701 - The Archeologists' Dilemma

[temper_3243]

Hi,
This is about problem 701 (Archaeologists dilemma).
http://acm.uva.es/p/v7/701.html


If i am given k digits of a number and the missing digits n-k > k , how do i find the smallest exponent x such that first k digits of 2^x =k . I have been trying out an algorithm but couldn't find one. I had searched the forum but no algorithms are present only sample inputs and outputs.

Can someone help me out or explain me how to solve this problem fast ?

Regards
Terry

[brunokbcao]

Thanks... I was using strings

[atif_ilm]

I don't remember exactly but main idea is to use logs. We're searching x, and there must be

N*10^k <= 2^x and 2^x < (N+1)*10^k, k -- number of missed digits, so
log2(N*10^k) <= log2(2^x) and log2(2^x) < log2((N+1)*10^k)
log2(N) + k*log2(10) <= x and x < log2(N+1) + k*log2(10)

With these "equations" and brute-forcing on k we can found x fast enough.

(May be I've missed something, sorry, I've solved this one a long time ago).

Well with these equations we can find that E or x lies between a certain range.
But how can we select a particular value of E within this range. I m sorry but i m not completely getting this point.

my code is like this in this code

Code: Select all

void ProcessNumber(int n)
{
int k = 0, temp = n;
double  min = 0, max = 0;
		
		while (temp > 0) {
			temp /= 10;
			++k;
		}
		
		k+= 1;
		
		for (k; k < INT_MAX; ++k) {

			min = (log10(n) + k) / log10(2);
			max = (log10(n + 1) + k) / log10(2);
 /* this is the minium and maxmim range for E  but how can i choose a  particular E */
                         

		}
		
}

[phoenix7]

can any one answer what the range of output is?

[ranban282]

I'm following the log approach as suggest. Can anyone tell me what is wrong, or why i'm getting floating point errors?
Here's the code:

Code: Select all

#include<iostream>
#include<cstdio>
#include<vector>
#include<string>
#include<algorithm>
#include<cmath>
#include<list>
#include<queue>
#include<cctype>
#include<stack>
#include<map>
#include<set>
using namespace std;
int n_digits(int n)
{
	int i=0;
	while(n>=10)
	{
		n/=10;
		i++;
	}
return i;
}
int main()
{
	
	unsigned int  n;
	long double lg=log((long double)10.0)/log((long double)2.0),lg2=log((long double)2.0);
	while(cin>>n)
	{
		int i=n_digits(n);
		for(long double j=i+2.0;;j++)
		{
			if(ceil(log((long double)n)/lg2+j*lg)==floor(log((long double)n+1.0)/lg2+j*lg))
			{
				printf("%.0Lf\n",ceil(log((long double)n)/lg2+j*lg));
				break;
			}
		}	
	}	
	return 0;
}

[ranban282]

Hi,
I changed everything to long double as suggested. Can anyone tell me the error in my code? If anyone has got AC can they post or mail me the code?
If you could have a look at my code and tell me whether it is a logical error or a precision error, I would be infinitely grateful to you.
Here's the code:

Code: Select all

#include<iostream>
#include<cstdio>
#include<vector>
#include<string>
#include<algorithm>
#include<cmath>
#include<list>
#include<queue>
#include<cctype>
#include<stack>
#include<map>
#include<set>
using namespace std;
long double n_digits(long double n)
{
	long double i=0.0;
	while(n>=10.0)
	{
		n/=10.0;
		i+=1.0;
	}
return i;
}
int main()
{
	
	long double  n;
	while(cin>>n)
	{
		long double i=n_digits(n);
		for(long double j=i+2.0;;j+=1.0)
		{
			long double arg1= n;
			long double arg2=n+1.0;
			long double log2=log10(2.0);
			if(ceil((log10(arg1)+j)/log2)==floor((log10(arg2)+j)/log2))
			{
				printf("%.0Lf\n",ceil((log10(arg1)+j)/log2));
				break;
			}
		}	
	}	
	return 0;
}

[ranban282]

WHAT IS THE PROBLEM WITH MY CODE? WHY DO I GET WA?????
EITHER GIVE ME TEST CASES FOR WHICH I GET WA OR HOW TO ELIMINATE PRECISION ERROR?

here's the code:

Code: Select all

#include<iostream>
#include<cstdio>
#include<vector>
#include<string>
#include<algorithm>
#include<cmath>
#include<list>
#include<queue>
#include<cctype>
#include<stack>
#include<map>
#include<set>
using namespace std;


int main()
{
	
	long double  n;
	while(cin>>n)
	{
		long double i=floor(log10(n));
		long double log2=log10((long double)2.0);
		for(long double j=i+2.0;;j+=1.0)
		{
			long double arg1= n;
			long double arg2=n+1.0;
			
			
			if(ceil((log10(arg1)+j)/log2)==floor((log10(arg2)+j)/log2))
			{
				printf("%.0Lf\n",ceil((log10(arg1)+j)/log2));
				break;
			}
		}	
	}	
	return 0;
}

[vahid sanei]

I got time limit for this tescase n = 2147483647
but i got Acc
could you help me for solving this test case ?
(" I know we can`t prove that "n is`nt a part of power of 2" because of numbers are infinity so we should`nt print "no power of 2"
Thanks in advance

[vahid sanei]

to ranban
I got Acc with your code
you can use ceill instead of ceil

[DD]

I think there will be an answer for this problem so it means we don't need to print "no power of 2".

[metaflow]

It look like there are a lot of inputs that leads to "no power of 2" answer that is valid. Try 309305843 for example.
I have no formal proof but I have checked that if we take in account only 8 first digits then set of possible prefixes of numbers is significantly less than 10^8-1 (at 2^6432187 = 16777216.. new cycle starts)

[brianfry713]

You can get AC on the judge for this problem without ever printing "no power of 2", so the judge's input doesn't contain cases like this.

Can you provide a proof or your code that you used to come up with the cycle length? Why did you only take into account the first 8 digits when N can be up to 10 digits long?

[brianfry713]

http://math.stackexchange.com/questions ... le-numbers

There are no cases where you should print no power of two, but some inputs may require a large output and take some time to run.

[Zwergesel]

This problem seems to work only because there are no tricky judge inputs.

Can someone give, for example, the answer for the input 2147333666?