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Introduction to Algorithms. 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

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algorithms
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
finding the smallest element in an array
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

find the smallest and make it the first
Find the Smallest and Make it the First
  • 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
find the smallest element and its index
Find the smallest element and its 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

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

min = data(k)

minIndex = k

End If

Next k

switching the values of variables problem
Switching the Values of Variables(PROBLEM!!)
  • Consider the following code:

varA = 1

varB = 4

varA = varB

varB = varA

  • What are the values of varA and varB after I do this?
switching the values of variables
Switching the Values of Variables
  • 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?
now using the array
Now Using the Array

‘*** 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

idea for sorting
Idea for Sorting
  • 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
algorithm picture 1
Algorithm picture (1)
  • 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.
algorithm picture 2
Algorithm picture (2)
  • 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.
algorithm picture 3
Algorithm picture (3)
  • Consider elements 3-5. The smallest is in position 5.
  • Let’s switch with the element in index 3
algorithm picture 4
Algorithm picture (4)
  • 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?)
algorithm structure
Algorithm Structure
  • 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.
loop setup
Loop Setup
  • 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

in general
In general…
  • 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!

the nested loop
The Nested Loop
  • Here’s what the structure looks like

Forj = 1 Ton - 1

Fork = j + 1 Ton

<code>

Next k

Next j

here s the code
Here’s the Code

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

tricky bits
Tricky Bits
  • 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?
other ways of sorting
Other Ways of Sorting
  • 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
which method is best
Which Method is Best?
  • 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?
time for selection sort
Time for Selection Sort
  • 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
importance of algorithms
Importance of Algorithms
  • 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
Software Engineering
  • 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
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