c语言Huffman树构建的时间复杂度

Huffman树的构建算法是基于贪心策略的，时间复杂度为O(nlogn)，其中n为待编码的字符个数。在构建Huffman树时，需要进行多次的合并操作，每次选取最小的两个节点进行合并，因此需要进行n-1次合并操作。每次合并操作都需要在已排序的节点集合中寻找最小的两个节点，这可以通过使用最小堆来实现，每次插入和删除操作的时间复杂度为logn。因此，构建Huffman树的总时间复杂度为O(nlogn)。

相关问题

c语言实现构建huffman树算法

下面是用C语言实现构建Huffman树算法的示例代码：

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

#define MAX_SIZE 1000

// 定义Huffman树节点结构体
typedef struct {
    int weight; // 权值
    int parent; // 父节点下标
    int left;   // 左子节点下标
    int right;  // 右子节点下标
} HuffNode, *HuffTree;

// 定义Huffman编码结构体
typedef struct {
    char ch;        // 字符
    char code[MAX_SIZE]; // 编码
} HuffCode, *HuffCodeTable;

// 选择权值最小的两个节点
void select(HuffTree tree, int n, int *s1, int *s2) {
    int i;
    int min1, min2;
    min1 = min2 = MAX_SIZE;

    // 找出第一个权值最小的节点
    for (i = 0; i < n; i++) {
        if (tree[i].weight < min1 && tree[i].parent == -1) {
            min1 = tree[i].weight;
            *s1 = i;
        }
    }

    // 找出第二个权值最小的节点
    for (i = 0; i < n; i++) {
        if (tree[i].weight < min2 && tree[i].parent == -1 && i != *s1) {
            min2 = tree[i].weight;
            *s2 = i;
        }
    }
}

// 构建Huffman树
void buildHuffTree(HuffTree tree, int n) {
    if (n <= 1) return;

    int i, s1, s2;
    for (i = 0; i < 2 * n - 1; i++) {
        tree[i].parent = -1;
        tree[i].left = -1;
        tree[i].right = -1;
    }

    // 构建n-1个节点的Huffman树
    for (i = n; i < 2 * n - 1; i++) {
        select(tree, i, &s1, &s2);
        tree[s1].parent = i;
        tree[s2].parent = i;
        tree[i].left = s1;
        tree[i].right = s2;
        tree[i].weight = tree[s1].weight + tree[s2].weight;
    }
}

// 获取Huffman编码
void getHuffCode(HuffTree tree, HuffCodeTable table, int n) {
    int i, j;
    char *code = (char *)malloc(n * sizeof(char));
    code[n - 1] = '\0';

    for (i = 0; i < n; i++) {
        int start = n - 1;
        int c = i;
        int p = tree[c].parent;
        while (p != -1) {
            if (tree[p].left == c) {
                code[--start] = '0';
            } else {
                code[--start] = '1';
            }
            c = p;
            p = tree[c].parent;
        }
        strcpy(table[i].code, &code[start]);
    }
    free(code);
}

int main() {
    int n;
    printf("输入字符个数：");
    scanf("%d", &n);

    // 输入字符及其权值
    HuffTree tree = (HuffTree)malloc(sizeof(HuffNode) * (2 * n - 1));
    HuffCodeTable table = (HuffCodeTable)malloc(sizeof(HuffCode) * n);
    int i;
    for (i = 0; i < n; i++) {
        printf("输入第%d个字符及其权值：", i + 1);
        scanf("%s %d", &table[i].ch, &tree[i].weight);
    }

    // 构建Huffman树
    buildHuffTree(tree, n);

    // 获取Huffman编码
    getHuffCode(tree, table, n);

    // 输出Huffman编码
    printf("Huffman编码如下：\n");
    for (i = 0; i < n; i++) {
        printf("%c: %s\n", table[i].ch, table[i].code);
    }

    free(tree);
    free(table);
    return 0;
}

这段代码实现了从用户输入字符及其权值，构建Huffman树，获取Huffman编码，输出Huffman编码的功能。在构建Huffman树时，采用的是贪心策略，每次选择权值最小的两个节点进行合并。在获取Huffman编码时，采用的是自底向上的遍历方式，从叶子节点到根节点依次获取每个字符的编码。

c语言 huffman编码解码

Huffman编码是一种无损压缩算法，常用于数据传输和存储中。下面是C语言实现Huffman编码和解码的基本思路：

Huffman编码：

1. 统计字符出现频率，构建字符频率表；
2. 将字符频率表中的字符和频率信息构建成一棵Huffman树；
3. 遍历Huffman树，给每个字符生成对应的Huffman编码；
4. 将编码写入文件。

Huffman解码：

1. 读取压缩文件中的编码信息和字符频率表；
2. 根据字符频率表构建Huffman树；
3. 遍历Huffman树，根据编码信息还原原始字符串。

以下是C语言实现Huffman编码和解码的参考代码：

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

#define MAX_TREE_HT 100

// Huffman树节点结构体
struct MinHeapNode {
    char data;
    unsigned freq;
    struct MinHeapNode *left, *right;
};

// Huffman编码表结构体
struct HuffmanCode {
    char data;
    char *code;
};

// 最小优先队列结构体
struct MinHeap {
    unsigned size;
    unsigned capacity;
    struct MinHeapNode **array;
};

// 创建一个新的Huffman树节点
struct MinHeapNode* newNode(char data, unsigned freq)
{
    struct MinHeapNode* node = (struct MinHeapNode*)malloc(sizeof(struct MinHeapNode));
    node->left = node->right = NULL;
    node->data = data;
    node->freq = freq;
    return node;
}

// 创建一个最小优先队列
struct MinHeap* createMinHeap(unsigned capacity)
{
    struct MinHeap* minHeap = (struct MinHeap*)malloc(sizeof(struct MinHeap));
    minHeap->size = 0;
    minHeap->capacity = capacity;
    minHeap->array = (struct MinHeapNode**)malloc(minHeap->capacity * sizeof(struct MinHeapNode*));
    return minHeap;
}

// 交换两个节点
void swapMinHeapNode(struct MinHeapNode** a, struct MinHeapNode** b)
{
    struct MinHeapNode* t = *a;
    *a = *b;
    *b = t;
}

// 最小堆化
void minHeapify(struct MinHeap* minHeap, int idx)
{
    int smallest = idx;
    int left = 2 * idx + 1;
    int right = 2 * idx + 2;
    if (left < minHeap->size && minHeap->array[left]->freq < minHeap->array[smallest]->freq)
        smallest = left;
    if (right < minHeap->size && minHeap->array[right]->freq < minHeap->array[smallest]->freq)
        smallest = right;
    if (smallest != idx) {
        swapMinHeapNode(&minHeap->array[smallest], &minHeap->array[idx]);
        minHeapify(minHeap, smallest);
    }
}

// 检查堆大小是否为1
int isSizeOne(struct MinHeap* minHeap)
{
    return (minHeap->size == 1);
}

// 取出堆顶元素
struct MinHeapNode* extractMin(struct MinHeap* minHeap)
{
    struct MinHeapNode* temp = minHeap->array[0];
    minHeap->array[0] = minHeap->array[minHeap->size - 1];
    --minHeap->size;
    minHeapify(minHeap, 0);
    return temp;
}

// 插入新的节点
void insertMinHeap(struct MinHeap* minHeap, struct MinHeapNode* minHeapNode)
{
    ++minHeap->size;
    int i = minHeap->size - 1;
    while (i && minHeapNode->freq < minHeap->array[(i - 1) / 2]->freq) {
        minHeap->array[i] = minHeap->array[(i - 1) / 2];
        i = (i - 1) / 2;
    }
    minHeap->array[i] = minHeapNode;
}

// 构建最小优先队列
void buildMinHeap(struct MinHeap* minHeap)
{
    int n = minHeap->size - 1;
    int i;
    for (i = (n - 1) / 2; i >= 0; --i)
        minHeapify(minHeap, i);
}

// 判断是否是叶子节点
int isLeaf(struct MinHeapNode* root)
{
    return !(root->left) && !(root->right);
}

// 创建一个最小优先队列并构建Huffman树
struct MinHeap* createAndBuildMinHeap(char data[], int freq[], int size)
{
    int i;
    struct MinHeap* minHeap = createMinHeap(size);
    for (i = 0; i < size; ++i)
        minHeap->array[i] = newNode(data[i], freq[i]);
    minHeap->size = size;
    buildMinHeap(minHeap);
    return minHeap;
}

// 构建Huffman树
struct MinHeapNode* buildHuffmanTree(char data[], int freq[], int size)
{
    struct MinHeapNode *left, *right, *top;
    struct MinHeap* minHeap = createAndBuildMinHeap(data, freq, size);
    while (!isSizeOne(minHeap)) {
        left = extractMin(minHeap);
        right = extractMin(minHeap);
        top = newNode('$', left->freq + right->freq);
        top->left = left;
        top->right = right;
        insertMinHeap(minHeap, top);
    }
    return extractMin(minHeap);
}

// 生成Huffman编码
void generateCodes(struct MinHeapNode* root, char *code, int top, struct HuffmanCode *huffmanCode)
{
    if (root->left) {
        code[top] = '0';
        generateCodes(root->left, code, top + 1, huffmanCode);
    }
    if (root->right) {
        code[top] = '1';
        generateCodes(root->right, code, top + 1, huffmanCode);
    }
    if (isLeaf(root)) {
        huffmanCode[root->data].data = root->data;
        huffmanCode[root->data].code = (char*)malloc((top + 1) * sizeof(char));
        memcpy(huffmanCode[root->data].code, code, (top + 1) * sizeof(char));
    }
}

// 打印Huffman编码表
void printHuffmanCodes(struct HuffmanCode *huffmanCode, int size)
{
    int i;
    printf("Huffman编码表:\n");
    for (i = 0; i < size; ++i)
        printf("%c: %s\n", huffmanCode[i].data, huffmanCode[i].code);
}

// 压缩文件
void compressFile(char *fileName, struct HuffmanCode *huffmanCode, int size)
{
    FILE *fpIn, *fpOut;
    char ch, *code;
    int i, j, len;
    fpIn = fopen(fileName, "r");
    if (fpIn == NULL) {
        printf("打开文件失败!\n");
        return;
    }
    fpOut = fopen("compressed.dat", "wb");
    fwrite(&size, sizeof(int), 1, fpOut);
    for (i = 0; i < size; ++i) {
        fwrite(&huffmanCode[i].data, sizeof(char), 1, fpOut);
        len = strlen(huffmanCode[i].code);
        fwrite(&len, sizeof(int), 1, fpOut);
        fwrite(huffmanCode[i].code, sizeof(char), len, fpOut);
    }
    code = (char*)malloc(MAX_TREE_HT * sizeof(char));
    while ((ch = fgetc(fpIn)) != EOF) {
        for (i = 0; i < size; ++i) {
            if (huffmanCode[i].data == ch) {
                len = strlen(huffmanCode[i].code);
                for (j = 0; j < len; ++j)
                    code[j] = huffmanCode[i].code[j];
                fwrite(code, sizeof(char), len, fpOut);
                break;
            }
        }
    }
    fclose(fpIn);
    fclose(fpOut);
    printf("压缩成功!\n");
}

// 解压文件
void decompressFile(char *fileName)
{
    FILE *fpIn, *fpOut;
    struct HuffmanCode huffmanCode[256];
    char ch, *code;
    int i, j, len, size;
    fpIn = fopen(fileName, "rb");
    if (fpIn == NULL) {
        printf("打开文件失败!\n");
        return;
    }
    fpOut = fopen("decompressed.txt", "w");
    fread(&size, sizeof(int), 1, fpIn);
    for (i = 0; i < size; ++i) {
        fread(&huffmanCode[i].data, sizeof(char), 1, fpIn);
        fread(&len, sizeof(int), 1, fpIn);
        huffmanCode[i].code = (char*)malloc((len + 1) * sizeof(char));
        fread(huffmanCode[i].code, sizeof(char), len, fpIn);
        huffmanCode[i].code[len] = '\0';
    }
    code = (char*)malloc(MAX_TREE_HT * sizeof(char));
    struct MinHeapNode *root = buildHuffmanTree(NULL, NULL, 0);
    struct MinHeapNode *curr = root;
    while ((ch = fgetc(fpIn)) != EOF) {
        for (i = 0; i < 8; ++i) {
            if (ch & (1 << (7 - i))) // ASCII码从高位开始读
                curr = curr->right;
            else
                curr = curr->left;
            if (isLeaf(curr)) {
                fputc(curr->data, fpOut);
                curr = root;
            }
        }
    }
    fclose(fpIn);
    fclose(fpOut);
    printf("解压成功!\n");
}

int main()
{
    char data[] = {'a', 'b', 'c', 'd', 'e', 'f'};
    int freq[] = {5, 9, 12, 13, 16, 45};
    int size = sizeof(data) / sizeof(data[0]);
    struct HuffmanCode huffmanCode[256];
    struct MinHeapNode* root = buildHuffmanTree(data, freq, size);
    char code[MAX_TREE_HT];
    int top = 0;
    generateCodes(root, code, top, huffmanCode);
    printHuffmanCodes(huffmanCode, size);
    compressFile("input.txt", huffmanCode, size);
    decompressFile("compressed.dat");
    return 0;
}

上述代码中，我们使用了最小优先队列来构建Huffman树。在生成Huffman编码时，我们使用了递归的方法遍历树，并记录下每个叶子节点的编码。在压缩文件时，我们将字符的Huffman编码写入输出文件中。在解压文件时，我们读取压缩文件中的编码信息和字符频率表，根据它们构建Huffman树，并根据编码信息还原原始字符串。