【Codeforces】-445C-DZY Loves Physics(几何,数学)
2016-10-31 22:36
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C. DZY Loves Physics
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output
DZY loves Physics, and he enjoys calculating density.
Almost everything has density, even a graph. We define the density of a non-directed graph (nodes and edges of the graph have some values) as follows:
where v is the sum of the values of the nodes, e is
the sum of the values of the edges.
Once DZY got a graph G, now he wants to find a connected induced subgraph G' of
the graph, such that the density of G' is as large as possible.
An induced subgraph G'(V', E') of a graph G(V, E) is
a graph that satisfies:
;
edge
if
and only if
,
and edge
;
the value of an edge in G' is the same as the value of the corresponding edge in G,
so as the value of a node.
Help DZY to find the induced subgraph with maximum density. Note that the induced subgraph you choose must be connected.
Input
The first line contains two space-separated integers n (1 ≤ n ≤ 500),
.
Integer n represents the number of nodes of the graph G, m represents
the number of edges.
The second line contains n space-separated integers xi (1 ≤ xi ≤ 106),
where xi represents
the value of the i-th node. Consider the graph nodes are numbered from 1 to n.
Each of the next m lines contains three space-separated integers ai, bi, ci (1 ≤ ai < bi ≤ n; 1 ≤ ci ≤ 103),
denoting an edge between node ai and bi with
value ci.
The graph won't contain multiple edges.
Output
Output a real number denoting the answer, with an absolute or relative error of at most 10 - 9.
Examples
input
output
input
output
input
output
Note
In the first sample, you can only choose an empty subgraph, or the subgraph containing only node 1.
In the second sample, choosing the whole graph is optimal.
题意:求一个图的最大密度,密度定义:为顶点的价值和/边的价值和
题解:一个图最大的密度只由两个顶点构成。
证明:V1和V2边价值为a,构成图的最大密度。
若加上与V2相连的一个顶点V3最大,边的价值为b,则要求:V3/b>(V1+V2)/a;那么(V2+V3)/b,就是较大的。所以最大密度就是由两个顶点比他们的边得到的。
#include<cstdio>
#include<cstring>
#include<algorithm>
#include<vector>
using namespace std;
int main()
{
int n,m;
while(~scanf("%d%d",&n,&m))
{
double s[520];
for(int i=1;i<=n;i++)
scanf("%lf",&s[i]);
double maxx=0;
for(int i=1;i<=m;i++)
{
int a,b,c;
scanf("%d%d%d",&a,&b,&c);
maxx=max(maxx,(s[a]+s[b])/c);
}
printf("%.15lf\n",maxx);
}
return 0;
}
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output
DZY loves Physics, and he enjoys calculating density.
Almost everything has density, even a graph. We define the density of a non-directed graph (nodes and edges of the graph have some values) as follows:
where v is the sum of the values of the nodes, e is
the sum of the values of the edges.
Once DZY got a graph G, now he wants to find a connected induced subgraph G' of
the graph, such that the density of G' is as large as possible.
An induced subgraph G'(V', E') of a graph G(V, E) is
a graph that satisfies:
;
edge
if
and only if
,
and edge
;
the value of an edge in G' is the same as the value of the corresponding edge in G,
so as the value of a node.
Help DZY to find the induced subgraph with maximum density. Note that the induced subgraph you choose must be connected.
Input
The first line contains two space-separated integers n (1 ≤ n ≤ 500),
.
Integer n represents the number of nodes of the graph G, m represents
the number of edges.
The second line contains n space-separated integers xi (1 ≤ xi ≤ 106),
where xi represents
the value of the i-th node. Consider the graph nodes are numbered from 1 to n.
Each of the next m lines contains three space-separated integers ai, bi, ci (1 ≤ ai < bi ≤ n; 1 ≤ ci ≤ 103),
denoting an edge between node ai and bi with
value ci.
The graph won't contain multiple edges.
Output
Output a real number denoting the answer, with an absolute or relative error of at most 10 - 9.
Examples
input
1 0 1
output
0.000000000000000
input
2 1 1 2 1 2 1
output
3.000000000000000
input
5 6 13 56 73 98 17 1 2 56 1 3 29 1 4 42 2 3 95 2 4 88 3 4 63
output
2.965517241379311
Note
In the first sample, you can only choose an empty subgraph, or the subgraph containing only node 1.
In the second sample, choosing the whole graph is optimal.
题意:求一个图的最大密度,密度定义:为顶点的价值和/边的价值和
题解:一个图最大的密度只由两个顶点构成。
证明:V1和V2边价值为a,构成图的最大密度。
若加上与V2相连的一个顶点V3最大,边的价值为b,则要求:V3/b>(V1+V2)/a;那么(V2+V3)/b,就是较大的。所以最大密度就是由两个顶点比他们的边得到的。
#include<cstdio>
#include<cstring>
#include<algorithm>
#include<vector>
using namespace std;
int main()
{
int n,m;
while(~scanf("%d%d",&n,&m))
{
double s[520];
for(int i=1;i<=n;i++)
scanf("%lf",&s[i]);
double maxx=0;
for(int i=1;i<=m;i++)
{
int a,b,c;
scanf("%d%d%d",&a,&b,&c);
maxx=max(maxx,(s[a]+s[b])/c);
}
printf("%.15lf\n",maxx);
}
return 0;
}
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