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Programming for Engineers in

Python

Autumn 2011-12

Lecture 9: Sorting, Searching and Time Complexity Analysis

Design a recursive algorithm by1. Solving big instances using the solution to smaller

instances2. Solving directly the base cases

Recursive algorithms have1. Stopping criteria2. Recursive case(s)3. Construction of a solution using solution to

smaller instances 2

Lecture 8: Highlights

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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

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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

1 2 5 678 139

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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

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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)

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Naïve Search in a General Array

• Find location of a value in a given array

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Binary Search (requires a sorted array)

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• 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]

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Example

-5-30481122565797value

index 0 2 3 4 51 6 7 8 9

Searching for 56

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Example

-5-30481122565797value

index 0 2 3 4 51 6 7 8 9

Searching for 4

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Code –Binary Search

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Binary Search – 2nd Try

Time Complexity

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• Worst case: • Array size decreases with every recursive call

• Every step is extremely fast (constant number of operations - c)

• There are at most log2(n) steps

• Total of approximately c*log2(n)

• For n = 1,000,000 binary search will take 20 steps - much faster than the naive search

Time Complexityעל רגל אחת

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• 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 (http://en.wikipedia.org/wiki/Big_O_notation)

• Example: n = 1,000,000• O(n2) = constant * trillion (Tera)

• O(n) = constant * million (Mega)

• O(log2(n)) = constant * 20

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Order of Magnitude

nlog2 nn log2 nn2

1001

16464256

25682,04865,536

4,0961249,15216,777,216

65,536161,048,5654,294,967,296

1,048,5762020,971,5201,099,511,627,776

16,777,21624402,653,183281,474,976,710,656

Graphical Comparison

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Code – Iterative Binary Search

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Testing EfficiencyPreparations

http://www.doughellmann.com/PyMOTW/timeit/

(Tutorial for timeit: http://www.doughellmann.com/PyMOTW/timeit/)

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Testing Efficiency

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Results

Until now we assumed that the array is sorted…

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How to sort an array efficiently?

Bubble Sortמיון בועות

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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.

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7 2 8 5 4

2 7 8 5 4

2 7 8 5 4

2 7 5 8 4

2 7 5 4 8

2 7 5 4 8

2 5 7 4 8

2 5 4 7 8

2 7 5 4 8

2 5 4 7 8

2 4 5 7 8

2 5 4 7 8

2 4 5 7 8

2 4 5 7 8

)done(

Bubble Sort Example

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Code – Bubble Sort

n iterations

i iterations

constant

)n-1 + n-2 + n-3 + …. + 1 * (const ~ ½ * n2

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זמן סיבוכיות לחישוב דוגמאותריצה

מצא ערך מקסימלי במערך לא ממויין•מצא ערך מקסימלי במערך ממויין•מצא את הערך החמישי הכי גדול במערך •

ממוייןמצא ערך מסויים במערך לא ממויין•מצא ערך מסויים במערך ממויין• "שאלות פיבונאצ'י"nענה על •

בסדרת פיבונאצ'י?Kשאלת פיבונאצ'י: מהו הערך ה-•MAX מוגבל להיות קטן מ-Kנניח ש-•

We showed that it is possible to sort in O(n2)Can we do it faster?

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Yes! – Merge Sort

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Comparing Bubble Sort with sorted

The Idea Behind Merge Sort

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• 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

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• 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

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