We are given a stick of length L and N possible cut positions. Choose exactly K of those positions, dividing the stick into K + 1 pieces, and maximize the length of the shortest piece.
For a candidate minimum length d, scan the possible cut positions from left to right. Greedily make a cut as soon as it is at least d from the previous cut (starting at position 0), and stop after K cuts. This places each cut as early as possible, leaving the most room for later pieces; consequently, if any selection of K cuts works, these earliest greedy cuts work too. The candidate is feasible exactly when K cuts can be made and the remaining suffix from the last selected cut to L is at least d. When K = 0, no cuts are needed, so feasibility is simply L >= d.
Feasibility is monotone: if a minimum length d is feasible, every smaller positive length is feasible too. Binary search the largest feasible d in [0, L]. The greedy scan costs O(N) per check, so the total time is O(N log L) and the extra space is O(N) for the input positions.
Java
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import java.io.BufferedInputStream;
import java.io.IOException;
public class Main {
public static void main(String[] args) throws IOException {
FastScanner input = new FastScanner();
int n = input.nextInt();
int length = input.nextInt();
int k = input.nextInt();
int[] positions = new int[n];
for (int i = 0; i < n; i++) {
positions[i] = input.nextInt();
}
int low = 0;
int high = length + 1;
while (high - low > 1) {
int middle = low + (high - low) / 2;
if (canAchieve(middle, positions, k, length)) {
low = middle;
} else {
high = middle;
}
}
System.out.println(low);
}
private static boolean canAchieve(int minimumLength, int[] positions, int k, int length) {
if (k == 0) {
return length >= minimumLength;
}
int previousCut = 0;
int cuts = 0;
for (int position : positions) {
if (position - previousCut >= minimumLength) {
previousCut = position;
if (++cuts == k) {
return length - previousCut >= minimumLength;
}
}
}
return false;
}
private static final class FastScanner {
private final BufferedInputStream input = new BufferedInputStream(System.in);
private final byte[] buffer = new byte[1 << 16];
private int length;
private int position;
private int read() throws IOException {
if (position == length) {
length = input.read(buffer);
position = 0;
if (length == -1) {
return -1;
}
}
return buffer[position++];
}
int nextInt() throws IOException {
int c;
do {
c = read();
} while (c <= ' ' && c != -1);
int value = 0;
while (c > ' ') {
value = value * 10 + c - '0';
c = read();
}
return value;
}
}
}