Lecture 2 data structures & algorithms - sorting techniques

Description: A detailed discussion about algorithms and their
measures, and understanding sorting.
Duration: 90 minutes
Starts at: Saturday 11th May 2013, 11:00AM
-by Dharmendra Prasad
1

Table of Contents
1. A detailed talk on algorithms.
2. Measuring the effectiveness of an algorithm.
3. Sorting Algorithms (In memory)
4. Some real world sorting scenarios.
2

Algorithm:
1. A step by step procedure to solve a given problem.
2. It takes an input and applies the sequence of steps to arrive
on a desired output.
3. Represented as a Pseudo Codes, which are English like
statements.
3

Analysis of Algorithms:
1. The theoretical study of computer program performance
and resource usage.
2. Things more important than the performance and resource
usage are:
1. Correctness
2. Simplicity
3. Features
4. User friendliness
5. Maintainability
6. Security
7. Algorithms
4

Why study performance if other things are more important than
performance?
Performance measures the line between the feasible or in feasible:
 E.g. : If something takes un limited time, it is not useful to us.
 If it takes a lot of space which is more than available , it is infeasible.
 Real time constraints, need the result on time else the result is not useful.
We are the decision makers when have to choose between performance and other
factors like user experience, maintainability etc.
Analyzing an algorithm means studying its performance in terms of running time,
space required, etc.
5

Sorting: The classic problem
Problem definition:
Input: We have a sequence <a1, a2, … , an> of numbers.
Output expected: A permutation <a1’, a2; …, an’> such that
a1’ <= a2’ <= a3’ … <=an’
Basic definition: Take a bunch of numbers and put them in order. Order
necessarily doesn’t mean increasing order. It can be
ascending/descending etc.
6

Pseudo Code:
Insertion-Sort(A, n) // sorts A[1..n]
for j <- 2 to n
do key <- A[j];
i <- j-1
while i > 0 and A[i] > key
do A[i+1] <- A[i]
i = i -1
A[i+1] = key
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A->
sorted
J
key

Example:
8 2 4 9 3 6
2 8 4 9 3 6
2 4 8 9 3 6
2 4 8 9 3 6
2 3 4 8 9 6
2 3 4 6 8 9
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Analyzing the above algorithm:
Running Time:
• Depends on the input (e.g. if it is already sorted)
• Depends on input size ( e.g. 10 elements vs. 10,000,000)
If the input is already sorted, Insertion Sort has nothing much to do.( This is
called the best case)
If the input is reverse sorted, Insertion sort will shuffle in each step and takes
the maximum possible time. ( This is called the worst case)
If the input is random that will be called as the average case.
Time taken in the above algorithm = Sum of the work inside the outer loop
T(n) =∑ f( j ) which is proportional to j²
You can also say as T(n) = θ(n²)
Can we say that insertion sort is fast?
9

Merge Sort:
Merge Sort A[1 .. n]
Step 1: if n = 1, done;
Step 2: Recursively sort A[1..(n/2)] and A[n/2 +1, n]
Step 3: Merge both the sorted lists.
Merge A, B
10
A B
1 2 7 9 11 12 13 20

How much time merging took ?
It just took n steps or we can say the work done is proportional to
n , hence the time taken will be:
T(n) proportional to n where n is the total number of elements in
the sorted array.
T(n) = θ(n)
Analyzing merge sort:
Step 1: How much this step takes?
Constant time, you just need to find the middle index of the
array. This means it is not dependent on the input size.
Also called θ(1)
Step 2: How much this step takes?
Time taken in sorting A[1, n/2] + Time taken in sorting A[n/2+1,
n]
Lets say that each of the two steps take T[n/2] time.
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Step 3: The merge step took θ(n).
So the total time taken by the merge sort is a sum of all the
three steps. That can be defined as below:
θ (1) , if n =1
T(n) =
2T(n/2) + θ (n) if n > 1
Hence, T(n) = 2T(n/2) + c.n, where c > 0
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Height of the above tree is : log n
T(n) = cn (log n) + something proportional to n
T(n) = cn (log n) + θ (n)
Hence T(n) = θ (n log n)
Comparing Insertion Sort and Merge Sort
Insertion sort time T(n) = θ(n²)
Merge sort time T’(n) = θ (n log n)
Which one do you think is more time?
14

Real life usage of sorting algorithm:
1) A cable operators personalized service.
2) Alphabetical arrangement of list of items on a shopping
website.
3) Offering admissions to applicants with highest grades.
And many more…
15

Lecture 2 data structures & algorithms - sorting techniques

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