Algorithm

• An algorithm is a step-by-step set of operations to be performed.• Real-life example: a recipe• Computer science example: determining the mode in an array

Sorting• Sorting is the process of arranging the elements in

an array in a particular order

• The sorting process is based on specific values – examples:

• sorting a list of test scores in ascending numeric order• sorting a list of people alphabetically by last

name

• There are many algorithms for sorting an array

• These algorithms vary in efficiency & speed

Big-O Notation• The efficiency/performance or “time complexity” of a sorting algorithm is

measured in “Big O” notation

• For example: if an algorithm has O(n^3) efficiency, that means that the maximum number of “touches” of the elements of the array necessary to sort it equals the cube of the number of elements in the array.

• So, for example, if an array holds 5 numbers, then in the worst-case scenario, it would take this particular sorting algorithm 125 different comparisons or moves of those numbers in order to sort it.

• If a sorting algorithm has efficiency O(n), this is called “linear time,” meaning that the algortithm’s efficiency is directly proportional to the size of the array.

You need to know the basics of these 4 sorting algorithms:1) Selection Sort

2) Insertion Sort

3) Merge Sort

4) Quick Sort

Selection Sort

1. find the smallest element in the array2. swap it with the first element3. find the next-smallest element in the array4. swap it with the second element5. repeat until all values are in their proper places

Efficiency: O(n^2)

Example:

original array: 3 9 6 1 2

smallest is 1: 1 9 6 3 2

smallest is 2: 1 2 6 3 9

smallest is 3: 1 2 3 6 9

smallest is 6: 1 2 3 6 9

Insertion Sort1) pick an item and insert it into its proper place in a sorted

sublist2) repeat until all items have been inserted

Efficiency: O(n)

An example:

original: 9 3 6 1 2

insert 3: 3 9 6 1 2

insert 6: 3 6 9 1 2

insert 1: 1 3 6 9 2

insert 2: 1 2 3 6 9

Merge Sort

• Divides a list in half, recursively sorts each half, and then combines the two lists

• At the deepest level of recursion, one-element lists are reached

• A one-element list is already sorted

• The work of the sort comes in when the sorted sublists are merged together

• Efficiency: O(n log n)

Quick Sort

• Chooses a pivot value, then partitions the list into two sublists, (one list contains everything smaller than the pivot, the other contains everything larger), then recursively sorts each sublist

• Unlike a merge sort, a quick sort does most of its work when it divides the list

• It doesn’t need to combine sublists after the recursive steps; the list is already sorted at that point

• Also, unlike a merge sort, the 2 sublists do not have to be the same size

• Efficiency: O(n log n)

Review

1. Selection Sort: Swaps numbers in a list until the list is sorted2. Insertion Sort: Sorts the first 2 #s in the list, then the first 3, then

the first 4, etc…3. Merge Sort: cuts list in half, sorts them, then combines the 2 lists4. Quick Sort: cuts list in half using a pivot value; sorts each sublist,

no need to combine the lists at the end since they are already sorted

Assignments

• Finish Battleship, Connect3, etc…• AP Exam takers: work on practice test