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716 lines
18 KiB
Markdown
716 lines
18 KiB
Markdown
# 堆
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「堆 Heap」是一种满足特定条件的完全二叉树，可分为两种类型：
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- 「大顶堆 Max Heap」，任意节点的值 $\geq$ 其子节点的值；
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- 「小顶堆 Min Heap」，任意节点的值 $\leq$ 其子节点的值；
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堆作为完全二叉树的一个特例，具有以下特性：
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- 最底层节点靠左填充，其他层的节点都被填满。
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- 我们将二叉树的根节点称为「堆顶」，将底层最靠右的节点称为「堆底」。
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- 对于大顶堆（小顶堆），堆顶元素（即根节点）的值分别是最大（最小）的。
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## 堆常用操作
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需要指出的是，许多编程语言提供的是「优先队列 Priority Queue」，这是一种抽象数据结构，定义为具有优先级排序的队列。
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实际上，**堆通常用作实现优先队列，大顶堆相当于元素按从大到小顺序出队的优先队列**。从使用角度来看，我们可以将「优先队列」和「堆」看作等价的数据结构。因此，本书对两者不做特别区分，统一使用「堆」来命名。
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堆的常用操作见下表，方法名需要根据编程语言来确定。
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<div class="center-table" markdown>
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| 方法名 | 描述 | 时间复杂度 |
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| --------- | ------------------------------------------ | ----------- |
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| push() | 元素入堆 | $O(\log n)$ |
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| pop() | 堆顶元素出堆 | $O(\log n)$ |
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| peek() | 访问堆顶元素（大 / 小顶堆分别为最大 / 小值） | $O(1)$ |
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| size() | 获取堆的元素数量 | $O(1)$ |
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| isEmpty() | 判断堆是否为空 | $O(1)$ |
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</div>
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在实际应用中，我们可以直接使用编程语言提供的堆类（或优先队列类）。
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!!! tip
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类似于排序算法中的“从小到大排列”和“从大到小排列”，我们可以通过修改 Comparator 来实现“小顶堆”与“大顶堆”之间的转换。
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=== "Java"
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```java title="heap.java"
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/* 初始化堆 */
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// 初始化小顶堆
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Queue<Integer> minHeap = new PriorityQueue<>();
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// 初始化大顶堆（使用 lambda 表达式修改 Comparator 即可）
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Queue<Integer> maxHeap = new PriorityQueue<>((a, b) -> b - a);
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/* 元素入堆 */
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maxHeap.offer(1);
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maxHeap.offer(3);
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maxHeap.offer(2);
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maxHeap.offer(5);
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maxHeap.offer(4);
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/* 获取堆顶元素 */
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int peek = maxHeap.peek(); // 5
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/* 堆顶元素出堆 */
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// 出堆元素会形成一个从大到小的序列
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peek = heap.poll(); // 5
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peek = heap.poll(); // 4
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peek = heap.poll(); // 3
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peek = heap.poll(); // 2
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peek = heap.poll(); // 1
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/* 获取堆大小 */
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int size = maxHeap.size();
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/* 判断堆是否为空 */
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boolean isEmpty = maxHeap.isEmpty();
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/* 输入列表并建堆 */
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minHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4));
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```
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=== "C++"
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```cpp title="heap.cpp"
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/* 初始化堆 */
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// 初始化小顶堆
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priority_queue<int, vector<int>, greater<int>> minHeap;
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// 初始化大顶堆
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priority_queue<int, vector<int>, less<int>> maxHeap;
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/* 元素入堆 */
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maxHeap.push(1);
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maxHeap.push(3);
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maxHeap.push(2);
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maxHeap.push(5);
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maxHeap.push(4);
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/* 获取堆顶元素 */
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int peek = maxHeap.top(); // 5
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/* 堆顶元素出堆 */
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// 出堆元素会形成一个从大到小的序列
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maxHeap.pop(); // 5
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maxHeap.pop(); // 4
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maxHeap.pop(); // 3
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maxHeap.pop(); // 2
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maxHeap.pop(); // 1
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/* 获取堆大小 */
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int size = maxHeap.size();
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/* 判断堆是否为空 */
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bool isEmpty = maxHeap.empty();
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/* 输入列表并建堆 */
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vector<int> input{1, 3, 2, 5, 4};
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priority_queue<int, vector<int>, greater<int>> minHeap(input.begin(), input.end());
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```
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=== "Python"
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```python title="heap.py"
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# 初始化小顶堆
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min_heap, flag = [], 1
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# 初始化大顶堆
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max_heap, flag = [], -1
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# Python 的 heapq 模块默认实现小顶堆
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# 考虑将“元素取负”后再入堆，这样就可以将大小关系颠倒，从而实现大顶堆
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# 在本示例中，flag = 1 时对应小顶堆，flag = -1 时对应大顶堆
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# 元素入堆
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heapq.heappush(max_heap, flag * 1)
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heapq.heappush(max_heap, flag * 3)
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heapq.heappush(max_heap, flag * 2)
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heapq.heappush(max_heap, flag * 5)
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heapq.heappush(max_heap, flag * 4)
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# 获取堆顶元素
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peek: int = flag * max_heap[0] # 5
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# 堆顶元素出堆
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# 出堆元素会形成一个从大到小的序列
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val = flag * heapq.heappop(max_heap) # 5
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val = flag * heapq.heappop(max_heap) # 4
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val = flag * heapq.heappop(max_heap) # 3
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val = flag * heapq.heappop(max_heap) # 2
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val = flag * heapq.heappop(max_heap) # 1
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# 获取堆大小
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size: int = len(max_heap)
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# 判断堆是否为空
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is_empty: bool = not max_heap
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# 输入列表并建堆
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min_heap: List[int] = [1, 3, 2, 5, 4]
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heapq.heapify(min_heap)
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```
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=== "Go"
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```go title="heap.go"
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// Go 语言中可以通过实现 heap.Interface 来构建整数大顶堆
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// 实现 heap.Interface 需要同时实现 sort.Interface
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type intHeap []any
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// Push heap.Interface 的方法，实现推入元素到堆
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func (h *intHeap) Push(x any) {
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// Push 和 Pop 使用 pointer receiver 作为参数
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// 因为它们不仅会对切片的内容进行调整，还会修改切片的长度。
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*h = append(*h, x.(int))
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}
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// Pop heap.Interface 的方法，实现弹出堆顶元素
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func (h *intHeap) Pop() any {
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// 待出堆元素存放在最后
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last := (*h)[len(*h)-1]
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*h = (*h)[:len(*h)-1]
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return last
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}
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// Len sort.Interface 的方法
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func (h *intHeap) Len() int {
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return len(*h)
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}
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// Less sort.Interface 的方法
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func (h *intHeap) Less(i, j int) bool {
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// 如果实现小顶堆，则需要调整为小于号
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return (*h)[i].(int) > (*h)[j].(int)
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}
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// Swap sort.Interface 的方法
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func (h *intHeap) Swap(i, j int) {
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(*h)[i], (*h)[j] = (*h)[j], (*h)[i]
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}
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// Top 获取堆顶元素
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func (h *intHeap) Top() any {
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return (*h)[0]
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}
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/* Driver Code */
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func TestHeap(t *testing.T) {
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/* 初始化堆 */
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// 初始化大顶堆
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maxHeap := &intHeap{}
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heap.Init(maxHeap)
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/* 元素入堆 */
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// 调用 heap.Interface 的方法，来添加元素
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heap.Push(maxHeap, 1)
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heap.Push(maxHeap, 3)
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heap.Push(maxHeap, 2)
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heap.Push(maxHeap, 4)
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heap.Push(maxHeap, 5)
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/* 获取堆顶元素 */
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top := maxHeap.Top()
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fmt.Printf("堆顶元素为 %d\n", top)
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/* 堆顶元素出堆 */
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// 调用 heap.Interface 的方法，来移除元素
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heap.Pop(maxHeap) // 5
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heap.Pop(maxHeap) // 4
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heap.Pop(maxHeap) // 3
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heap.Pop(maxHeap) // 2
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heap.Pop(maxHeap) // 1
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/* 获取堆大小 */
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size := len(*maxHeap)
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fmt.Printf("堆元素数量为 %d\n", size)
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/* 判断堆是否为空 */
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isEmpty := len(*maxHeap) == 0
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fmt.Printf("堆是否为空 %t\n", isEmpty)
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}
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```
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=== "JavaScript"
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```javascript title="heap.js"
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// JavaScript 未提供内置 Heap 类
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```
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=== "TypeScript"
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```typescript title="heap.ts"
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// TypeScript 未提供内置 Heap 类
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```
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=== "C"
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```c title="heap.c"
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// C 未提供内置 Heap 类
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```
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=== "C#"
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```csharp title="heap.cs"
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/* 初始化堆 */
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// 初始化小顶堆
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PriorityQueue<int, int> minHeap = new PriorityQueue<int, int>();
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// 初始化大顶堆（使用 lambda 表达式修改 Comparator 即可）
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PriorityQueue<int, int> maxHeap = new PriorityQueue<int, int>(Comparer<int>.Create((x, y) => y - x));
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/* 元素入堆 */
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maxHeap.Enqueue(1, 1);
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maxHeap.Enqueue(3, 3);
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maxHeap.Enqueue(2, 2);
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maxHeap.Enqueue(5, 5);
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maxHeap.Enqueue(4, 4);
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/* 获取堆顶元素 */
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int peek = maxHeap.Peek();//5
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/* 堆顶元素出堆 */
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// 出堆元素会形成一个从大到小的序列
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peek = maxHeap.Dequeue(); // 5
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peek = maxHeap.Dequeue(); // 4
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peek = maxHeap.Dequeue(); // 3
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peek = maxHeap.Dequeue(); // 2
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peek = maxHeap.Dequeue(); // 1
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/* 获取堆大小 */
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int size = maxHeap.Count;
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/* 判断堆是否为空 */
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bool isEmpty = maxHeap.Count == 0;
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/* 输入列表并建堆 */
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minHeap = new PriorityQueue<int, int>(new List<(int, int)> { (1, 1), (3, 3), (2, 2), (5, 5), (4, 4), });
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```
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=== "Swift"
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```swift title="heap.swift"
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// Swift 未提供内置 Heap 类
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```
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=== "Zig"
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```zig title="heap.zig"
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```
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## 堆的实现
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下文实现的是大顶堆。若要将其转换为小顶堆，只需将所有大小逻辑判断取逆（例如，将 $\geq$ 替换为 $\leq$ ）。感兴趣的读者可以自行实现。
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### 堆的存储与表示
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我们在二叉树章节中学习到，完全二叉树非常适合用数组来表示。由于堆正是一种完全二叉树，**我们将采用数组来存储堆**。
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当使用数组表示二叉树时，元素代表节点值，索引代表节点在二叉树中的位置。**节点指针通过索引映射公式来实现**。
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具体而言，给定索引 $i$ ，其左子节点索引为 $2i + 1$ ，右子节点索引为 $2i + 2$ ，父节点索引为 $(i - 1) / 2$（向下取整）。当索引越界时，表示空节点或节点不存在。
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我们可以将索引映射公式封装成函数，方便后续使用。
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=== "Java"
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```java title="my_heap.java"
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[class]{MaxHeap}-[func]{left}
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[class]{MaxHeap}-[func]{right}
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[class]{MaxHeap}-[func]{parent}
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```
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=== "C++"
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```cpp title="my_heap.cpp"
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[class]{MaxHeap}-[func]{left}
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[class]{MaxHeap}-[func]{right}
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[class]{MaxHeap}-[func]{parent}
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```
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=== "Python"
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```python title="my_heap.py"
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[class]{MaxHeap}-[func]{left}
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[class]{MaxHeap}-[func]{right}
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[class]{MaxHeap}-[func]{parent}
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```
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=== "Go"
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```go title="my_heap.go"
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[class]{maxHeap}-[func]{left}
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[class]{maxHeap}-[func]{right}
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[class]{maxHeap}-[func]{parent}
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```
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=== "JavaScript"
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```javascript title="my_heap.js"
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[class]{MaxHeap}-[func]{#left}
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[class]{MaxHeap}-[func]{#right}
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[class]{MaxHeap}-[func]{#parent}
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```
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=== "TypeScript"
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```typescript title="my_heap.ts"
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[class]{MaxHeap}-[func]{left}
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[class]{MaxHeap}-[func]{right}
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[class]{MaxHeap}-[func]{parent}
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```
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=== "C"
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```c title="my_heap.c"
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[class]{maxHeap}-[func]{left}
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[class]{maxHeap}-[func]{right}
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[class]{maxHeap}-[func]{parent}
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```
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=== "C#"
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```csharp title="my_heap.cs"
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[class]{MaxHeap}-[func]{left}
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[class]{MaxHeap}-[func]{right}
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[class]{MaxHeap}-[func]{parent}
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```
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=== "Swift"
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```swift title="my_heap.swift"
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[class]{MaxHeap}-[func]{left}
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[class]{MaxHeap}-[func]{right}
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[class]{MaxHeap}-[func]{parent}
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```
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=== "Zig"
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```zig title="my_heap.zig"
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[class]{MaxHeap}-[func]{left}
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[class]{MaxHeap}-[func]{right}
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[class]{MaxHeap}-[func]{parent}
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```
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### 访问堆顶元素
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堆顶元素即为二叉树的根节点，也就是列表的首个元素。
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=== "Java"
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```java title="my_heap.java"
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[class]{MaxHeap}-[func]{peek}
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```
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=== "C++"
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```cpp title="my_heap.cpp"
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[class]{MaxHeap}-[func]{peek}
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```
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=== "Python"
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```python title="my_heap.py"
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[class]{MaxHeap}-[func]{peek}
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```
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=== "Go"
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```go title="my_heap.go"
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[class]{maxHeap}-[func]{peek}
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```
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=== "JavaScript"
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```javascript title="my_heap.js"
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[class]{MaxHeap}-[func]{peek}
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```
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=== "TypeScript"
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```typescript title="my_heap.ts"
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[class]{MaxHeap}-[func]{peek}
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```
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=== "C"
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```c title="my_heap.c"
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[class]{maxHeap}-[func]{peek}
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```
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=== "C#"
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```csharp title="my_heap.cs"
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[class]{MaxHeap}-[func]{peek}
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```
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=== "Swift"
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```swift title="my_heap.swift"
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[class]{MaxHeap}-[func]{peek}
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```
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=== "Zig"
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```zig title="my_heap.zig"
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[class]{MaxHeap}-[func]{peek}
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```
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### 元素入堆
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给定元素 `val` ，我们首先将其添加到堆底。添加之后，由于 val 可能大于堆中其他元素，堆的成立条件可能已被破坏。因此，**需要修复从插入节点到根节点的路径上的各个节点**，这个操作被称为「堆化 Heapify」。
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考虑从入堆节点开始，**从底至顶执行堆化**。具体来说，我们比较插入节点与其父节点的值，如果插入节点更大，则将它们交换。然后继续执行此操作，从底至顶修复堆中的各个节点，直至越过根节点或遇到无需交换的节点时结束。
|
||
|
||
=== "<1>"
|
||
|
||
|
||
=== "<2>"
|
||
|
||
|
||
=== "<3>"
|
||
|
||
|
||
=== "<4>"
|
||
|
||
|
||
=== "<5>"
|
||
|
||
|
||
=== "<6>"
|
||
|
||
|
||
设节点总数为 $n$ ，则树的高度为 $O(\log n)$ 。由此可知，堆化操作的循环轮数最多为 $O(\log n)$ ，**元素入堆操作的时间复杂度为 $O(\log n)$** 。
|
||
|
||
=== "Java"
|
||
|
||
```java title="my_heap.java"
|
||
[class]{MaxHeap}-[func]{push}
|
||
|
||
[class]{MaxHeap}-[func]{siftUp}
|
||
```
|
||
|
||
=== "C++"
|
||
|
||
```cpp title="my_heap.cpp"
|
||
[class]{MaxHeap}-[func]{push}
|
||
|
||
[class]{MaxHeap}-[func]{siftUp}
|
||
```
|
||
|
||
=== "Python"
|
||
|
||
```python title="my_heap.py"
|
||
[class]{MaxHeap}-[func]{push}
|
||
|
||
[class]{MaxHeap}-[func]{sift_up}
|
||
```
|
||
|
||
=== "Go"
|
||
|
||
```go title="my_heap.go"
|
||
[class]{maxHeap}-[func]{push}
|
||
|
||
[class]{maxHeap}-[func]{siftUp}
|
||
```
|
||
|
||
=== "JavaScript"
|
||
|
||
```javascript title="my_heap.js"
|
||
[class]{MaxHeap}-[func]{push}
|
||
|
||
[class]{MaxHeap}-[func]{#siftUp}
|
||
```
|
||
|
||
=== "TypeScript"
|
||
|
||
```typescript title="my_heap.ts"
|
||
[class]{MaxHeap}-[func]{push}
|
||
|
||
[class]{MaxHeap}-[func]{siftUp}
|
||
```
|
||
|
||
=== "C"
|
||
|
||
```c title="my_heap.c"
|
||
[class]{maxHeap}-[func]{push}
|
||
|
||
[class]{maxHeap}-[func]{siftUp}
|
||
```
|
||
|
||
=== "C#"
|
||
|
||
```csharp title="my_heap.cs"
|
||
[class]{MaxHeap}-[func]{push}
|
||
|
||
[class]{MaxHeap}-[func]{siftUp}
|
||
```
|
||
|
||
=== "Swift"
|
||
|
||
```swift title="my_heap.swift"
|
||
[class]{MaxHeap}-[func]{push}
|
||
|
||
[class]{MaxHeap}-[func]{siftUp}
|
||
```
|
||
|
||
=== "Zig"
|
||
|
||
```zig title="my_heap.zig"
|
||
[class]{MaxHeap}-[func]{push}
|
||
|
||
[class]{MaxHeap}-[func]{siftUp}
|
||
```
|
||
|
||
### 堆顶元素出堆
|
||
|
||
堆顶元素是二叉树的根节点，即列表首元素。如果我们直接从列表中删除首元素，那么二叉树中所有节点的索引都会发生变化，这将使得后续使用堆化修复变得困难。为了尽量减少元素索引的变动，我们采取以下操作步骤：
|
||
|
||
1. 交换堆顶元素与堆底元素（即交换根节点与最右叶节点）；
|
||
2. 交换完成后，将堆底从列表中删除（注意，由于已经交换，实际上删除的是原来的堆顶元素）；
|
||
3. 从根节点开始，**从顶至底执行堆化**；
|
||
|
||
顾名思义，**从顶至底堆化的操作方向与从底至顶堆化相反**，我们将根节点的值与其两个子节点的值进行比较，将最大的子节点与根节点交换；然后循环执行此操作，直到越过叶节点或遇到无需交换的节点时结束。
|
||
|
||
=== "<1>"
|
||
|
||
|
||
=== "<2>"
|
||
|
||
|
||
=== "<3>"
|
||
|
||
|
||
=== "<4>"
|
||
|
||
|
||
=== "<5>"
|
||
|
||
|
||
=== "<6>"
|
||
|
||
|
||
=== "<7>"
|
||
|
||
|
||
=== "<8>"
|
||
|
||
|
||
=== "<9>"
|
||
|
||
|
||
=== "<10>"
|
||
|
||
|
||
与元素入堆操作相似，堆顶元素出堆操作的时间复杂度也为 $O(\log n)$ 。
|
||
|
||
=== "Java"
|
||
|
||
```java title="my_heap.java"
|
||
[class]{MaxHeap}-[func]{pop}
|
||
|
||
[class]{MaxHeap}-[func]{siftDown}
|
||
```
|
||
|
||
=== "C++"
|
||
|
||
```cpp title="my_heap.cpp"
|
||
[class]{MaxHeap}-[func]{pop}
|
||
|
||
[class]{MaxHeap}-[func]{siftDown}
|
||
```
|
||
|
||
=== "Python"
|
||
|
||
```python title="my_heap.py"
|
||
[class]{MaxHeap}-[func]{pop}
|
||
|
||
[class]{MaxHeap}-[func]{sift_down}
|
||
```
|
||
|
||
=== "Go"
|
||
|
||
```go title="my_heap.go"
|
||
[class]{maxHeap}-[func]{pop}
|
||
|
||
[class]{maxHeap}-[func]{siftDown}
|
||
```
|
||
|
||
=== "JavaScript"
|
||
|
||
```javascript title="my_heap.js"
|
||
[class]{MaxHeap}-[func]{pop}
|
||
|
||
[class]{MaxHeap}-[func]{#siftDown}
|
||
```
|
||
|
||
=== "TypeScript"
|
||
|
||
```typescript title="my_heap.ts"
|
||
[class]{MaxHeap}-[func]{pop}
|
||
|
||
[class]{MaxHeap}-[func]{siftDown}
|
||
```
|
||
|
||
=== "C"
|
||
|
||
```c title="my_heap.c"
|
||
[class]{maxHeap}-[func]{pop}
|
||
|
||
[class]{maxHeap}-[func]{siftDown}
|
||
```
|
||
|
||
=== "C#"
|
||
|
||
```csharp title="my_heap.cs"
|
||
[class]{MaxHeap}-[func]{pop}
|
||
|
||
[class]{MaxHeap}-[func]{siftDown}
|
||
```
|
||
|
||
=== "Swift"
|
||
|
||
```swift title="my_heap.swift"
|
||
[class]{MaxHeap}-[func]{pop}
|
||
|
||
[class]{MaxHeap}-[func]{siftDown}
|
||
```
|
||
|
||
=== "Zig"
|
||
|
||
```zig title="my_heap.zig"
|
||
[class]{MaxHeap}-[func]{pop}
|
||
|
||
[class]{MaxHeap}-[func]{siftDown}
|
||
```
|
||
|
||
## 堆常见应用
|
||
|
||
- **优先队列**：堆通常作为实现优先队列的首选数据结构，其入队和出队操作的时间复杂度均为 $O(\log n)$ ，而建队操作为 $O(n)$ ，这些操作都非常高效。
|
||
- **堆排序**：给定一组数据，我们可以用它们建立一个堆，然后不断地执行元素出堆操作，从而得到有序数据。当然，堆排序还有一种更优雅的实现，详见后续的堆排序章节。
|
||
- **获取最大的 $k$ 个元素**：这是一个经典的算法问题，同时也是一种典型应用，例如选择热度前 10 的新闻作为微博热搜，选取销量前 10 的商品等。
|