SortingAlgorithm Analysis

Sorting

Sorting is important! Things that would be much more difficult

without sorting:– finding a phone number in the phone book– looking up a word in the dictionary– finding a book in the library– buying a cd/dvd– renting a video – buying groceries

Any more ideas???

How to sort?

Sorting the student roster:

Alex Guttler Matt Simon Alexandra Eurdolian Katie Blaszak Joseph Kelly An Xuan Rebecca Davis

Sorting a hand of cards

Comparison-based sorting

Ordering decision based on comparison of elements

Requires that elements are comparable– have a natural order

• numerical numbers

• lexicographic (alphabetical) characters

• chronological dates

Arrays

Used to store a collection or list of elements of the same type

Arrays are indexed Example: an array A of integers

4 is the element of the array at index 1 A[1]=4

8 76914

0 1 2 3 4 5

arrayelements

arrayindices

A =

Insertion Sort

Similar to sorting a hand of cards Good for sorting a small number of elements Idea:

– n is number of elements or input size– start at element A[0]

• already sorted– for k = 1, 2, …, n

sort elements A[0] through A[k] by comparing A[k] with each element A[k-1], A[k-2], …, A[1], A[0]

Example

1. Start: 8 is sorted2. Compare 4,8

Swap 4,83. Compare 1,8

Swap 1,8Compare 1,4Swap 1,4

4. Compare 9,85. Compare 6,9

Swap 6,9 Compare 6,8 Swap 6,8 Compare 6,4

8 6914

4 6918

4 6981

1 6984

1 6984

1 9684

1 9864

Algorithm analysis

Determine the amount of resources an algorithm requires to run– computation time, space in memory

Running time of an algorithm is the number of basic operations performed – additions, multiplications, comparisons– usually grows with the size of the input– faster to add 2 numbers than to add 2,000,000!

Example: Adding n numbers takes linear time– requires n –1 basic operations (additions)– number of basic operations is proportional to the size

of the input

Running times

Worst-case running time– upper bound on the running time– guarantee the algorithm will never take

longer to run Average-case running time

– time it takes the algorithm to run on average (expected value)

Best-case running time– lower bound on the running time– guarantee the algorithm will not run faster

Analysis of Insertion Sort

We compare each element with previous elements until the ordering is correct

In the worst case, we compare each element with all of the previous elements– A[1] is compared with A[0]– A[2] is compared with A[1], A[0] – A[3] is compared with A[2], A[1], A[0] – A[4] is compared with A[3], A[2], A[1], A[0]

– A[n-1] is compared with A[n-2], …, A[1], A[0]

…

Number of comparisons (worst case)

Element k requires k comparisons Total number of comparisons:

0+1+2+ … + n-1 = ½ (n)(n-1) = ½ (n2-n)

Running time of insertion sort in the worst case is quadratic

Worst case behavior occurs when the array is in reverse sorted order

9 14678

Number of comparisons (best case)

What if the array is already sorted?

How many comparisons per element will be made by insertion sort?

1 98764

Running time of Insertion Sort

Best case running time is linear Worst case running time is quadratic Average case running time is quadratic

Insertion sort is only practical for small input– 100 operations to sort 10 elements– 10000 operations to sort 100 elements– 1000000 operations to sort 1000 elements

There are more efficient sorting algorithms!

The divide-and-conquer approach

Insertion sort uses an incremental approach– puts elements in correct place one at a time

We can design more efficient sorting algorithms using the divide-and-conquer approach:– Divide the problem into a number of

subproblems– Conquer the subproblems by solving them

recursively until the subproblems are small enough to solve directly

– Merge the solutions for the subproblems into the solution for the original problem

Mergesort

Divide-and-conquer sorting algorithm

Given an array of n elements– Divide the array into two subarrays each

with n/2 items– Conquer (solve) each subarray by sorting

it recursively– Merge the solutions to the subarrays by

merging them into a single sorted array

Example

27 10 12 20

27 10 12 20

12 2027 10

10 27 12 20

10 12 20 27

divide

divide

divide

merge

mergemerge

Merging two sorted arrays

10 27 12 20 10

10 27

10 27

10 27

12 20

12 20

12 20

10 12

10 12 20

10 12 20 27

result of mergesecond arrayfirst array