Soluție HackerRank pentru Counting Equations, subdomeniul Algebra, în C++14. Include cerința formatată, exemple, explicația pașilor și cod sursă.
- Problemă: Counting Equations
- Domeniu: Algebra
- Limbaj: C++14
Challenge: Counting Equations
Subdomeniu: Algebra (algebra)
Scor cont: 100.0 / 100
Submission status: Accepted
Submission score: 1.0
Submission ID: 464775176
Limbaj: cpp14
Link challenge: https://www.hackerrank.com/challenges/counting-equations/problem
Cerință
[Sevenkplus](http://sevenkplus.com/) was interested in contributing a challenge to hackerrank and he came up with this problem.
You are given a linear congruence system with `n` variables and `m` equations:
a<sub>11</sub> x<sub>1</sub> + a<sub>12</sub> x<sub>2</sub> + ... + a<sub>1n</sub> x<sub>n</sub> = b<sub>1</sub> (mod p)
a<sub>21</sub> x<sub>1</sub> + a<sub>22</sub> x<sub>2</sub> + ... + a<sub>2n</sub> x<sub>n</sub> = b<sub>2</sub> (mod p)
...
a<sub>m1</sub> x<sub>1</sub> + a<sub>m2</sub> x<sub>2</sub> + ... + a<sub>mn</sub> x<sub>n</sub> = b<sub>m</sub> (mod p)
where,
p is a prime number
0 <= a<sub>ij</sub> < p
0 <= x<sub>i</sub> < p
0 <= b<sub>i</sub> < p
Given integers `n`, `m`, `p`, `a`, `b`, count the number of solutions to this equation. Since the output can be large, please output your answer modulo `10^9+7`.
He writes the standard solution and a test data generator without difficulty, and generates some test data.
However, when he attempts to remove hidden folders from the problem folder before uploading, he accidentally deletes the input file.
Luckily, the output file remains and he still remembers some features of the input. He remembers `n`, `m`, `p` and that `w` entries of `a` are zero. However, he cannot recall more about the input.
He wants to count how many possible inputs are there that will result in the desired output `S` (number of solutions to the equation system) output modulo `10^9+7`. Can you help Sevenkplus?
Input Format
The first line contains an integer T. T testcases follow.
For each test case, the first line contains five numbers, `m`, `n`, `p`, `S`, `w`. separated by a single space.
`w` lines follow. Each line contains two numbers `x`, `y`, which indicates that a<sub>xy</sub>=0.
Output Format
For each test case, output one line in the format `Case #t: ans`, where `t` is the case number (starting from 1), and `ans` is the answer.
Constraints
1 ≤ T ≤ 33
1 <= m, n <= 1000
p <= 10^9, p is a prime number
0 <= S < 10^9+7
w <= 17
1 <= x <= m
1 <= y <= n
In any test case, one pair (x, y) will not occur more than once.
Sample Input
6
2 2 2 0 1
1 1
2 2 2 1 1
1 1
2 2 2 2 1
1 1
2 2 2 3 1
1 1
2 2 2 4 1
1 1
488 629 183156769 422223791 10
350 205
236 164
355 8
3 467
355 164
350 467
3 479
72 600
17 525
223 370
Sample Output
Case #1: 13
Case #2: 8
Case #3: 10
Case #4: 0
Case #5: 1
Case #6: 225166925
Explanation
For test case 1, the 13 possible equations are:
a11 a12 b1 a21 a22 b2
0 0 0 0 0 1
0 0 1 0 0 0
0 0 1 0 0 1
0 0 1 0 1 0
0 0 1 0 1 1
0 0 1 1 0 0
0 0 1 1 0 1
0 0 1 1 1 0
0 0 1 1 1 1
0 1 0 0 0 1
0 1 0 0 1 1
0 1 1 0 0 1
0 1 1 0 1 0
Timelimits
Timelimits for this challenge is given [here](https://www.hackerrank.com/environment)
Cod sursă
#include <cstdio>
#include <cstring>
#include <cmath>
#include <cassert>
#include <ctime>
#include <set>
#include <map>
#include <vector>
#include <string>
#include <sstream>
#include <numeric>
#include <iostream>
#include <algorithm>
using namespace std;
typedef long long ll;
typedef pair<int, int> PII;
typedef pair<PII, int> PI3;
typedef pair<PI3, vector<int> > PIV;
#define fi first
#define se second
#define mp make_pair
#define pb push_back
#define mp3(x, y, z) mp(mp(x, y), z)
#define fi3 fi.fi
#define se3 fi.se
#define th3 se
#define pct __builtin_popcount
#define ctz __builtin_ctz
#define P 1000000007
#define N 1010
int n, m, p, S0, wn;
vector<PII> w;
int px[N], py[N], qx[N], qy[N];
int sx[N], sy[N];
int p0[N]; int mn, mm;
bool cmpx(int x, int y) { return sx[x] > sx[y]; }
bool cmpy(int x, int y) { return sy[x] > sy[y]; }
int F0[2010][1010];
map<PIV, int> F1;
int rem(int x, int w) {
return (x >> w+1 << w) + (x & ((1 << w)-1));
}
int rem(int x, int w1, int w2) {
if (w1 > w2) swap(w1, w2);
return rem(rem(x, w1), w2-1);
}
int gg(int n, int m, vector<int> a, int r) {
if (r < 0 || r > n || r > m) return 0;
if (r == 0 || n == 0 || m == 0) return 1;
PIV wp = mp(mp3(n, m, r), a);
if (F1.find(wp) != F1.end()) return F1[wp];
int &S = F1[wp];
for (int j = 2; j >= 0; j --) {
for (int i = n-1; i >= 0; i --)
if (pct(a[i]) <= j || pct(a[i]) >= m-j) {
swap(a[i], a[n-1]);
break;
}
}
int pc = pct(a[n-1]);
if (pc == m) {
vector<int> a1;
for (int i = 0; i < n-1; i ++) a1.pb(a[i]);
S = gg(n-1, m, a1, r);
} else
if (pc == 0) {
vector<int> a1;
for (int i = 0; i < n-1; i ++) a1.pb(a[i]);
int S1 = gg(n-1, m, a1, r);
int S2 = gg(n-1, m, a1, r-1);
(S += (ll)p0[r]*S1%P) %= P;
(S += (ll)(p0[m]-p0[r-1]+P)%P*S2%P) %= P;
} else
if (pc == m-1) {
vector<int> a1;
for (int i = 0; i < n-1; i ++) a1.pb(a[i]);
(S += gg(n-1, m, a1, r)) %= P;
int inva = (1 << m) - 1 - a[n-1];
int w = ctz(inva), s = 0;
for (int i = 0; i < n-1; i ++) {
if (~(a1[i] >> w)&1) s ++;
a1[i] = rem(a1[i], w);
}
(S += (ll)(p-1)*p0[s]%P*gg(n-1, m-1, a1, r-1)%P) %= P;
} else
if (pc == 1) {
vector<int> a1, a2;
for (int i = 0; i < n-1; i ++) a1.pb(a[i]);
int w = ctz(a[n-1]);
for (int i = 0; i < n-1; i ++)
a2.pb(rem(a[i], w));
int S1 = gg(n-1, m-1, a2, r);
int S2 = gg(n-1, m-1, a2, r-1);
int S3 = (gg(n-1, m, a1, r)-S1+P)%P;
int S4 = (gg(n-1, m, a1, r-1)-S2+P)%P;
(S += (ll)p0[r]*S1%P) %= P;
(S += (ll)(p0[m-1]-p0[r-1]+P)%P*S2%P) %= P;
(S += (ll)p0[r-1]*S3%P) %= P;
if (r >= 2) (S += (ll)(p0[m-1]-p0[r-2]+P)%P*S4%P) %= P;
} else
if (pc == m-2) {
vector<int> a1;
for (int i = 0; i < n-1; i ++) a1.pb(a[i]);
(S += gg(n-1, m, a1, r)) %= P;
int inva = (1 << m) - 1 - a[n-1];
int w1 = ctz(inva), w2 = ctz(inva^(1 << w1));
int s1 = 0, s2 = 0;
for (int i = 0; i < n-1; i ++) {
if (~(a[i] >> w1) & 1) s1 ++;
if (~(a[i] >> w2) & 1) s2 ++;
}
for (int i = 0; i < n-1; i ++)
a1[i] = rem(a[i], w1);
(S += (ll)(p-1)*p0[s1]%P*gg(n-1, m-1, a1, r-1)%P) %= P;
for (int i = 0; i < n-1; i ++)
a1[i] = rem(a[i], w2);
(S += (ll)(p-1)*p0[s2]%P*gg(n-1, m-1, a1, r-1)%P) %= P;
int s3 = 0;
vector<int> a2;
for (int i = 0; i < n-1; i ++) {
if ((~(a[i] >> w1) & 1) && (~(a[i] >> w2) &1)) s3 ++;
int x = a[i];
if (~(x >> w1)&1)
if ((x >> w2) & 1) x ^= (1 << w2);
a2.pb(rem(x, w1));
}
(S += (ll)(p-1)*(p-1)%P*p0[s3]%P*gg(n-1, m-1, a2, r-1)%P) %= P;
} else
if (pc == 2) {
int w1 = ctz(a[n-1]), w2 = ctz(a[n-1] ^ (1 << w1));
vector<int> a0, a1, a2, a3, a4;
for (int i = 0; i < n-1; i ++)
a0.pb(rem(a[i], w1, w2));
int S1 = gg(n-1, m-2, a0, r);
int S2 = gg(n-1, m-2, a0, r-1);
(S += (ll)p0[r]*S1%P) %= P;
(S += (ll)(p0[m-2]-p0[r-1]+P)%P*S2%P) %= P;
for (int i = 0; i < n-1; i ++) {
a1.pb(rem(a[i], w1));
a2.pb(rem(a[i], w2));
}
int S3 = (gg(n-1, m-1, a1, r)-S1+P)%P;
int S4 = (gg(n-1, m-1, a1, r-1)-S2+P)%P;
int S5 = (gg(n-1, m-1, a2, r)-S1+P)%P;
int S6 = (gg(n-1, m-1, a2, r-1)-S2+P)%P;
(S += (ll)p0[r-1]*S3%P) %= P;
if (r >= 2) (S += (ll)(p0[m-2]-p0[r-2]+P)%P*S4%P) %= P;
(S += (ll)p0[r-1]*S5%P) %= P;
if (r >= 2) (S += (ll)(p0[m-2]-p0[r-2]+P)%P*S6%P) %= P;
for (int i = 0; i < n-1; i ++) {
int x = a[i];
if ((x >> w1) &1) x |= (1 << w2);
a3.pb(rem(x, w1));
}
int S7 = (ll)(p-1)*(gg(n-1, m-1, a3, r)-S1+P)%P;
int S8 = (ll)(p-1)*(gg(n-1, m-1, a3, r-1)-S2+P)%P;
(S += (ll)p0[r-1]*S7%P) %= P;
if (r >= 2)(S += (ll)(p0[m-2]-p0[r-2]+P)%P*S8%P) %= P;
for (int i = 0; i < n-1; i ++) a4.pb(a[i]);
int S9 = (gg(n-1, m, a4, r)-((ll)S1+S3+S5+S7)%P+P)%P;
int S10 = (gg(n-1, m, a4, r-1)-((ll)S2+S4+S6+S8)%P+P)%P;
if (r >= 2) (S += (ll)p0[r-2]*S9%P) %= P;
if (r >= 3) (S += (ll)(p0[m-2]-p0[r-3]+P)%P*S10%P) %= P;
} else {
vector<int> b;
for (int i = 0; i < m; i ++) b.pb(0);
for (int i = 0; i < n; i ++)
for (int j = 0; j < m; j ++)
if ((a[i] >> j)&1) b[j] |= (1 << i);
S = gg(m, n, b, r);
}
return S;
}
int ff(int n, int m, int r) {
if (r < 0 || r > n || r > m) return 0;
if (r == 0 || n == 0 || m == 0) return 1;
if (F0[n+m][r] != -1) return F0[n+m][r];
int &S = F0[n+m][r] = 0;
if (n > mn) {
(S += (ll)p0[r]*ff(n-1, m, r)%P) %= P;
(S += (ll)(p0[m]-p0[r-1]+P)%P*ff(n-1, m, r-1)%P) %= P;
} else
if (m > mm) {
(S += (ll)p0[r]*ff(n, m-1, r)%P) %= P;
(S += (ll)(p0[n]-p0[r-1]+P)%P*ff(n, m-1, r-1)%P) %= P;
} else {
vector<int> a;
for (int i = 0; i < n; i ++) a.pb(0);
for (int i = 0; i < wn; i ++)
a[w[i].fi] |= (1 << w[i].se);
S = gg(n, m, a, r);
}
return S;
}
void solve() {
static int _ = 0; _ ++;
memset(F0, -1, sizeof F0);
F1.clear();
p0[0] = 1;
for (int i = 1; i < N; i ++)
p0[i] = (ll)p0[i-1]*p%P;
memset(sx, 0, sizeof sx);
memset(sy, 0, sizeof sy);
w.clear();
for (int i = 0; i < wn; i ++) {
int x, y;
scanf("%d%d", &x, &y);
--x, --y;
w.pb(mp(x, y));
sx[x] ++; sy[y] ++;
}
for (int i = 0; i < n; i ++) px[i] = i;
for (int i = 0; i < m; i ++) py[i] = i;
sort(px, px+n, cmpx);
sort(py, py+m, cmpy);
for (int i = 0; i < n; i ++) qx[px[i]] = i;
for (int i = 0; i < m; i ++) qy[py[i]] = i;
for (int i = 0; i < wn; i ++)
w[i] = mp(qx[w[i].fi], qy[w[i].se]);
for (int i = 0; i < wn; i ++) {
mn = max(mn, w[i].fi + 1);
mm = max(mm, w[i].se + 1);
}
int S = 0;
for (int i = 0; i <= min(n, m); i ++)
if (S0 == 0) {
(S += (ll)(p0[n]-p0[i]+P)%P*ff(n, m, i)%P) %= P;
} else
if (p0[m-i] == S0) {
(S += (ll)p0[i]*ff(n, m, i)%P) %= P;
}
printf ("Case #%d: %d\n", _, S);
}
int main() {
int _ = 0;
int test;
scanf("%d", &test);
while (scanf("%d%d%d%d%d", &n, &m, &p, &S0, &wn) != EOF) {
solve();
}
return 0;
}