【CLRS】《算法导论》读书笔记（一）：堆排序（Heapsort）

堆排序（Heapsort）

维基百科：http://en.wikipedia.org/wiki/Heapsort

时间复杂度：O(n log n)

示例：

[6, 5, 3, 1, 8, 7, 2, 4]

 

1、堆（Heap）

维基百科：http://en.wikipedia.org/wiki/Heap_(data_structure)

In computer science, a heap is a specialized tree-based data structure that satisfies the heap property: If A is a parent node of B then key(A) is ordered with respect to key(B) with the same ordering applying across the heap.

 

PARENT(i)

　　return i / 2

LEFT(i)

　　return 2 * i

RIGHT(i)

　　2 * i + 1

 

2、Heapify

最大堆为例，伪代码：

MAX-HEAPIFY(A, i)

　　l = LIFT(i)

　　r = RIGHT(i)

　　if l <= A.heapsize and A[l] > A[i]

　　　　largest = l

　　else largest = i

　　if r <= A.heapsize and A[r] > A[largest]

　　　　largest = r

　　if largest != i

　　　　exchage A[i] with A[largest]

　　　　MAX-HEAPIFY(A, largest)

 

3、Build Heap

最大堆为例，伪代码：

BUILD-MAX-HEAP(A)

　　A.heap-size = A.length

　　for A.length / 2 downto 1

　　　　MAX-HEAPIFY(A, i)

 

4、Heapsort

最大堆为例，伪代码：

HEAPSORT(A)

　　BUILD-MAX-HEAP(A)

　　for i = A.length downto 2

　　　　exchange A[1] with A[i]

　　　　A.heap-size = A.heap-size - 1

　　　　MAX-HEAPIFY(A, 1)

 

5、Priority Queue

最大堆为例，伪代码：

HEAP-MAXIMUM(A)

　　return A[1]

 

HEAP-EXTRACT-MAX(A)

　　if A.heap-size < 1

　　　　error "heap underflow"

　　max = A[1]

　　A[1] = A[A.heap-size]

　　A.heap-size = A.heap-size - 1

　　MAX-HEAPIFY(A, 1)

　　return max

 

HEAP-INCREASE-KEY(A, i, key)

　　if key < A[i]

　　　　error "new key is smaller than current key"

　　A[i] = key

　　while i > 1 and A[PARENT(i)] < A[i] 

　　　　exchange A[i] with A[PARENT(i)]

　　　　i = PARENT(i)

 

MAX-HEAP-INSERT(A, key)

　　A.heap-size = A.heap-size + 1

　　A[A.heap-size] = - INFINITY

　　HEAP-INCREASE-KEY(A, A.heap-size, key)

 

C语言实现

heap.h

typedef struct {
    int *array;
    int length;
    int heap_size;
} Heap;

int heap_parent(int index);
int heap_left_child(int index);
int heap_right_child(int index);

 

heap.c

int heap_parent(int index) {
    return (index - 1) / 2;
}

int heap_left_child(int index) {
    return 2 * index + 1;
}

int heap_right_child(int index) {
    return 2 * (index + 1);
}

 

heap_sort.h

#import "heap.h"

void max_heap_sort(Heap heap);

 

heap_sort.c

#import "heap_sort.h"

void exchange(int *a, int *b) {
    int temp = *a;
    *a = *b;
    *b = temp;
}

void max_heapify(Heap heap, int index) {
    int largest = index;

    int left = heap_left_child(index);
    int right = heap_right_child(index);

    if (left < heap.heap_size && heap.array[left] > heap.array[right]) {
        largest = left;
    }

    if (right < heap.heap_size && heap.array[right] > heap.array[largest]) {
        largest = right;
    }

    if (largest != index) {
        exchange(heap.array + largest, heap.array + index);

        max_heapify(heap, largest);
    }

}

void build_max_heap(Heap heap) {
    int i;
    for (i = heap.length / 2 - 1; i >= 0; i--) {
        max_heapify(heap, i);
    }
}

void max_heap_sort(Heap heap) {
    int i;
    build_max_heap(heap);
    for (i = heap.length - 1; i > 1; i--) {
        exchange(heap.array, heap.array + i);

        heap.heap_size -= 1;

        max_heapify(heap, 0);
    }
}

 

main.c

#import <stdio.h>
#import "heap_sort.h"

int main() {
    int i;
    int array[8] = {6, 5, 3, 1, 8, 7, 2, 4};
    Heap heap = {array, 8, 8};

    max_heap_sort(heap);

    for(i = 0; i < heap.length; i++) {
        printf("%d \n", heap.array[i]);
    }

    return 0;
}

 

运行结果：