oj.uz

Submission #624110

# Submission time Handle Problem Language Result Execution time Memory
624110 2022-08-07T09:16:02 Z jophyyjh Packing Biscuits (IOI20_biscuits) C++14
77 / 100
 1000 ms 1332 KB 
/**
 * Binary. Hmm, interesting. I started with [S2], which I solved with recursion in
 * O(kq). The basic idea is to treat two 2^i biscuits in the bag as one 2^(i+1).
 * 
 * A few greedy observations:
 * (1)  Given x, can we determine if it's possible? Well, we can simply iterate i
 *      from large to small. If the additional tastiness x we still need for a
 *      certain bag satisfies x >= 2^i, put a biscuit 2^i in the bag if there's one.
 * (2)  Biscuits of the same type can be distributed as evenly as possible, i.e.
 *      there is an optimal partition s.t. for every i
 *              |#2^i_biscuits_in_bag_1 - #2^i_biscuits_in_big_2| <= 1.
 * (3)  Unlike (1), (2), we consider from small to large. If x is odd, each bag needs
 *      at least 1 2^0 biscuit. After that, every 2 2^O biscuits can be grouped to
 *      form 1 2^1 biscuit, so we can add this num to the original num of 2^1
 *      biscuits. Well, let's write this recursively:
 *           def find():
 *               total = 0
 *               if #2^0 >= x:
 *                   add floor((#2^0 - x) / 2) to #2^1 and recurse downdwards
 *               add floor(#2^0 / 2) to 2^1 and recurse downwards
 *               ...
 * 
 * Can we AC this problem with (3)? I guess that the num of configurations given to
 * find() isn't too many, so we can optimize it with memoization. Indeed, we can
 * prove that in each level of find(), the num. of unique calls the next level of
 * find() is bounded by x. This is because x >= x/2 + x/4 + x/8 + x/16 + .... [S1-3]
 * are solved.
 * 
 * In impl1, the function search(p, first) doesn't decrease as first increases.
 * Furthermore, if search(p, w) and search(p, z) are both called, |w-z| <= x. (For
 * each p, the val first only varies in a range with len <= x.) Next, one can prove
 * that within this range search() can take at most 3 distinct values (by induction).
 * We can binary search for the largest / smallest value for first that attains a
 * certain search(p, ..).
 * 
 * Time Complexity: O(kq * log(x) * log(k))
 * Implementation 1.5       (Full solution (?!), greedy + binary search + maths)
*/

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

typedef long long	ll;

ll wrap(int p, ll first);       // wraper function for search()
ll search(int p, ll first);


ll x;
std::vector<ll> a;

std::vector<std::map<ll, ll>> cache1;
std::vector<std::map<ll, ll, std::greater<ll>>> cache2;

ll wrap(int p, ll first) {
    auto v1 = cache1[p].lower_bound(first);
    auto v2 = cache2[p].lower_bound(first);
    if (v1 != cache1[p].end() && v2 != cache2[p].end()) {
        if (v1->second == v2->second)
            return v1->second;
    }
    ll val = search(p, first);
    ll high = first, low = first;
    for (ll step = x / 2 + 1; step >= 1; step /= 2) {
        while (search(p, high + step) == val)
            high += step;
        while (low - step >= 0 && search(p, low - step) == val)
            low -= step;
    }
    cache1[p][high] = cache2[p][low] = val;
    return val;
}

ll search(int p, ll first) {
    int k = a.size();
    if (p == k - 1)
        return first / x + 1;
    ll total = wrap(p + 1, a[p + 1] + first / 2);
    if (first >= x)
        total += wrap(p + 1, a[p + 1] + (first - x) / 2);
    return total;
}

ll count_tastiness(ll _x, std::vector<ll> _a) {
    x = _x, a = _a;
    int k = _a.size();
    cache1.assign(k, std::map<ll, ll>());
    cache2.assign(k, std::map<ll, ll, std::greater<ll>>());
    return wrap(0, a[0]);
}

Subtask #1
9.0 / 9.0

# Verdict Execution time Memory Grader output
1 Correct 0 ms 212 KB Output is correct
2 Correct 1 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 0 ms 212 KB Output is correct
11 Correct 1 ms 212 KB Output is correct
12 Correct 1 ms 212 KB Output is correct
13 Correct 1 ms 212 KB Output is correct
14 Correct 1 ms 212 KB Output is correct

Subtask #2
12.0 / 12.0

# Verdict Execution time Memory Grader output
1 Correct 0 ms 212 KB Output is correct
2 Correct 1 ms 212 KB Output is correct
3 Correct 1 ms 212 KB Output is correct
4 Correct 1 ms 212 KB Output is correct
5 Correct 1 ms 212 KB Output is correct
6 Correct 1 ms 340 KB Output is correct
7 Correct 1 ms 212 KB Output is correct
8 Correct 1 ms 340 KB Output is correct
9 Correct 1 ms 212 KB Output is correct

Subtask #3
21.0 / 21.0

# 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 1 ms 212 KB Output is correct
5 Correct 0 ms 212 KB Output is correct
6 Correct 1 ms 340 KB Output is correct
7 Correct 0 ms 212 KB Output is correct
8 Correct 3 ms 340 KB Output is correct
9 Correct 0 ms 212 KB Output is correct
10 Correct 3 ms 340 KB Output is correct
11 Correct 8 ms 384 KB Output is correct
12 Correct 13 ms 340 KB Output is correct
13 Correct 14 ms 392 KB Output is correct

Subtask #4
35.0 / 35.0

# Verdict Execution time Memory Grader output
1 Correct 1 ms 340 KB Output is correct
2 Correct 663 ms 872 KB Output is correct
3 Correct 636 ms 1324 KB Output is correct
4 Correct 611 ms 1236 KB Output is correct
5 Correct 574 ms 1236 KB Output is correct
6 Correct 824 ms 1200 KB Output is correct
7 Correct 862 ms 1212 KB Output is correct
8 Correct 804 ms 1200 KB Output is correct
9 Correct 789 ms 1216 KB Output is correct
10 Correct 555 ms 1324 KB Output is correct

Subtask #5
0 / 23.0

# Verdict Execution time Memory Grader output
1 Correct 0 ms 212 KB Output is correct
2 Correct 1 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 0 ms 212 KB Output is correct
11 Correct 1 ms 212 KB Output is correct
12 Correct 1 ms 212 KB Output is correct
13 Correct 1 ms 212 KB Output is correct
14 Correct 1 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 1 ms 212 KB Output is correct
18 Correct 1 ms 212 KB Output is correct
19 Correct 1 ms 212 KB Output is correct
20 Correct 1 ms 340 KB Output is correct
21 Correct 1 ms 212 KB Output is correct
22 Correct 1 ms 340 KB Output is correct
23 Correct 1 ms 212 KB Output is correct
24 Correct 0 ms 212 KB Output is correct
25 Correct 0 ms 212 KB Output is correct
26 Correct 0 ms 212 KB Output is correct
27 Correct 1 ms 212 KB Output is correct
28 Correct 0 ms 212 KB Output is correct
29 Correct 1 ms 340 KB Output is correct
30 Correct 0 ms 212 KB Output is correct
31 Correct 3 ms 340 KB Output is correct
32 Correct 0 ms 212 KB Output is correct
33 Correct 3 ms 340 KB Output is correct
34 Correct 8 ms 384 KB Output is correct
35 Correct 13 ms 340 KB Output is correct
36 Correct 14 ms 392 KB Output is correct
37 Correct 1 ms 340 KB Output is correct
38 Correct 663 ms 872 KB Output is correct
39 Correct 636 ms 1324 KB Output is correct
40 Correct 611 ms 1236 KB Output is correct
41 Correct 574 ms 1236 KB Output is correct
42 Correct 824 ms 1200 KB Output is correct
43 Correct 862 ms 1212 KB Output is correct
44 Correct 804 ms 1200 KB Output is correct
45 Correct 789 ms 1216 KB Output is correct
46 Correct 555 ms 1324 KB Output is correct
47 Correct 2 ms 340 KB Output is correct
48 Correct 58 ms 1332 KB Output is correct
49 Correct 2 ms 312 KB Output is correct
50 Correct 46 ms 1176 KB Output is correct
51 Correct 99 ms 1328 KB Output is correct
52 Correct 5 ms 340 KB Output is correct
53 Correct 77 ms 1168 KB Output is correct
54 Execution timed out 1045 ms 1236 KB Time limit exceeded
55 Halted 0 ms 0 KB -