוב סיבוכיות זמן ריצה מצא ערך מקסימלי במערך לא ממויין מצא ערך מקסימלי במערך ממויין מצא את הערך החמישי הכי גדול במערך ממויין מצא ערך מסויים במערך לא ממויין מצא ערך מסויים במערך ממויין ענה על n " שאלות פיבונאצ ' י " שאלת פיבונאצ ' י : מהו הערך ה -K בסדרת פיבונאצ ' י ? נניח ש -K מוגבל להיות קטן מ -MAX

We showed that it is possible to sort in O(n 2 ) Can we do it faster? 27 Yes! – Merge Sort

28 Comparing Bubble Sort with sorted

The Idea Behind Merge Sort 29 Sorting a short array is much faster than a long array Two sorted arrays can be merged to a combined sorted array quite fast (O(n))

Generic Sorting 30 We would like to sort all kinds of data types Ascending / descending order What is the difference between the different cases? Same algorithm! Are we required to duplicate the same algorithm for each data type?

The Idea Behind Generic Sorting Write a single function that will be able to sort all types in ascending and descending order What are the parameters? The list Ascending/descending order A comparative function 31