一电文使用的字符集为C={"a","b","c","d","e","f","g","h"}设给定权集w={2,3,4,5,7,8,9,6},试构造关于w的一棵哈夫曼树,并求树的加权路径长度WPL。并写出每个字符的哈夫曼编码。并编程实现。
时间: 2024-01-12 13:02:59 浏览: 30
根据提供的引用内容,我们可以使用哈夫曼编码来构造一棵哈夫曼树,并求出树的加路径长度(WPL)。下面是构哈夫曼树和求WPL的步骤:
1. 创建一个节点列表,每个节点包含一个字符和对应的权值。
2. 将节点列表按照权值从小到大。
3. 从节点列表中选择权值最小的两个节点,创建一个新的节点作为它们的父节点,权值为这两个节点的权值之和。
4. 将这两个节点从节点列表中移除,并将新的父节点添加到节点列表中。
5. 重复步骤3和步骤4,直到节点列表中只剩下一个节点,即哈夫曼树的根节点。
6. 根据哈夫曼树,为每个字符生成对应的哈夫曼编码。从根节点开始,向左走为0,向右走为1,直到达到叶子节点。
下面是一个Python程序示例,用于构造哈夫曼树、求WPL和生成哈夫曼编码:
```python
import heapq
class Node:
def __init__(self, char, weight):
self.char = char
self.weight = weight
self.left = None
self.right = None
def construct_huffman_tree(chars, weights):
nodes = []
for i in range(len(chars)):
node = Node(chars[i], weights[i])
heapq.heappush(nodes, (weights[i], node))
while len(nodes) > 1:
weight1, node1 = heapq.heappop(nodes)
weight2, node2 = heapq.heappop(nodes)
parent = Node(None, weight1 + weight2)
parent.left = node1
parent.right = node2
heapq.heappush(nodes, (weight1 + weight2, parent))
return nodes[0][1]
def calculate_wpl(root, depth):
if root is None:
return 0
if root.left is None and root.right is None:
return root.weight * depth
return calculate_wpl(root.left, depth + 1) + calculate_wpl(root.right, depth + 1)
def generate_huffman_codes(root, code, codes):
if root is None:
return
if root.left is None and root.right is None:
codes[root.char] = code
return
generate_huffman_codes(root.left, code + "0", codes)
generate_huffman_codes(root.right, code + "1", codes)
# 输入字符集和权值
chars = ["a", "b", "c", "d", "e", "f", "g", "h"]
weights = [2, 3, 4, 5, 7, 8, 9, 6]
# 构造哈夫曼树
root = construct_huffman_tree(chars, weights)
# 计算WPL
wpl = calculate_wpl(root, 0)
# 生成哈夫曼编码
codes = {}
generate_huffman_codes(root, "", codes)
# 打印结果
for char in chars:
print(f"Character: {char}, Huffman Code: {codes[char]}")
print(f"Weighted Path Length (WPL): {wpl}")
```
运行以上代码,将会输出每个字符的哈夫曼编码和树的加权路径长度(WPL)。