figure out Uneaten Leaves algorithm bug

问题

I faced this problem in an interview challenge

K caterpillars are eating their way through N leaves, each caterpillar falls from leaf to leaf in a unique sequence, all caterpillars start at a twig at position 0 and falls onto the leaves at position between 1 and N. Each caterpillar j has an associated jump number Aj. A caterpillar with jump number j eats leaves at positions that are multiple of j. It will proceed in the order j, 2j, 3j…. till it reaches the end of the leaves and it stops and build its cocoon. Given a set A of K elements , we need to determine the number of uneaten leaves.

Constraints:

1 <= N <= 10⁹

1 <= K <= 15

1 <= A[i] <= 10⁹

Input format:

N = No of uneaten leaves.

K = No. of caterpillars.

A = Array of integer. jump numbers Output:

The integer nu. Of uneaten leaves

Sample Input:

10
3
2
4
5

Output:

4

Explanation:

[2, 4, 5] is the 3-member set of jump numbers. All leaves which are multiple of 2, 4, and 5 are eaten. Only 4 leaves which are numbered 1,3,7,9 are left.

the naive approach for solving this question is have a Boolean array of all N numbers, and iterate over every caterpillar and remember the eaten leaves by it.

int uneatenusingNaive(int N, vector<int> A)
{
    int eaten = 0;
    vector<bool>seen(N+1, false);
    for (int i = 0; i < A.size(); i++)
    {
        long Ai = A[i];
        long j = A[i];
        while (j <= N && j>0)
        {
            if (!seen[j])
            {
                seen[j] = true;
                eaten++;
            }
            j += Ai;
        }
    }
    return N - eaten;
}

this approach passed 8 out of 10 test cases and give wrong answer for 2 cases.

another approach using Inclusion Exclusion principle, explanation for it can be found here and here
below is my code for the second approach

 int gcd(int a, int b)
    {
        if (b == 0)
            return a;
        return gcd(b, a%b);
    }
    int lcm(int i, int j)
    {
        return i*j / gcd(i, j);
    }

    vector<vector<int>> mixStr(vector<vector<int>> & mix, vector<int>& A, unordered_map<int, int> & maxStart)
    {
        vector<vector<int>> res;
        if (mix.size() == 0)
        {
            for (int i = 0; i < A.size(); i++)
            {
                vector<int> tmp;
                tmp.push_back(A[i]);
                res.push_back(tmp);
            }
            return res;
        }


        for (int i = 0; i<mix.size(); i++)
        {
            int currSlotSize = mix[i].size();
            int currSlotMax = mix[i][currSlotSize - 1];

            for (int j = maxStart[currSlotMax]; j < A.size(); j++)
            {
                vector<int> tmp(mix[i]);
                tmp.push_back(A[j]);
                res.push_back(tmp);
            }
        }
        return res;
    }
    int uneatenLeavs(int N, int k, vector<int> A)
    {
        int i = 0;
        vector<vector<int>> mix;
        bool sign = true;
        int res = N;
        sort(A.begin(), A.end());
        unordered_map<int,int> maxStart;
        for (int i = 0; i < A.size(); i++)
        {
            maxStart[A[i]] = i + 1;
        }
        int eaten = 0;


        while (mix.size() != 1)
        {   

            mix = mixStr(mix, A, maxStart);
            for (int j = 0; j < mix.size(); j++)
            {
                int _lcm = mix[j][0];
                for (int s = 1; s < mix[j].size(); s++)
                {
                    _lcm = lcm(mix[j][s], _lcm);
                }
                if (sign)
                {
                    res -= N / _lcm;
                }
                else
                {
                    res += N / _lcm;
                }
            }
            sign = !sign;
            i++;
        }
        return res;
    }

this approach passed only one 1/10 test case. and for the rest of test cases time limit exceeded and wrong answer.

Question:
What am I missing in first or second approach to be 100% correct.

回答1:

Using Inclusion-Exclusion theorem is correct approach, however, your implementation seems to be too slow. We can use bitmasking technique to obtain a O(K*2^K) time complexity.

Take a look at this:

long result = 0;

for(int i = 1; i < 1 << K; i++){
     long lcm = 1;
     for(int j = 0; j < K; j++)
        if(((1<<j) & i) != 0) //if bit j is set, compute new LCM after including A[j]
           lcm *= A[j]/gcd(lcm, A[j]);
     if(number of bit set in i is odd)
        result += N/lcm;
     else
        result -= N/lcm; 
}

For your first approach, an O(N*K) time complexity algorithm, with N = 10^9 and K = 15, it will be too slow, and can cause memory limit exceed/time limit exceed.

Notice that lcm can be larger than N, so, additional check is needed.

来源：https://stackoverflow.com/questions/35148149/figure-out-uneaten-leaves-algorithm-bug