11090 - Going in Cycle!!

[sclo]

Can this problem be done in O(n^3)?
My AC solution is O(n^4) in the worse case. More precisely, my solution is O(mn^2)

[kalinov]

A bit faster algorithm exists.
There is a way to check for some mean M if there is a cycle with mean at least M in O(n*m). So you can do a binary search on the mean to find the result.

[sclo]

I never thought about using binary search here, thanks for the idea.

[FAQ]

Please help me some I/O please? I still got WA

[Hadi]

There exist "many" O(mn) algorithms for this problem. See http://citeseer.ist.psu.edu/dasdan98experimental.html

[kp]

It also can be solved using binary search + Floyd.

[kp]

Please help me some I/O please? I still got WA

Be sure you handle cases with muliple edges from a to b correctly.

Test:

1
3 5
1 2 1
2 3 5
3 1 1
3 1 5
2 3 1

Answer:
Case #1: 1.00

[FAQ]

I got AC !! Thanks kp

[cpphamza]

There exist "many" O(mn) algorithms for this problem. See http://citeseer.ist.psu.edu/dasdan98experimental.html

I followed the link but couldn't get to any implementation (or pseudo code) for any of them, can you help with a link to the steps or implementation of any of these algorithms?

[cpphamza]

It also can be solved using binary search + Floyd.

I think you binary search for the mean value, but can you explain how you use floyd warshall to test if a cycle with this mean (or less) exists?

[Hadi]

1. To get an idea how to do the binary-search approach, see http://www.topcoder.com/stat?c=problem_ ... 50&rd=9984 and its editorial at http://www.topcoder.com/tc?module=Stati ... s_analysis

2. There are some pseudo-codes at the end of that article

[Hadi]

I think you binary search for the mean value, but can you explain how you use floyd warshall to test if a cycle with this mean (or less) exists?

My previous post and its link should help you.

[cpphamza]

Thanks alot, I tried to solve the prob using the pseudo code for karp's algorithm but got WA, the only different thing i changed from the pseudo code is replacing max by min in line 12 of the pseudo code (I suspect this is a mistake in the pseudo code).

here is my program

Code: Select all

#include<iostream>
using namespace std;

int INF = 100000000;
const int MAX = 50;
int weight[MAX][MAX];
int n, m;

double min(double x, double y){
	return x < y ? x : y;
}
double max(double x, double y){
	return x > y ? x : y;
}


int d[MAX+1][MAX];

double MMC(){
	
	//Initialize
	int k, u, v, s = 0;
	for(k = 0 ; k <= n ; k++)
		for(u = 0 ; u < n ; u++)
			d[k][u] = INF;
	d[0][s] = 0;

	//Compute the distances
	for(k = 1 ; k <= n ; k++)
		for(v = 0 ; v < n ; v++)
			for(u = 0 ; u < n ; u++)
				if(weight[u][v] < INF)
					d[k][v] = min(d[k][v], d[k-1][u]+weight[u][v]);

	//Compute lamda using karp's theorem
	double lamda = INF;
	for(u = 0 ; u < n ; u++){
		double currentLamda = INF;
		for(int k = 0 ; k < n ; k++)
			if(d[n][u] < INF && d[k][u] < INF)
				currentLamda = min(currentLamda, 1.0*(d[n][u]-d[k][u])/(n-k) );
		
		lamda = min(lamda, currentLamda);
	}
		
	return lamda;
}

int main(){

	freopen("1.in", "r", stdin);

	int tt; cin >> tt;
	for(int t = 0 ;  t < tt ; t++){

		cin >> n >> m;

		int i;
		for(i = 0 ; i < n ; i++)
			for(int j = 0 ; j < n ; j++)
				weight[i][j] = INF;
		for(i = 0 ; i < m ; i++){
			int l, r, c;
			cin >> l >>r >> c;
			weight[l-1][r-1] = min(c,weight[l-1][r-1]);
		}

		cout << "Case #" << t+1 << ": ";

		double l = MMC();
		if(l < INF){
			cout.setf(ios::fixed);
			cout.setf(ios::showpoint);
			cout.precision(2);
			cout << l;
		}
		else
			cout << "No cycle found.";
		cout << endl;
	}

	return 0;
}

it also fails a testcase like this,
1
3 4
1 2 2
2 3 4
3 1 6
1 3 6

any ideas?

[Hadi]

I haven't implemented the code myself, but I guess I know what's your mistake. You should note that the input to Karp's algorithm is a "strongly connected directed graph" not a "general directed graph". This means that there should be a path from every vertex u to every vertex v in the graph.

To use it for a general graph, first step is detecting the strongly connected components which can be done using 2 dfs's in O(v+e). Then run the algorithm for each of the strongly connected components. If you don't know the algorithm for detecting Strongly Connected Components, tell.

I hope this can help ...

[cpphamza]

Thanks alot it helped very much, I got AC after getting over a couple of things,

1- The Karp's pseudo code in the paper has a mistake in line 10, we should set lamda to a small value instead of INF.

2- I've applied Kosaraju's Algorithm for SCC as you've advised.

by the way do you know if these algorithms has extensions to cases more than finding a cycle, for example can we use them for solving the TopCoder problem you posted a link to?

also are these algorithms applied to find the maximum mean cycle?