如何通过指定函数找到数组从初始到目标排列的最短操作路径？

Alright, let's break down how to solve this problem—finding the shortest sequence of operations to turn arr_start into arr_finish using the given cut_arr and faro_shuffle functions. We'll also cover how to scale this approach for longer arrays and custom functions later on.

1. Problem Setup #

First, let's recap the given inputs and functions:

Input Arrays #

• Starting array: arr_start = [1,2,3,4,5,6]
• Target array: arr_finish = [5,3,6,1,4,2]

Operation Functions #

Here's the code for the two functions, with quick explanations:

def cut_arr(n,arr):
    r=[1]*len(arr)
    for i in range(len(arr)):
        if (i+n)<len(arr):
            r[i] = arr[i+n]
        else:
            r[i] = arr[i+n-len(arr)]
    return r

cut_arr(n, arr) performs a cyclic cut: it shifts the array so that the element at position i+n (wrapping around if it exceeds the array length) moves to position i.

def faro_shuffle(arr):
    r = []
    for (a, b) in zip(arr[0:3], arr[3:]):
        r.append(a)
        r.append(b)
    arr = r
    return r

faro_shuffle(arr) does a perfect shuffle: it splits the array into two equal halves (first 3 and last 3 elements for our 6-length array) and interleaves them, taking one element from each half in order.

2. Verified Shortest Sequence #

As noted in the problem statement, the shortest sequence takes 2 operations and works like this:

1. First, run faro_shuffle on arr_start:
   • Split [1,2,3,4,5,6] into [1,2,3] and [4,5,6], then interleave to get [1,4,2,5,3,6].
2. Next, run cut_arr(3) on the result:
   • For each index i in the 6-length array, we take the element at i+3 (wrapping around when needed):
     • i=0 → index 3 → value 5
     • i=1 → index 4 → value 3
     • i=2 → index 5 → value 6
     • i=3 → index 0 (3+3-6=0) → value 1
     • i=4 → index 1 (4+3-6=1) → value 4
     • i=5 → index 2 (5+3-6=2) → value 2
   • The final array is [5,3,6,1,4,2], which matches arr_finish.

So the exact sequence is:

• faro_shuffle(arr_start)
• cut_arr(3, <result from previous step>)

3. General Approach to Find Shortest Sequences #

For any problem like this—where you need the minimal number of state transitions—Breadth-First Search (BFS) is the way to go. Here's why and how it works:

How BFS Applies Here #

• Treat each unique array state as a node in a graph.
• Treat each possible operation (e.g., faro_shuffle, cut_arr(1), cut_arr(2), etc.) as an edge connecting one node to another.
• Start BFS from arr_start, exploring all possible next states level by level. The first time you reach arr_finish, the path taken to get there is the shortest sequence of operations.

Key Implementation Tips #

• Track visited states: Use a set to store states you've already checked (convert arrays to tuples or strings for hashability) to avoid looping infinitely.
• Limit valid operations: For cut_arr, skip n=0 or n=len(arr)—these don't change the array, so they're useless for making progress.
• Record the path: For each state, keep track of the operations that led to it, so you can reconstruct the sequence once you hit the target.

4. Scaling to Longer Arrays & Custom Functions #

When you're working with longer arrays or adding more custom functions, here's how to keep the solution efficient:

• Optimize state storage: For long arrays, storing the full list as a state is memory-heavy. Convert arrays to tuples (hashable, so they can be stored in sets/dictionaries) or compact strings (e.g., "1,2,3,4,5,6") to save space.
• Precompute transformations: If you have functions with repeated parameters (like cut_arr(n) for multiple n), precompute their effects once instead of recalculating every time.
• Heuristic-guided search (A)*: If BFS is too slow for huge state spaces, use the A* algorithm. Add a heuristic (like counting how many elements are out of place compared to the target) to prioritize states that are closer to the finish line.
• Parallelize search: Split the BFS into multiple threads/processes to explore different branches at the same time, speeding up the search for the target state.

内容的提问来源于stack exchange，提问作者sarasataco