二进制搜索给出的结果略有错误

Question

The problem I am trying to solve is the following: I get N rectangular paper strips with 1cm width and length C . I need to cut the strips at a height where the sum of the areas of the cut strip is equal to A . You can see an example bellow for which N = 5, the strips are of length, 5,3,6,2 and 3 cm and A = 3cm where the cut is made at 4cm.

Note that I'm looking here for the red area.

The input is given as follows. The first line in each case begins with two integers N (1 ≤ N ≤ 10^5) and A (1 ≤ A ≤ 10^9) representing respectively the number of strips and the expected resulting area. The next line contains N integers, representing the length C_i (1 <= C_i <= 10^4) of each strip. The input ends with A = C = 0, which should not be processed.

For each test case, output a single line, the height H of the cut that must be done so that the sum of the area of the cut strips is equal to A cm². Print the answer with 4 decimal places. Output ":D" if no cutting is required, or "-.-" if it's impossible.

This problem can be found here

My idea for solving this problem was to use a binary search where I pick a height in the middle of the strips and make it larger or smaller depending on whether my cut was too high or too low. My implementation of the problem is given bellow:

#include <iostream>
#include <vector>
#include <iomanip>
#include <algorithm>

using namespace std;

int main(){
    vector<int> v;  // Vector that holds paper heights
    int n;          // Number of papers
    double h,       // Height of the cut
           sum,     // Area sum
           min_n,   // Minimum height for cut to happen
           max_n,   // Maximum height for cut to happen
           a;       // Desired final area

    // Set desired output
    cout << fixed << setprecision(4);

    /* Get number of papers and desired area,
       terminates if N = A = 0
    */
    while(cin >> n >> a && (n||a)){
        v.resize(n); // Resize vector to fit all papers
        // Get all paper sizes
        for(int i=0;i<n;i++){
            cin >> v[i];
        }
        /* Sort the vector in decreasing order to
           simplify the search
        */
        sort(v.begin(),v.end(),greater<int>());
        max_n = v[0]; // Largest possible cut is at the height of the largest paper
        min_n = 0; // Smallest possible cut is at the base with height 0

        // Iterate until answer is found
        while(true){
            // Initialize cut height as the average of smallest and largest cut sizes
            h = (min_n + max_n)/2;

            /* The area sum is equal to the sum of the areas of each cut, which is
               given by the height of the paper minus the cut height. If the cut is
               higher than the paper, the cut has area 0.
            */
            sum = 0;
            // Using mascoj sugenstion, a few changes were added
            int s; // Temporary variable to hold number of time h is subtracted
            for(int i=0; i<n;i++){
                if(v[i] <= h) break; // From here onward cut area is 0 and there is no point adding
                sum += v[i]; // Removed the subtraction inside of the for loop
                s++; // Count how many paper strips were used
            }
            sum -= h*s // Subtracts the area cut from the s paper strips

            // If the error is smaller than the significant value, cut height is printed
            if(std::abs(sum-a) < 1e-5){
                // If no cut is needed print :D else print cut height
                (h < 1e-4 ? cout << ":D" << endl : cout << h << endl);
                break;
            }
            // If max_n is "equal" to min_n and no answer was found, there is no answer
            else if(max_n - min_n < 1e-7){
                cout << "-.-" << endl;
                break;
            }
            // Reduces search interval
            sum < a ? max_n = h : min_n = h;
        }
    }
    return 0;
}

The problem is, after submitting my answer I keep getting a 10% error. The website has a tool for comparing the output of you program with the expected output so I ran a test file with over 1000 randomly generated test cases and when I compared both I got a rounding error on the 4th decimal case, unfortunately, I don't have the file nor the script to generate test cases for me anymore. I tried changing the acceptable error to a smaller one but that didn't work. I can't seem to find the error, does any of you have an idea of what is happening?

ps: Although the problem doesn't say on the description, you can get cuts with fractions as heights

Answer 1

Might be your problem, maybe not: This line is exacerbating floating point error: sum += v[i]-h;

Floating points are only so accurate and compounding this error over a larger summation adds up. I would try using multiplication on h and subtracting that from the total sum of applicable lengths. Should be well within the range of the double precision format so I wouldn't worry about overrunning the format.

Answer 2

Not sure to understand your algorithm but I think that can be done a lot simpler using a map instead a vector.

In the following example the map mp memorize how much (the value) strips are of a given lenght (the key).

An advantage of the map is ordered.

Next you can see how much you have to save (not to cat) and calculate the level of the cut starting from zero, adding 1 when appropriate and adding a fraction when neccessary.

Hope the following example can help

#include <map>
#include <iomanip>
#include <iostream>

int main()
 {
   int                         n;
   int                         n2;
   int                         v;
   int                         cut;
   int                         a;
   std::map<int, std::size_t>  mp;
   long long int               sum;
   long long int               ts;


   std::cout << std::fixed << std::setprecision(4);

   while( (std::cin >> n >> a) && ( n || a ) )
    {
      mp.clear();

      n2  = 0;
      sum = 0LL;

      for ( auto i = 0 ; i < n ; ++i )
       {
         std::cin >> v;

         if ( v > 0 )
          {
            sum += v;

            ++mp[v];
            ++n2;
          }
       }

      // mp is a map, so the values are ordered

      // ts is "to save"; sum of lenghts minus a
      ts  = sum - a;

      // cut level
      cut = 0;

      // while we can add a full cm to the cut level
      while ( (ts > 0LL) && (n2 > 0) && (ts >= n2) )
       {
         ++cut;

         ts -= n2;

         if ( cut >= mp.cbegin()->first )
          {
            n2 -= mp.cbegin()->second;

            mp.erase(mp.cbegin());
          }
       }

      if ( (ts == 0LL) && (cut == 0) )
         std::cout << ":D" << std::endl; // no cut required (?)
      else if ( n2 == 0 )
         std::cout << "-.-" << std::endl; // impossible (?)
      else
         std::cout << (cut + double(ts) / n2) << std::endl;
    }
 }

ps: observe that a is defined as an integer in the page that you link.