十一. Minimum Spaning Tree 1 Krusk

Minimum Spaning Tree: 在我看来，就是在图G中包含所有点的tree，并且tree中所有边的权加起来最小。

Kruskal's algorithm：用于求出无向连通图中最小生成🌲---Minimum Spaning Tree（当然也可以求最大生成🌲）
或者在图不连通的情况下，求出最小（最大）森林。

过程：

1. 设置 每一个点都是一棵树
2. 先将所有cut(边)由小到大进行sort
3. 尝试所有的边
   A. 如果两个端点分别位于两颗树，那么连接两颗树，形成一条边。
   B. 如果两个点都在一颗树内，产生了一条环，那么就舍弃。

练习：
有二十个地方，每个地方相互连通，每个连通公路造价各不同，求最小工程造价和具体的施工方案。

import random

class node():
    def __init__(self, num):
        self.value = num



def build_map(nums):
    prices = {}
    for j in range(0, nums):
        for i in range(0, nums):
            if i != j and j < i:
                weight = 10 * random.random()
                name = (i, j)
                prices[name] = weight
    return prices

def quick_sort(array_price, connection_list, low, high):
    if low < high:
        middle = find_pivot(array_price, connection_list, low, high)
        quick_sort(array_price, connection_list, middle + 1, high)
        quick_sort(array_price, connection_list, low, middle - 1)

def find_pivot(array_prices, connection_list, low, high):
    pivot = high
    leftwall = low

    for i in range(low, high):
        if array_prices[pivot] > array_prices[i]:
            array_prices[leftwall], array_prices[i] = array_prices[i], array_prices[leftwall]
            connection_list[leftwall], connection_list[i] = connection_list[i], connection_list[leftwall]
            leftwall += 1

    array_prices[high], array_prices[leftwall] = array_prices[leftwall], array_prices[high]
    connection_list[high], connection_list[leftwall] = connection_list[leftwall], connection_list[high]

    return leftwall

def kruskal(prices, nums):
    nodes = []
    total_prices = 0
    add_order = []
    # create the node
    for j in range(0, nums):
        nodes.append(node(j))


    prices_list = list(prices.values())
    connection_list = list(prices.keys())

    # sort
    quick_sort(prices_list, connection_list, 0, len(prices_list)-1)

    for i in range(0, len(prices_list)):
        node_1 = connection_list[i][0]
        node_2 = connection_list[i][1]

        if nodes[node_1].value != nodes[node_2].value:
            if node_1 > node_2:
                value = nodes[node_1].value
                nodes[node_1].value = nodes[node_2].value
                # refresh represent of those nodes
                for j in range(0, nums):
                    if nodes[j].value == value:
                        nodes[j].value = nodes[node_2].value

            else:
                value = nodes[node_2].value
                nodes[node_2].value = nodes[node_1].value
                # refresh represent of those nodes
                for j in range(0, nums):
                    if nodes[j].value == value:
                        nodes[j].value = nodes[node_1].value


            total_prices += prices_list[i]
            add_order.append(connection_list[i])

    print(total_prices, add_order)
    return total_prices, add_order

prices = build_map(20)

kruskal(prices, 20)

求MST的方法不光是kruskal，Prim也是可以的。都是贪心思想。
区别：
prim：一个优先队列，每次选择距离当前部分最近的节点加入，直到所有节点都加入。适合稠密图，多用邻接矩阵。
Kruskal：并查集，每次总是选择权重最小的边加入，直到加入n-1条边为止。适合稀疏图，多用领接表。