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Introduction to Algorithms

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Introduction to Algorithms

- Algorithms are ways of solving problems. There is a technical definition that basically says an algorithm is a clear set of instructions which if followed will lead to a correct problem solution in a finite amount of time
- Sorting is a very important problem and has been studied extensively. We begin by looking at a simple sorting algorithm
- We build up to the algorithm by starting with finding the smallest element in an array

min = data(1) 'the smallest so far

Fork = 2 TolastIndex

'test each element to see if it is the min so far

Ifdata(k) < min Then'new min found

min = data(k)

End If

Next

- What if we want to find the smallest element and put it first in the array?
- We’ll do this by switching it with the first element
- We need to know the index of the smallest element to do this
- A slight modification of our code for finding the smallest element will let us do this

min = data(1) 'the smallest so far

minIndex = 1

For k = 2 TolastNdx

'test each element to see if it is the min so far

If data(k) < min Then'new min found

min = data(k)

minIndex = k

End If

Next k

- Consider the following code:
varA = 1

varB = 4

varA = varB

varB = varA

- What are the values of varA and varB after I do this?

- Doing it right:
varA = 1

varB = 4

temp = varB

varB = varA

varA = temp

- What are the values of varA and varB after I do this?

‘*** find the smallest element, value and index

min = data(1) 'the smallest so far

minIndex = 1

For k = 2 TolastNdx

'test each element to see if it is the min so far

Ifdata(k) < min Then'new min found

min = data(k)

minIndex = k

End If

Next

‘*** exchange the smallest element with the first

temp = data(1)

data(1) = data(minIndex)

data(minIndex) = temp

- Find the smallest element in the array, and switch it with the first one
- Find the smallest element in the rest of the array, and switch it with the second one
- Etc.
- This is called selection sort

- Here’s an initial array:
- The smallest element is in index 3. If we switch it with the element in index 1, we get:
- We now know the first element is the smallest.

- Looking at elements 2-5, the smallest is in index 5
- Let’s switch with the element in index 2
- Now we know the first two are smallest, and are in the right order.

- Consider elements 3-5. The smallest is in position 5.
- Let’s switch with the element in index 3

- Consider elements 4-5. The smallest is in position 5.
- Let’s switch with the element in index 4
- This finishes sorting the array (why?)

- We want to work on the whole array, then the array without the first element, then the array without the second element, etc.
- If we work on a whole array of n elements, that’s a loop from 1 to n.
- If we work on a whole array minus the first element, that loop is from 2 to n.
- Next we do 3 to n, etc.

- A loop from 1 to n looks like:
Fork = 1 Ton

<code>

Next k

- Here’s a loop from 2 to n:
Fork = 2 Ton

<code>

Next k

- We need a loop that looks like this:
Fork = jTon

<code>

Next k

for each j going from 1 to n-1. This we can do by using another loop!

- Here’s what the structure looks like
Forj = 1 Ton - 1

Fork = j + 1 Ton

<code>

Next k

Next j

For j = 1 To lastNdx – 1 ‘start with element j

min = data(j) 'the largest so far

minIndex = j

For k = j + 1 TolastNdx‘look at elements that follow j

Ifdata(k) < min Then

min = data(k)

minIndex = k

End If

Next k

temp = data(j) ‘exchange the smallest element with element j

data(j) = data(minIndex)

data(minIndex) = temp

Next j

- Note the -1 and +1 in the loop limits. Getting those right takes some thought
- Does the code work on arrays with just one element? With two elements? With no elements? (Nothing to sort, but we want to avoid a runtime error.) What if the data is already sorted?

Demo: Simple Sort

- There are actually many ways of sorting items
- Sorting is very important so people have put a lot of thought into it
- Some well-known methods:
- Bubble sort
- Quicksort
- Heapsort
- Mergesort
- Bucket sort

- With small data sets, the best method is usually the easiest one to program
- With large data sets, speed becomes an issue
- We could measure the time with a stopwatch, but the essential factor is the functional form of the time: if n is the length of the list of data, is the time proportional to n? n log n? n2?

- The number of comparisons of data elements in a sorting algorithm is usually proportional to the time
- On the first loop in Selection Sort, we do n-1 comparisons. The second loop does n-2, etc; the last loop does 1
- So the time is roughly proportional to
(n-1) + (n-2) + … + 1 = (n^2 – n)/2

- The largest power of n is n^2 which dominates the time for this algorithm
- This means Selection Sort is actually too slow to use on large amounts of data

- You now know the basics for programming: assignment statements, conditionals, procedures and functions, loops, and arrays
- This is like knowing the rules for chess or go
- What you have only started to learn are the tactics and strategies to use these tools effectively
- Algorithms are the tactics for how to accomplish tasks quickly and correctly

- Software engineering is about the strategies to control the complexity of designing large programs
- We’ve been learning a few of these strategies (e.g. naming conventions, principles of program structure, requirements and specifications)
- Good software engineering allows one person or a large group to produce a complex program that is correct and cost-effective