简介

堆是一种完全二叉树，有最大堆和最小堆两种。

• 最大堆: 每个节点，都比叶子节点大，如：

  
  

• 最小堆：和最大堆相反

堆的特性

堆是一种完全二叉树，具备二叉树的特性：

• 父节点下标：parent = int((i-1) / 2) # 取整
• 左节点下标：left = 2 * i + 1
• 右节点下标：right = 2 * i + 2
  

如：节点60的父节点下标是1，本身节点下标是3，左下标7，右下标8

堆的表示

堆可以用数组表示，如
[10,7,2,5,1]

堆的python实现

class Array(object):

    def __init__(self, size=32):
        self._size = size
        self._items = [None] * size

    def __getitem__(self, index):
        return self._items[index]

    def __setitem__(self, index, value):
        self._items[index] = value

    def __len__(self):
        return self._size

    def clear(self, value=None):
        for i in range(len(self._items)):
            self._items[i] = value

    def __iter__(self):
        for item in self._items:
            yield item

class MaxHeap(object):

    def __init__(self, maxsize=None):
        self.maxsize = maxsize
        self._elements = Array(maxsize)
        self._count = 0

    def __len__(self):
        return self._count

    def add(self, value):
        if self._count >= self.maxsize:
            raise Exception('full')
        self._elements[self._count] = value
        self._count += 1
        self._siftup(self._count-1)  # 维持堆的特性

    def _siftup(self, ndx):
        if ndx > 0:
            parent = int((ndx-1)/2)
            if self._elements[ndx] > self._elements[parent]:    # 如果插入的值大于 parent，一直交换
                self._elements[ndx], self._elements[parent] = self._elements[parent], self._elements[ndx]
                self._siftup(parent)    # 递归

    def extract(self):
        if self._count <= 0:
            raise Exception('empty')
        value = self._elements[0]    # 保存 root 值
        self._count -= 1
        self._elements[0] = self._elements[self._count]    # 最右下的节点放到root后siftDown
        self._siftdown(0)    # 维持堆特性
        return value

    def _siftdown(self, ndx):
        left = 2 * ndx + 1
        right = 2 * ndx + 2
        # determine which node contains the larger value
        largest = ndx
        if (left < self._count and     # 有左孩子
                self._elements[left] >= self._elements[largest] and
                self._elements[left] >= self._elements[right]):  # 原书这个地方没写实际上找的未必是largest
            largest = left
        elif right < self._count and self._elements[right] >= self._elements[largest]:
            largest = right
        if largest != ndx:
            self._elements[ndx], self._elements[largest] = self._elements[largest], self._elements[ndx]
            self._siftdown(largest)


def test_maxheap():
    import random
    n = 5
    h = MaxHeap(n)
    for i in range(n):
        h.add(i)
    for i in reversed(range(n)):
        assert i == h.extract()


def heapsort_reverse(array):
    length = len(array)
    maxheap = MaxHeap(length)
    for i in array:
        maxheap.add(i)
    res = []
    for i in range(length):
        res.append(maxheap.extract())
    return res


def test_heapsort_reverse():
    import random
    l = list(range(10))
    random.shuffle(l)
    assert heapsort_reverse(l) == sorted(l, reverse=True)


def heapsort_use_heapq(iterable):
    from heapq import heappush, heappop
    items = []
    for value in iterable:
        heappush(items, value)
    return [heappop(items) for i in range(len(items))]


def test_heapsort_use_heapq():
    import random
    l = list(range(10))
    random.shuffle(l)
    assert heapsort_use_heapq(l) == sorted(l)

• Array 实现一个数组，用于填充堆数据
• 实现一个最大堆MaxHeap
• add方法用于增加堆节点时，每次增加后会调用_siftup进行调整。
• _siftup传入增加节点的最后一个下标作为参数，这个节点的值与父节点对比，根据堆性质看是否需要交换。
• extract提取最大值，并_siftdown进行重整，维持堆特性。
• _siftdown进行比较，交换，递归之。

python heapd模块

python自带的内置heapd模块，用于实现堆的相关操作。