Aggressive Cows/Chess Tournament

There is a new barn with N stalls and C cows. The stalls are located on a straight line at positions x1,….,xN (0 <= xi <= 1,000,000,000). We want to assign the cows to the stalls, such that the minimum distance between any two of them is as large as possible. What is the largest minimum distance?

Brute force

• For every distance between 1 to maxDist try to find the minimum distance between any two cows by placing cows d distance apart.
• TC: O(n*c)
• SC: O(1)

Code

bool isPossible(vector<int> inp, int dist, int cows)
{
    int n = inp.size();
    int k = inp[0];
    cows--;
    for (int i = 1; i < n; i++) {
        if (inp[i] - k >= dist) {
            cows--;
            if (!cows) {
                return true;
            }
            k = inp[i];
        }
    }
    return false;
}

int chessTournament(vector<int> positions, int n, int c)
{
    int ans = INT_MIN;
    sort(positions.begin(), positions.end());
    int maxDist = positions[n - 1] - positions[0];

    for (int d = 1; d <= maxDist; d++) {
        bool possible = isPossible(positions, d, c);
        if (possible) {
            ans = max(ans, d);
        }
    }
    return ans;
}

Binary Search

• TC: O(n*log(c))

Code

bool isPossible(vector<int> inp, int dist, int cows)
{
    int n = inp.size();
    int k = inp[0];
    cows--;
    for (int i = 1; i < n; i++) {
        if (inp[i] - k >= dist) {
            cows--;
            if (!cows) {
                return true;
            }
            k = inp[i];
        }
    }
    return false;
}

int chessTournament(vector<int> positions, int n, int c)
{
    sort(positions.begin(), positions.end());

    int low = 1, high = positions[n - 1] - positions[0];
    while (low <= high) {
        int mid = (low + high) >> 1;
        if (isPossible(positions, mid, c)) {
            low = mid + 1;
        } else {
            high = mid - 1;
        }
    }
    return high;
}