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답안 #991230

# 제출 시각 아이디 문제 언어 결과 실행 시간 메모리
991230 2024-06-01T15:32:12 Z model_code Fire (BOI24_fire) C++17
100 / 100
 214 ms 41496 KB 
/* TASK: The Holy Fire (fire)
 * Task proposal by: Sandra Schumann and Ahto Truu (EST)
 * Solution code by: Aldas Lenkšas (LTU SC)
 * Task also prepared by: Daumilas Ardickas (LTU SC)
 *
 * The task is to select the minimum number of intervals, such that the full circle
 * is covered by them. Circle length is M.
 *
 * Useful observation is that we can safely remove all the intervals that are fully covered 
 * by some other interval. They are not needed since we can always take the longer interval 
 * instead (that would cover the same interval, and possibly some more points). Note that if 
 * we have two same intervals (same beginning, same end), then we should keep one of them.
 *
 * This is not needed, but for simplicity, this solution changes intervals on the circle to the
 * intervals on the line (it makes some implementation parts easier). If the interval [s[i], e[i]]
 * is such that e[i] < s[i], then on the line it is the interval [s[i], e[i] + M]. Also, we
 * duplicate each of the intervals: each original interval [s[i], e[i]] gets a duplicate interval
 * [s[i] + M, e[i] + M]. To get the original answer (minimum number of intervals, such that the
 * full circle is covered), now it is enough to find minimum number of intervals on this line,
 * such that some continuous segment of length M is covered.
 *
 * The further solution assumes we are dealing with intervals on the line and no interval is 
 * fully covered by some other interval.
 *
 * Observe that for each interval i that we could take, there is some next best interval that
 * we would take. We can denote it as next[i].
 *
 * O(n^2) solution would start at some interval a, and then take b=next[a], then c=next[b], and so
 * on, until the segment of length M is taken. Then we can try another starting interval a.
 * We check all intervals as starting ones, and out of that we choose what we found to be the
 * minimum number of intervals to cover the segment of length M. 
 *
 * If by doing so we couldn't continuously cover a segment of length M, the answer is -1.
 *
 * To speed up this solution, we can add binary lifting to this. Now for each starting position,
 * we do (same as in binary search) checking "where would be the end, if we start from interval a, 
 * and take K intervals". We can find the minimum such K. This yields O(n log n) solution.
 *
 * Note: here we did not explain the binary lifting in great detail. You can check a better explanation
 * here: https://cp-algorithms.com/graph/lca_binary_lifting.html
 */


#include <bits/stdc++.h>

using namespace std;

const int MAXN = 4e5; // twice the contraints (since we duplicate intervals in modifyIntervals())
const int MAXK = 21; // ~log(MAXN), for binary lifting

long long n, m;
long long s[MAXN], e[MAXN];
int nextBest[MAXN];   // precalculated next best interval
int lift[MAXN][MAXK]; // precalculated binary lifting. lift[i][k] is taking 2^k intervals, starting from i-th one

void readInput() {
	cin >> n >> m;
	for (int i = 0; i < n; i++) {
		cin >> s[i] >> e[i];
	}
}

void removeUnnecessaryIntervals() {
	// interval is unnecessary if it is covered fully by other intervals

	int cnt = 0;

	// sort the intervals
	vector<pair<long long, long long>> intervals;
	for (int i = 0; i < n; i++) 
		intervals.push_back({s[i], -e[i]}); // this will help sort by increasing s[i], 
						    // and then by decreasing e[i]

	sort(intervals.begin(), intervals.end());

	long long furthestEnd = -1;

	for (auto interval : intervals) {
		long long start = interval.first;
		long long end = -interval.second;

		if (end <= furthestEnd) { 
			continue; // this interval is unnecessary
		}

		furthestEnd = end;
		s[cnt] = start;
		e[cnt] = end;
		cnt++;
	}

	n = cnt; // update interval count
}

void modifyIntervals() {
	// to make implementation a bit easier
	
	// deal with the cases of s > e
	for (int i = 0; i < n; i++) {
		if (s[i] > e[i]) e[i] += m;
	}

	// duplicate all intervals: (s, e) -> (s+m, e+m)
	for (int i = 0; i < n; i++) {
		s[i+n] = s[i] + m;
		e[i+n] = e[i] + m;
	}

	n *= 2; // since we duplicated

	removeUnnecessaryIntervals(); // remove covered fully by other intervals
}

void precalcNexts() {
	// sort intervals by their starts
	vector<pair<long long, int>> intervals; // vector of pairs {start, id}
	for (int i = 0; i < n; i++) 
		intervals.push_back({s[i], i});
	sort(intervals.begin(), intervals.end());

	// calculate nexts with two pointers technique
	int i = 0;

	for (auto interval : intervals) {
		int id = interval.second;
		
		while (i < n) {
			int intervalId = intervals[i].second;
			if (s[intervalId] > e[id])  // not overlapping
				break;
			i++;
		}

		nextBest[id] = intervals[i - 1].second;
	}
}

void precalcBinaryLifting() {
	for (int i = 0; i < n; i++) 
		lift[i][0] = nextBest[i];

	for (int k = 1; k < MAXK; k++) for (int i = 0; i < n; i++)
		lift[i][k] = lift[ lift[i][k-1] ][k-1];
}

int tryFrom(int startAdmin) {
	long long startTime = s[startAdmin];

	// do binary lifting (binary search) to find the furthest reachable admin
	// such that it would still not cover full circle
	
	int cntJumps = 0;
	int curAdmin = startAdmin;

	for (int i = MAXK-1; i >= 0; i--) {
		int jumpedToId = lift[curAdmin][i];
		if (e[jumpedToId] < (startTime + m)) { // does not cover full circle
			cntJumps += (1<<i); // 2^i
			curAdmin = jumpedToId;
		}
	}

	// we found the last before we get to full circle
	cntJumps++;
	curAdmin = lift[curAdmin][0];

	if (e[curAdmin] < (startTime + m)) {
		return -1; // impossible since we did not cover full circle
	}

	return cntJumps + 1;
}

int main() {
	readInput();
	modifyIntervals(); // see the explanation at the top
	precalcNexts(); 
	precalcBinaryLifting();

	int ans = -1;
	for (int i = 0; i < n; i++) {
		int curTaken = tryFrom(i);

		// update ans
		if (curTaken == -1) continue; 
		if (ans == -1 || curTaken < ans) ans = curTaken;
	}

	cout << ans << endl;


	return 0;
}

Subtask #1
14.0 / 14.0

# 결과 실행 시간 메모리 Grader output
1 Correct 1 ms 348 KB Output is correct
2 Correct 0 ms 428 KB Output is correct
3 Correct 0 ms 348 KB Output is correct
4 Correct 0 ms 348 KB Output is correct
5 Correct 0 ms 348 KB Output is correct
6 Correct 0 ms 348 KB Output is correct
7 Correct 0 ms 348 KB Output is correct
8 Correct 0 ms 348 KB Output is correct
9 Correct 0 ms 348 KB Output is correct
10 Correct 0 ms 348 KB Output is correct
11 Correct 0 ms 348 KB Output is correct
12 Correct 0 ms 348 KB Output is correct
13 Correct 1 ms 344 KB Output is correct
14 Correct 0 ms 344 KB Output is correct
15 Correct 0 ms 348 KB Output is correct
16 Correct 0 ms 348 KB Output is correct
17 Correct 0 ms 348 KB Output is correct
18 Correct 0 ms 348 KB Output is correct
19 Correct 0 ms 348 KB Output is correct
20 Correct 0 ms 348 KB Output is correct
21 Correct 0 ms 348 KB Output is correct
22 Correct 0 ms 348 KB Output is correct
23 Correct 0 ms 348 KB Output is correct

Subtask #2
17.0 / 17.0

# 결과 실행 시간 메모리 Grader output
1 Correct 1 ms 348 KB Output is correct
2 Correct 0 ms 428 KB Output is correct
3 Correct 0 ms 348 KB Output is correct
4 Correct 0 ms 348 KB Output is correct
5 Correct 0 ms 348 KB Output is correct
6 Correct 0 ms 348 KB Output is correct
7 Correct 0 ms 348 KB Output is correct
8 Correct 0 ms 348 KB Output is correct
9 Correct 0 ms 348 KB Output is correct
10 Correct 0 ms 348 KB Output is correct
11 Correct 0 ms 348 KB Output is correct
12 Correct 0 ms 348 KB Output is correct
13 Correct 1 ms 344 KB Output is correct
14 Correct 0 ms 344 KB Output is correct
15 Correct 0 ms 348 KB Output is correct
16 Correct 0 ms 348 KB Output is correct
17 Correct 0 ms 348 KB Output is correct
18 Correct 0 ms 348 KB Output is correct
19 Correct 0 ms 348 KB Output is correct
20 Correct 0 ms 348 KB Output is correct
21 Correct 0 ms 348 KB Output is correct
22 Correct 0 ms 348 KB Output is correct
23 Correct 0 ms 348 KB Output is correct
24 Correct 0 ms 348 KB Output is correct
25 Correct 0 ms 348 KB Output is correct
26 Correct 1 ms 344 KB Output is correct
27 Correct 0 ms 348 KB Output is correct
28 Correct 1 ms 348 KB Output is correct
29 Correct 1 ms 348 KB Output is correct
30 Correct 0 ms 348 KB Output is correct
31 Correct 1 ms 348 KB Output is correct
32 Correct 0 ms 348 KB Output is correct
33 Correct 1 ms 348 KB Output is correct
34 Correct 1 ms 344 KB Output is correct
35 Correct 1 ms 344 KB Output is correct
36 Correct 1 ms 344 KB Output is correct
37 Correct 0 ms 348 KB Output is correct
38 Correct 0 ms 348 KB Output is correct

Subtask #3
9.0 / 9.0

# 결과 실행 시간 메모리 Grader output
1 Correct 1 ms 348 KB Output is correct
2 Correct 0 ms 428 KB Output is correct
3 Correct 0 ms 348 KB Output is correct
4 Correct 0 ms 348 KB Output is correct
5 Correct 0 ms 348 KB Output is correct
6 Correct 0 ms 348 KB Output is correct
7 Correct 0 ms 348 KB Output is correct
8 Correct 0 ms 348 KB Output is correct
9 Correct 0 ms 348 KB Output is correct
10 Correct 0 ms 348 KB Output is correct
11 Correct 0 ms 348 KB Output is correct
12 Correct 0 ms 348 KB Output is correct
13 Correct 1 ms 344 KB Output is correct
14 Correct 0 ms 344 KB Output is correct
15 Correct 0 ms 348 KB Output is correct
16 Correct 0 ms 348 KB Output is correct
17 Correct 0 ms 348 KB Output is correct
18 Correct 0 ms 348 KB Output is correct
19 Correct 0 ms 348 KB Output is correct
20 Correct 0 ms 348 KB Output is correct
21 Correct 0 ms 348 KB Output is correct
22 Correct 0 ms 348 KB Output is correct
23 Correct 0 ms 348 KB Output is correct
24 Correct 0 ms 348 KB Output is correct
25 Correct 0 ms 348 KB Output is correct
26 Correct 1 ms 344 KB Output is correct
27 Correct 0 ms 348 KB Output is correct
28 Correct 1 ms 348 KB Output is correct
29 Correct 1 ms 348 KB Output is correct
30 Correct 0 ms 348 KB Output is correct
31 Correct 1 ms 348 KB Output is correct
32 Correct 0 ms 348 KB Output is correct
33 Correct 1 ms 348 KB Output is correct
34 Correct 1 ms 344 KB Output is correct
35 Correct 1 ms 344 KB Output is correct
36 Correct 1 ms 344 KB Output is correct
37 Correct 0 ms 348 KB Output is correct
38 Correct 0 ms 348 KB Output is correct
39 Correct 4 ms 860 KB Output is correct
40 Correct 4 ms 1324 KB Output is correct
41 Correct 5 ms 1564 KB Output is correct
42 Correct 4 ms 1560 KB Output is correct
43 Correct 3 ms 860 KB Output is correct
44 Correct 5 ms 1564 KB Output is correct
45 Correct 5 ms 1564 KB Output is correct
46 Correct 5 ms 1564 KB Output is correct
47 Correct 4 ms 1116 KB Output is correct
48 Correct 3 ms 860 KB Output is correct
49 Correct 2 ms 1080 KB Output is correct
50 Correct 4 ms 1564 KB Output is correct

Subtask #4
13.0 / 13.0

# 결과 실행 시간 메모리 Grader output
1 Correct 0 ms 352 KB Output is correct
2 Correct 0 ms 348 KB Output is correct
3 Correct 0 ms 348 KB Output is correct
4 Correct 0 ms 348 KB Output is correct
5 Correct 0 ms 348 KB Output is correct
6 Correct 0 ms 348 KB Output is correct
7 Correct 0 ms 348 KB Output is correct
8 Correct 0 ms 348 KB Output is correct
9 Correct 0 ms 348 KB Output is correct
10 Correct 1 ms 348 KB Output is correct
11 Correct 0 ms 348 KB Output is correct
12 Correct 1 ms 348 KB Output is correct
13 Correct 0 ms 344 KB Output is correct
14 Correct 0 ms 344 KB Output is correct
15 Correct 5 ms 1396 KB Output is correct
16 Correct 4 ms 1564 KB Output is correct
17 Correct 4 ms 860 KB Output is correct
18 Correct 5 ms 1564 KB Output is correct
19 Correct 4 ms 904 KB Output is correct
20 Correct 59 ms 15012 KB Output is correct
21 Correct 141 ms 15036 KB Output is correct
22 Correct 139 ms 14964 KB Output is correct
23 Correct 186 ms 41324 KB Output is correct
24 Correct 180 ms 33508 KB Output is correct
25 Correct 135 ms 14980 KB Output is correct
26 Correct 169 ms 31036 KB Output is correct
27 Correct 201 ms 41240 KB Output is correct
28 Correct 142 ms 15016 KB Output is correct

Subtask #5
21.0 / 21.0

# 결과 실행 시간 메모리 Grader output
1 Correct 0 ms 344 KB Output is correct
2 Correct 1 ms 348 KB Output is correct
3 Correct 0 ms 348 KB Output is correct
4 Correct 0 ms 348 KB Output is correct
5 Correct 0 ms 348 KB Output is correct
6 Correct 0 ms 348 KB Output is correct
7 Correct 0 ms 348 KB Output is correct
8 Correct 1 ms 348 KB Output is correct
9 Correct 1 ms 348 KB Output is correct
10 Correct 1 ms 348 KB Output is correct
11 Correct 0 ms 348 KB Output is correct
12 Correct 5 ms 1536 KB Output is correct
13 Correct 4 ms 1564 KB Output is correct
14 Correct 4 ms 1564 KB Output is correct
15 Correct 57 ms 14928 KB Output is correct
16 Correct 191 ms 41096 KB Output is correct
17 Correct 214 ms 41260 KB Output is correct
18 Correct 197 ms 41248 KB Output is correct
19 Correct 198 ms 41496 KB Output is correct
20 Correct 179 ms 41136 KB Output is correct

Subtask #6
26.0 / 26.0

# 결과 실행 시간 메모리 Grader output
1 Correct 1 ms 348 KB Output is correct
2 Correct 0 ms 428 KB Output is correct
3 Correct 0 ms 348 KB Output is correct
4 Correct 0 ms 348 KB Output is correct
5 Correct 0 ms 348 KB Output is correct
6 Correct 0 ms 348 KB Output is correct
7 Correct 0 ms 348 KB Output is correct
8 Correct 0 ms 348 KB Output is correct
9 Correct 0 ms 348 KB Output is correct
10 Correct 0 ms 348 KB Output is correct
11 Correct 0 ms 348 KB Output is correct
12 Correct 0 ms 348 KB Output is correct
13 Correct 1 ms 344 KB Output is correct
14 Correct 0 ms 344 KB Output is correct
15 Correct 0 ms 348 KB Output is correct
16 Correct 0 ms 348 KB Output is correct
17 Correct 0 ms 348 KB Output is correct
18 Correct 0 ms 348 KB Output is correct
19 Correct 0 ms 348 KB Output is correct
20 Correct 0 ms 348 KB Output is correct
21 Correct 0 ms 348 KB Output is correct
22 Correct 0 ms 348 KB Output is correct
23 Correct 0 ms 348 KB Output is correct
24 Correct 0 ms 348 KB Output is correct
25 Correct 0 ms 348 KB Output is correct
26 Correct 1 ms 344 KB Output is correct
27 Correct 0 ms 348 KB Output is correct
28 Correct 1 ms 348 KB Output is correct
29 Correct 1 ms 348 KB Output is correct
30 Correct 0 ms 348 KB Output is correct
31 Correct 1 ms 348 KB Output is correct
32 Correct 0 ms 348 KB Output is correct
33 Correct 1 ms 348 KB Output is correct
34 Correct 1 ms 344 KB Output is correct
35 Correct 1 ms 344 KB Output is correct
36 Correct 1 ms 344 KB Output is correct
37 Correct 0 ms 348 KB Output is correct
38 Correct 0 ms 348 KB Output is correct
39 Correct 4 ms 860 KB Output is correct
40 Correct 4 ms 1324 KB Output is correct
41 Correct 5 ms 1564 KB Output is correct
42 Correct 4 ms 1560 KB Output is correct
43 Correct 3 ms 860 KB Output is correct
44 Correct 5 ms 1564 KB Output is correct
45 Correct 5 ms 1564 KB Output is correct
46 Correct 5 ms 1564 KB Output is correct
47 Correct 4 ms 1116 KB Output is correct
48 Correct 3 ms 860 KB Output is correct
49 Correct 2 ms 1080 KB Output is correct
50 Correct 4 ms 1564 KB Output is correct
51 Correct 0 ms 352 KB Output is correct
52 Correct 0 ms 348 KB Output is correct
53 Correct 0 ms 348 KB Output is correct
54 Correct 0 ms 348 KB Output is correct
55 Correct 0 ms 348 KB Output is correct
56 Correct 0 ms 348 KB Output is correct
57 Correct 0 ms 348 KB Output is correct
58 Correct 0 ms 348 KB Output is correct
59 Correct 0 ms 348 KB Output is correct
60 Correct 1 ms 348 KB Output is correct
61 Correct 0 ms 348 KB Output is correct
62 Correct 1 ms 348 KB Output is correct
63 Correct 0 ms 344 KB Output is correct
64 Correct 0 ms 344 KB Output is correct
65 Correct 5 ms 1396 KB Output is correct
66 Correct 4 ms 1564 KB Output is correct
67 Correct 4 ms 860 KB Output is correct
68 Correct 5 ms 1564 KB Output is correct
69 Correct 4 ms 904 KB Output is correct
70 Correct 59 ms 15012 KB Output is correct
71 Correct 141 ms 15036 KB Output is correct
72 Correct 139 ms 14964 KB Output is correct
73 Correct 186 ms 41324 KB Output is correct
74 Correct 180 ms 33508 KB Output is correct
75 Correct 135 ms 14980 KB Output is correct
76 Correct 169 ms 31036 KB Output is correct
77 Correct 201 ms 41240 KB Output is correct
78 Correct 142 ms 15016 KB Output is correct
79 Correct 0 ms 344 KB Output is correct
80 Correct 1 ms 348 KB Output is correct
81 Correct 0 ms 348 KB Output is correct
82 Correct 0 ms 348 KB Output is correct
83 Correct 0 ms 348 KB Output is correct
84 Correct 0 ms 348 KB Output is correct
85 Correct 0 ms 348 KB Output is correct
86 Correct 1 ms 348 KB Output is correct
87 Correct 1 ms 348 KB Output is correct
88 Correct 1 ms 348 KB Output is correct
89 Correct 0 ms 348 KB Output is correct
90 Correct 5 ms 1536 KB Output is correct
91 Correct 4 ms 1564 KB Output is correct
92 Correct 4 ms 1564 KB Output is correct
93 Correct 57 ms 14928 KB Output is correct
94 Correct 191 ms 41096 KB Output is correct
95 Correct 214 ms 41260 KB Output is correct
96 Correct 197 ms 41248 KB Output is correct
97 Correct 198 ms 41496 KB Output is correct
98 Correct 179 ms 41136 KB Output is correct
99 Correct 153 ms 20032 KB Output is correct
100 Correct 143 ms 14848 KB Output is correct
101 Correct 198 ms 40668 KB Output is correct
102 Correct 182 ms 34524 KB Output is correct
103 Correct 196 ms 41048 KB Output is correct
104 Correct 154 ms 15032 KB Output is correct
105 Correct 128 ms 31068 KB Output is correct