ACM/ICPC 之 差分约束系统两道(ZOJ2770-POJ1201)

当对问题建立数学模型后，发现其是一个差分方程组，那么问题可以转换为最短路问题，一下分别选用Bellmanford-SPFA解题

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ZOJ2770-Burn the Linked Camp

//差分约束方程组-转换为最短路问题
//d[v] <= d[u] + dis[u][v]	->	d[v] - d[u] <= dis[u][v]
//Time:110Ms	Memory:12116
#include<iostream>
#include<cstring>
#include<cstdio>
using namespace std;

#define MAXN 1005
#define INF 0x3f3f3f3f

struct Edge {
	int u, v, w;
	Edge(){}
	Edge(int uu,int vv,int ww):u(uu),v(vv),w(ww){}
}e[MAXN*MAXN];

int n, m, le;
int sd[MAXN];	//从1-i兵营总共最多多少人
int d[MAXN];

bool bellmanford(int x)
{
	memset(d, INF, sizeof(d));
	d[x] = 0;
	for (int i = 0; i < n; i++)
	{
		for (int j = 0; j < le; j++)
		{
			int u = e[j].u, v = e[j].v;
			int w = e[j].w;
			if (d[v] > d[u] + w)
				d[v] = d[u] + w;
		}
	}
	for (int i = 0; i < le; i++)
		if (d[e[i].v] > d[e[i].u] + e[i].w)
			return false;
	return true;
}

int main()
{
	//freopen("in.txt", "r", stdin);
	while (scanf("%d%d", &n, &m) != EOF)
	{
		int a, b, w;
		le = 0;
		for (int i = 1; i <= n; i++)
		{
			scanf("%d", &sd[i]);
			sd[i] += sd[i - 1];
			e[le++] = Edge(i, i - 1, 0);
			e[le++] = Edge(i - 1, i, sd[i] - sd[i-1]);
		}
		for (int i = 1; i <= m; i++)
		{
			scanf("%d%d%d", &a, &b, &w);
			e[le++] = Edge(b, a - 1, -w);
			//e[le++] = Edge(a - 1, b, sd[b] - sd[a - 1]);
		}
		if (!bellmanford(n))	printf("Bad Estimations
");
		else	printf("%d
", -d[0]);
	}
	return 0;
}

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POJ1201_Intervals

//差分约束系统-SPFA
//求一个集合Z中最少有多少个数
//给定Z与n个区间的最小交集个数
//Time:407Ms	Memory:2624K
#include<iostream>
#include<cstring>
#include<cstdio>
#include<algorithm>
#include<queue>
using namespace std;

#define MAX 50005
#define INF 0x3f3f3f3f

struct Edge {
	int u, w, next;
	Edge(){}
	Edge(int uu,int ww,int nn):u(uu),w(ww),next(nn){}
}e[3*MAX];

int l, r, m, le;
int h[MAX];
int d[MAX];
bool v[MAX];

void spfa(int x)
{
	memset(d, INF, sizeof(d));
	memset(v, false, sizeof(v));
	d[x] = 0;
	queue<int> q;
	q.push(x);	v[x] = true;
	while (!q.empty()) {
		int cur = q.front();
		q.pop();	v[cur] = false;
		for (int i = h[cur]; i != -1; i = e[i].next)
		{
			int u = e[i].u, w = e[i].w;
			if (d[u] > d[cur] + w) {
				d[u] = d[cur] + w;
				if (!v[u]) {
					v[u] = true;	q.push(u);
				}
			}
		}
	}
}

int main()
{
	memset(h, -1, sizeof(h));
	scanf("%d", &m);
	l = INF;	r = 0;
	for (int i = 1; i <= m; i++)
	{
		int a, b, w;
		scanf("%d%d%d", &a, &b, &w);
		r = max(b, r);
		l = min(a, l);
		e[le] = Edge(a - 1, -w, h[b]);
		h[b] = le++;
	}
	for (int i = l; i <= r; i++)
	{
		e[le] = Edge(i - 1, 0, h[i]);
		h[i] = le++;
		e[le] = Edge(i, 1, h[i - 1]);
		h[i - 1] = le++;
	}

	spfa(r);

	printf("%d
", d[r] - d[l-1]);

	return 0;
}