PAT甲级 Dijkstra 相关题_C++题解

Dijkstra

PAT (Advanced Level) Practice Dijkstra 相关题

目录

• 《算法笔记》重点摘要
• 1003 Emergency (25)

《算法笔记》 10.4.1 Dijkstra 重点摘要

对任意给出的图 G(V,E) 和 起点 S，终点 T，求 S 到 T 的最短路径

1. 简介

• 解决单源最短路问题
• 只能处理所有边权均非负的情况

  若出现负数，最好使用 SPFA 算法

2. 邻接矩阵

const int MAXV = 1000;
const int INF = 0x3fffffff;
int n, G[MAXV][MAXV], d[MAXV], pre[MAXV];
bool vis[MAXV] = {false};
void Dijkstra(int s){
    fill(d, d + MAXV, INF);
    d[s] = 0;
    for (int i = 0; i < n; i++) pre[i] = i;
    for (int i = 0; i < n; i++){
        int u = -1, MIN = INF;
        for (int j = 0; j < n; j++){    // 找未访问结点中 d[] 最小的
            if (!vis[j] && d[j] < MIN){
                u = j;
                MIN = d[j];
            }
        }
        if (u == -1) return;    // 找不到 d[u] < INF 点，说明剩下的点与起点 s 不连通
        vis[u] = true;
        for (int v = 0; v < n; v++){
            if (!vis[v] && G[u][v] != INF && d[u] + G[u][v] < d[v]){
                d[v] = d[u] + G[u][v];
                pre[v] = u;
            }
        }
    }
}
void DFS(int s, intv){
    if (v == s){
        printf("%d", s);
        return;
    }
    DFS(s,pre[v]);
    printf(" %d", v);
}

3. 第二标尺

第一标尺为距离

(1) 新增边权

如边的花费

int cost[MAXV][MAXV], c[MAXV];
fill(c, c + MAXV, INF);
c[s] = 0;
for (int v = 0; v < n; v++){
    if (!vis[v] && G[u][v] != INF){
        if (d[u] + G[u][v] == d[v] && c[u] + cost[u][v] < c[v]){
            pre[v] = u;
            c[v] = c[u] + cost[u][v];
        }
        else if (d[u] + G[u][v] < d[v]){
            pre[v] = u;
            d[v] = d[u] + G[u][v];
            c[v] = c[u] + cost[u][v];
        }
    }
}

(2) 新增点权

如点的权重

int weight[MAXV], w[MAXV] = {0};
w[s] = weight[s];
for (int v = 0; v < n; v++){
    if (!vis[v] && G[u][v] != INF){
        if (d[u] + G[u][v] == d[v] && w[u] + weight[v] > w[v]){
            pre[v] = u;
            w[v] = w[u] + weight[v];
        }
        else if (d[u] + G[u][v] < d[v]){
            pre[v] = u;
            d[v] = d[u] + G[u][v];
            w[v] = w[u] + weight[v];
        }
    }
}

(3) 求最短路径条数

int num[MAXV] = {0};
num[s] = 1;
for (int v = 0; v < n; v++){
    if (!vis[v] && G[u][v] != INF){
        if (d[u] + G[u][v] == d[v]{
            pre[v] = u;
            num[v] += num[u];
        }
        else if (d[u] + G[u][v] < d[v]){
            pre[v] = u;
            d[v] = d[u] + G[u][v];
            num[v] = num[u];
        }
    }
}

1003 Emergency (25)

题目思路

• 两个标尺：距离，点权；且要求最短路径数

#include<iostream>
using namespace std;
const int MAXN = 500, INF = 0x3fffffff;
int n, s, t, G[MAXN][MAXN], weight[MAXN], d[MAXN], pathnum[MAXN], w[MAXN] = {0};
bool vis[MAXN] = {false};
void Dijkstra(){
	fill(d, d + MAXN, INF);
	d[s] = 0;
	pathnum[s] = 1;
	w[s] = weight[s];
	for (int i = 0; i < n; i++){
		int u = -1, MIN = INF;
		for (int j = 0; j < n; j++){
			if (!vis[j] && d[j] < MIN){
				u = j;
				MIN = d[j];
			}
		}
		vis[u] = true;
		for (int v = 0; v < n; v++){
			if (!vis[v] && G[u][v] != INF){
				if (d[v] == d[u] + G[u][v]){
					pathnum[v] += pathnum[u];
					if (w[v] < w[u] + weight[v]) w[v] = w[u] + weight[v];
				}
				else if (d[v] > d[u] + G[u][v]){
					d[v] = d[u] + G[u][v];
					w[v] = w[u] + weight[v];
					pathnum[v] = pathnum[u];
				}
			}
		}
	}
}
int main()
{
	int m, u, v, len;
	scanf("%d%d%d%d", &n, &m, &s, &t);
	for (int i = 0; i < n; i++) scanf("%d", &weight[i]);
	fill(G[0], G[0] + MAXN * MAXN, INF);
	for (int i = 0; i < m; i++){
		scanf("%d%d%d", &u, &v, &len);
		G[u][v] = len;
		G[v][u] = len;
	}
	Dijkstra();
	printf("%d %d", pathnum[t], w[t]);
	return 0;
}

• fill(G[0], G[0] + MAXN * MAXN, INF) 注意二维数组要取首地址作为指针类型不能直接用数组名，因为它相当于是一维数组的指针，而参数要求是指针