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The push_swap problem from 42 Network is a coding challenge that involves sorting a stack of integers using a limited set of operations, aiming to use the fewest operations possible. A powerful approach to solve this problem is using the three-way, three-pivot quicksort algorithm.
Here’s a step-by-step explanation of how this algorithm is implemented to solve the push_swap problem:

• Initialization
• Choosing Pivots
• Partitioning
• Recursive Sorting
• Combining Results

1. Stacks Setup: You start with two stacks, a and b.
   Initially, stack a contains all the unsorted integers, and stack b is empty.

2. Operation Set: The allowed operations are:

   • sa: swap the first two elements of stack a
   • sb: swap the first two elements of stack b
   • ss: sa and sb simultaneously
   • pa: push the top element from stack b to stack a
   • pb: push the top element from stack a to stack b
   • ra: rotate stack a upwards (first element becomes the last)
   • rb: rotate stack b upwards (first element becomes the last)
   • rr: ra and rb simultaneously
   • rra: rotate stack a downwards (last element becomes the first)
   • rrb: rotate stack b downwards (last element becomes the first)
   • rrr: rra and rrb simultaneously

1. Finding Pivots:
   For three-way partitioning, you need to choose three pivot values.

   • pivot1: value at one-third position in the sorted array
   • pivot2: value at two-thirds position in the sorted array
   • pivot3: median value in the sorted array

2. Dividing the Array: These pivots divide the array into four parts:

   • Elements less than pivot1
   • Elements between pivot1 and pivot2
   • Elements between pivot2 and pivot3
   • Elements greater than pivot3

1. Initial Partitioning: Use the pivots to partition the elements in stack a into four parts. During partitioning:

   • Elements less than pivot1 are pushed to stack b.
   • Elements between pivot1 and pivot2 are rotated to the bottom of stack a.
   • Elements between pivot2 and pivot3 remain in their place.
   • Elements greater than pivot3 are pushed to stack b.

2. Operations:

   • Use pb to move elements to stack b.
   • Use ra and rra to rotate stack a.

1. Sort the Subarrays:

   • Recursively apply the same partitioning process to the subarrays created by the pivots.

2. Handle Each Subarray:

   • Sort elements in stack b (those less than pivot1 and greater than pivot3) using similar partitioning and recursive sorting.

1. Merging Sorted Sections:
   • After sorting the elements recursively, merge them back together.
   • Use pa to move elements from stack b to stack a.
2. Final Adjustments:
   • Ensure that all elements are sorted and in the correct order in stack a.

• Efficiency: This approach is efficient because it reduces the problem size with each recursive call.
• Edge Cases: Handle edge cases where the stack has very few elements differently to avoid unnecessary operations.
• Optimization: Further optimize by balancing the number of operations and minimizing the depth of recursion.