题目:求一个无向图的严格次小生成树(即次小生成树的边权和严格小于最小生成树的边权和)

首先求出图中的最小生成树。任意加一条树外边都会导致环的出现。我们现在目标是在树外边集合B中,找到边b∈B,a∈b所在环,b->weight - a->weight最小且不为0。

首先,依题意,a->weight应当是环内所有边中最大或第二大(最大可能a->weight==b->weight)的。如何找呢?我们采用树上倍增的方法。定义cur->Elder[k]为cur的第k辈祖先,MaxW[k]为cur与cur->Elder[k]路径中的最长边,MaxW2[k]为cur与cur->Elder[k]路径中的严格次长边(MaxW2[k]<MaxW[k])。枚举b时,求b->From和b->To的最近公共祖先。因为求LCA的过程基础便是cur=cur->MaxW[k],于是取在此过程中MaxW与MaxW2的最大值,便可求出答案。

如何求MaxW和MaxW2?有递归式:

cur->MaxW[i] = max(cur->MaxW[i - ], cur->Elder[i - ]->MaxW[i - ]);

        if (cur->MaxW[i - ] == cur->Elder[i - ]->MaxW[i - ])
cur->MaxW2[i] = max(cur->MaxW2[i - ], cur->Elder[i - ]->MaxW2[i - ]);
if (cur->MaxW[i - ] < cur->Elder[i - ]->MaxW[i - ])
cur->MaxW2[i] = max(cur->MaxW[i - ], cur->Elder[i - ]->MaxW2[i - ]);
if (cur->MaxW[i - ] > cur->Elder[i - ]->MaxW[i - ])
cur->MaxW2[i] = max(cur->MaxW2[i - ], cur->Elder[i - ]->MaxW[i - ]);

初值:

cur->MaxW[] = cur->ToFa ? cur->ToFa->Weight : ;
cur->MaxW2[] = -INF;

完整代码:

#include <cstdio>
#include <cstring>
#include <cassert>
#include <algorithm>
#include <cmath>
using namespace std; #define LOOP(i,n) for(int i=1; i<=n; i++)
const int MAX_NODE = , MAX_EDGE = * , MAX_FA = ,
INF = 0x3f3f3f3f; struct Node;
struct Edge; struct Node
{
int Id, Depth;
Edge *Head, *ToFa;
Node *Elder[MAX_FA], *Prev;
int MaxW[MAX_FA], MaxW2[MAX_FA];
}_nodes[MAX_NODE];
Node *GRoot;
int _vCount; struct Edge
{
Node *From, *To;
Edge *Next, *Rev;
int Weight;
bool InTree;
Edge(){}
Edge(Node *from, Node *to, Edge *next, int weight)
:From(from),To(to),Next(next),Weight(weight),InTree(false){}
}*_edges[MAX_EDGE];
int _eCount; int Log2(int x)
{
int cnt = ;
while (x /= )
cnt++;
return cnt;
} void Init(int vCount)
{
_eCount = ;
_vCount = vCount;
GRoot = + _nodes;
memset(_nodes, , sizeof(_nodes));
} Edge *AddEdge(Node *from, Node *to, int w)
{
Edge *e = _edges[++_eCount] = new Edge(from, to, from->Head, w);
from->Head = e;
return e;
} void Build(int uId, int vId, int w)
{
Node *u = uId + _nodes, *v = vId + _nodes;
u->Id = uId;
v->Id = vId;
Edge *e1 = AddEdge(u, v, w), *e2 = AddEdge(v, u, w);
e1->Rev = e2;
e2->Rev = e1;
} Node *GetRoot(Node *cur)
{
return cur->Prev ? cur->Prev = GetRoot(cur->Prev) : cur;
} void Join(Node *a, Node *b)
{
a->Prev = b;
} bool CmpEdge(Edge *a, Edge *b)
{
return a->Weight < b->Weight;
} long long MinW;
void Kruskal()
{
MinW = ;
sort(_edges + , _edges + _eCount + , CmpEdge);
int ans = , cnt = ;
LOOP(i, _eCount)
{
if (cnt == _vCount)
break;
Edge *e = _edges[i];
Node *root1 = GetRoot(e->From), *root2 = GetRoot(e->To);
if (root1 != root2)
{
cnt++;
e->InTree = true;
MinW += (long long)e->Weight;
Join(root1, root2);
}
}
} void Dfs(Node *cur)
{
cur->MaxW[] = cur->ToFa ? cur->ToFa->Weight : ;
cur->MaxW2[] = -INF;
int topFa = Log2(cur->Depth);
for (int i = ; i <= topFa && cur->Elder[i - ]; i++)
{
cur->Elder[i] = cur->Elder[i - ]->Elder[i - ];
cur->MaxW[i] = max(cur->MaxW[i - ], cur->Elder[i - ]->MaxW[i - ]); if (cur->MaxW[i - ] == cur->Elder[i - ]->MaxW[i - ])
cur->MaxW2[i] = max(cur->MaxW2[i - ], cur->Elder[i - ]->MaxW2[i - ]);
else if (cur->MaxW[i - ] < cur->Elder[i - ]->MaxW[i - ])
cur->MaxW2[i] = max(cur->MaxW[i - ], cur->Elder[i - ]->MaxW2[i - ]);
else if (cur->MaxW[i - ] > cur->Elder[i - ]->MaxW[i - ])
cur->MaxW2[i] = max(cur->MaxW2[i - ], cur->Elder[i - ]->MaxW[i - ]);
}
for (Edge *e = cur->Head; e; e = e->Next)
{
if (!e->To->Depth && (e->InTree || e->Rev->InTree))
{
e->To->ToFa = e;
e->To->Elder[] = cur;
e->To->Depth = cur->Depth + ;
Dfs(e->To);
}
}
} void GetReady()
{
GRoot->Depth = ;
Dfs(GRoot);
} void Update(int& ans, Node *a, int i, int w)
{
ans = max(ans, a->MaxW[i] < w ? a->MaxW[i] : a->MaxW2[i]);
} int GetAltEdgeW(Edge *e)
{
int ans = -INF, w = e->Weight;
Node *a = e->From, *b = e->To;
if (a->Depth < b->Depth)
swap(a, b);
for (int i = Log2(a->Depth-b->Depth); i >= ; i--)
{
if (a->Elder[i] && a->Elder[i]->Depth >= b->Depth)
{
Update(ans, a, i, w);
a = a->Elder[i];
}
}
assert(a->Depth == b->Depth);
if (a == b)
return ans;
for (int i = Log2(a->Depth); i >= ; i--)
{
if (a->Elder[i] && a->Elder[i] != b->Elder[i])
{
Update(ans, a, i, w);
Update(ans, b, i, w);
a = a->Elder[i];
b = b->Elder[i];
}
}
Update(ans, a, , w);
Update(ans, b, , w);
return ans;
} long long Proceed()
{
int delta = INF;
for (int i = ; i <= _eCount; i++)
if (!_edges[i]->InTree && !_edges[i]->Rev->InTree)
delta = min(delta, _edges[i]->Weight - GetAltEdgeW(_edges[i]));
return (long long)delta + MinW;
} int main()
{
#ifdef _DEBUG
freopen("c:\\noi\\source\\input.txt", "r", stdin);
#endif
int totNode, totEdge, uId, vId, w;
scanf("%d%d", &totNode, &totEdge);
Init(totNode);
for (int i = ; i <= totEdge; i++)
{
scanf("%d%d%d", &uId, &vId, &w);
Build(uId, vId, w);
}
Kruskal();
GetReady();
printf("%lld\n", Proceed());
return ;
}

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