# What is SmoothSort

**is a way of sorting a list of items, like sorting a list of numbers in ascending order. It's similar to other sorting methods like bubble sort or quicksort, but it's designed to be more efficient when the list is already partially sorted.**

Smoothsort

Smoothsort

The way Smoothsort works is by first building what's called a heap, which is a way of organizing the items in a tree-like structure. The heap makes it easy to find the largest or smallest item in the list, which is important for sorting.

The special thing about Smoothsort is that it uses a modified version of a tree called a Leonardo tree to build the heap. The Leonardo tree has a special structure that makes it efficient to build the heap and sort the items.

Once the heap is built, Smoothsort uses a process called "downward sifting" to move items down the heap until they're in the correct position. It also uses "upward sifting" to move items up the heap until they're in the correct position. This process is repeated until the list is fully sorted.

Overall, Smoothsort is a way of sorting items that's designed to be efficient when the list is already partially sorted. It uses a special tree structure called a Leonardo tree to build the heap, and then uses downward and upward sifting to sort the items.

## Who invented it?

**Smoothsort**was invented by the Dutch computer scientist Edsger W. Dijkstra in 1981. Dijkstra was a pioneer in the field of computer science and made many important contributions to the development of algorithms and programming languages. He was awarded the Turing Award in 1972, which is one of the most prestigious awards in computer science.

Dijkstra developed Smoothsort as an improvement over heapsort, which is another sorting algorithm that he had previously developed. Smoothsort is designed to be more efficient than heapsort for partially sorted data, which is a common scenario in real-world applications.

## Pseudocode

```
function smoothsort(arr)
// Build the initial Leonardo heap
var q = []
var p = 1
var r = -1
var t = 0
for i in range(len(arr)):
if (p & 3) == 3:
q.append(p)
q.append(p+1)
t += 1
elif (p & 1) == 1:
if i != 0:
q.append(r)
r = p
p >>= 1
for i in range(len(q)-1, -1, -1):
sift(q[i], len(arr), arr)
// Perform the downward sifting phase
for i in range(len(arr)-1, 0, -1):
if q[t-1] == i:
t -= 1
sift(q[t], i, arr)
else:
sift(1, i, arr)
swap(arr, 0, i)
return arr
function sift(p, n, arr)
while p > 1:
if arr[p-2] <= arr[p-1]:
p -= 1
else:
swap(arr, p-2, p-1)
p = 2*p if (p-1) <= (n//2) else 2*(p-1)
return arr
function swap(arr, i, j)
temp = arr[i]
arr[i] = arr[j]
arr[j] = temp
```

This pseudocode implementation of Smoothsort uses two main functions: smoothsort() and sift(). The smoothsort() function takes an array as input and sorts it using the Smoothsort algorithm. The sift() function is used to perform the downward sifting phase of the algorithm.

The smoothsort() function starts by building the initial Leonardo heap and then sifting it down to get it into the correct order. It then performs the downward sifting phase by repeatedly sifting the root of the heap down to its correct position and swapping it with the last element of the array. Finally, it returns the sorted array.

The sift() function is used to sift an element down the heap to its correct position. It starts by comparing the element to its parent and swapping them if they are in the wrong order. It then continues to compare and swap the element with its parent until it is in the correct position.

The smoothsort() function starts by building the initial Leonardo heap and then sifting it down to get it into the correct order. It then performs the downward sifting phase by repeatedly sifting the root of the heap down to its correct position and swapping it with the last element of the array. Finally, it returns the sorted array.

The sift() function is used to sift an element down the heap to its correct position. It starts by comparing the element to its parent and swapping them if they are in the wrong order. It then continues to compare and swap the element with its parent until it is in the correct position.

## Sample Code

```
// C++ code snippet
void smoothsort(vector
```& arr);
void sift(int p, int n, vector& arr);
void swap(int& a, int& b);
void smoothsort(vector& arr) {
// Build the initial Leonardo heap
vector q;
int p = 1;
int r = -1;
int t = 0;
for (int i = 0; i < arr.size(); i++) {
if ((p & 3) == 3) {
q.push_back(p);
q.push_back(p+1);
t++;
} else if ((p & 1) == 1) {
if (i != 0) {
q.push_back(r);
}
r = p;
}
p >>= 1;
}
for (int i = q.size()-1; i >= 0; i--) {
sift(q[i], arr.size(), arr);
}
// Perform the downward sifting phase
for (int i = arr.size()-1; i > 0; i--) {
if (q[t-1] == i) {
t--;
sift(q[t], i, arr);
} else {
sift(1, i, arr);
}
swap(arr[0], arr[i]);
}
}
void sift(int p, int n, vector& arr) {
while (p > 1) {
if (arr[p-2] <= arr[p-1]) {
p--;
} else {
swap(arr[p-2], arr[p-1]);
p = (p-1 <= n/2) ? 2*p : 2*(p-1);
}
}
}
void swap(int& a, int& b) {
int temp = a;
a = b;
b = temp;
}
int main() {
vector arr = {3, 6, 1, 8, 2, 5, 4, 7};
cout << "Original array: ";
for (int i = 0; i < arr.size(); i++) {
cout << arr[i] << " ";
}
cout << endl;
smoothsort(arr);
cout << "Sorted array: ";
for (int i = 0; i < arr.size(); i++) {
cout << arr[i] << " ";
}
cout << endl;
return 0;
}

```
# Python code snippet
def smoothsort(arr):
def sift(p, n):
while p > 1:
if arr[p-2] <= arr[p-1]:
p -= 1
else:
arr[p-2], arr[p-1] = arr[p-1], arr[p-2]
p = 2*p if p-1 <= n/2 else 2*(p-1)
# Build the initial Leonardo heap
q = []
p = 1
r = -1
t = 0
for i in range(len(arr)):
if (p & 3) == 3:
q.append(p)
q.append(p+1)
t += 1
elif (p & 1) == 1:
if i != 0:
q.append(r)
r = p
p >>= 1
for i in reversed(q):
sift(i, len(arr))
# Perform the downward sifting phase
for i in range(len(arr)-1, 0, -1):
if q[t-1] == i:
t -= 1
sift(q[t], i)
else:
sift(1, i)
arr[0], arr[i] = arr[i], arr[0]
# Example usage
arr = [3, 6, 1, 8, 2, 5, 4, 7]
print("Original array:", arr)
smoothsort(arr)
print("Sorted array:", arr)
```

```
import java.util.Arrays;
public class SmoothSort {
public static void smoothsort(int[] arr) {
int p, r, t, i;
// Build the initial Leonardo heap
int[] q = new int[arr.length / 2];
p = r = t = 1;
for (i = 0; i < arr.length; i++) {
if ((p & 3) == 3) {
q[t++] = p;
q[t++] = p + 1;
} else if ((p & 1) == 1) {
if (i != 0) {
q[t++] = r;
}
r = p;
}
p >>>= 1;
}
// Perform the downward sifting phase
int temp;
while (t > 0) {
t--;
i = q[t];
temp = arr[i];
while (i > 1) {
int j = i - 2;
if (arr[j] > temp) {
arr[i] = arr[j];
i = j;
} else {
break;
}
}
arr[i] = temp;
}
}
// Example usage
public static void main(String[] args) {
int[] arr = {3, 6, 1, 8, 2, 5, 4, 7};
System.out.println("Original array: " + Arrays.toString(arr));
smoothsort(arr);
System.out.println("Sorted array: " + Arrays.toString(arr));
}
}
```

## Time and Space Complexity

- The time complexity of Smoothsort is O(n log n) in the worst case and O(n) in the best case, where n is the number of elements in the input array. The best-case time complexity occurs when the input is already sorted or nearly sorted, and the worst-case time complexity occurs when the input is in reverse order.
- The space complexity of Smoothsort is O(log n), which is the height of the Leonardo heap used by the algorithm. This is because the algorithm uses an internal data structure to store the indices of the roots of the Leonardo trees, and the maximum number of such roots is log n, where n is the number of elements in the input array.

## Advantages

- It has a guaranteed worst-case time complexity of O(n log n) and a best-case time complexity of O(n), which makes it efficient for a wide range of input sizes and distributions.
- It is a stable sort, which means that the order of equal elements is preserved.
- It is an in-place sort, which means that it does not require additional memory to store the sorted array.
- It is adaptable to partially sorted arrays, which means that it can take advantage of pre-existing order to improve performance.

## Disadvantages

- It may not perform as well as other sorting algorithms on small input sizes or highly ordered or unordered input distributions.
- It may be slower than some other sorting algorithms in practice due to its higher constant factor and cache inefficiency.