C++图论算法实战精解

图论基础理论与实际应用详解（C++实现）

一、基本概念

图 $G$ 由顶点集 $V$ 和边集 $E$ 构成，记为 $G=(V,E)$。

• 有向图：边有方向，$E \\subseteq V \\times V$
• 无向图：边无方向，$E$ 为无序对集合
• 权重图：边附加权值函数 $w: E \\to \\mathbb{R}$

存储结构：

const int MAXN = 1000;
int graph[MAXN][MAXN]; // 权重存于矩阵

#include <vector>
using namespace std;

struct Edge {
int to;
int weight;
};

vector<vector<Edge>> adjList; // adjList[i]存储顶点i的邻接边

二、图的遍历算法

深度优先搜索（DFS）

vector<bool> visited;

void dfs(int u) {
visited[u] = true;
for (auto& edge : adjList[u]) {
if (!visited[edge.to]) {
dfs(edge.to);
}
}
}

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应用场景：连通分量检测、拓扑排序、回溯法

广度优先搜索（BFS）

#include <queue>
void bfs(int start) {
queue<int> q;
q.push(start);
visited[start] = true;

while (!q.empty()) {
int u = q.front();
q.pop();
for (auto& edge : adjList[u]) {
if (!visited[edge.to]) {
visited[edge.to] = true;
q.push(edge.to);
}
}
}
}

应用场景：最短路径（无权图）、层级遍历

三、经典问题与算法

1. 最短路径（Dijkstra算法）

核心思想：贪心策略，适用于非负权图 $$d[v] = \\min(d[u] + w(u,v))$$

#include <queue>
#include <climits>

vector<int> dijkstra(int src, int n) {
vector<int> dist(n, INT_MAX);
dist[src] = 0;
priority_queue<pair<int, int>, vector<pair<int, int>>, greater<pair<int, int>>> pq;
pq.push({0, src});

while (!pq.empty()) {
auto [d, u] = pq.top();
pq.pop();
if (d != dist[u]) continue; // 旧数据跳过
for (auto& edge : adjList[u]) {
int newDist = d + edge.weight;
if (newDist < dist[edge.to]) {
dist[edge.to] = newDist;
pq.push({newDist, edge.to});
}
}
}
return dist;
}

2. 拓扑排序（Kahn算法）

适用条件：有向无环图（DAG） $$ \\text{入度} \\operatorname{deg}^{-}(v) = 0 \\implies \\text{可移除} $$

vector<int> topologicalSort(int n) {
vector<int> indegree(n, 0);
vector<int> result;
queue<int> q;

// 计算入度
for (int u = 0; u < n; u++)
for (auto& edge : adjList[u])
indegree[edge.to]++;

// 入度为0入队
for (int i = 0; i < n; i++)
if (indegree[i] == 0) q.push(i);

while (!q.empty()) {
int u = q.front();
q.pop();
result.push_back(u);
for (auto& edge : adjList[u]) {
if (–indegree[edge.to] == 0)
q.push(edge.to);
}
}
return result.size() == n ? result : vector<int>{}; // 判断环
}

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3. 最小生成树（Prim算法）

目标：连接所有顶点的最小权重边集 $$ T \\subseteq E, \\quad \\min \\sum_{e \\in T} w(e) $$

int prim(int n) {
vector<bool> visited(n, false);
vector<int> minDist(n, INT_MAX);
minDist[0] = 0;
int total = 0;

for (int i = 0; i < n; i++) {
int u = -1;
for (int j = 0; j < n; j++) // 选最小未访问点
if (!visited[j] && (u == -1 || minDist[j] < minDist[u]))
u = j;

if (minDist[u] == INT_MAX) return -1; // 不连通
visited[u] = true;
total += minDist[u];
for (auto& edge : adjList[u])
if (edge.weight < minDist[edge.to])
minDist[edge.to] = edge.weight;
}
return total;
}

四、实际应用场景

五、复杂问题扩展

• 负权处理：Bellman-Ford算法（动态规划）
• 全源最短路径：Floyd-Warshall算法（三重循环）
• 图的连通性：Tarjan强连通分量算法

提示：实际编码需处理边界条件（如不连通图、自环边），建议使用C++ STL的priority_queue优化Dijkstra，时间复杂度降至 $O(|E|\\log|V|)$。