mirror of
https://github.com/zenorogue/hyperrogue.git
synced 2024-11-27 06:27:17 +00:00
685 lines
19 KiB
C++
685 lines
19 KiB
C++
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namespace fieldpattern {
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extern int subpathid;
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extern int subpathorder;
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bool isprime(int n) {
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for(int k=2; k<n; k++) if(n%k == 0) return false;
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return true;
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}
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struct matrix {
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int a[3][3];
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int* operator [] (int k) { return a[k]; }
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const int* operator [] (int k) const { return a[k]; }
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};
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bool operator == (const matrix& A, const matrix& B) {
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for(int i=0; i<3; i++) for(int j=0; j<3; j++)
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if(A[i][j] != B[i][j]) return false;
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return true;
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}
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bool operator != (const matrix& A, const matrix& B) {
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for(int i=0; i<3; i++) for(int j=0; j<3; j++)
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if(A[i][j] != B[i][j]) return true;
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return false;
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}
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bool operator < (const matrix& A, const matrix& B) {
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for(int i=0; i<3; i++) for(int j=0; j<3; j++)
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if(A[i][j] != B[i][j]) return A[i][j] < B[i][j];
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return false;
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}
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int btspin(int id, int d) {
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return 7*(id/7) + (id + d) % 7;
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}
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struct fpattern {
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int Prime, wsquare, Field;
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// we perform our computations in the field Z_Prime[w] where w^2 equals wsquare
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// (or simply Z_Prime for wsquare == 0)
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#define EASY
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// 'easy' assumes that all elements of the field actually used
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// are of form n or mw (not n+mw), and cs and ch are both of form n
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// by experimentation, such cs and ch always exist
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// many computations are much simpler under that assumption
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#ifndef EASY
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static int neasy;
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int m(int x) { x %= Prime; if(x<0) x+= Prime; return x; }
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#endif
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int sub(int a, int b) {
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#ifdef EASY
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return (a + b * (Prime-1)) % Prime;
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#else
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return m(a%Prime-b%Prime) + Prime * m(a/Prime-b/Prime);
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#endif
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}
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int add(int a, int b) {
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#ifdef EASY
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return (a+b)%Prime;
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#else
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return m(a%Prime+b%Prime) + Prime * m(a/Prime+b/Prime);
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#endif
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}
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int mul(int tx, int ty) {
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#ifdef EASY
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return (tx*ty*((tx<0&&ty<0)?wsquare:1)) % Prime;
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#else
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if(tx >= Prime && tx % Prime) neasy++;
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if(ty >= Prime && ty % Prime) neasy++;
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int x[2], y[2], z[3];
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for(int i=0; i<3; i++) z[i] = 0;
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for(int i=0; i<2; i++)
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x[i] = tx%Prime, tx /= Prime;
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for(int i=0; i<2; i++)
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y[i] = ty%Prime, ty /= Prime;
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for(int i=0; i<2; i++)
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for(int j=0; j<2; j++)
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z[i+j] = (z[i+j] + x[i] * y[j]) % Prime;
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z[0] += z[2] * wsquare;
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return m(z[0]) + Prime * m(z[1]);
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#endif
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}
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int sqr(int x) { return mul(x,x); }
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matrix mmul(const matrix& A, const matrix& B) {
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matrix res;
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for(int i=0; i<3; i++) for(int k=0; k<3; k++) {
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#ifdef EASY
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res[i][k] =
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(mul(A[i][0], B[0][k]) + mul(A[i][1], B[1][k]) + mul(A[i][2], B[2][k])) % Prime;
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#else
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int t=0;
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for(int j=0; j<3; j++) t = add(t, mul(A[i][j], B[j][k]));
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res[i][k] = t;
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#endif
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}
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return res;
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}
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map<matrix, int> matcode;
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vector<matrix> matrices;
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vector<string> qpaths;
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vector<matrix> qcoords;
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matrix Id, R, P;
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matrix strtomatrix(string s) {
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matrix res = Id;
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matrix m = Id;
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for(int i=size(s)-1; i>=0; i--)
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if(s[i] == 'R') res = mmul(R, res);
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else if (s[i] == 'P') res = mmul(P, res);
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else if (s[i] == 'x') { m[0][0] = -1; res = mmul(m, res); m[0][0] = +1; }
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else if (s[i] == 'y') { m[1][1] = -1; res = mmul(m, res); m[1][1] = +1; }
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else if (s[i] == 'z') { m[2][2] = -1; res = mmul(m, res); m[2][2] = +1; }
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return res;
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}
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void addas(const matrix& M, int i) {
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if(!matcode.count(M)) {
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matcode[M] = i;
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for(int j=0; j<size(qcoords); j++)
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addas(mmul(M, qcoords[j]), i);
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}
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}
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void add(const matrix& M) {
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if(!matcode.count(M)) {
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int i = matrices.size();
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matcode[M] = i, matrices.push_back(M);
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for(int j=0; j<size(qcoords); j++)
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addas(mmul(M, qcoords[j]), i);
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add(mmul(R, M));
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}
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}
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#define MXF 1000000
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vector<int> connections;
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vector<int> inverses;
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vector<int> rrf; // rrf[i] equals gmul(i, 6)
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vector<int> rpf; // rpf[i] equals gmul(i, 7)
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matrix mpow(matrix M, int N) {
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while((N&1) == 0) N >>= 1, M = mmul(M, M);
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matrix res = M;
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N >>= 1;
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while(N) {
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M = mmul(M,M); if(N&1) res = mmul(res, M);
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N >>= 1;
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}
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return res;
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}
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int gmul(int a, int b) { return matcode[mmul(matrices[a], matrices[b])]; }
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int gpow(int a, int N) { return matcode[mpow(matrices[a], N)]; }
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pair<int,bool> gmul(pair<int, bool> a, int b) {
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return make_pair(gmul(a.first,b), a.second);
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}
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int order(const matrix& M) {
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int cnt = 1;
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matrix Po = M;
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while(Po != Id) Po = mmul(Po, M), cnt++;
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return cnt;
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}
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string decodepath(int i) {
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string s;
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while(i) {
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if(i % 7) i--, s += 'R';
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else i = connections[i], s += 'P';
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}
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return s;
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}
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int orderstats();
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int cs, sn, ch, sh;
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int solve() {
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for(int a=0; a<3; a++) for(int b=0; b<3; b++) Id[a][b] = a==b?1:0;
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if(!isprime(Prime)) {
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return 1;
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}
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for(int pw=1; pw<3; pw++) {
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if(pw>3) break;
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Field = pw==1? Prime : Prime*Prime;
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if(pw == 2) {
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for(wsquare=1; wsquare<Prime; wsquare++) {
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int roots = 0;
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for(int a=0; a<Prime; a++) if((a*a)%Prime == wsquare) roots++;
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if(!roots) break;
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}
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} else wsquare = 0;
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#ifdef EASY
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int sqrts[Prime];
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for(int k=0; k<Prime; k++) sqrts[k] = 0;
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for(int k=1-Prime; k<Prime; k++) sqrts[sqr(k)] = k;
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int fmax = Prime;
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#else
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int sqrts[Field];
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for(int k=0; k<Field; k++) sqrts[sqr(k)] = k;
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int fmax = Field;
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#endif
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if(Prime == 13 && wsquare) {
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for(int i=0; i<Prime; i++) printf("%3d", sqrts[i]);
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printf("\n");
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}
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for(int i=0; i<3; i++) for(int j=0; j<3; j++)
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R[i][j] = P[i][j] = i==j ? 1 : 0;
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for(cs=0; cs<fmax; cs++) {
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int sb = sub(1, sqr(cs));
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sn = sqrts[sb];
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R[0][0] = cs; R[1][1] = cs;
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R[0][1] = sn; R[1][0] = sub(0, sn);
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matrix Z = R;
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for(int i=0; i<6; i++) Z = mmul(Z, R);
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if(Z != Id) continue;
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if(R[0][0] == 1) continue;
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for(ch=2; ch<fmax; ch++) {
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int chx = sub(mul(ch,ch), 1);
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sh = sqrts[chx];
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P[0][0] = sub(0, ch);
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P[0][2] = sub(0, sh);
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P[1][1] = Prime-1;
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P[2][0] = sh;
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P[2][2] = ch;
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matrix Z = mmul(P, R);
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Z = mmul(Z, mmul(Z, Z));
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if(Z == Id) return 0;
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}
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}
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}
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return 2;
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}
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void build() {
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for(int i=0; i<size(qpaths); i++) {
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matrix M = strtomatrix(qpaths[i]);
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qcoords.push_back(M);
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printf("Solved %s as matrix of order %d\n", qpaths[i].c_str(), order(M));
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}
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matcode.clear(); matrices.clear();
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add(Id);
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if(matrices.size() != 7) { printf("Error: rotation crash\n"); exit(1); }
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connections.clear();
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for(int i=0; i<(int)matrices.size(); i++) {
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matrix M = matrices[i];
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matrix PM = mmul(P, M);
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add(PM);
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if(matrices.size() % 7) { printf("Error: rotation crash\n"); exit(1); }
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if(!matcode.count(PM)) { printf("Error: not marked\n"); exit(1); }
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connections.push_back(matcode[PM]);
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}
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DEBB(DF_FIELD, (debugfile, "Computing inverses...\n"));
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int N = size(matrices);
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DEBB(DF_FIELD, (debugfile, "Number of heptagons: %d\n", N));
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rrf.resize(N); rrf[0] = 6;
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for(int i=0; i<N; i++)
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rrf[btspin(i,1)] = btspin(rrf[i], 1),
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rrf[connections[i]] = connections[rrf[i]];
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rpf.resize(N); rpf[0] = 7;
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for(int i=0; i<N; i++)
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rpf[btspin(i,1)] = btspin(rpf[i], 1),
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rpf[connections[i]] = connections[rpf[i]];
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inverses.resize(N);
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inverses[0] = 0;
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for(int i=0; i<N; i++) // inverses[i] = gpow(i, N-1);
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inverses[btspin(i,1)] = rrf[inverses[i]], // btspin(inverses[i],6),
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inverses[connections[i]] = rpf[inverses[i]];
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int errs = 0;
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for(int i=0; i<N; i++) if(gmul(i, inverses[i])) errs++;
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if(errs) printf("errs = %d\n", errs);
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if(0) for(int i=0; i<size(matrices); i++) {
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printf("%5d/%4d", connections[i], inverses[i]);
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if(i%7 == 6) printf("\n");
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}
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DEBB(DF_FIELD, (debugfile, "Built.\n"));
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}
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vector<char> disthep;
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vector<char> disthex;
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vector<char> distwall, distriver, distwall2, distriverleft, distriverright, distflower;
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vector<eItem> markers;
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int getdist(pair<int,bool> a, vector<char>& dists) {
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if(!a.second) return dists[a.first];
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int m = 60;
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int ma = dists[a.first];
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int mb = dists[connections[btspin(a.first, 3)]];
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int mc = dists[connections[btspin(a.first, 4)]];
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m = min(m, 1 + ma);
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m = min(m, 1 + mb);
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m = min(m, 1 + mc);
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if(m <= 2 && ma+mb+mc <= m*3-2) return m-1; // special case
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m = min(m, 2 + dists[connections[btspin(a.first, 2)]]);
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m = min(m, 2 + dists[connections[btspin(a.first, 5)]]);
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m = min(m, 2 + dists[connections[btspin(connections[btspin(a.first, 3)], 5)]]);
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return m;
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}
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int getdist(pair<int,bool> a, pair<int,bool> b) {
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if(a.first == b.first) return a.second == b.second ? 0 : 1;
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if(b.first) a.first = gmul(a.first, inverses[b.first]), b.first = 0;
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return getdist(a, b.second ? disthex : disthep);
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}
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int maxdist, otherpole, circrad, wallid, wallorder, riverid;
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int dijkstra(vector<char>& dists, vector<int> indist[64]) {
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int N = connections.size();
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dists.resize(N);
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for(int i=0; i<N; i++) dists[i] = 60;
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int maxd = 0;
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for(int i=0; i<64; i++) while(!indist[i].empty()) {
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int at = indist[i].back();
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indist[i].pop_back();
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if(dists[at] <= i) continue;
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maxd = i;
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dists[at] = i;
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for(int q=0; q<7; q++) {
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dists[at] = i;
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if(purehepta)
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indist[i+1].push_back(connections[at]);
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else {
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indist[i+2].push_back(connections[at]);
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indist[i+3].push_back(connections[btspin(connections[at], 2)]);
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}
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at = btspin(at, 1);
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}
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}
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return maxd;
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}
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void analyze() {
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DEBB(DF_FIELD, (debugfile, "purehepta = %d\n", purehepta));
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int N = connections.size();
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markers.resize(N);
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vector<int> indist[64];
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indist[0].push_back(0);
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int md0 = dijkstra(disthep, indist);
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indist[1].push_back(0);
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indist[1].push_back(connections[3]);
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indist[1].push_back(connections[4]);
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indist[2].push_back(connections[btspin(connections[3], 5)]);
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indist[2].push_back(connections[2]);
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indist[2].push_back(connections[5]);
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int md1 = dijkstra(disthex, indist);
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maxdist = max(md0, md1);
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otherpole = 0;
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for(int i=0; i<N; i+=7) {
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int mp = 0;
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for(int q=0; q<7; q++) if(disthep[connections[i+q]] < disthep[i]) mp++;
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if(mp == 7) {
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bool eq = true;
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for(int q=0; q<7; q++) if(disthep[connections[i+q]] != disthep[connections[i]]) eq = false;
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if(eq) {
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// for(int q=0; q<7; q++) printf("%3d", disthep[connections[i+q]]);
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// printf(" (%2d) at %d\n", disthep[i], i);
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if(disthep[i] > disthep[otherpole]) otherpole = i;
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// for(int r=0; r<7; r++) {
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// printf("Matrix: "); for(int a=0; a<3; a++) for(int b=0; b<3; b++)
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// printf("%4d", matrices[i+r][a][b]); printf("\n");
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// }
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}
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}
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}
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circrad = 99;
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for(int i=0; i<N; i++) for(int u=2; u<4; u++) if(disthep[i] < circrad)
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if(disthep[connections[i]] < disthep[i] && disthep[connections[btspin(i,u)]] < disthep[i])
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circrad = disthep[i];
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DEBB(DF_FIELD, (debugfile, "maxdist = %d otherpole = %d circrad = %d\n", maxdist, otherpole, circrad));
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matrix PRRR = strtomatrix("PRRR");
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matrix PRRPRRRRR = strtomatrix("PRRPRRRRR");
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matrix PRRRP = strtomatrix("PRRRP");
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matrix PRP = strtomatrix("PRP");
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matrix PR = strtomatrix("PR");
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matrix Wall = strtomatrix("RRRPRRRRRPRRRP");
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wallorder = order(Wall);
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wallid = matcode[Wall];
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DEBB(DF_FIELD, (debugfile, "wall order = %d\n", wallorder));
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#define SETDIST(X, d, it) {int c = matcode[X]; indist[d].push_back(c); if(it == itNone) ; else if(markers[c] && markers[c] != it) markers[c] = itBuggy; else markers[c] = it; }
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matrix W = Id;
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for(int i=0; i<wallorder; i++) {
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SETDIST(W, 0, itAmethyst)
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W = mmul(W, Wall);
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}
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W = P;
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for(int i=0; i<wallorder; i++) {
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SETDIST(W, 0, itEmerald)
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W = mmul(W, Wall);
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}
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int walldist = dijkstra(distwall, indist);
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DEBB(DF_FIELD, (debugfile, "wall dist = %d\n", walldist));
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W = strtomatrix("RRRRPR");
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for(int j=0; j<wallorder; j++) {
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W = mmul(W, Wall);
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for(int i=0; i<wallorder; i++) {
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SETDIST(W, 0, itNone)
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SETDIST(mmul(PRRR, W), 1, itNone)
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W = mmul(Wall, W);
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}
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}
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|
dijkstra(distwall2, indist);
|
|
|
|
int rpushid = matcode[PRRPRRRRR];
|
|
riverid = 0;
|
|
|
|
for(int i=0; i<N; i++) {
|
|
int j = i;
|
|
int ipush = gmul(rpushid, i);
|
|
for(int k=0; k<wallorder; k++) {
|
|
if(ipush == j) {
|
|
DEBB(DF_FIELD, (debugfile, "River found at %d:%d\n", i, k));
|
|
riverid = i;
|
|
goto riveridfound;
|
|
}
|
|
j = gmul(j, wallid);
|
|
}
|
|
}
|
|
|
|
riveridfound: ;
|
|
|
|
W = strtomatrix("RRRRPR");
|
|
for(int j=0; j<wallorder; j++) {
|
|
W = mmul(W, Wall);
|
|
for(int i=0; i<wallorder; i++) {
|
|
if(i == 7) SETDIST(W, 0, itCoast)
|
|
if(i == 3) SETDIST(mmul(PRRRP, W), 0, itWhirlpool)
|
|
W = mmul(Wall, W);
|
|
}
|
|
}
|
|
dijkstra(purehepta ? distriver : distflower, indist);
|
|
|
|
W = matrices[riverid];
|
|
for(int i=0; i<wallorder; i++) {
|
|
SETDIST(W, 0, itStatue)
|
|
W = mmul(W, Wall);
|
|
}
|
|
W = mmul(P, W);
|
|
for(int i=0; i<wallorder; i++) {
|
|
SETDIST(W, 0, itSapphire)
|
|
W = mmul(W, Wall);
|
|
}
|
|
W = mmul(PRP, matrices[riverid]);
|
|
for(int i=0; i<wallorder; i++) {
|
|
SETDIST(W, 1, itShard)
|
|
W = mmul(W, Wall);
|
|
}
|
|
W = mmul(PR, matrices[riverid]);
|
|
for(int i=0; i<wallorder; i++) {
|
|
SETDIST(W, 1, itGold)
|
|
W = mmul(W, Wall);
|
|
}
|
|
int riverdist = dijkstra(purehepta ? distflower : distriver, indist);
|
|
DEBB(DF_FIELD, (debugfile, "river dist = %d\n", riverdist));
|
|
|
|
if(!purehepta) {
|
|
W = matrices[riverid];
|
|
for(int i=0; i<wallorder; i++) {
|
|
SETDIST(W, 0, itStatue)
|
|
W = mmul(W, Wall);
|
|
}
|
|
W = mmul(PR, matrices[riverid]);
|
|
for(int i=0; i<wallorder; i++) {
|
|
SETDIST(W, 0, itGold)
|
|
W = mmul(W, Wall);
|
|
}
|
|
W = mmul(P, matrices[riverid]);
|
|
for(int i=0; i<wallorder; i++) {
|
|
SETDIST(W, 1, itSapphire)
|
|
W = mmul(W, Wall);
|
|
}
|
|
dijkstra(distriverleft, indist);
|
|
W = mmul(PRP, matrices[riverid]);
|
|
for(int i=0; i<wallorder; i++) {
|
|
SETDIST(W, 0, itShard)
|
|
W = mmul(W, Wall);
|
|
}
|
|
W = mmul(P, matrices[riverid]);
|
|
for(int i=0; i<wallorder; i++) {
|
|
SETDIST(W, 0, itSapphire)
|
|
W = mmul(W, Wall);
|
|
}
|
|
W = matrices[riverid];
|
|
for(int i=0; i<wallorder; i++) {
|
|
SETDIST(W, 1, itStatue)
|
|
W = mmul(W, Wall);
|
|
}
|
|
dijkstra(distriverright, indist);
|
|
}
|
|
else {
|
|
W = strtomatrix("RRRRPR");
|
|
for(int j=0; j<wallorder; j++) {
|
|
W = mmul(W, Wall);
|
|
for(int i=0; i<wallorder; i++) {
|
|
if(i == 7) SETDIST(W, 0, itCoast)
|
|
W = mmul(Wall, W);
|
|
}
|
|
}
|
|
dijkstra(distriverleft, indist);
|
|
W = strtomatrix("RRRRPR");
|
|
for(int j=0; j<wallorder; j++) {
|
|
W = mmul(W, Wall);
|
|
for(int i=0; i<wallorder; i++) {
|
|
if(i == 3) SETDIST(mmul(PRRRP, W), 0, itWhirlpool)
|
|
W = mmul(Wall, W);
|
|
}
|
|
}
|
|
dijkstra(distriverright, indist);
|
|
}
|
|
|
|
DEBB(DF_FIELD, (debugfile, "wall-river distance = %d\n", distwall[riverid]));
|
|
DEBB(DF_FIELD, (debugfile, "river-wall distance = %d\n", distriver[0]));
|
|
}
|
|
|
|
bool easy(int i) {
|
|
return i < Prime || !(i % Prime);
|
|
}
|
|
|
|
// 11 * 25
|
|
// (1+z+z^3) * (1+z^3+z^4) ==
|
|
// 1+z+z^7 == 1+z+z^2(z^5) == 1+z+z^2(1+z^2) = 1+z+z^2+z^4
|
|
|
|
void init(int p) {
|
|
Prime = p;
|
|
if(solve()) {
|
|
printf("error: could not solve the fieldpattern\n");
|
|
exit(1);
|
|
}
|
|
build();
|
|
}
|
|
|
|
fpattern(int p) {
|
|
if(!p) return;
|
|
init(p);
|
|
}
|
|
|
|
void findsubpath() {
|
|
int N = size(matrices);
|
|
for(int i=1; i<N; i++)
|
|
if(gpow(i, Prime) == 0) {
|
|
subpathid = i;
|
|
subpathorder = Prime;
|
|
DEBB(DF_FIELD, (debugfile, "Subpath found: %s\n", decodepath(i).c_str()));
|
|
return;
|
|
}
|
|
}
|
|
};
|
|
|
|
int fpattern::orderstats() {
|
|
int N = size(matrices);
|
|
|
|
#define MAXORD 10000
|
|
int ordcount[MAXORD];
|
|
int ordsample[MAXORD];
|
|
|
|
for(int i=0; i<MAXORD; i++) ordcount[i] = 0;
|
|
|
|
for(int i=0; i<N; i++) {
|
|
int cnt = order(matrices[i]);
|
|
|
|
if(cnt < MAXORD) {
|
|
if(!ordcount[cnt]) ordsample[cnt] = i;
|
|
ordcount[cnt]++;
|
|
}
|
|
}
|
|
|
|
printf("Listing:\n");
|
|
for(int i=0; i<MAXORD; i++) if(ordcount[i])
|
|
printf("Found %4d matrices of order %3d: %s\n", ordcount[i], i, decodepath(ordsample[i]).c_str());
|
|
|
|
return ordsample[Prime];
|
|
}
|
|
|
|
|
|
fpattern fp43(43);
|
|
|
|
void info() {
|
|
fpattern fp(0);
|
|
int cases = 0, hard = 0;
|
|
for(int p=0; p<500; p++) {
|
|
fp.Prime = p;
|
|
if(fp.solve() == 0) {
|
|
printf("%4d: wsquare=%d cs=%d sn=%d ch=%d sh=%d\n",
|
|
p, fp.wsquare, fp.cs, fp.sn, fp.ch, fp.sh);
|
|
cases++;
|
|
if(!fp.easy(fp.cs) || !fp.easy(fp.sn) || !fp.easy(fp.ch) || !fp.easy(fp.sn))
|
|
hard++;
|
|
#ifndef EASY
|
|
neasy = 0;
|
|
#endif
|
|
fp.build();
|
|
#ifndef EASY
|
|
printf("Not easy: %d\n", neasy);
|
|
#endif
|
|
int N = size(fp.matrices);
|
|
int left = N / fp.Prime;
|
|
printf("Prime decomposition: %d = %d", N, fp.Prime);
|
|
for(int p=2; p<=left; p++) while(left%p == 0) printf("*%d", p), left /= p;
|
|
printf("\n");
|
|
printf("Order of RRP is: %d\n", fp.order(fp.strtomatrix("RRP")));
|
|
printf("Order of RRRP is: %d\n", fp.order(fp.strtomatrix("RRRP")));
|
|
printf("Order of RRRPRRRRRPRRRP is: %d\n", fp.order(fp.strtomatrix("RRRPRRRRRPRRRP")));
|
|
}
|
|
}
|
|
printf("cases found = %d (%d hard)\n", cases, hard);
|
|
}
|
|
|
|
}
|
|
|
|
using fieldpattern::fp43;
|