oj.uz

Submission #203005

# Submission time Handle Problem Language Result Execution time Memory
203005 2020-02-19T01:57:03 Z Noam527 Snake Escaping (JOI18_snake_escaping) C++17
100 / 100
 1089 ms 46680 KB 
#include <bits/stdc++.h>
#define finish(x) return cout << x << endl, 0
typedef long long ll;
typedef long double ldb;
const int md = 1e9 + 7;
const ll inf = 4e18;
const int OO = 1;
const int OOO = 1;
using namespace std;

/*
Solution idea:
first compute through dp, the values F[mask] = sum(a[S]) for S contained in mask, G[mask] = sum(a[S]) for S contained in mask.
this can be done in O(L * 2^L) (through a nontrivial dp state).
if the string would contain only (0, ?), then the answer is F[mask], mask is binary representation of ?'s.
if the string would contain only (1, ?), same thing but G[mask].

so we use both arrays to cut the time complexity for the general case.
consider the character with minimum frequency in the query string, out of (0, 1, ?). The minimum frequency is between 0 and floor(L / 3).

if the minimum is ?: solve the query naively - try to replace each ? with either 0 or 1
if the minimum is 0: we get rid of 0's, and turn them to 1's and ?'s:
	for the query string x0y (x and y can be any part of the query string), its answer is x?y - x1y.
	once we end up with no 0's (on base cases of this recursive procedure), we can use array G[] as described above.
if the minimum is 1: we turn 1's to 0's and ?'s, in the same manner as above, then use F[].

in all cases we produce 2 possibilities for each character from the lowest frequency, and solve each possibility in O(1), so the complexity is:
time: O(2^L * L + Q * 2^floor(L / 3)).
memory: although the dp has O(2^L * L) states, we eventually only care about the last layer for all queries, so memory is O(2^L).

the problem was very interesting so I decided to write this little description.
the code may not be clear though (and I did not explain the dp); you are welcome to send questions to my codeforces account, Noam527.
*/

const int N = 1 << 20;

int L, L2, q, a[N];
int Z[2][N], O[2][N];

int MP[256] = {};

// cur = binary rep. by 1's
int work_question(int left, int cur) {
	if (!left) return a[cur];
	int x = left & -left;
	return work_question(left ^ x, cur) + work_question(left ^ x, cur ^ x);
}

// gets to 0's.
// cur = binary rep. of ?'s
int work_0(int left, int cur) {
	if (!left) return Z[L & 1][cur];
	int x = left & -left;
	return work_0(left ^ x, cur ^ x) - work_0(left ^ x, cur);
}

// gets to 1's.
// cur = binary rep. of ?'s
int work_1(int left, int cur) {
	if (!left) return O[L & 1][L2 - 1 - cur];
	int x = left & -left;
	return work_1(left ^ x, cur ^ x) - work_1(left ^ x, cur);
}

int main() {
	ios::sync_with_stdio(0), cin.tie(0);
	MP['0'] = 0, MP['1'] = 1, MP['?'] = 2;

	cin >> L >> q;
	L2 = (1 << L);
	for (int i = 0; i < L2; i++) {
		char c;
		cin >> c;
		a[i] = c - '0';
		Z[0][i] = O[0][i] = a[i];
	}

	for (int i = 1; i <= L; i++) {
		for (int j = 0; j < L2; j++) {
			int x = j ^ (1 << (i - 1));
			if (j & (1 << (i - 1))) {
				Z[i & 1][j] = Z[i & 1 ^ 1][j] + Z[i & 1 ^ 1][x];
				O[i & 1][j] = O[i & 1 ^ 1][j];
			}
			else {
				Z[i & 1][j] = Z[i & 1 ^ 1][j];
				O[i & 1][j] = O[i & 1 ^ 1][j] + O[i & 1 ^ 1][x];
			}
		}
	}

	while (q--) {
		string s;
		cin >> s;
		int cnt[3] = {}, rep[3] = {};
		for (int i = 0; i < L; i++) {
			cnt[MP[s[i]]]++;
			rep[MP[s[i]]] |= 1 << (L - 1 - i);
		}
		int at = 0;
		if (cnt[1] < cnt[at]) at = 1;
		if (cnt[2] < cnt[at]) at = 2;

		if (at == 2) {
			cout << work_question(rep[at], rep[1]) << '\n';
		}
		else {
			if (at == 0)
				cout << work_1(rep[at], rep[2]) << '\n';
			else
				cout << work_0(rep[at], rep[2]) << '\n';
		}
	}
}

Compilation message

snake_escaping.cpp: In function 'int main()':
snake_escaping.cpp:82:23: warning: suggest parentheses around arithmetic in operand of '^' [-Wparentheses]
     Z[i & 1][j] = Z[i & 1 ^ 1][j] + Z[i & 1 ^ 1][x];
                     ~~^~~
snake_escaping.cpp:82:41: warning: suggest parentheses around arithmetic in operand of '^' [-Wparentheses]
     Z[i & 1][j] = Z[i & 1 ^ 1][j] + Z[i & 1 ^ 1][x];
                                       ~~^~~
snake_escaping.cpp:83:23: warning: suggest parentheses around arithmetic in operand of '^' [-Wparentheses]
     O[i & 1][j] = O[i & 1 ^ 1][j];
                     ~~^~~
snake_escaping.cpp:86:23: warning: suggest parentheses around arithmetic in operand of '^' [-Wparentheses]
     Z[i & 1][j] = Z[i & 1 ^ 1][j];
                     ~~^~~
snake_escaping.cpp:87:23: warning: suggest parentheses around arithmetic in operand of '^' [-Wparentheses]
     O[i & 1][j] = O[i & 1 ^ 1][j] + O[i & 1 ^ 1][x];
                     ~~^~~
snake_escaping.cpp:87:41: warning: suggest parentheses around arithmetic in operand of '^' [-Wparentheses]
     O[i & 1][j] = O[i & 1 ^ 1][j] + O[i & 1 ^ 1][x];
                                       ~~^~~
snake_escaping.cpp:97:15: warning: array subscript has type 'char' [-Wchar-subscripts]
    cnt[MP[s[i]]]++;
               ^
snake_escaping.cpp:98:15: warning: array subscript has type 'char' [-Wchar-subscripts]
    rep[MP[s[i]]] |= 1 << (L - 1 - i);
               ^

Subtask #1
5.0 / 5.0

# Verdict Execution time Memory Grader output
1 Correct 5 ms 376 KB Output is correct
2 Correct 5 ms 376 KB Output is correct
3 Correct 5 ms 376 KB Output is correct
4 Correct 5 ms 376 KB Output is correct
5 Correct 5 ms 376 KB Output is correct
6 Correct 5 ms 376 KB Output is correct
7 Correct 5 ms 376 KB Output is correct
8 Correct 5 ms 376 KB Output is correct
9 Correct 5 ms 376 KB Output is correct
10 Correct 5 ms 376 KB Output is correct

Subtask #2
7.0 / 7.0

# Verdict Execution time Memory Grader output
1 Correct 5 ms 376 KB Output is correct
2 Correct 5 ms 376 KB Output is correct
3 Correct 5 ms 376 KB Output is correct
4 Correct 5 ms 376 KB Output is correct
5 Correct 5 ms 376 KB Output is correct
6 Correct 5 ms 376 KB Output is correct
7 Correct 5 ms 376 KB Output is correct
8 Correct 5 ms 376 KB Output is correct
9 Correct 5 ms 376 KB Output is correct
10 Correct 5 ms 376 KB Output is correct
11 Correct 230 ms 8952 KB Output is correct
12 Correct 225 ms 8700 KB Output is correct
13 Correct 250 ms 7928 KB Output is correct
14 Correct 252 ms 7928 KB Output is correct
15 Correct 246 ms 8956 KB Output is correct
16 Correct 262 ms 8440 KB Output is correct
17 Correct 259 ms 8060 KB Output is correct
18 Correct 221 ms 9976 KB Output is correct
19 Correct 217 ms 7164 KB Output is correct
20 Correct 233 ms 8696 KB Output is correct

Subtask #3
10.0 / 10.0

# Verdict Execution time Memory Grader output
1 Correct 5 ms 376 KB Output is correct
2 Correct 5 ms 376 KB Output is correct
3 Correct 5 ms 376 KB Output is correct
4 Correct 5 ms 376 KB Output is correct
5 Correct 5 ms 376 KB Output is correct
6 Correct 5 ms 376 KB Output is correct
7 Correct 5 ms 376 KB Output is correct
8 Correct 5 ms 376 KB Output is correct
9 Correct 5 ms 376 KB Output is correct
10 Correct 5 ms 376 KB Output is correct
11 Correct 230 ms 8952 KB Output is correct
12 Correct 225 ms 8700 KB Output is correct
13 Correct 250 ms 7928 KB Output is correct
14 Correct 252 ms 7928 KB Output is correct
15 Correct 246 ms 8956 KB Output is correct
16 Correct 262 ms 8440 KB Output is correct
17 Correct 259 ms 8060 KB Output is correct
18 Correct 221 ms 9976 KB Output is correct
19 Correct 217 ms 7164 KB Output is correct
20 Correct 233 ms 8696 KB Output is correct
21 Correct 252 ms 9152 KB Output is correct
22 Correct 250 ms 9208 KB Output is correct
23 Correct 280 ms 8184 KB Output is correct
24 Correct 318 ms 8184 KB Output is correct
25 Correct 272 ms 9976 KB Output is correct
26 Correct 309 ms 8568 KB Output is correct
27 Correct 299 ms 8784 KB Output is correct
28 Correct 248 ms 11000 KB Output is correct
29 Correct 233 ms 7032 KB Output is correct
30 Correct 255 ms 9208 KB Output is correct
31 Correct 280 ms 9080 KB Output is correct
32 Correct 332 ms 9080 KB Output is correct
33 Correct 267 ms 7928 KB Output is correct
34 Correct 303 ms 8068 KB Output is correct
35 Correct 305 ms 8532 KB Output is correct
36 Correct 245 ms 7140 KB Output is correct
37 Correct 244 ms 8952 KB Output is correct
38 Correct 261 ms 7032 KB Output is correct
39 Correct 277 ms 8184 KB Output is correct
40 Correct 303 ms 7928 KB Output is correct

Subtask #4
53.0 / 53.0

# Verdict Execution time Memory Grader output
1 Correct 5 ms 376 KB Output is correct
2 Correct 5 ms 376 KB Output is correct
3 Correct 5 ms 376 KB Output is correct
4 Correct 5 ms 376 KB Output is correct
5 Correct 5 ms 376 KB Output is correct
6 Correct 5 ms 376 KB Output is correct
7 Correct 5 ms 376 KB Output is correct
8 Correct 5 ms 376 KB Output is correct
9 Correct 5 ms 376 KB Output is correct
10 Correct 5 ms 376 KB Output is correct
11 Correct 91 ms 23288 KB Output is correct
12 Correct 91 ms 23160 KB Output is correct
13 Correct 103 ms 23116 KB Output is correct
14 Correct 103 ms 23032 KB Output is correct
15 Correct 89 ms 23160 KB Output is correct
16 Correct 113 ms 23160 KB Output is correct
17 Correct 110 ms 23160 KB Output is correct
18 Correct 84 ms 23288 KB Output is correct
19 Correct 90 ms 23032 KB Output is correct
20 Correct 91 ms 23176 KB Output is correct
21 Correct 96 ms 23164 KB Output is correct
22 Correct 109 ms 23160 KB Output is correct
23 Correct 89 ms 23032 KB Output is correct
24 Correct 121 ms 23032 KB Output is correct
25 Correct 106 ms 23160 KB Output is correct
26 Correct 83 ms 23032 KB Output is correct
27 Correct 89 ms 23160 KB Output is correct
28 Correct 83 ms 23032 KB Output is correct
29 Correct 103 ms 23080 KB Output is correct
30 Correct 104 ms 23032 KB Output is correct
31 Correct 95 ms 23032 KB Output is correct
32 Correct 120 ms 23164 KB Output is correct
33 Correct 116 ms 23160 KB Output is correct
34 Correct 86 ms 23032 KB Output is correct
35 Correct 104 ms 23160 KB Output is correct
36 Correct 105 ms 23064 KB Output is correct
37 Correct 106 ms 23288 KB Output is correct
38 Correct 107 ms 23160 KB Output is correct
39 Correct 101 ms 23160 KB Output is correct
40 Correct 103 ms 23160 KB Output is correct

Subtask #5
25.0 / 25.0

# Verdict Execution time Memory Grader output
1 Correct 5 ms 376 KB Output is correct
2 Correct 5 ms 376 KB Output is correct
3 Correct 5 ms 376 KB Output is correct
4 Correct 5 ms 376 KB Output is correct
5 Correct 5 ms 376 KB Output is correct
6 Correct 5 ms 376 KB Output is correct
7 Correct 5 ms 376 KB Output is correct
8 Correct 5 ms 376 KB Output is correct
9 Correct 5 ms 376 KB Output is correct
10 Correct 5 ms 376 KB Output is correct
11 Correct 230 ms 8952 KB Output is correct
12 Correct 225 ms 8700 KB Output is correct
13 Correct 250 ms 7928 KB Output is correct
14 Correct 252 ms 7928 KB Output is correct
15 Correct 246 ms 8956 KB Output is correct
16 Correct 262 ms 8440 KB Output is correct
17 Correct 259 ms 8060 KB Output is correct
18 Correct 221 ms 9976 KB Output is correct
19 Correct 217 ms 7164 KB Output is correct
20 Correct 233 ms 8696 KB Output is correct
21 Correct 252 ms 9152 KB Output is correct
22 Correct 250 ms 9208 KB Output is correct
23 Correct 280 ms 8184 KB Output is correct
24 Correct 318 ms 8184 KB Output is correct
25 Correct 272 ms 9976 KB Output is correct
26 Correct 309 ms 8568 KB Output is correct
27 Correct 299 ms 8784 KB Output is correct
28 Correct 248 ms 11000 KB Output is correct
29 Correct 233 ms 7032 KB Output is correct
30 Correct 255 ms 9208 KB Output is correct
31 Correct 280 ms 9080 KB Output is correct
32 Correct 332 ms 9080 KB Output is correct
33 Correct 267 ms 7928 KB Output is correct
34 Correct 303 ms 8068 KB Output is correct
35 Correct 305 ms 8532 KB Output is correct
36 Correct 245 ms 7140 KB Output is correct
37 Correct 244 ms 8952 KB Output is correct
38 Correct 261 ms 7032 KB Output is correct
39 Correct 277 ms 8184 KB Output is correct
40 Correct 303 ms 7928 KB Output is correct
41 Correct 91 ms 23288 KB Output is correct
42 Correct 91 ms 23160 KB Output is correct
43 Correct 103 ms 23116 KB Output is correct
44 Correct 103 ms 23032 KB Output is correct
45 Correct 89 ms 23160 KB Output is correct
46 Correct 113 ms 23160 KB Output is correct
47 Correct 110 ms 23160 KB Output is correct
48 Correct 84 ms 23288 KB Output is correct
49 Correct 90 ms 23032 KB Output is correct
50 Correct 91 ms 23176 KB Output is correct
51 Correct 96 ms 23164 KB Output is correct
52 Correct 109 ms 23160 KB Output is correct
53 Correct 89 ms 23032 KB Output is correct
54 Correct 121 ms 23032 KB Output is correct
55 Correct 106 ms 23160 KB Output is correct
56 Correct 83 ms 23032 KB Output is correct
57 Correct 89 ms 23160 KB Output is correct
58 Correct 83 ms 23032 KB Output is correct
59 Correct 103 ms 23080 KB Output is correct
60 Correct 104 ms 23032 KB Output is correct
61 Correct 95 ms 23032 KB Output is correct
62 Correct 120 ms 23164 KB Output is correct
63 Correct 116 ms 23160 KB Output is correct
64 Correct 86 ms 23032 KB Output is correct
65 Correct 104 ms 23160 KB Output is correct
66 Correct 105 ms 23064 KB Output is correct
67 Correct 106 ms 23288 KB Output is correct
68 Correct 107 ms 23160 KB Output is correct
69 Correct 101 ms 23160 KB Output is correct
70 Correct 103 ms 23160 KB Output is correct
71 Correct 447 ms 26100 KB Output is correct
72 Correct 463 ms 26360 KB Output is correct
73 Correct 623 ms 24696 KB Output is correct
74 Correct 803 ms 25084 KB Output is correct
75 Correct 497 ms 27156 KB Output is correct
76 Correct 954 ms 27896 KB Output is correct
77 Correct 897 ms 25272 KB Output is correct
78 Correct 414 ms 28920 KB Output is correct
79 Correct 421 ms 22904 KB Output is correct
80 Correct 478 ms 26360 KB Output is correct
81 Correct 643 ms 26104 KB Output is correct
82 Correct 809 ms 24956 KB Output is correct
83 Correct 497 ms 24188 KB Output is correct
84 Correct 1089 ms 24936 KB Output is correct
85 Correct 953 ms 25152 KB Output is correct
86 Correct 394 ms 22880 KB Output is correct
87 Correct 431 ms 25976 KB Output is correct
88 Correct 450 ms 23176 KB Output is correct
89 Correct 623 ms 24696 KB Output is correct
90 Correct 697 ms 24952 KB Output is correct
91 Correct 489 ms 24056 KB Output is correct
92 Correct 1070 ms 25232 KB Output is correct
93 Correct 936 ms 25080 KB Output is correct
94 Correct 397 ms 23032 KB Output is correct
95 Correct 799 ms 25208 KB Output is correct
96 Correct 788 ms 25208 KB Output is correct
97 Correct 797 ms 25164 KB Output is correct
98 Correct 819 ms 40568 KB Output is correct
99 Correct 804 ms 46600 KB Output is correct
100 Correct 824 ms 46680 KB Output is correct