oj.uz

Submission #684363

# Submission time Handle Problem Language Result Execution time Memory
684363 2023-01-21T04:33:20 Z jophyyjh Mountains (IOI17_mountains) C++14
100 / 100
 32 ms 16092 KB 
/**
 * It took me long before I can even find 1 dp solution. At some point, I started to
 * consider the heights in decreasing order. Let the height of the highest
 * mountain(s) be h, and let m_1, m_2, ..., m_k be the mountain(s) attaining h from
 * left to right. Clearly, {m_1, ..., m_k} can see each other, so at most one of them
 * can be chosen. Now, the remaining mountains are partitioned into (k+1) contigous
 * segments, and we know that mountains from different segments can't see each other
 * (cuz they're separated by a mountain with height h)! In other words, the segments
 * are "somewhat independent", prompting us to consider range dp.
 * 
 * Let's do some rigorous reasoning. Suppose that we're finding the max num of deevs
 * that can be imprisoned with mountains [i,j]. If mountain i is not chosen, the ans
 * is the same as [i+1,j]'s. If mountain i (s) is chosen, we notice sth~
 * Let m_1, m_2, ..., m_l be the mountains (in [i+1,j] from left to right) that can
 * be seen by mount i. Here're some observations:
 *  (1) "Seeing" is a mutual relationship, i.e. a can see b iff b can see a;
 *  (2) Lines sm_1, sm_2, sm_3, ..., sm_l have non-decreasing slope;
 *  (3) None of m_1, m_2, ..., m_l can be chosen;
 *  (4) Similarly, after neglecting m_1, ..., m_l, mountains [i+1,j] are partitioned
 *      into several contigous segments. We claim that each segment is independent
 *      of each other, which means that each segment adds a dp ans of a smaller range
 *      to the dp val of [i,j]. This is true because of [Lemma].
 * [Lemma]  If mountain a can see mountain b and c, and a cannot see any mountains
 *          between b, c (a is to the left of b and b is to the left of c), then a
 *          cannot see any mountains between b and c.
 * Note that the statement implys that from mountain a, b and c are "adjacent seeable
 * mountains".
 * 
 * In impl2, we slightly modify the order of doing range dp: we compute the ranges
 * by sorting them inn decreasing order of i and increasing order of j. This
 * simplifies the process of finding "seeable" mountains.
 * 
 * Time Complexity: O(n^3)          (Full solution)
 * Implementation 2                 (Range dp)
*/

#include <bits/stdc++.h>
#include "mountains.h"

typedef std::vector<int>	vec;
typedef std::complex<int>   complex_t;
typedef long long   ll;

const int INF = 2 * 1e9;

inline ll cross(const complex_t& c1, const complex_t& c2) {
    return ll(c1.real()) * ll(c2.imag()) - ll(c1.imag()) * ll(c2.real());
}


int maximum_deevs(vec values) {
    int n = values.size();
    std::vector<complex_t> mounts(n);
    for (int k = 0; k < n; k++)
        mounts[k] = complex_t{k, values[k]};
    
    std::vector<vec> max_d(n, vec(n));
    for (int i = n - 1; i >= 0; i--) {
        vec seen;   // the mountains that can be seen by i and are on its right
        complex_t last_vec = {0, -INF};
        int prefix_sum = 0;
        for (int j = i; j < n; j++) {
            if (cross(mounts[j] - mounts[i], last_vec) <= 0LL) {
                if (!seen.empty() && seen.back() + 1 <= j - 1)
                    prefix_sum += max_d[seen.back() + 1][j - 1];
                seen.push_back(j);
                last_vec = mounts[j] - mounts[i];
            }
            max_d[i][j] = 1 + prefix_sum;
            if (seen.back() + 1 <= j)
                max_d[i][j] += max_d[seen.back() + 1][j];
            if (i < j)
                max_d[i][j] = std::max(max_d[i][j], max_d[i + 1][j]);   // skip i
        }
    }
    return max_d[0][n - 1];
}

Subtask #1
20.0 / 20.0

# Verdict Execution time Memory Grader output
1 Correct 0 ms 212 KB Output is correct
2 Correct 0 ms 212 KB Output is correct
3 Correct 0 ms 212 KB Output is correct
4 Correct 0 ms 212 KB Output is correct
5 Correct 0 ms 212 KB Output is correct
6 Correct 0 ms 212 KB Output is correct
7 Correct 1 ms 212 KB Output is correct
8 Correct 1 ms 212 KB Output is correct
9 Correct 0 ms 212 KB Output is correct
10 Correct 0 ms 212 KB Output is correct
11 Correct 0 ms 212 KB Output is correct
12 Correct 1 ms 212 KB Output is correct
13 Correct 0 ms 212 KB Output is correct
14 Correct 0 ms 212 KB Output is correct
15 Correct 0 ms 212 KB Output is correct
16 Correct 1 ms 212 KB Output is correct

Subtask #2
20.0 / 20.0

# Verdict Execution time Memory Grader output
1 Correct 0 ms 212 KB Output is correct
2 Correct 0 ms 212 KB Output is correct
3 Correct 0 ms 212 KB Output is correct
4 Correct 0 ms 212 KB Output is correct
5 Correct 0 ms 212 KB Output is correct
6 Correct 0 ms 212 KB Output is correct
7 Correct 1 ms 212 KB Output is correct
8 Correct 1 ms 212 KB Output is correct
9 Correct 0 ms 212 KB Output is correct
10 Correct 0 ms 212 KB Output is correct
11 Correct 0 ms 212 KB Output is correct
12 Correct 1 ms 212 KB Output is correct
13 Correct 0 ms 212 KB Output is correct
14 Correct 0 ms 212 KB Output is correct
15 Correct 0 ms 212 KB Output is correct
16 Correct 1 ms 212 KB Output is correct
17 Correct 0 ms 212 KB Output is correct
18 Correct 0 ms 212 KB Output is correct
19 Correct 0 ms 212 KB Output is correct
20 Correct 0 ms 212 KB Output is correct
21 Correct 0 ms 212 KB Output is correct
22 Correct 0 ms 212 KB Output is correct
23 Correct 0 ms 212 KB Output is correct
24 Correct 0 ms 212 KB Output is correct
25 Correct 1 ms 212 KB Output is correct
26 Correct 0 ms 212 KB Output is correct
27 Correct 0 ms 212 KB Output is correct
28 Correct 0 ms 212 KB Output is correct
29 Correct 1 ms 212 KB Output is correct
30 Correct 0 ms 212 KB Output is correct
31 Correct 0 ms 212 KB Output is correct
32 Correct 1 ms 212 KB Output is correct
33 Correct 0 ms 212 KB Output is correct
34 Correct 0 ms 212 KB Output is correct
35 Correct 0 ms 212 KB Output is correct
36 Correct 0 ms 212 KB Output is correct
37 Correct 1 ms 248 KB Output is correct
38 Correct 0 ms 212 KB Output is correct

Subtask #3
30.0 / 30.0

# Verdict Execution time Memory Grader output
1 Correct 0 ms 212 KB Output is correct
2 Correct 0 ms 212 KB Output is correct
3 Correct 0 ms 212 KB Output is correct
4 Correct 0 ms 212 KB Output is correct
5 Correct 0 ms 212 KB Output is correct
6 Correct 0 ms 212 KB Output is correct
7 Correct 1 ms 212 KB Output is correct
8 Correct 1 ms 212 KB Output is correct
9 Correct 0 ms 212 KB Output is correct
10 Correct 0 ms 212 KB Output is correct
11 Correct 0 ms 212 KB Output is correct
12 Correct 1 ms 212 KB Output is correct
13 Correct 0 ms 212 KB Output is correct
14 Correct 0 ms 212 KB Output is correct
15 Correct 0 ms 212 KB Output is correct
16 Correct 1 ms 212 KB Output is correct
17 Correct 0 ms 212 KB Output is correct
18 Correct 0 ms 212 KB Output is correct
19 Correct 0 ms 212 KB Output is correct
20 Correct 0 ms 212 KB Output is correct
21 Correct 0 ms 212 KB Output is correct
22 Correct 0 ms 212 KB Output is correct
23 Correct 0 ms 212 KB Output is correct
24 Correct 0 ms 212 KB Output is correct
25 Correct 1 ms 212 KB Output is correct
26 Correct 0 ms 212 KB Output is correct
27 Correct 0 ms 212 KB Output is correct
28 Correct 0 ms 212 KB Output is correct
29 Correct 1 ms 212 KB Output is correct
30 Correct 0 ms 212 KB Output is correct
31 Correct 0 ms 212 KB Output is correct
32 Correct 1 ms 212 KB Output is correct
33 Correct 0 ms 212 KB Output is correct
34 Correct 0 ms 212 KB Output is correct
35 Correct 0 ms 212 KB Output is correct
36 Correct 0 ms 212 KB Output is correct
37 Correct 1 ms 248 KB Output is correct
38 Correct 0 ms 212 KB Output is correct
39 Correct 1 ms 468 KB Output is correct
40 Correct 1 ms 596 KB Output is correct
41 Correct 1 ms 596 KB Output is correct
42 Correct 1 ms 596 KB Output is correct
43 Correct 1 ms 596 KB Output is correct
44 Correct 2 ms 596 KB Output is correct
45 Correct 1 ms 596 KB Output is correct
46 Correct 1 ms 596 KB Output is correct
47 Correct 1 ms 596 KB Output is correct
48 Correct 1 ms 596 KB Output is correct
49 Correct 1 ms 596 KB Output is correct
50 Correct 1 ms 596 KB Output is correct
51 Correct 1 ms 596 KB Output is correct
52 Correct 1 ms 596 KB Output is correct
53 Correct 1 ms 596 KB Output is correct
54 Correct 1 ms 596 KB Output is correct
55 Correct 1 ms 596 KB Output is correct
56 Correct 2 ms 552 KB Output is correct
57 Correct 1 ms 596 KB Output is correct
58 Correct 1 ms 596 KB Output is correct
59 Correct 2 ms 596 KB Output is correct
60 Correct 1 ms 596 KB Output is correct
61 Correct 1 ms 596 KB Output is correct

Subtask #4
30.0 / 30.0

# Verdict Execution time Memory Grader output
1 Correct 0 ms 212 KB Output is correct
2 Correct 0 ms 212 KB Output is correct
3 Correct 0 ms 212 KB Output is correct
4 Correct 0 ms 212 KB Output is correct
5 Correct 0 ms 212 KB Output is correct
6 Correct 0 ms 212 KB Output is correct
7 Correct 1 ms 212 KB Output is correct
8 Correct 1 ms 212 KB Output is correct
9 Correct 0 ms 212 KB Output is correct
10 Correct 0 ms 212 KB Output is correct
11 Correct 0 ms 212 KB Output is correct
12 Correct 1 ms 212 KB Output is correct
13 Correct 0 ms 212 KB Output is correct
14 Correct 0 ms 212 KB Output is correct
15 Correct 0 ms 212 KB Output is correct
16 Correct 1 ms 212 KB Output is correct
17 Correct 0 ms 212 KB Output is correct
18 Correct 0 ms 212 KB Output is correct
19 Correct 0 ms 212 KB Output is correct
20 Correct 0 ms 212 KB Output is correct
21 Correct 0 ms 212 KB Output is correct
22 Correct 0 ms 212 KB Output is correct
23 Correct 0 ms 212 KB Output is correct
24 Correct 0 ms 212 KB Output is correct
25 Correct 1 ms 212 KB Output is correct
26 Correct 0 ms 212 KB Output is correct
27 Correct 0 ms 212 KB Output is correct
28 Correct 0 ms 212 KB Output is correct
29 Correct 1 ms 212 KB Output is correct
30 Correct 0 ms 212 KB Output is correct
31 Correct 0 ms 212 KB Output is correct
32 Correct 1 ms 212 KB Output is correct
33 Correct 0 ms 212 KB Output is correct
34 Correct 0 ms 212 KB Output is correct
35 Correct 0 ms 212 KB Output is correct
36 Correct 0 ms 212 KB Output is correct
37 Correct 1 ms 248 KB Output is correct
38 Correct 0 ms 212 KB Output is correct
39 Correct 1 ms 468 KB Output is correct
40 Correct 1 ms 596 KB Output is correct
41 Correct 1 ms 596 KB Output is correct
42 Correct 1 ms 596 KB Output is correct
43 Correct 1 ms 596 KB Output is correct
44 Correct 2 ms 596 KB Output is correct
45 Correct 1 ms 596 KB Output is correct
46 Correct 1 ms 596 KB Output is correct
47 Correct 1 ms 596 KB Output is correct
48 Correct 1 ms 596 KB Output is correct
49 Correct 1 ms 596 KB Output is correct
50 Correct 1 ms 596 KB Output is correct
51 Correct 1 ms 596 KB Output is correct
52 Correct 1 ms 596 KB Output is correct
53 Correct 1 ms 596 KB Output is correct
54 Correct 1 ms 596 KB Output is correct
55 Correct 1 ms 596 KB Output is correct
56 Correct 2 ms 552 KB Output is correct
57 Correct 1 ms 596 KB Output is correct
58 Correct 1 ms 596 KB Output is correct
59 Correct 2 ms 596 KB Output is correct
60 Correct 1 ms 596 KB Output is correct
61 Correct 1 ms 596 KB Output is correct
62 Correct 8 ms 4180 KB Output is correct
63 Correct 32 ms 15960 KB Output is correct
64 Correct 32 ms 16076 KB Output is correct
65 Correct 31 ms 16084 KB Output is correct
66 Correct 31 ms 16084 KB Output is correct
67 Correct 31 ms 16076 KB Output is correct
68 Correct 30 ms 16080 KB Output is correct
69 Correct 30 ms 16016 KB Output is correct
70 Correct 31 ms 16084 KB Output is correct
71 Correct 28 ms 16084 KB Output is correct
72 Correct 31 ms 16060 KB Output is correct
73 Correct 31 ms 16084 KB Output is correct
74 Correct 30 ms 16080 KB Output is correct
75 Correct 32 ms 16084 KB Output is correct
76 Correct 30 ms 15956 KB Output is correct
77 Correct 27 ms 16084 KB Output is correct
78 Correct 27 ms 16084 KB Output is correct
79 Correct 28 ms 16084 KB Output is correct
80 Correct 28 ms 16088 KB Output is correct
81 Correct 28 ms 16088 KB Output is correct
82 Correct 32 ms 16084 KB Output is correct
83 Correct 31 ms 16040 KB Output is correct
84 Correct 30 ms 15956 KB Output is correct
85 Correct 31 ms 16072 KB Output is correct
86 Correct 28 ms 16092 KB Output is correct