图论模板

1. 黑白染色法

黑白染色法可用于判断奇圈，从而判定是否为二分图。

package main

import (
	"fmt"
)

const (
	Black = -1
	Gray  = 0
	White = 1
)

const (
	maxVertexNum = 110
	maxEdgeNum   = maxVertexNum * maxVertexNum
)

type Edge struct {
	u, v, next int
}

// 存储图相关信息
var head [maxVertexNum]int
var edges [maxEdgeNum]Edge
var edgeNum int

// InitGraph 初始化构建
func InitGraph() {
	edgeNum = 0
	for i := 0; i < maxVertexNum; i++ {
		head[i] = -1
	}
}

// AddEdge 有向图 或 无向图
func AddEdge(u, v int) {
	// 正向边
	edges[edgeNum] = Edge{u: u, v: v, next: head[u]}
	head[u] = edgeNum
	edgeNum++

	// 反向边
	edges[edgeNum] = Edge{u: v, v: u, next: head[v]}
	head[v] = edgeNum
	edgeNum++
}

// IsExistOddCycle
func IsExistOddCycle(n int) bool {

	color := make([]int, n+1)
	var dfs func(u, c int) bool
	dfs = func(u, c int) bool {
		color[u] = c
		for i := head[u]; i != -1; i = edges[i].next {
			if color[edges[i].v] == Gray {
				return dfs(edges[i].v, -c)
			}
			if color[u] == color[edges[i].v] {
				return true
			}
		}
		return false
	}

	var isExist bool
	for i := 1; i <= n; i++ {
		if color[i] == Gray {
			isExist = isExist || dfs(i, Black)
		}
		if isExist {
			break
		}
	}
	return isExist
}

func main() {
	// 1. 奇环
	InitGraph()
	AddEdge(1, 2)
	AddEdge(2, 3)
	AddEdge(3, 1)
	fmt.Println(IsExistOddCycle(3))

	// 2. 偶环
	InitGraph()
	AddEdge(1, 2)
	AddEdge(2, 1)
	fmt.Println(IsExistOddCycle(2))

	// 3. 无环
	InitGraph()
	AddEdge(1, 2)
	AddEdge(2, 3)
	fmt.Println(IsExistOddCycle(3))

	// 4. 奇环 + 偶环
	InitGraph()
	AddEdge(1, 2)
	AddEdge(2, 3)
	AddEdge(3, 1)
	AddEdge(4, 5)
	AddEdge(5, 6)
	fmt.Println(IsExistOddCycle(6))
}

2. 次短路径（Dijkstra + Heap）

package main

import (
	"container/heap"
	"fmt"
)

type Node struct {
	d int // 距离 d
	v int // 顶点 v
}

// NodeHeap 顶点堆
type NodeHeap []Node

// Len 长度
func (sh *NodeHeap) Len() int { return len(*sh) }

// Less 比较函数
// < : 小顶堆
// > : 大顶堆
func (sh *NodeHeap) Less(i, j int) bool { return (*sh)[i].d < (*sh)[j].d }

// Swap 交换函数
func (sh *NodeHeap) Swap(i, j int) { (*sh)[i], (*sh)[j] = (*sh)[j], (*sh)[i] }

// Pop heap pop
func (sh *NodeHeap) Pop() interface{} {
	n := len(*sh)
	x := (*sh)[n-1]
	*sh = (*sh)[:n-1]
	return x
}

// Push heap push
func (sh *NodeHeap) Push(x interface{}) {
	*sh = append(*sh, x.(Node))
}

// heap.Init()
// heap.Pop()
// heap.Push()

const (
	maxN = 1010
	maxE = 1010 * 1010 / 2
	inf  = 0x3f3f3f3f
)

type Edge struct {
	u    int
	v    int
	w    int
	next int
}

var (
	dis     [maxN][2]int
	edges   = [maxE]Edge{}
	head    = [maxN]int{}
	edgeNum int
)

// n : 顶点个数，编号 0 ~ n-1
func initGraph(n int) {
	edgeNum = 0
	for i := 0; i < n; i++ {
		head[i] = -1
	}
}

// 无向图加边
// 有向图加边
func addEdge(u, v, w int) {

	edges[edgeNum] = Edge{u: u, v: v, w: w}
	edges[edgeNum].next = head[u]
	head[u] = edgeNum
	edgeNum++

	edges[edgeNum] = Edge{u: v, v: u, w: w}
	edges[edgeNum].next = head[v]
	head[v] = edgeNum
	edgeNum++
}

// n: 顶点编号 0 ~ n-1
// src: 源点
// 注意: 次短路径的求解过程中，路径可以重复走
func solve(n int, src int) {

	var (
		ndHeap = &NodeHeap{}
		nd     Node
		u      int
		v      int
		d      int
	)

	for i := 0; i < n; i++ {
		dis[i][0] = inf // 最短路径
		dis[i][1] = inf // 次短路径
	}

	// dis[src][0], dis[src][1] = 0, 0
	dis[src][0] = 0
	*ndHeap = append(*ndHeap, Node{0, src})
	heap.Init(ndHeap)
	for len(*ndHeap) > 0 {
		nd = heap.Pop(ndHeap).(Node)
		if dis[nd.v][1] < nd.d { // 注意: 此处只能取 < ，而不能取 <=
			continue
		}
		u = nd.v
		for i := head[u]; i != -1; i = edges[i].next {
			v = edges[i].v
			d = nd.d + edges[i].w
			if dis[v][0] > d {
				// TODO: 维护最短路径上前继节点
				dis[v][0], d = d, dis[v][0]
				heap.Push(ndHeap, Node{dis[v][0], v})
			}
			if dis[v][0] <= d && dis[v][1] > d { // 注意: dis[v][0] <= d 而不是 dis[v][0] < d
				// TODO: 维护次短路径上前继节点
				dis[v][1] = d
				heap.Push(ndHeap, Node{dis[v][1], v})
			}
		}
	}
	return
}

func main() {

	// 1. 最短路径唯一
	initGraph(3)
	addEdge(0, 1, 1)
	addEdge(1, 2, 2)
	solve(3, 0)
	for i := 0; i < 3; i++ {
		fmt.Println("Test: ", i, dis[i][0], dis[i][1])
	}

	fmt.Println("--------------------------")
	// 2. 最短路径和次短路径长度不相同
	initGraph(4)
	addEdge(0, 1, 1)
	addEdge(0, 2, 2)
	addEdge(1, 3, 1)
	addEdge(2, 3, 1)
	solve(4, 0)
	for i := 0; i < 4; i++ {
		fmt.Println("Test: ", i, dis[i][0], dis[i][1])
	}

	fmt.Println("--------------------------")
	// 3. 最短路径和次短路径长度相同
	initGraph(4)
	addEdge(0, 1, 1)
	addEdge(0, 2, 1)
	addEdge(1, 3, 1)
	addEdge(2, 3, 1)
	solve(4, 0)
	for i := 0; i < 4; i++ {
		fmt.Println("Test: ", i, dis[i][0], dis[i][1])
	}
}