Push Swap - Sorting a stack

1. Goal

Develop a program that display on the standard input the smallest number of push_swap actions to sort (from smallest to biggest) a stack of integer received.

2. Rules of the games

• The program can use 2 stacks (will cal a the input one and b the "support" one).
• Stack a can contain a random amount of unique integers (positive and/or negative)
• Stack b starts empty.
• The program will receive the stack a as an input formatted as a list of integers where the first integer is at the top.
• The followign are the only actions available (or push_swap actions):
  • sa : swap a - swap the first 2 elements at the top of stack a. Do nothing if there is only one or no elements).
  • sb : swap b - swap the first 2 elements at the top of stack b. Do nothing if there is only one or no elements).
  • ss : sa and sb at the same time.
  • pa : push a - take the first element at the top of b and put it at the top of a. Do nothing if b is empty.
  • pb : push b - take the first element at the top of a and put it at the top of b. Do nothing if a is empty.
  • ra : rotate a - shift up all elements of stack a by 1. The first element becomes the last one.
  • rb : rotate b - shift up all elements of stack b by 1. The first element becomes the last one.
  • rr : ra and rb at the same time.
  • rra : reverse rotate a - shift down all elements of stack a by 1. The last element becomes the first one.
  • rrb : reverse rotate b - shift down all elements of stack b by 1. The last element becomes the first one.
  • rrr : rra and rrb at the same time.

3. Implementation

To develop the program I went through the following steps

1. Chose the structure of my stack. It could have been either an array of integer or a linked list (as described here in Wikipedia); I opted for the second option.
2. Implemented functions to execute the push_swap actions.
3. Develop algorithm to sort "small stacks" (no more than 5 numbers). For this step I first developed an algorithm to sort a stack of 3 number. To do so, I considered the limited combination of 3 unsorted integers (5 cases) and for each of them identified the shortest list of push_swap actions to sort them. For stack with more than 3 elements, I implemented an algorithm that (1) pushes elements to stack b until stack a is size 3 and then (2) insert the element from stack b back to sorted stack a in their correct position.
4. Develop alogrithm for "big" stacks. After studying several possible algorithm, I chose to develop a radix sort algorithm since it was efficient enough and easy to implement. Here is a short video to visualize the logic behind the algorithm. This algorithm only works for positive-only arrays, therefore I used the position of each number in the final sorted arrays instead of the original inputs. For example, instead of -234 -670 10 5 -3, I considered 1 0 4 3 2, since the following is the sorted list of number -670 -234 -3 5 10. Moreover, since the available stacks where only two, I implemented the same logic explained in the video using the binary representation of the numbers.