# Sort a stack using recursion

Given a stack, sort it using recursion. Use of any loop constructs like while, for..etc is not allowed. We can only use the following ADT functions on Stack S:

```is_empty(S)  : Tests whether stack is empty or not.
push(S)         : Adds new element to the stack.
pop(S)         : Removes top element from the stack.
top(S)         : Returns value of the top element. Note that this
function does not remove element from the stack.
```

Example:

```Input:  -3  <--- Top
14
18
-5
30

Output: 30  <--- Top
18
14
-3
-5
```

This problem is mainly a variant of Reverse stack using recursion.
The idea of the solution is to hold all values in Function Call Stack until the stack becomes empty. When the stack becomes empty, insert all held items one by one in sorted order. Here sorted order is important.
Algorithm
We can use below algorithm to sort stack elements:

```sortStack(stack S)
if stack is not empty:
temp = pop(S);
sortStack(S);
sortedInsert(S, temp);
```

Below algorithm is to insert element is sorted order:

```sortedInsert(Stack S, element)
if stack is empty OR element > top element
push(S, elem)
else
temp = pop(S)
sortedInsert(S, element)
push(S, temp)
```

Illustration:

```Let given stack be
-3    <-- top of the stack
14
18
-5
30
```

Let us illustrate sorting of stack using above example:
First pop all the elements from the stack and store poped element in variable ‘temp’. After poping all the elements function’s stack frame will look like:

```temp = -3    --> stack frame #1
temp = 14    --> stack frame #2
temp = 18    --> stack frame #3
temp = -5    --> stack frame #4
temp = 30       --> stack frame #5
```

Now stack is empty and ‘insert_in_sorted_order()’ function is called and it inserts 30 (from stack frame #5) at the bottom of the stack. Now stack looks like below:

```30    <-- top of the stack
```

Now next element i.e. -5 (from stack frame #4) is picked. Since -5 < 30, -5 is inserted at the bottom of stack. Now stack becomes:

```30    <-- top of the stack
-0
```

Next 18 (from stack frame #3) is picked. Since 18 < 30, 18 is inserted below 30. Now stack becomes:

```30    <-- top of the stack
18
-5
```

Next 14 (from stack frame #2) is picked. Since 14 < 30 and 14 < 18, it is inserted below 18. Now stack becomes:

```30    <-- top of the stack
18
14
-5
```

Now -3 (from stack frame #1) is picked, as -3 < 30 and -3 < 18 and -3 < 14, it is inserted below 14. Now stack becomes:

```30    <-- top of the stack
18
14
-3
-5
```

Implementation:

Below is the implementation of the above algorithm.

## C++

 `// C++ program to sort a stack using recursion ` `#include ` `using` `namespace` `std; ` ` `  `// Stack is represented using linked list ` `struct` `stack { ` `    ``int` `data; ` `    ``struct` `stack* next; ` `}; ` ` `  `// Utility function to initialize stack ` `void` `initStack(``struct` `stack** s) { *s = NULL; } ` ` `  `// Utility function to chcek if stack is empty ` `int` `isEmpty(``struct` `stack* s) ` `{ ` `    ``if` `(s == NULL) ` `        ``return` `1; ` `    ``return` `0; ` `} ` ` `  `// Utility function to push an item to stack ` `void` `push(``struct` `stack** s, ``int` `x) ` `{ ` `    ``struct` `stack* p = (``struct` `stack*)``malloc``(``sizeof``(*p)); ` ` `  `    ``if` `(p == NULL) { ` `        ``fprintf``(stderr, ``"Memory allocation failed.\n"``); ` `        ``return``; ` `    ``} ` ` `  `    ``p->data = x; ` `    ``p->next = *s; ` `    ``*s = p; ` `} ` ` `  `// Utility function to remove an item from stack ` `int` `pop(``struct` `stack** s) ` `{ ` `    ``int` `x; ` `    ``struct` `stack* temp; ` ` `  `    ``x = (*s)->data; ` `    ``temp = *s; ` `    ``(*s) = (*s)->next; ` `    ``free``(temp); ` ` `  `    ``return` `x; ` `} ` ` `  `// Function to find top item ` `int` `top(``struct` `stack* s) { ``return` `(s->data); } ` ` `  `// Recursive function to insert an item x in sorted way ` `void` `sortedInsert(``struct` `stack** s, ``int` `x) ` `{ ` `    ``// Base case: Either stack is empty or newly inserted ` `    ``// item is greater than top (more than all existing) ` `    ``if` `(isEmpty(*s) or x > top(*s)) { ` `        ``push(s, x); ` `        ``return``; ` `    ``} ` ` `  `    ``// If top is greater, remove the top item and recur ` `    ``int` `temp = pop(s); ` `    ``sortedInsert(s, x); ` ` `  `    ``// Put back the top item removed earlier ` `    ``push(s, temp); ` `} ` ` `  `// Function to sort stack ` `void` `sortStack(``struct` `stack** s) ` `{ ` `    ``// If stack is not empty ` `    ``if` `(!isEmpty(*s)) { ` `        ``// Remove the top item ` `        ``int` `x = pop(s); ` ` `  `        ``// Sort remaining stack ` `        ``sortStack(s); ` ` `  `        ``// Push the top item back in sorted stack ` `        ``sortedInsert(s, x); ` `    ``} ` `} ` ` `  `// Utility function to print contents of stack ` `void` `printStack(``struct` `stack* s) ` `{ ` `    ``while` `(s) { ` `        ``cout << s->data << ``" "``; ` `        ``s = s->next; ` `    ``} ` `    ``cout << ``"\n"``; ` `} ` ` `  `// Driver code ` `int` `main(``void``) ` `{ ` `    ``struct` `stack* top; ` ` `  `    ``initStack(&top); ` `    ``push(&top, 30); ` `    ``push(&top, -5); ` `    ``push(&top, 18); ` `    ``push(&top, 14); ` `    ``push(&top, -3); ` ` `  `    ``cout << ``"Stack elements before sorting:\n"``; ` `    ``printStack(top); ` ` `  `    ``sortStack(&top); ` `    ``cout << ``"\n"``; ` ` `  `    ``cout << ``"Stack elements after sorting:\n"``; ` `    ``printStack(top); ` ` `  `    ``return` `0; ` `} ` ` `  `// This code is contributed by SHUBHAMSINGH10`

## C

 `// C program to sort a stack using recursion ` `#include ` `#include ` ` `  `// Stack is represented using linked list ` `struct` `stack { ` `    ``int` `data; ` `    ``struct` `stack* next; ` `}; ` ` `  `// Utility function to initialize stack ` `void` `initStack(``struct` `stack** s) { *s = NULL; } ` ` `  `// Utility function to chcek if stack is empty ` `int` `isEmpty(``struct` `stack* s) ` `{ ` `    ``if` `(s == NULL) ` `        ``return` `1; ` `    ``return` `0; ` `} ` ` `  `// Utility function to push an item to stack ` `void` `push(``struct` `stack** s, ``int` `x) ` `{ ` `    ``struct` `stack* p = (``struct` `stack*)``malloc``(``sizeof``(*p)); ` ` `  `    ``if` `(p == NULL) { ` `        ``fprintf``(stderr, ``"Memory allocation failed.\n"``); ` `        ``return``; ` `    ``} ` ` `  `    ``p->data = x; ` `    ``p->next = *s; ` `    ``*s = p; ` `} ` ` `  `// Utility function to remove an item from stack ` `int` `pop(``struct` `stack** s) ` `{ ` `    ``int` `x; ` `    ``struct` `stack* temp; ` ` `  `    ``x = (*s)->data; ` `    ``temp = *s; ` `    ``(*s) = (*s)->next; ` `    ``free``(temp); ` ` `  `    ``return` `x; ` `} ` ` `  `// Function to find top item ` `int` `top(``struct` `stack* s) { ``return` `(s->data); } ` ` `  `// Recursive function to insert an item x in sorted way ` `void` `sortedInsert(``struct` `stack** s, ``int` `x) ` `{ ` `    ``// Base case: Either stack is empty or newly inserted ` `    ``// item is greater than top (more than all existing) ` `    ``if` `(isEmpty(*s) || x > top(*s)) { ` `        ``push(s, x); ` `        ``return``; ` `    ``} ` ` `  `    ``// If top is greater, remove the top item and recur ` `    ``int` `temp = pop(s); ` `    ``sortedInsert(s, x); ` ` `  `    ``// Put back the top item removed earlier ` `    ``push(s, temp); ` `} ` ` `  `// Function to sort stack ` `void` `sortStack(``struct` `stack** s) ` `{ ` `    ``// If stack is not empty ` `    ``if` `(!isEmpty(*s)) { ` `        ``// Remove the top item ` `        ``int` `x = pop(s); ` ` `  `        ``// Sort remaining stack ` `        ``sortStack(s); ` ` `  `        ``// Push the top item back in sorted stack ` `        ``sortedInsert(s, x); ` `    ``} ` `} ` ` `  `// Utility function to print contents of stack ` `void` `printStack(``struct` `stack* s) ` `{ ` `    ``while` `(s) { ` `        ``printf``(``"%d "``, s->data); ` `        ``s = s->next; ` `    ``} ` `    ``printf``(``"\n"``); ` `} ` ` `  `// Driver code ` `int` `main(``void``) ` `{ ` `    ``struct` `stack* top; ` ` `  `    ``initStack(&top); ` `    ``push(&top, 30); ` `    ``push(&top, -5); ` `    ``push(&top, 18); ` `    ``push(&top, 14); ` `    ``push(&top, -3); ` ` `  `    ``printf``(``"Stack elements before sorting:\n"``); ` `    ``printStack(top); ` ` `  `    ``sortStack(&top); ` `    ``printf``(``"\n\n"``); ` ` `  `    ``printf``(``"Stack elements after sorting:\n"``); ` `    ``printStack(top); ` ` `  `    ``return` `0; ` `}`

## Java

 `// Java program to sort a Stack using recursion ` `// Note that here predefined Stack class is used ` `// for stack operation ` ` `  `import` `java.util.ListIterator; ` `import` `java.util.Stack; ` ` `  `class` `Test  ` `{ ` `    ``// Recursive Method to insert an item x in sorted way ` `    ``static` `void` `sortedInsert(Stack s, ``int` `x) ` `    ``{ ` `        ``// Base case: Either stack is empty or newly ` `        ``// inserted item is greater than top (more than all ` `        ``// existing) ` `        ``if` `(s.isEmpty() || x > s.peek())  ` `        ``{ ` `            ``s.push(x); ` `            ``return``; ` `        ``} ` ` `  `        ``// If top is greater, remove the top item and recur ` `        ``int` `temp = s.pop(); ` `        ``sortedInsert(s, x); ` ` `  `        ``// Put back the top item removed earlier ` `        ``s.push(temp); ` `    ``} ` ` `  `    ``// Method to sort stack ` `    ``static` `void` `sortStack(Stack s) ` `    ``{ ` `        ``// If stack is not empty ` `        ``if` `(!s.isEmpty())  ` `        ``{ ` `            ``// Remove the top item ` `            ``int` `x = s.pop(); ` ` `  `            ``// Sort remaining stack ` `            ``sortStack(s); ` ` `  `            ``// Push the top item back in sorted stack ` `            ``sortedInsert(s, x); ` `        ``} ` `    ``} ` ` `  `    ``// Utility Method to print contents of stack ` `    ``static` `void` `printStack(Stack s) ` `    ``{ ` `        ``ListIterator lt = s.listIterator(); ` ` `  `        ``// forwarding ` `        ``while` `(lt.hasNext()) ` `            ``lt.next(); ` ` `  `        ``// printing from top to bottom ` `        ``while` `(lt.hasPrevious()) ` `            ``System.out.print(lt.previous() + ``" "``); ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``Stack s = ``new` `Stack<>(); ` `        ``s.push(``30``); ` `        ``s.push(-``5``); ` `        ``s.push(``18``); ` `        ``s.push(``14``); ` `        ``s.push(-``3``); ` ` `  `        ``System.out.println( ` `            ``"Stack elements before sorting: "``); ` `        ``printStack(s); ` ` `  `        ``sortStack(s); ` ` `  `        ``System.out.println( ` `            ``" \n\nStack elements after sorting:"``); ` `        ``printStack(s); ` `    ``} ` `}`

## Python3

 `# Python program to sort a stack using recursion ` ` `  `# Recursive method to insert element in sorted way ` ` `  ` `  `def` `sortedInsert(s, element): ` ` `  `    ``# Base case: Either stack is empty or newly inserted ` `    ``# item is greater than top (more than all existing) ` `    ``if` `len``(s) ``=``=` `0` `or` `element > s[``-``1``]: ` `        ``s.append(element) ` `        ``return` `    ``else``: ` ` `  `        ``# Remove the top item and recur ` `        ``temp ``=` `s.pop() ` `        ``sortedInsert(s, element) ` ` `  `        ``# Put back the top item removed earlier ` `        ``s.append(temp) ` ` `  `# Method to sort stack ` ` `  ` `  `def` `sortStack(s): ` ` `  `    ``# If stack is not empty ` `    ``if` `len``(s) !``=` `0``: ` ` `  `        ``# Remove the top item ` `        ``temp ``=` `s.pop() ` ` `  `        ``# Sort remaining stack ` `        ``sortStack(s) ` ` `  `        ``# Push the top item back in sorted stack ` `        ``sortedInsert(s, temp) ` ` `  `# Printing contents of stack ` ` `  ` `  `def` `printStack(s): ` `    ``for` `i ``in` `s[::``-``1``]: ` `        ``print``(i, end``=``" "``) ` `    ``print``() ` ` `  ` `  `# Driver Code ` `if` `__name__ ``=``=` `'__main__'``: ` `    ``s ``=` `[] ` `    ``s.append(``30``) ` `    ``s.append(``-``5``) ` `    ``s.append(``18``) ` `    ``s.append(``14``) ` `    ``s.append(``-``3``) ` ` `  `    ``print``(``"Stack elements before sorting: "``) ` `    ``printStack(s) ` ` `  `    ``sortStack(s) ` ` `  `    ``print``(``"\nStack elements after sorting: "``) ` `    ``printStack(s) ` ` `  `# This code is contributed by Muskan Kalra. `

## C#

 `// C# program to sort a Stack using recursion ` `// Note that here predefined Stack class is used ` `// for stack operation ` `using` `System; ` `using` `System.Collections; ` ` `  `public` `class` `GFG  ` `{ ` `    ``// Recursive Method to insert an item x in sorted way ` `    ``static` `void` `sortedInsert(Stack s, ``int` `x) ` `    ``{ ` `        ``// Base case: Either stack is empty or ` `        ``// newly inserted item is greater than top ` `        ``// (more than all existing) ` `        ``if` `(s.Count == 0 || x > (``int``)s.Peek()) { ` `            ``s.Push(x); ` `            ``return``; ` `        ``} ` ` `  `        ``// If top is greater, remove ` `        ``// the top item and recur ` `        ``int` `temp = (``int``)s.Peek(); ` `        ``s.Pop(); ` `        ``sortedInsert(s, x); ` ` `  `        ``// Put back the top item removed earlier ` `        ``s.Push(temp); ` `    ``} ` ` `  `    ``// Method to sort stack ` `    ``static` `void` `sortStack(Stack s) ` `    ``{ ` `        ``// If stack is not empty ` `        ``if` `(s.Count > 0) { ` `            ``// Remove the top item ` `            ``int` `x = (``int``)s.Peek(); ` `            ``s.Pop(); ` ` `  `            ``// Sort remaining stack ` `            ``sortStack(s); ` ` `  `            ``// Push the top item back in sorted stack ` `            ``sortedInsert(s, x); ` `        ``} ` `    ``} ` ` `  `    ``// Utility Method to print contents of stack ` `    ``static` `void` `printStack(Stack s) ` `    ``{ ` `        ``foreach``(``int` `c ``in` `s) { Console.Write(c + ``" "``); } ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `Main(String[] args) ` `    ``{ ` `        ``Stack s = ``new` `Stack(); ` `        ``s.Push(30); ` `        ``s.Push(-5); ` `        ``s.Push(18); ` `        ``s.Push(14); ` `        ``s.Push(-3); ` ` `  `        ``Console.WriteLine( ` `            ``"Stack elements before sorting: "``); ` `        ``printStack(s); ` ` `  `        ``sortStack(s); ` ` `  `        ``Console.WriteLine( ` `            ``" \n\nStack elements after sorting:"``); ` `        ``printStack(s); ` `    ``} ` `} ` ` `  `// This code is Contibuted by Arnab Kundu`

Output:

```Stack elements before sorting:
-3 14 18 -5 30

Stack elements after sorting:
30 18 14 -3 -5 ```

Complexity Analysis:

• Time Complexity: O(n2).
In the worst case for every sortstack(), sortedinsert() is called for ‘N’ times recursively for putting element to the right place
• Auxiliary Space: O(N)
Use of stack data structure for storing values

Exercise: Modify above code to reverse stack in descending order.