Python: tree 树 Heap Sort
打印Full binary tree:
import math def reveal(vail: list | tuple): """ 利用Python居中打印 :param vail: :return: """ length = len(vail) height = math.ceil(math.log2(length + 1)) width = 2 ** height - 1 width = 2 * width print(f'height: {height}') print(f'width: {width}') print() index = 0 for b in range(1, height + 1): nodes = 2 ** (b - 1) for p in range(nodes): if index >= length: # 其实放前放后无所谓(假设计算height=4, 则第四次至少有一个node) break node = f'{vail[index]:0>2}' if p == nodes - 1: print(f'{node:^{width}}', end='') else: print(f'{node:^{width}}', end=' ' * 2) index += 1 # print(' ' * 10, width) width = (width - 2) // 2 # 每次减node宽度即可完美整除2 print() reveal(list(range(1, 20)))
构建大顶堆测试代码
import e # vail = [30, 20, 80, 40, 50, 10, 60, 70, 90] vail = [30, 20, 80, 40, 50, 10, 10, 70, 90] # 测试 break vail.insert(0, 0) total = len(vail[1:]) print(total, total // 2) print(f'{"~" * 10}Initial State{"~" * 10}') e.reveal(vail[1:]) print('{0}Initial State{0}'.format("~" * 10)) def heapify(n, i, vale: list): """ heapify current node 调整的起点在 n//2, 保证所有待调整节点都有孩子节点 :param n: 待比较个数 :param i: 当前节点下标 :param vale: 待排序数据 :return: None """ while 2 * i <= n: # 2*i==n or 2*i+1==n left_child_index = 2 * i max_child_index = left_child_index # n > left_child_index: n==2*i+1(存在右子树) if n > left_child_index and vale[left_child_index + 1] > vale[left_child_index]: max_child_index = left_child_index + 1 if vale[max_child_index] > vale[i]: vale[i], vale[max_child_index] = vale[max_child_index], vale[i] i = max_child_index e.reveal(vale[1:]) else: print('\nelse') e.reveal(vale[1:]) break # 因为已经构建完大顶堆, 故一旦 vale[i] > vale[max_child_index] 即可停止 # heapify(total, 1, vail) def init_heap(vale: list): total = len(vale[1:]) for i in range(total // 2, 0, -1): # 从最后一个分支节点 -> 第一个分支节点 vale[1] print(f'i = {i}') heapify(total, i, vale) return vale init_heap(vail)
跟树相关的 时间复杂度一个都有 log(n) , 一般前面还有n
n & log(n)
import math from typing import Sequence def reveal(vail: Sequence): """ *** 假定元素是三位数以内的 *** length=len(vail) 元素个数 2^k-1=length (Full Binary Tree) 树高: height=math.ceil(math.log2(length+1)) width: 2^height-1 所有元素排在一行的宽度 最后一行元素个数: final=length-(2^(height-1)-1) 每行元素个数: for b in range(1, height+1): n=2**(b-1) if b == height: return final else: return n 一个元素会产生2个interval space, 从第二行开始还要保留上面行的元素位置(递归) 每行组成: space_size=(width-(2^height-1)) / intervals height nodes intervals space_size 1 1 2 (15-1) / 2 = 7 2 2 4 (15-3) / 4 = 3 3 4 8 (15-7) / 8 = 1 4 8 16 (15-15) / 16 = 0 :param vail: :return: """ length = len(vail) height = math.ceil(math.log2(length + 1)) width = 2 ** height - 1 final = length - (2 ** (height - 1) - 1) print(f'height: {height}') print(f'width: {width}') print(f'final: {final}') print() index = 0 # 记录元素在vail的位置 space = ' ' * 2 # 假定元素的是三位数以内, 宽度为2 for b in range(1, height + 1): if b == height: nodes = final # nodes 每行元素个数 else: nodes = 2 ** (b - 1) space_size = int((width - (2 ** b - 1)) / (2 * nodes)) # 需要为int, 才能进行space * space_size # space_size = (width - (2 ** h - 1)) // (nodes * 2) # print(f'height: {b}') # print(f'current nodes: {nodes}') # print(f'total nodes: {2 ** b - 1}') # print(f'space_size: {space_size}') for node in range(nodes): if b == height and node > final - 1: # 也可在index += 1 判断 break if node == nodes - 1: # 每行最后一个node print(f'{space * space_size}{vail[index]:0>2}{space * space_size}', end='') # 最后一个元素无须补上一行的元素 else: print(f'{space * space_size}{vail[index]:0>2}{space * space_size}', end=' ' * 2) # 空格为了补上一行元素位置 index += 1 # if index > length - 1: # break print() # 可注释, 在node == nodes - 1最后一个元素位置直接打印换行 reveal(range(1, 20))
import e # vail = [30, 20, 80, 40, 50, 10, 60, 70, 90] vail = [30, 20, 80, 40, 50, 10, 10, 70, 90] # 测试 break vail.insert(0, 0) total = len(vail[1:]) print(total, total // 2) print(f'{"~" * 10}Initial State{"~" * 10}') e.reveal(vail[1:]) print('{0}Initial State{0}'.format("~" * 10)) def heapify(n, i, vale: list): """ heapify current node 调整的起点在 n//2, 保证所有待调整节点都有孩子节点 :param n: 待比较个数 :param i: 当前节点下标 :param vale: 待排序数据 :return: None """ while 2 * i <= n: # 2*i==n or 2*i+1==n left_child_index = 2 * i max_child_index = left_child_index # n > left_child_index: n==2*i+1(存在右子树) if n > left_child_index and vale[left_child_index + 1] > vale[left_child_index]: max_child_index = left_child_index + 1 if vale[max_child_index] > vale[i]: vale[i], vale[max_child_index] = vale[max_child_index], vale[i] i = max_child_index e.reveal(vale[1:]) else: print('\nelse') e.reveal(vale[1:]) break # 因为已经构建完大顶堆, 故一旦 vale[i] > vale[max_child_index] 即可停止 # heapify(total, 1, vail) def init_heap(vale: list): total = len(vale[1:]) for i in range(total // 2, 0, -1): # 从最后一个分支节点 -> 第一个分支节点 vale[1] print(f'i = {i}') heapify(total, i, vale) return vale init_heap(vail) print('{0}Heap sort{0}'.format('~' * 10)) def heap_sort(vale: list): total = len(vale[1:]) while total > 1: vale[1], vale[total] = vale[total], vale[1] print('{0}After Swap{0}'.format('~' * 20)) e.reveal(vale[1:]) print('{0}After Swap XXX{0}'.format('~' * 20)) print('\n' * 2) total -= 1 if total == 2 and vale[total] >= vale[total - 1]: # total-1==1 break heapify(total, 1, vale) return vale[1:] vail = heap_sort(vail) print(vail)
import c tug = [30, 20, 80, 40, 50, 10, 60, 70, 90] def tinker(length: int, node: int, vale: list) -> None: """ 大顶堆单节点向下调整算法 root: i, left child: 2*i+1, right child: 2*i+2 :param length: 待排序列表长度 :param node: 需调整的节点 :param vale: 需排序的列表 :return: None """ while 2 * node + 1 < length: # 存在左孩子进行调整, 不存在左孩子时, 调整到了叶子节点, 循环结束 if 2 * node + 2 < length and vale[2 * node + 2] > vale[2 * node + 1]: # 存在右孩子 且 右孩子大于左孩子 max_child_index = 2 * node + 2 else: max_child_index = 2 * node + 1 if vale[max_child_index] > vale[node]: # 左右孩子大于其根节点, 进行交换 vale[node], vale[max_child_index] = vale[max_child_index], vale[node] node = max_child_index # 隐式递归, 对交换后的叶子节点继续调整 c.reveal(vale) else: # 根节点比children都大时, node没有进行调整, 需跳出循环 print(f'\ntinker done, node >>{node}<< is bigger than children') c.reveal(vale) break print(f'{"~" * 20}Initial State{"~" * 20}') c.reveal(tug) print('{0}Initial State{0}'.format('~' * 20)) def heapify(vale: list): n = ((len(vale) - 1) - 1) // 2 # 最后一个非叶子节点, 即最后一个叶子节点的父节点, 从此处开始倒序->0, 调整为大顶堆 for i in range(n, -1, -1): tinker(len(vale), i, vale) def heap_sort(vale: list): heapify(vale) # 调整为大顶堆 print(f'{"~" * 20}Max Heap{"~" * 20}') c.reveal(vale) print('{0}Max Heap{0}'.format('~' * 20)) length = len(vale) while length > 1: vale[0], vale[length - 1] = vale[length - 1], vale[0] # 交换堆顶和最后一个元素 print('{0}After Swap{0}'.format('~' * 20)) c.reveal(vale) print('{0}After Swap XXX{0}'.format('~' * 20)) print('\n' * 2) length -= 1 # 堆变小 if length == 2 and vale[length - 1] > vale[length - 2]: break tinker(length, 0, vale) # 对交换后的堆, 从堆顶继续调整为大顶堆 heap_sort(tug) print(tug)
交换过程
C:\Users\pretentious\PycharmProjects\pythonProject\venv\Scripts\python.exe C:/Users/pretentious/PycharmProjects/pythonProject/f.py 9 4 ~~~~~~~~~~Initial State~~~~~~~~~~ 30 20 80 40 50 10 10 70 90 ~~~~~~~~~~Initial State~~~~~~~~~~ i = 4 30 20 80 90 50 10 10 70 40 i = 3 else 30 20 80 90 50 10 10 70 40 i = 2 30 90 80 20 50 10 10 70 40 30 90 80 70 50 10 10 20 40 i = 1 90 30 80 70 50 10 10 20 40 90 70 80 30 50 10 10 20 40 90 70 80 40 50 10 10 20 30 ~~~~~~~~~~Heap sort~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~After Swap~~~~~~~~~~~~~~~~~~~~ 30 70 80 40 50 10 10 20 90 ~~~~~~~~~~~~~~~~~~~~After Swap XXX~~~~~~~~~~~~~~~~~~~~ 80 70 30 40 50 10 10 20 90 else 80 70 30 40 50 10 10 20 90 ~~~~~~~~~~~~~~~~~~~~After Swap~~~~~~~~~~~~~~~~~~~~ 20 70 30 40 50 10 10 80 90 ~~~~~~~~~~~~~~~~~~~~After Swap XXX~~~~~~~~~~~~~~~~~~~~ 70 20 30 40 50 10 10 80 90 70 50 30 40 20 10 10 80 90 ~~~~~~~~~~~~~~~~~~~~After Swap~~~~~~~~~~~~~~~~~~~~ 10 50 30 40 20 10 70 80 90 ~~~~~~~~~~~~~~~~~~~~After Swap XXX~~~~~~~~~~~~~~~~~~~~ 50 10 30 40 20 10 70 80 90 50 40 30 10 20 10 70 80 90 ~~~~~~~~~~~~~~~~~~~~After Swap~~~~~~~~~~~~~~~~~~~~ 10 40 30 10 20 50 70 80 90 ~~~~~~~~~~~~~~~~~~~~After Swap XXX~~~~~~~~~~~~~~~~~~~~ 40 10 30 10 20 50 70 80 90 40 20 30 10 10 50 70 80 90 ~~~~~~~~~~~~~~~~~~~~After Swap~~~~~~~~~~~~~~~~~~~~ 10 20 30 10 40 50 70 80 90 ~~~~~~~~~~~~~~~~~~~~After Swap XXX~~~~~~~~~~~~~~~~~~~~ 30 20 10 10 40 50 70 80 90 ~~~~~~~~~~~~~~~~~~~~After Swap~~~~~~~~~~~~~~~~~~~~ 10 20 10 30 40 50 70 80 90 ~~~~~~~~~~~~~~~~~~~~After Swap XXX~~~~~~~~~~~~~~~~~~~~ 20 10 10 30 40 50 70 80 90 ~~~~~~~~~~~~~~~~~~~~After Swap~~~~~~~~~~~~~~~~~~~~ 10 10 20 30 40 50 70 80 90 ~~~~~~~~~~~~~~~~~~~~After Swap XXX~~~~~~~~~~~~~~~~~~~~ [10, 10, 20, 30, 40, 50, 70, 80, 90] Process finished with exit code 0
