Programming for Engineers in Python
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1 Programming for Engineers in Python Lecture 9: Sorting, Searching and Time Complexity Analysis Autumn Lecture 8: Highlights 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 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 3 There are about 20,000,000,000 web pages in the internet Information 4 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 5 6 1
2 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) Naïve Search in a General Array Find location of a value in a given array 7 8 Binary Search (requires a sorted array) Example 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] Searching for 56 index value Example Code Binary Search Searching for 4 index value
3 Binary Search 2 nd Try Time Complexity 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 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 Order of Magnitude n log 2 n n log 2 n n ,048 65,536 4, ,152 16,777,216 65, ,048,565 4,294,967, ,971,520 1,099,511,627,776 1,048, ,653, ,474,976,710,656 16,777, Graphical Comparison Code Iterative Binary Search
4 Testing Efficiency Preparations Testing Efficiency (Tutorial for timeit: Results Until now we assumed that the array is sorted How to sort an array efficiently? Bubble Sort מיון בועות Bubble Sort Example 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)
5 Code Bubble Sort constant (n-1 + n-2 + n ) * const ~ ½ * n 2 n iterations i iterations 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 Comparing Bubble Sort with sorted We showed that it is possible to sort in O(n 2 ) Can we do it faster? Yes! Merge Sort The Idea Behind Merge Sort 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 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?
6 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 6
Programming for Engineers in Python
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