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QOJ

IDProblemSubmitterResultTimeMemoryLanguageFile sizeSubmit timeJudge time
#87754#4808. Great PartycodiconWA 3ms3596kbC++206.7kb2023-03-14 09:14:312023-03-14 09:14:35

Judging History

你现在查看的是最新测评结果

  • [2023-08-10 23:21:45]
  • System Update: QOJ starts to keep a history of the judgings of all the submissions.
  • [2023-03-14 09:14:35]
  • 评测
  • 测评结果:WA
  • 用时:3ms
  • 内存:3596kb
  • [2023-03-14 09:14:31]
  • 提交

answer

//
// Created by codicon on 3/13/2023 at 11:21 AM.
//

// O(N sqrt(Q))

/*
 * First, for a given set of piles, S, who wins?
 *
 * Observation:
 *      1 pile:
 *           First player wins
 *      2 piles:
 *           Let's say that player A performs an operation and now there is only 1 pile left.
 *           Then, by our solution for 1 pile, player B wins.
 *
 *           Denote the piles where we remove 1 stone from each pile as S'.
 *           Thus, the winner of the game is the winner of nim on S'.
 *
 *           This is because either:
 *               The game of nim is played to completion (all piles have 1 stone now).
 *               Then, the loser removes the final stone from some pile. Now there is 1 pile left.
 *               or
 *               The loser completely removes a pile before the game of nim is completed.
 *               In this case, there is also 1 pile left.
 *      3 piles:
 *           WLOG, say the piles have stones: a < b < c.
 *           Then, player 1 can simply remove stones from c until there are b-a left.
 *           Then, they merge those b-a stones into a. Now the piles are (b, b).
 *
 *           Thus, player 1 wins since the game has reduced to the 2 pile case and
 *           player 1 wins nim on (b-1, b-1).
 *      4 piles:
 *           Again, if player A removes a pile, the game reduces to the 3 pile case and
 *           the player A loses.
 *
 *           Thus, winner of the game on S' is the winner.
 *
 * Proof of Full Strategy via Induction:
 *      It seems like player 1 wins when there are an odd number of piles and
 *      if there are an even number of piles, the winner is the winner of nim on S'.
 *
 *      Base Case:
 *          For n = 1 the statement is true.
 *
 *      Inductive Hypothesis:
 *          Let n >= 1 such that the statement is true for n.
 *
 *      Inductive Step:
 *          Let's prove the statement for n+1
 *
 *          If n is odd:
 *              No player wants to remove any pile and thus the winner is the winner of nim on S'.
 *
 *          If n is even:
 *              Let's prove that it is always possible for the first player to
 *              remove stones from some pile and then merge the remaining stones into
 *              some other pile such that they would win nim on S'.
 *
 *              For player 1 to win nim on S', the xor-sum after they move must be 0.
 *
 *              Consider how the operations affect the pile sizes and thus xor-sum on S':
 *              We have to pick some pile X (has X+1 stones in S). Then, remove X and add
 *              up to X stones to some pile A (has A+1 stones in S). We need the xor-sum = 0.
 *              Thus, we must change A to A xor (S' xor X). Here S' xor X denotes xor-sum(S') xor X.
 *
 *              This means we can do so iff A <= (A xor (S' xor X)) <= A + X.
 *
 *              Realize that xoring A by (S' xor X) can increase A by at most (S' xor X).
 *              Accordingly, let's start by finding a pile X such that X >= S' xor X.
 *              Thus, we will at least be able to ensure:
 *              A xor (S' xor X) <= A + X.
 *
 *                  Let sum(i), be the sum of the ith bits of all piles in S'.
 *
 *                  Then, iterate i from the most significant bit to the least.
 *                  If sum(i) is even, continue.
 *                  Otherwise, sum(i) is odd, so find any pile X such that
 *                  the ith bit is on.
 *
 *                  This means X >= S' xor X since they have the same bits in positions > i
 *                  and X has the ith bit on whilst S' xor X does not.
 *
 *                  If we reach i = 1 and sum(i) is even, then that just means
 *                  X = S' xor X for any X, so X >= S' xor X.
 *
 *              Now, to satisfy the entire condition: A <= A xor (S' xor X) <= A + X,
 *              we need to satisfy A <= A xor (S' xor X).
 *              Now, the cool part is that since n is even, we can find a pile A
 *              that satisfies the condition for any X!
 *
 *                  Let i be the most significant on bit in (S' xor X)
 *                  Since n is even, there must exist at least one pile A such that
 *                  the ith bit is off.
 *                  Thus, A < A xor (S' xor X), so A <= A xor (S' xor X).
 *
 *                  If there is no most significant on bit in (S' xor X), then (S' xor X) = 0.
 *                  Thus, A xor (S' xor X) = A, so A <= A xor (S' xor X).
 *
 *              Thus, we've proved that if n is even, we can find pile X and A in S' such that
 *              we can remove some stones from pile X and add the remaining to A such that xor-sum(S') = 0.
 *
 * Full Solution:
 *      To actually answer queries, we just need to use Mo's algorithm to maintain:
 *          1. The number of odd length subarrays
 *          2. The number of even length subarrays with xor-sum != 0.
 */


#include <bits/stdc++.h>

using namespace std;

const int MAXN = 1e5, MAXXOR = 1.1e6, SQRTN = 300;

int n, q;
int a[MAXN + 1];
long long ans[MAXN];

int pxor[MAXN + 1];
int pxor_cnt[2][MAXXOR + 1];

struct query {
    int i, l, r;

    bool operator < (const query& other) const {
        if (l/SQRTN == other.l/SQRTN) {
            return r < other.r;
        }
        else {
            return l/SQRTN < other.l/SQRTN;
        }
    }
};

vector<query> queries;

int main() {
    cin.tie(0)->sync_with_stdio(0);

    cin >> n >> q;

    for (int i = 1; i <= n; i++) {
        cin >> a[i];
    }

    for (int i = 0; i < q; i++) {
        int l, r; cin >> l >> r;

        queries.push_back({i, l, r});
    }

    for (int i = 1; i <= n; i++) {
        pxor[i] = a[i] ^ pxor[i-1];
    }

    // Mo's
    sort(queries.begin(), queries.end());

    int l = 1, r = 0;
    pxor_cnt[0][0]++;
    long long cans = 0;

    auto add = [&](int idx, bool left) {
        cans += (r-l+1)/2 - pxor_cnt[idx-left&1][pxor[idx-left]]++; // even length, diff xor
        cans += (r-l+2)/2; // odd length
    };
    auto remove = [&](int idx, bool left) {
        cans -= (r-l+1)/2 - --pxor_cnt[idx-left&1][pxor[idx-left]]; // even length, diff xor
        cans -= (r-l+2)/2; // odd length
    };

    for (auto q: queries) {
        while (q.l < l) {
            l--;
            add(l, true);
        }
        while (q.r > r) {
            r++;
            add(r, false);
        }
        while (q.l > l) {
            remove(l, true);
            l++;
        }
        while (q.r < r) {
            remove(r, false);
            r--;
        }

        ans[q.i] = cans;
    }

    for (int i = 0; i < q; i++) {
        cout << ans[i] << '\n';
    }
}

Details

Tip: Click on the bar to expand more detailed information

Test #1:

score: 100
Accepted
time: 2ms
memory: 3412kb

input:

4 5
1 2 2 4
1 2
2 3
3 4
1 3
2 4

output:

3
2
3
5
5

result:

ok 5 number(s): "3 2 3 5 5"

Test #2:

score: 0
Accepted
time: 2ms
memory: 3412kb

input:

4 5
5 6 7 8
1 2
2 3
3 4
1 3
2 4

output:

3
3
3
6
6

result:

ok 5 number(s): "3 3 3 6 6"

Test #3:

score: 0
Accepted
time: 0ms
memory: 3324kb

input:

10 10
3 7 3 1 6 6 10 3 3 3
9 10
5 10
3 7
5 6
5 6
9 10
3 10
1 4
6 6
1 4

output:

2
18
14
2
2
2
33
10
1
10

result:

ok 10 numbers

Test #4:

score: 0
Accepted
time: 2ms
memory: 3408kb

input:

10 8
91 63 1 34 50 11 10 73 96 67
5 9
2 7
2 5
4 7
1 10
3 3
1 4
5 10

output:

15
21
10
10
55
1
10
21

result:

ok 8 numbers

Test #5:

score: 0
Accepted
time: 2ms
memory: 3360kb

input:

10 6
9 539 285 408 615 861 951 413 319 368
4 4
8 10
1 7
3 9
2 3
2 10

output:

1
6
28
28
3
45

result:

ok 6 numbers

Test #6:

score: 0
Accepted
time: 0ms
memory: 3596kb

input:

10 6
1348 7002 4687 6325 8253 5750 2464 5509 6543 8704
3 9
4 8
8 8
8 9
2 9
9 10

output:

28
15
1
3
36
3

result:

ok 6 numbers

Test #7:

score: 0
Accepted
time: 3ms
memory: 3412kb

input:

10 8
59041 28802 92255 14246 65768 79252 70656 81265 98363 85237
1 6
9 10
4 7
6 8
9 10
1 2
1 3
4 5

output:

21
3
10
6
3
3
6
3

result:

ok 8 numbers

Test #8:

score: 0
Accepted
time: 2ms
memory: 3432kb

input:

10 7
28607 249948 373828 584253 989446 308313 199311 253174 283937 133758
2 4
1 2
4 9
7 8
7 8
2 6
1 1

output:

6
3
21
3
3
15
1

result:

ok 7 numbers

Test #9:

score: -100
Wrong Answer
time: 1ms
memory: 3364kb

input:

100 98
6 9 6 10 8 10 3 4 7 5 4 10 2 10 4 5 2 1 7 1 3 1 4 1 1 2 6 9 3 10 2 5 3 2 6 2 1 7 7 6 5 4 2 5 3 2 7 2 6 2 9 7 10 7 4 2 9 3 3 7 9 1 4 9 6 1 5 5 8 3 7 5 8 3 9 5 8 7 8 6 6 3 2 3 8 1 8 1 5 9 1 8 6 3 3 7 10 6 5 5
48 72
14 46
23 28
37 84
1 65
45 72
9 19
9 81
37 53
47 50
25 26
26 88
51 54
53 69
22 94...

output:

316
536
20
1144
2068
395
65
2611
148
10
3
1956
10
146
2611
33
167
6
27
316
604
2977
369
4596
2830
132
1226
53
164
2974
131
3059
798
1496
90
1946
225
65
76
423
335
646
266
1088
907
963
2682
2205
452
149
224
5
3783
1323
102
1229
576
2201
291
677
362
962
1
226
149
723
117
103
3370
911
132
1
90
1143
200...

result:

wrong answer 2nd numbers differ - expected: '534', found: '536'