C949 - Sorting Algorithms

Selection Sort Definition

a sorting algorithm that treats the input as two parts, sorted and unsorted, and repeatedly selects the proper next value to move from the unsorted part to the end of the sorted part.

Selection Sort Algorithm

for i in range(len(numbers) - 1):

# Find index of smallest remaining element

index_smallest = i

for j in range(i + 1, len(numbers)):

if numbers[j] < numbers[index_smallest]:

index_smallest = j

# Swap numbers[i] and numbers[index_smallest]

temp = numbers[i]

numbers[i] = numbers[index_smallest]

numbers[index_smallest] = temp

Insertion Sort Definition

a sorting algorithm that treats the input as two parts, sorted and unsorted, and repeatedly inserts the next value from the unsorted part into the correct location in the sorted part.

Insertion Sort Algorithm

for i in range(1, len(numbers)):

j = i

# Insert numbers[i] into sorted part

# stopping once numbers[i] in correct position

while j > 0 and numbers[j] < numbers[j - 1]:

# Swap numbers[j] and numbers[j - 1]

temp = numbers[j]

numbers[j] = numbers[j - 1]

numbers[j - 1] = temp

j = j - 1

Quick Sort Definition

a sorting algorithm that repeatedly partitions the input into low and high parts (each unsorted), and then recursively sorts each of those parts.

Quick Sort Algorithm

def partition(numbers, i, k):

# Pick middle value as pivot

Midpoint = i + (k - i) // 2

Pivot = numbers[midpoint]

# Initialize variables

done = False

l = i

h = k

while not done:

# Increment l while numbers[l] < pivot

while numbers[l] < pivot:

l = l + 1

# Decrement h while pivot < numbers[h]

while pivot < numbers[h]:

h = h - 1

# If there are zero or one items remaining,

# all numbers are partitioned. Return h

if l >= h:

done = True

else:

# Swap numbers[l] and numbers[h],

# update l and h

temp = numbers[l]

numbers[l] = numbers[h]

numbers[h] = temp

l = l + 1

h = h - 1

return h

Merge Sort Definition

a sorting algorithm that divides a list into two halves, recursively sorts each half, and then merges the sorted halves to produce a sorted list.

Merge Sort Algorithm

def algorithm(numbers, i, j, k):

# Create temporary list merged_numbers

# Initialize position variables

# Add smallest element to merged numbers

while left_pos <= j and right_pos <= k:

if numbers[left_pos] <= numbers[right_pos]:

merged_numbers[merge_pos] = numbers[left_pos]

left_pos = left_pos + 1

else:

merged_numbers[merge_pos] = numbers[right_pos]

right_pos = right_pos + 1

merge_pos = merge_pos + 1

# If left partition not empty, add remaining elements

while left_pos <= j:

merged_numbers[merge_pos] = numbers[left_pos]

left_pos = left_pos + 1

merge_pos = merge_pos + 1

# If right partition not empty, add remaining elements

while right_pos <= k:

merged_numbers[merge_pos] = numbers[right_pos]

right_pos = right_pos + 1

merge_pos = merge_pos + 1

# Copy merge number back to numbers

for merge_pos in range(merged_size):

numbers[i + merge_pos] = merged_numbers[merge_pos]

Radix Sort Definition

a sorting algorithm designed specifically for integers. The algorithm makes use of a concept called buckets and is a type of bucket sort.

Radix Sort Algorithm

Algorithm(array, arraySize) {

buckets = create array of 10 buckets

// Find the max length, in number of digits

maxDigits = RadixGetMaxLength(array, arraySize)

// Start with the least significant digit

pow10 = 1

for (digitIndex = 0; digitIndex < maxDigits; digitIndex++) {

for (i = 0; i < arraySize; i++) {

bucketIndex = abs(array[i] / pow10) % 10

Append array[i] to buckets[bucketIndex]

}

arrayIndex = 0

for (i = 0; i < 10; i++) {

for (j = 0; j < buckets[i]⇢size(); j++)

array[arrayIndex++] = buckets[i][j]

}

pow10 = 10 * pow10

Clear all buckets

}

}

Shell Sort Definition

a sorting algorithm that treats the input as a collection of interleaved lists, and sorts each list individually with a variant of the insertion sort algorithm. Uses gap values to determine the number of interleaved lists.

Shell Sort Algorithm

def algorithm(numbers, start_index, gap):

for i in range(start_index + gap, len(numbers), gap):

j = i

while (j - gap >= start_index) and (numbers[j] < numbers[j - gap]):

temp = numbers[j]

numbers[j] = numbers[j - gap]

numbers[j - gap] = temp

j = j - gap

Bubble Sort Definition

a sorting algorithm that iterates through a list, comparing and swapping adjacent elements if the second element is less than the first element. Uses nested loops.

Bubble Sort Algorithm

Algorithm(numbers, numbersSize) {

for (i = 0; i < numbersSize - 1; i++) {

for (j = 0; j < numbersSize - i - 1; j++) {

if (numbers[j] > numbers[j+1]) {

temp = numbers[j]

numbers[j] = numbers[j + 1]

numbers[j + 1] = temp

}

}

}

}

Quickselect

an algorithm that selects the k^th smallest element in a list. Ex: Running on the list (15, 73, 5, 88, 9) with k = 0, returns the smallest element in the list, or 5.

Quickselect algorithm

def algorithm(numbers, start_index, end_index, k):

if start_index >= end_index:

return numbers[start_index]

low_last_index = partition(numbers, start_index, end_index)

if k <= low_last_index:

return quickselect(numbers, start_index, low_last_index, k)

return quickselect(numbers, low_last_index + 1, end_index, k)

Heap Sort

a sorting algorithm that takes advantage of a max-heap's properties by repeatedly removing the max and building a sorted array in reverse order.

Heap Sort Algorithm

Algorithm(numbers, numbersSize) {

//Heapify numbers

for (i = numbersSize / 2 - 1; i >= 0; i--) {

MaxHeapPercolateDown(i, numbers, numbersSize)

}

for (i = numbersSize - 1; i > 0; i--) {

Swap numbers[0] and numbers[i]

MaxHeapPercolateDown(0, numbers, i)

}

}

Bucket Sort

a numerical sorting algorithm that distributes numbers into buckets, sorts each bucket with an additional sorting algorithm, and then concatenates buckets together to build the sorted result.

Bucket Sort Algorithm

Algorithm(numbers, numbersSize, bucketCount) {

if (numbersSize < 1)

return

buckets = Create list of bucketCount buckets

// Find the maximum value

maxValue = numbers[0]

for (i = 1; i < numbersSize; i++) {

if (numbers[i] > maxValue)

maxValue = numbers[i]

}

// Put each number in a bucket

for each (number in numbers) {

index = floor(number * bucketCount / (maxValue + 1))

Append number to buckets[index]

}

// Sort each bucket

for each (bucket in buckets)

Sort(bucket)

// Combine all buckets back into numbers list

result = Concatenate all buckets together

Copy result to numbers

}

Pre-Order Traversal

In this traversal method, the root node is visited first, then the left subtree and finally the right subtree.

If a binary tree is traversed in this method, it can be used to duplicate the tree.

Post-order Traversal

In this traversal method, the root node is visited last, hence the name. First we traverse the left subtree, then the right subtree and finally the root node.

If a binary tree is traversed in this method, the tree can be deleted.

In-order Traversal

In this traversal method, the left subtree is visited first, then the root and later the right sub-tree. We should always remember that every node may represent a subtree itself.

If a binary tree is traversed in this method, the output will produce sorted key values in an ascending order.

Bubble Sort, Selection Sort, Insertion Sort Time Complexity

Average = Θ(n^2 )

Big-O worst = O(n^2 )

Fast = No

Quick Sort Time Complexity

Average = Θ(n log(n))

Big-O worst = O(n^2 )

Fast = Yes

Bucket Sort Time Complexity

Average = Θ(n)

Big-O worst = O(n^2 )

Fast = Yes

Heap Sort, Merge Sort Time Complexity

Average = Θ(n log(n))

Big-O worst = O(n log(n))

Fast = Yes

Radix Sort Time Complexity

Average = Θ(n)

Big-O worst = O(n)

Fast = Yes