POJ3204 Ikki's Story - Road Reconstruction 网络流图的关键割边

题目大意：一个有源有汇的城市，问最少增加城市中的多少道路可以增加源到汇上各个路径上可容纳的总车流量增加。

网络流关键割边集合指如果该边的容量增加，整个网络流图中的任意从原点到汇点的路径的流量便可增加。

从源点开始遍历未满流的边，这些边两端节点的集合称为S；同理再从汇点开始遍历，集合称为T；其余的点组成另一个集合。如果一个边是最小割，则其两端属于不同的集合。如果一个边是关键割边，则该边两端节点一个属于S，一个属于T。遍历每一个图中的边，看它是否满足该条件即可。

#include <cstdio>
#include <cstring>
#include <cassert>
#include <cmath>
#include <algorithm>
using namespace std;

const int NODE_MAX = 1100, EDGE_MAX = 10100, INF = 0x3f3f3f3f;

//below:for dinic-----------------------------------------------------
struct Node;
struct Edge;

struct Node
{
	int Id, Vis;
	int Level;
	Edge *Head;
};

struct Edge
{
	int Cap, Id;
	Edge* Rev, *Next;
	Node *From, *To;
};

Node _nodes[NODE_MAX];
Edge _edges[EDGE_MAX];
int _lastVId, _eCount = 0;
Node *StartNode, *TargetNode;

void Init()
{
	memset(_nodes, 0, sizeof(_nodes));
	memset(_edges, 0, sizeof(_edges));
	_eCount = 0;
}

Edge *NewEdge()
{
	return ++_eCount + _edges;
}

void SetST(int sId, int tId)
{
	StartNode = _nodes + sId;
	TargetNode = _nodes + tId;
}

void SetLastVId(int vId)
{
	_lastVId = vId;
}

Edge *AddEdge(Node *from, Node *to, int id, int cap)
{
	Edge *e = NewEdge();
	e->Id = id;
	e->From = from;
	e->To = to;
	e->Cap = cap;
	e->Next = e->From->Head;
	e->From->Head = e;
	return e;
}

void Build(int uId, int vId, int id, int cap)
{
	Node *u = uId + _nodes, *v = vId + _nodes;
	u->Id = uId;
	v->Id = vId;
	Edge *e1 = AddEdge(u, v, id, cap);
	Edge *e2 = AddEdge(v, u, -id, 0);
	e1->Rev = e2;
	e2->Rev = e1;
}

struct NodeQueue
{
	Node *q[NODE_MAX];
	int head, tail;
	void clear() { head = tail = 0; }
	void push(Node *v) { q[tail++] = v; }
	Node* front() { return q[head]; }
	void pop() { head++; }
	bool empty() { return head == tail; }
};

bool Bfs()
{
	for (int i = 0; i <= _lastVId; i++)
		_nodes[i].Level = 0;
	NodeQueue q;
	q.clear();
	StartNode->Level = 1;
	q.push(StartNode);
	while (!q.empty())
	{
		Node *cur = q.front();
		q.pop();
		for (Edge *e = cur->Head; e; e = e->Next)
		{
			if (!e->To->Level && e->Cap)
			{
				e->To->Level = cur->Level + 1;
				q.push(e->To);
			}
		}
	}
	return TargetNode->Level;
}

int Dfs(Node *cur, int limit)
{
	if (cur == TargetNode)
		return limit;
	if (limit == 0)
		return 0;
	int curTake = 0;
	for (Edge *e = cur->Head; e; e = e->Next)
	{
		if (e->To->Level == cur->Level + 1 && e->Cap)
		{
			int nextTake = Dfs(e->To, min(limit - curTake, e->Cap));
			e->Cap -= nextTake;
			e->Rev->Cap += nextTake;
			curTake += nextTake;
		}
	}
	return curTake;
}

int Dinic()
{
	int ans = 0;
	while (Bfs())
		ans += Dfs(StartNode, INF);
	return ans;
}
//above:for dinic-----------------------------------------------------

void Dfs1(Node *cur)
{
	assert(!cur->Vis);
	cur->Vis = 1;
	for (Edge *e = cur->Head; e; e = e->Next)
		if (e->Cap && !e->To->Vis && e->Id > 0)
			Dfs1(e->To);
}

void Dfs2(Node *cur)
{
	assert(cur->Vis != 1);
	cur->Vis = 2;
	for (Edge *e = cur->Head; e; e = e->Next)
		if (e->Rev->Cap && !e->To->Vis &&e->Rev->Id > 0)
			Dfs2(e->To);
}

int main()
{
	freopen("c:\\noi\\source\\input.txt", "r", stdin);
	int totCity, totEdge;
	scanf("%d%d", &totCity,&totEdge);
	Init();
	SetLastVId(totCity - 1);
	SetST(0, totCity - 1);
	int uId, vId, cap;
	for (int i = 1; i <= totEdge; i++)
	{
		scanf("%d%d%d", &uId, &vId, &cap);
		Build(uId, vId, i, cap);
	}
	Dinic();
	Dfs1(StartNode);
	Dfs2(TargetNode);
	int eId[EDGE_MAX], pe = 0;
	memset(eId, 0, sizeof(eId));
	for (int i = 1; i <= _eCount; i++)
		if (_edges[i].From->Vis == 1 && _edges[i].To->Vis == 2 && _edges[i].Id > 0)
			eId[pe++] = _edges[i].Id;
	sort(eId, eId + pe);
	printf("%d\n", pe);
	return 0;
}