如何将递归算法变成迭代算法

Question

我有我写的这个算法，但我真的不知道是否可以将它转换为迭代算法。我正在尝试以立方体形状的每个节点获取邻接节点。相邻节点必须满足两个条件：

1. 这是一个灰色节点。
2. 在distance的半径范围内

def find_continumm(seed, node, row, gray, xyz, distance):
"""
seed: the nodes we want to find the adjacent nodes for. 
node: the candidate nodes to be in the adjacency.
row:  save the nodes that are adjacent. 
gray: boolean array that tells if a node is a gray or not. 
xyz: the 3 dim of the shape. 
distance: the radius
"""
    node_ravel = np.ravel_multi_index(node, xyz)
    if node_ravel in row or ~gray[node_ravel] or math.dist(node, seed) > distance:
        return
    row.add(node_ravel)
    if node[0] < xyz[0]:
        node[0] = node[0] + 1
        find_continumm(seed, node, row, gray, xyz, distance)
        node[0] = node[0] - 1
    if node[0] > 0:
        node[0] = node[0] - 1
        find_continumm(seed, node, row, gray, xyz, distance)
        node[0] = node[0] + 1
    if node[1] < xyz[1]:
        node[1] = node[1] + 1
        find_continumm(seed, node, row, gray, xyz, distance)
        node[1] = node[1] - 1
    if node[1] > 0:
        node[1] = node[1] - 1
        find_continumm(seed, node, row, gray, xyz, distance)
        node[1] = node[1] + 1
    if node[2] < xyz[2]:
        node[2] = node[2] + 1
        find_continumm(seed, node, row, gray, xyz, distance)
        node[2] = node[2] - 1
    if node[2] > 0:
        node[2] = node[2] - 1
        find_continumm(seed, node, row, gray, xyz, distance)
        node[2] = node[2] + 1

Answer 1

是的，将递归算法转换为迭代算法总是可能的。执行此操作的一般过程是切换到continuation passing style，应用defunctionalization，然后应用tail-call elimination。这三个转换的组合会将递归函数变成迭代函数，可能需要堆栈。

在我将它应用到您的代码之前，我将简要地重写您的代码，如下所示：

def find_continumm(seed, node, row, gray, xyz, distance):
    def helper():
        node_ravel = np.ravel_multi_index(node, xyz)
        if node_ravel in row or ~gray[node_ravel] or math.dist(node, seed) > distance:
            return
        row.add(node_ravel)
        for i in range(3):
            if node[i] < xyz[i]:
                node[i] += 1
                helper()
                node[i] -= 1
            if node[i] > 0:
                node[i] -= 1
                helper()
                node[i] += 1
    helper()

您自己可以看到，这与您的代码版本是等价的。我将进行最后一次重写以使用while-loop 而不是for-loop：

def find_continumm(seed, node, row, gray, xyz, distance):
    def helper():
        node_ravel = np.ravel_multi_index(node, xyz)
        if node_ravel in row or ~gray[node_ravel] or math.dist(node, seed) > distance:
            return
        row.add(node_ravel)
        i = 0
        while i < 3:
            if node[i] < xyz[i]:
                node[i] += 1
                helper()
                node[i] -= 1
            if node[i] > 0:
                node[i] -= 1
                helper()
                node[i] += 1
            i += 1
    helper()

这极大地简化了代码并将其转换为迭代版本变得更加简单。

生成的迭代版本是：

beginning = 0
entering_loop = 1
finishing_first_call = 2
enter_second_if = 3
finishing_second_call = 4
increment_i = 5
# the actual values of the above variables don't matter
# so long as they're different

def find_continumm(seed, node, row, gray, xyz, distance):
    stack = []
    add_to_stack = lambda tag, data : stack.append((tag, data))
    back_to_beginning = lambda : add_to_stack(beginning, None)
    back_to_beginning()
    while stack:
        tag, i = stack.pop()
        
        if tag is beginning:
            node_ravel = np.ravel_multi_index(node, xyz)
            if node_ravel in row or ~gray[node_ravel] or math.dist(node, seed) > distance:
                pass
            else:
                row.add(node_ravel)
                add_to_stack(entering_loop, 0)
                
        elif tag is entering_loop:
            if i < 3:
                if node[i] < xyz[i]:
                    node[i] += 1
                    add_to_stack(finishing_first_call, i)
                    back_to_beginning()
                else:
                    add_to_stack(enter_second_if, i)
                
        elif tag is finishing_first_call:
            node[i] -= 1
            add_to_stack(enter_second_if, i)
            
        elif tag is enter_second_if:
            if node[i] > 0:
                node[i] += 1
                add_to_stack(finishing_second_call, i)
                back_to_beginning()
            else:
                add_to_stack(increment_i, i)
                
        elif tag is finishing_second_call:
            node[i] -= 1
            add_to_stack(increment_i, i)
            
        elif tag is increment_i:
            add_to_stack(entering_loop, i + 1)  

如果您查看迭代版本，您会发现它与带有while-loop 的递归版本非常接近。每个标签对应于我们“跳回”的递归版本中的特定代码行。