POJ 3258: River Hopscotch

POJ 3258: River Hopscotch

Problem Statement

River Hopscotch

Time Limit: 2000MS		Memory Limit: 65536K
Total Submissions: 16105		Accepted: 6765

Description

Every year the cows hold an event featuring a peculiar version of hopscotch that involves carefully jumping from rock to rock in a river. The excitement takes place on a long, straight river with a rock at the start and another rock at the end, L units away from the start (1 ≤ L ≤ 1,000,000,000). Along the river between the starting and ending rocks, N (0 ≤ N ≤ 50,000) more rocks appear, each at an integral distance D_i from the start (0 < D_i < L).
To play the game, each cow in turn starts at the starting rock and tries to reach the finish at the ending rock, jumping only from rock to rock. Of course, less agile cows never make it to the final rock, ending up instead in the river.
Farmer John is proud of his cows and watches this event each year. But as time goes by, he tires of watching the timid cows of the other farmers limp across the short distances between rocks placed too closely together. He plans to remove several rocks in order to increase the shortest distance a cow will have to jump to reach the end. He knows he cannot remove the starting and ending rocks, but he calculates that he has enough resources to remove up to M rocks (0 ≤ M ≤ N).
FJ wants to know exactly how much he can increase the shortest distance *before* he starts removing the rocks. Help Farmer John determine the greatest possible shortest distance a cow has to jump after removing the optimal set of M rocks.

Input

Line 1: Three space-separated integers: L, N, and M
Lines 2..N+1: Each line contains a single integer indicating how far some rock is away from the starting rock. No two rocks share the same position.

Output

Line 1: A single integer that is the maximum of the shortest distance a cow has to jump after removing M rocks

Sample Input

25 5 2
2
14
11
21
17

Sample Output

4

Hint

Before removing any rocks, the shortest jump was a jump of 2 from 0 (the start) to 2. After removing the rocks at 2 and 14, the shortest required jump is a jump of 4 (from 17 to 21 or from 21 to 25).

Simplified Problem

Given a array of N numbers, not including 0 and L, remove M elements such that the minimum difference between any two consecutive elements is maximized. What is that minimum, given that we use the optimal solution? The N numbers may or may not be in sorted order.

The Algorithm

The "minimum is maximized" hints that this is a binary search problem. What are we searching for? The minimum distance possible between two consecutive dots after removing M elements. How are we going to find that? Test each minimum out. And how exactly do we do that? Well, we first sort out the array, accounting 0 and L, if we didn't do so. Then, we iterate through the array, counting the difference of any consecutive elements. If the difference is more than our guess, we increment a "rocks removed" counter by 1. Else, we update the "previous element" counter to this element. When done, if the rocks removed counter is more than M, this minimum is unfeasible, else, it is.

My Code

#include <algorithm>

#include <cstdio>

using namespace std;

int locs[50005];

int last, rocks, trem;

int main()

{

scanf("%d %d %d", &last, &rocks, &trem);

locs[0]=0;

for(int i=0;i<rocks;i++)

scanf("%d", &locs[i+1]);

locs[rocks+1]=last;

sort(locs, locs+rocks+2);

int L=0, U=last;

while(L<U)

{

int m = (L+U+1)/2;

int curpos = 0, remd = 0;

for(int i=1;i<rocks+2;i++)

{

if(locs[i]-curpos < m)

remd++;

else

curpos = locs[i];

}

if(remd > trem) U = m-1;

else L = m;

}

printf("%d\n",L);

}