oj.uz

Submission #815399

# Submission time Handle Problem Language Result Execution time Memory
815399 2023-08-08T14:46:34 Z jophyyjh Ancient Books (IOI17_books) C++14
50 / 100
 100 ms 17228 KB 
/**
 * Cute problem~ I started by experimenting different examples. Nope. I noticed nothing,
 * and was stuck for a while. I initially guessed that maybe Aryan never moves a book to
 * an intermediate table, rather than directly to its destination, but I quickly dismissed
 * the thought because of (2,3,0,1). In fact, the min num of steps to solve (2,3,0,1) is 8,
 * not 10. Then, I wonder, can I prove that 8 is the minimum? Yep! That was how I found the
 * way. I chose to consider some monovariant/invariant. A common one for sorting is
 * sum{|i - p_i|}, i.e. the sum of distances between a book and its destination. This sum
 * is exactly 8 for (2,3,0,1), so 8 is indeed the min.
 * 
 * sum{|i-p_i|} is a minimum because every step of Aryan can only bring a book at most 1
 * table closer to its destination. The min is achieved iff Aryan takes a book towards its
 * right place at every step, regardless of whether a book has been put down in an
 * intermediate table. What's a convenient way to achieve sum{|i-p_i|}? Note that a
 * permutation can be seen as the collection of some cycles. For each cycle, we can just
 * pick the books in it, in order. Let's call such traversal the "min-cost traversal" of
 * the cycle. For example, if the perm is just one cycle of size n, then the sum is indeed
 * sum{|i-p_i|}, achieved with just one complete "min-cost traversal".
 * 
 * For convenience, let's call sum{|i-p_i|} the "principal cost". Note that the principal
 * cost isn't always the min num of steps. Let c_1, c_2, ..., c_k be all the non-trivial
 * cycles (len>1, containing elements that have to be moved) representing our perm. Let the
 * "repr range" of c_i be [smallest_j_in_c_i, greatest_j_in_c_i]. We've already
 * established that a single cycle can be handled with one min-cost traversal. If the repr
 * ranges of c_i and c_j intersect, it can be proved that their min-cost traversal can be
 * "combined" (with no additional steps needed), which means the k cycles' repr ranges can
 * now be seen as a few disjoint repr ranges. The gaps between these disjoint ranges are
 * definitely not covered in the principal cost, but Aryan must walk through each gap at
 * least twice (since he has to return to s too).
 * 
 * Finally, if p[i] == i, i.e. i isn't in any non-trivial cycle, we have to start from s
 * and walk to a j with p[j] != j. We can prove that this j can be/is the closest j to s
 * with p[j] != j. These 3 parts of cost make up the final cost.
 * 
 * Time Complexity: O(n * log(n))           (maths, permutation)
 * Implementation 1                 (Full solution)
*/

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

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


ll minimum_walk(vec perm, int s) {
    int n = perm.size();

    std::vector<bool> visited(n, false);
    vec ls, rs;
    int cs = 0, closest = n;
    ll cost = 0;
    for (int src = 0; src < n; src++) {
        if (visited[src] || src == perm[src])
            continue;
        int l = n, r = -1;
        for (int pt = src;; pt = perm[pt]) {
            visited[pt] = true;
            closest = std::min(closest, std::abs(s - pt));
            cost += std::abs(pt - perm[pt]), l = std::min(l, pt), r = std::max(r, pt);
            if (perm[pt] == src)
                break;
        }
        ls.push_back(l);
        rs.push_back(r);
        cs++;
    }
    if (cs == 0)
        return 0;
    cost += 2 * closest;
    
    std::sort(ls.begin(), ls.end());
    std::sort(rs.begin(), rs.end());
    for (int t = ls.front(), i = 0, j = 0, level = 0; t < rs.back(); t++) {
        while (i < cs && ls[i] <= t)
            level++, i++;
        while (j < cs && rs[j] <= t)
            level--, j++;
        if (level == 0)
            cost += 2;
    }
    return cost;
}

Subtask #1
12.0 / 12.0

# 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 0 ms 212 KB Output is correct
8 Correct 0 ms 212 KB Output is correct
9 Correct 0 ms 212 KB Output is correct
10 Correct 1 ms 300 KB Output is correct
11 Correct 1 ms 212 KB Output is correct
12 Correct 0 ms 212 KB Output is correct
13 Correct 0 ms 300 KB Output is correct
14 Correct 0 ms 212 KB Output is correct
15 Correct 0 ms 212 KB Output is correct
16 Correct 0 ms 340 KB Output is correct
17 Correct 0 ms 212 KB Output is correct
18 Correct 0 ms 212 KB Output is correct

Subtask #2
10.0 / 10.0

# 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 0 ms 212 KB Output is correct
8 Correct 0 ms 212 KB Output is correct
9 Correct 0 ms 212 KB Output is correct
10 Correct 1 ms 300 KB Output is correct
11 Correct 1 ms 212 KB Output is correct
12 Correct 0 ms 212 KB Output is correct
13 Correct 0 ms 300 KB Output is correct
14 Correct 0 ms 212 KB Output is correct
15 Correct 0 ms 212 KB Output is correct
16 Correct 0 ms 340 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 1 ms 212 KB Output is correct
22 Correct 0 ms 300 KB Output is correct
23 Correct 1 ms 296 KB Output is correct
24 Correct 1 ms 212 KB Output is correct
25 Correct 1 ms 304 KB Output is correct
26 Correct 1 ms 212 KB Output is correct
27 Correct 0 ms 300 KB Output is correct
28 Correct 0 ms 212 KB Output is correct
29 Correct 0 ms 212 KB Output is correct

Subtask #3
28.0 / 28.0

# 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 0 ms 212 KB Output is correct
8 Correct 0 ms 212 KB Output is correct
9 Correct 0 ms 212 KB Output is correct
10 Correct 1 ms 300 KB Output is correct
11 Correct 1 ms 212 KB Output is correct
12 Correct 0 ms 212 KB Output is correct
13 Correct 0 ms 300 KB Output is correct
14 Correct 0 ms 212 KB Output is correct
15 Correct 0 ms 212 KB Output is correct
16 Correct 0 ms 340 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 1 ms 212 KB Output is correct
22 Correct 0 ms 300 KB Output is correct
23 Correct 1 ms 296 KB Output is correct
24 Correct 1 ms 212 KB Output is correct
25 Correct 1 ms 304 KB Output is correct
26 Correct 1 ms 212 KB Output is correct
27 Correct 0 ms 300 KB Output is correct
28 Correct 0 ms 212 KB Output is correct
29 Correct 0 ms 212 KB Output is correct
30 Correct 91 ms 14924 KB Output is correct
31 Correct 92 ms 14972 KB Output is correct
32 Correct 74 ms 14952 KB Output is correct
33 Correct 91 ms 16704 KB Output is correct
34 Correct 90 ms 16708 KB Output is correct
35 Correct 91 ms 16840 KB Output is correct
36 Correct 91 ms 16716 KB Output is correct
37 Correct 81 ms 15208 KB Output is correct
38 Correct 86 ms 15028 KB Output is correct
39 Correct 78 ms 14944 KB Output is correct
40 Correct 91 ms 14940 KB Output is correct
41 Correct 87 ms 14936 KB Output is correct
42 Correct 86 ms 14924 KB Output is correct
43 Correct 100 ms 17228 KB Output is correct

Subtask #4
0 / 20.0

# Verdict Execution time Memory Grader output
1 Incorrect 0 ms 212 KB 3rd lines differ - on the 1st token, expected: '3304', found: '2744'
2 Halted 0 ms 0 KB -

Subtask #5
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 0 ms 212 KB Output is correct
8 Correct 0 ms 212 KB Output is correct
9 Correct 0 ms 212 KB Output is correct
10 Correct 1 ms 300 KB Output is correct
11 Correct 1 ms 212 KB Output is correct
12 Correct 0 ms 212 KB Output is correct
13 Correct 0 ms 300 KB Output is correct
14 Correct 0 ms 212 KB Output is correct
15 Correct 0 ms 212 KB Output is correct
16 Correct 0 ms 340 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 1 ms 212 KB Output is correct
22 Correct 0 ms 300 KB Output is correct
23 Correct 1 ms 296 KB Output is correct
24 Correct 1 ms 212 KB Output is correct
25 Correct 1 ms 304 KB Output is correct
26 Correct 1 ms 212 KB Output is correct
27 Correct 0 ms 300 KB Output is correct
28 Correct 0 ms 212 KB Output is correct
29 Correct 0 ms 212 KB Output is correct
30 Correct 91 ms 14924 KB Output is correct
31 Correct 92 ms 14972 KB Output is correct
32 Correct 74 ms 14952 KB Output is correct
33 Correct 91 ms 16704 KB Output is correct
34 Correct 90 ms 16708 KB Output is correct
35 Correct 91 ms 16840 KB Output is correct
36 Correct 91 ms 16716 KB Output is correct
37 Correct 81 ms 15208 KB Output is correct
38 Correct 86 ms 15028 KB Output is correct
39 Correct 78 ms 14944 KB Output is correct
40 Correct 91 ms 14940 KB Output is correct
41 Correct 87 ms 14936 KB Output is correct
42 Correct 86 ms 14924 KB Output is correct
43 Correct 100 ms 17228 KB Output is correct
44 Incorrect 0 ms 212 KB 3rd lines differ - on the 1st token, expected: '3304', found: '2744'
45 Halted 0 ms 0 KB -