Algorithms

Searching

[1, 5, 10, 20, 50, 100, 500]

[0] [1] [2] [3] [4] [5] [6]

Linear Search

for each  from left to right
    if 50 is in the 
        return true
return false

for each  from left to right
    if 50 is in the 
        return true
    else
        return false

for each  from left to right
    if 50 is in the 
        return true
return false

for i from 0 to n-1
    if 50 is in s[i]
        return true
return false

Binary Search

# Assuming the numbers are sorted

if 50 is in middle 
    return true

# Assuming the numbers are sorted

if 50 is in middle 
    return true
else if 50 < middle 
    search left half
else if 50 > middle 
    search right half   

# Assuming the numbers are sorted

if no 
    return false
if 50 is in middle 
    return true
else if 50 < middle 
    search left half
else if 50 > middle 
    search right half

# Assuming the numbers are sorted

if no 
    return false
if 50 is in s[middle]
    return true
else if 50 < s[middle]
    search s[0] through s[middle - 1]
else if 50 > s[middle]
    search s[middle + 1] through s[n - 1]

Running Time

O()

O()

O()

O()

O()

O()

O()

O()       linear search

O()

O()

O()

O()

O()       linear search

O()    binary search

O()

Ω()

Ω()

Ω()

Ω()

Ω()

Ω()

Ω()

Ω()

Ω()

Ω()       linear search, binary search

Sorting

[7, 2, 5, 4, 1, 6, 0, 3]

Selection Sort

for i from 0 to n-1
    find smallest number between numbers[i] and numbers[n-1]
    swap smallest number with numbers[i]

O()

O()       selection sort

O()

O()

O()

O()

Ω()       selection sort

Ω()

Ω()

Ω()

Ω()

Θ()       selection sort

Θ()

Θ()

Θ()

Θ()

Bubble Sort

[7, 2, 5, 4, 1, 6, 0, 3]

repeat n-1 times
    for i from 0 to n-2
        if numbers[i] and numbers[i+1] out of order
            swap the two

O()

O()       bubble sort

O()

O()

O()

O()

repeat n-1 times
    for i from 0 to n-2
        if numbers[i] and numbers[i+1] out of order
            swap the two
    if no swaps
        quit    

Ω()

Ω()

Ω()       bubble sort

Ω()

Ω()

Algorithm Visualization

• https://visualgo.net/en/sorting
• https://www.cs.usfca.edu/~galles/visualization/ComparisonSort.html

Recursion

# Assuming the numbers are already sorted

if no 
    return false
if 50 is in middle 
    return true
else if 50 < middle 
    search left half
else if 50 > middle 
    search right half   

# Assuming the numbers are already sorted

if no 
    return false
if 50 is in middle 
    return true
else if 50 < middle 
    search left half            # could use recursion
else if 50 > middle 
    search right half           # could use recursion

Merge Sort

if only one number
    quit
else
    sort left half of numbers
    sort right half of numbers
    merge sorted halves

[7, 2, 5, 4, 1, 6, 0, 3]

7 | 2 | 5 | 4 | 1 | 6 | 0 | 3

7 | 2 | 5 | 4 | 1 | 6 | 0 | 3

2 7 | 4 5 | 1 6 | 0 3

7 | 2 | 5 | 4 | 1 | 6 | 0 | 3

2 7 | 4 5 | 1 6 | 0 3

2 4 5 7 | 0 1 3 6

7 | 2 | 5 | 4 | 1 | 6 | 0 | 3

2 7 | 4 5 | 1 6 | 0 3

2 4 5 7 | 0 1 3 6

0 1 2 3 4 5 6 7

O()

O()       merge sort

O()

O()

O()

Ω()

Ω()       merge sort

Ω()

Ω()

Ω()

Θ()

Θ()       merge sort

Θ()

Θ()

Θ()

Time

Space