加权无向图最小生成树-Prim和Kruskal法

【最小生成树的前提条件】

# 必须是连通图

# 所有边的权重都不能相同

 

【参考】

代码实现参考这个： Prim算法和Kruskal算法_日积月累，天道酬勤-CSDN博客

理论知识参考这个： 图的最小生成树 - 智者侬哥 - 博客园 (cnblogs.com)

 

【Kruskal法】

# 先把所有边按权重排序
# 不断的选权重小的边来组成一颗树, 这棵树的权重和最小, 所以叫最小生成树

local Kruskal = {}
Kruskal.__index = Kruskal

function Kruskal.new(g)
    local obj = {}
    setmetatable(obj, Kruskal)

    obj:ctor(g)
    return obj
end

function Kruskal:ctor(g)
    self.graph = g
    self.mst = nil
    self.weight = 0
end

function Kruskal:Build()
    self.mst = {}
    self.weight = 0

    local queue = self.graph:GetEdges()
    local uf = UnionFind.new()
    function EdgeSort(a, b)
        return a:GetWeight() <= b:GetWeight()
    end
    table.sort(queue, EdgeSort)
    while #queue > 0 and #self.mst < self.graph:GetVertexCount() - 1 do
        local minEdge = queue[1]
        table.remove(queue, 1)

        local v1 = minEdge:GetV1()
        local v2 = minEdge:GetV2()
        if not uf:IsConnected(v1, v2) then
            self.weight = self.weight + minEdge:GetWeight()
            uf:Union(v1, v2)
            table.insert(self.mst, minEdge)
        end
    end
end

function Kruskal:GetMst()
    return self.mst
end

function Kruskal:GetWeight()
    return self.weight
end

测试代码：

function TestKruskal()
    local g = CreateGraph() --看加权无向图那片的这个函数

    local k = Kruskal.new(g)
    k:Build()

    local mst = k:GetMst()
    print("mst:", #mst)
    for i=1,#mst do
        local edge = mst[i]
        print(edge:GetV1(), edge:GetV2(), edge:GetWeight())
    end
end
TestKruskal()

 

【Prim法】

# 选一个顶点, 获取所有与他连接的边, 获得权重最小的边, 加入最小生成树
# 再取最小边的另一个点, 重复第1步

local Prim = {}
Prim.__index = Prim

function Prim.new(g)
    local obj = {}
    setmetatable(obj, Prim)

    obj:ctor(g)
    return obj
end

function Prim:ctor(g)
    self.graph = g
    self.mst = nil
    self.weight = 0
end

function Prim:Build(startVertex)
    self.mst = {}
    self.weight = 0
    local marked = {}
    local queue = {}

    function EdgeSort(a, b)
        return a:GetWeight() <= b:GetWeight()
    end

    function AddAdjacentEdges(v) --添加所有与顶点连接的边
        marked[v] = true
        local edgeList = self.graph:GetAdjacent(v)
        for i=1,#edgeList do
            local edge = edgeList[i]
            table.insert(queue, edge)
        end
        table.sort(queue, EdgeSort)
    end
    AddAdjacentEdges(startVertex)

    while #queue > 0 do
        local minEdge = queue[1]
        table.remove(queue, 1)

        local v2 = minEdge:GetV2()
        if not marked[v2] then
            table.insert(self.mst, minEdge)
            self.weight = self.weight + minEdge:GetWeight()

            AddAdjacentEdges(v2)
        end
    end
end

function Prim:GetMst()
    return self.mst
end

function Prim:GetWeight()
    return self.weight
end

测试代码：

function TestPrim1()
    local g = CreateGraph() --看加权无向图那篇的这个函数

    local prim = Prim.new(g)
    prim:Build("0")

    local mst = prim:GetMst()
    for i=1,#mst do
        local edge = mst[i]
        print(edge:GetV1(), edge:GetV2(), edge:GetWeight())
    end
end
TestPrim1()

 

上面两个用到的Graph如下：