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539 lines
16 KiB
C++
539 lines
16 KiB
C++
// RogueViz - Grigorchuk group
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// Copyright (C) 2011-2019 Zeno and Tehora Rogue, see 'hyper.cpp' for details
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/** \file rogueviz/grigorchuk.cpp
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* \brief Grigorchuk group
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*
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* This is a visualization of the Grigorchuk group. It is the first known group with
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* intermediate growth (i.e., superpolynomial and subexponential).
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*
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* The implementation is based on:
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*
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* Rostislav Grigorchuk, Igor Pak,
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* Groups of Intermediate Growth: an Introduction for Beginners
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* https://arxiv.org/pdf/math/0607384.pdf
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*
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* which presents the material in a simple way.
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*
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* This creates a map whose tiles correspond to the elements of the Grigorchuk group.
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* More precisely, the tiles correspond to the subgroup of index 2 generated by ac, ca, and b
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* (this is "more playable"). The three tiles adjacent to g are gb, gac, and gca.
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*
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* The 'lines' drawn split each tile into two halves, which correspond to the elements of the
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* actual Grigorchuk group (g and ga; ga is the one close to gac).
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*
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* Every element of the Grigorchuk group has finite order. Therefore, if you choose a specific
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* way of travelling (e.g. turn left, go, turn right, go) you will always eventually reach the
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* starting point.
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*
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* Command line options:
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*
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* -grigorchuk -- play on the Grigorchuk group
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* -grig-limit 100000 -canvas G -- color the tiles according to the distance from the starting point
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* (i.e., the neutral element), the number is the number of tiles colored
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* -grig-nolines -- show no splitting lines (also can be switched in the experiments menu)
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* -grig-nolabels -- show no labels (also can be switched in the experiments menu)
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*
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*/
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#include "rogueviz.h"
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namespace grigorchuk {
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using namespace hr;
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typedef tuple<bool, string, string> splitter;
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void add(string& s, char c) {
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if(s.size() == 0) s.push_back(c);
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else if(c == s.back()) s.pop_back();
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else if(c != 'a' && s.back() != 'a')
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s.back() = s.back() ^ c ^ 'd' ^ 'b' ^ 'c';
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else s.push_back(c);
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}
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splitter split(string s) {
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bool swapped = false;
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string s0, s1;
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for(char c: s) {
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if(c == 'b') add(s0, swapped?'a':'c'), add(s1, swapped?'c':'a');
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if(c == 'c') add(s0, swapped?'a':'d'), add(s1, swapped?'d':'a');
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if(c == 'd') add(swapped ? s1 : s0, 'b');
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if(c == 'a') swapped = !swapped;
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}
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return splitter{swapped, s0, s1};
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}
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splitter split_slow(string s) {
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bool swapped = false;
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string s0, s1;
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for(char c: s) {
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if(c == 'b') (s0 += swapped?'a':'c'), (s1 += swapped?'c':'a');
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if(c == 'c') (s0 += swapped?'a':'d'), (s1 += swapped?'d':'a');
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if(c == 'd') ((swapped ? s1 : s0) += 'b'), ((swapped ? s0 : s1) += '-');
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if(c == 'a') swapped = !swapped, s0 += '-', s1 += '-';
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}
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return splitter{swapped, s0, s1};
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}
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string reduce(const string& x) {
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string res;
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for(char c: x) add(res, c);
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return res;
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}
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#define Split(x) auto sw = split(x); auto swapped = get<0>(sw); auto s0 = get<1>(sw); auto s1 = get<2>(sw)
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bool empt(const string& x) {
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Split(x); // auto [swapped, s0, s1] = split(x);
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if(x == "") return true;
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if(x == "d") return false;
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if(swapped) return false;
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return empt(s0) && empt(s1);
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}
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bool empt_slow(const string& x) {
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Split(x); // auto [swapped, s0, s1] = split_slow(x);
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printf("%s -> %d %s %s\n", x.c_str(), swapped, s0.c_str(), s1.c_str());
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if(x == "") return true;
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if(x == "d") return false;
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if(swapped) return false;
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return empt(s0) && empt(s1);
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}
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typedef const struct rep* prep;
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struct rep {
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bool swapped;
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prep a0;
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prep a1;
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mutable char last;
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mutable bool visited = false;
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mutable int len;
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rep(bool s, prep a0, prep a1, char l, bool vis = false) : swapped(s), a0(a0), a1(a1), last(l), visited(vis), len(-1) {}
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};
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bool operator < (const rep a, const rep b) {
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return tie(a.swapped, a.a0, a.a1) < tie(b.swapped, b.a0, b.a1);
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}
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bool operator == (const rep a, const rep b) {
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return tie(a.swapped, a.a0, a.a1) == tie(b.swapped, b.a0, b.a1);
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}
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rep grig_I = rep{false, &grig_I, &grig_I, 0, false};
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extern rep grig_a, grig_b, grig_c, grig_d;
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rep grig_a = rep{true, &grig_I, &grig_I, 'a', false};
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rep grig_b = rep{false, &grig_a, &grig_c, 'b', false};
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rep grig_c = rep{false, &grig_a, &grig_d, 'c', false};
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rep grig_d = rep{false, &grig_I, &grig_b, 'd', false};
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// (ab) c = a (a,c) (a,d) = a(a,c) (a,d) =
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map<rep, char> all_reps; // = {grigid, &grigid};
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prep lookup(rep x) {
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if(x == grig_I) return &grig_I;
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else if(x == grig_a) return &grig_a;
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else if(x == grig_b) return &grig_b;
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else if(x == grig_c) return &grig_c;
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else if(x == grig_d) return &grig_d;
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else if(all_reps.count(x)) return &(all_reps.find(x)->first);
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else return &(all_reps.emplace(x, 0).first->first);
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}
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/*prep add_a(prep x) {
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return lookup({!x->swapped, x->a0, x->a1, 'a'});
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}
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prep add_d(prep x) {
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if(x == &grig_I) return &grig_d;
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if(x == &grig_d) return &grig_I;
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return lookup({x->swapped, x->swapped?add_d(x->a0):x->a0, x->swapped?x->a1:add_d(x->a1), 'd'});
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}
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prep add_c(prep x) { return lookup({x->swapped, (x->swapped?add_a:add_d)(x->a0), (x->swapped?add_d:add_a)(x->a1), 'c'}); }
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prep add_b(prep x) { return lookup({x->swapped, (x->swapped?add_a:add_c)(x->a0), (x->swapped?add_c:add_a)(x->a1), 'b'}); }
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*/
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/* ostream& operator << (ostream& os, prep x) {
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if(x == &grig_I) return os << "I";
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// else if(x == &grig_a) return os << "a";
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else if(x == &grig_b) return os << "b";
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else if(x == &grig_c) return os << "c";
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else if(x == &grig_d) return os << "d";
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else {
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if(x->swapped) os << "a";
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os << "(" << x->a0 << "," << x->a1 << ")";
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return os;
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}
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} */
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prep mul (prep x, prep y) {
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if(x == &grig_I) return y;
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if(y == &grig_I) return x;
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if(x == &grig_a && y == &grig_a) return &grig_I;
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if(x == &grig_b && y == &grig_b) return &grig_I;
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if(x == &grig_c && y == &grig_c) return &grig_I;
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if(x == &grig_d && y == &grig_d) return &grig_I;
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if(x == &grig_b && y == &grig_c) return &grig_d;
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if(x == &grig_c && y == &grig_b) return &grig_d;
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if(x == &grig_b && y == &grig_d) return &grig_c;
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if(x == &grig_d && y == &grig_b) return &grig_c;
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if(x == &grig_c && y == &grig_d) return &grig_b;
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if(x == &grig_d && y == &grig_c) return &grig_b;
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if(!y->swapped) return lookup(rep{x->swapped, mul(x->a0, y->a0), mul(x->a1, y->a1), y->last});
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else return lookup(rep{!x->swapped, mul(x->a1, y->a0), mul(x->a0, y->a1), y->last});
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}
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string encode(string s) {
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if(s == "") return "I";
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else if( s == "d") return "d";
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else {
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Split(s); // auto [swapped, s0, s1] = split(s);
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return (swapped ? "a(" : "(") + encode(s0) + "," + encode(s1) + ")";
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}
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}
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set<string> seen;
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void addmore(const string& s, int more) {
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if(more == 0) {
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string sr = s;
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reverse(sr.begin(), sr.end());
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for(string q: seen) {
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string qo = q;
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for(char cr: sr) add(q, cr);
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if(empt(q)) {
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// printf("%s = %s /%s\n", s.c_str(), qo.c_str(), sr.c_str());
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return;
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}
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}
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seen.insert(s);
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// printf("%s\n", s.c_str());
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return;
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}
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for(char c: {'a', 'b', 'c', 'd'}) {
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string s1 = s;
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add(s1, c);
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if(isize(s1) != isize(s)+1) continue;
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addmore(s1, more-1);
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}
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}
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string deform(prep x2) {
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string t = "";
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while(x2 != &grig_I) {
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if(x2->last == 'a') t += 'a', x2 = mul(x2, &grig_a);
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else if(x2->last == 'b') t += 'b', x2 = mul(x2, &grig_b);
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else if(x2->last == 'c') t += 'c', x2 = mul(x2, &grig_c);
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else if(x2->last == 'd') t += 'd', x2 = mul(x2, &grig_d);
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else if(x2->last == 'A') t += "ca", x2 = mul(mul(x2, &grig_c), &grig_a);
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else if(x2->last == 'C') t += "ac", x2 = mul(mul(x2, &grig_a), &grig_c);
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else return "?" + t + "?";
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}
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reverse(t.begin(), t.end());
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return t;
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}
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bool prepared = false;
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prep ac, ca;
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int grig_limit = 10000;
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int prepared_dists = 0;
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int next;
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int length = 0;
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vector<prep> all;
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void visit(prep x, char l, int d) {
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if(!x->visited) x->visited = true, x->last = l, all.push_back(x), x->len = d;
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}
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void prepare_to_next(bool verbose) {
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while(true) {
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int i = prepared_dists++;
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prep x = all[i];
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if(!x->visited) println(hlog, "visited or not");
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if(1) {
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// printf("%s\n", deform(x).c_str());
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}
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visit(mul(x, &grig_b), 'b', x->len + 1);
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visit(mul(x, ac), 'A', x->len + 1);
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visit(mul(x, ca), 'C', x->len + 1);
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if(i == next) {
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if(verbose)
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addMessage("there are "+its(i)+" elements in distance up to "+its(length));
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println(hlog, "Grigorchuk: ", tie(length, i));
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next = all.size(), length++;
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break;
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}
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}
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}
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void prepare() {
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prepared = true;
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// rep* grigid = lookup(rep { false, NULL, NULL });
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ac = mul(&grig_a, &grig_c);
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ca = mul(&grig_c, &grig_a);
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// prep where = &grig_I;
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string s = "";
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all.clear();
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/*
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for(int a=0; a<=32; a++) {
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// printf("%p -> %d %p %p\n", where, where->swapped, where->a0, where->a1);
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cout << where << " | " << encode(s) << "\n";
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where = ((a&1) ? add_a : add_b) (where);
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s += (a&1) ? 'a' : 'b';
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}
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string test = "ba";
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string pw = "";
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for(int i=0; i<=16; i++) {
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printf("%d: %d\n", i, empt(pw));
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pw += test;
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}
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*/
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// printf("TEST %s\n", encode("bcd").c_str());
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visit(&grig_I, 0, 0);
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length = 0;
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next = all.size();
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prepared_dists = 0;
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while(prepared_dists < grig_limit) prepare_to_next(false);
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prep test = &grig_b;
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test = mul(test, &grig_a);
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test = mul(test, &grig_d);
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test = mul(test, &grig_a);
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test = mul(test, &grig_d);
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printf("badad = %s\n", deform(test).c_str());
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}
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bool view_labels = true, view_lines = true;
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}
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namespace hr {
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extern ccolor::data grigorchuk_coloring;
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struct hrmap_grigorchuk : hrmap_standard {
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heptagon *origin;
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heptagon *getOrigin() override { return origin; }
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map<heptagon*, grigorchuk::prep> dec;
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map<grigorchuk::prep, heptagon*> enc;
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void gtie(heptagon* h, grigorchuk::prep p) {
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dec[h] = p;
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enc[p] = h;
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}
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hrmap_grigorchuk() {
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if(!grigorchuk::prepared) grigorchuk::prepare();
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origin = tailored_alloc<heptagon> (S7);
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origin->s = hsOrigin;
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origin->emeraldval = 0;
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origin->zebraval = 0;
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origin->fiftyval = 0;
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origin->fieldval = 0;
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origin->rval0 = origin->rval1 = 0;
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origin->cdata = NULL;
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origin->alt = NULL;
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origin->c7 = NULL;
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origin->distance = 0;
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origin->c7 = newCell(3, origin);
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gtie(origin, &grigorchuk::grig_I);
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}
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heptagon *create_step(heptagon *p, int d) override {
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auto pr = dec[p];
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// auto pr1 = pr;
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switch(d) {
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using namespace grigorchuk;
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case 0: pr = mul(mul(pr, &grig_a), &grig_c); break;
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case 1: pr = mul(mul(pr, &grig_c), &grig_a); break;
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case 2: pr = mul(pr, &grig_b); break;
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}
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heptagon *h;
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if(enc.count(pr)) {
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h = enc[pr];
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// println(hlog, deform(pr), "*", "acd"[d], " = ", deform(pr1));
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}
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else {
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if(!pr->visited) pr->last = "ACb" [d];
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h = tailored_alloc<heptagon> (S7);
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h->s = hsOrigin;
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h->emeraldval = 0;
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h->zebraval = 0;
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h->fiftyval = 0;
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h->fieldval = 0;
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h->rval0 = h->rval1 = 0;
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h->cdata = NULL;
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h->alt = NULL;
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h->c7 = newCell(3, h);
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h->distance = p->distance + 1;
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gtie(h, pr);
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}
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h->c.connect(d == 2 ? 2 : 1-d, p, d, false);;
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return h;
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}
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void draw_at(cell *at, const shiftmatrix& where) override {
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dq::clear_all();
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dq::enqueue_by_matrix(at->master, where * currentmap->master_relative(centerover, true));
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while(!dq::drawqueue.empty()) {
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auto& p = dq::drawqueue.front();
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heptagon *h = get<0>(p);
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shiftmatrix V = get<1>(p);
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dq::drawqueue.pop();
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cell *c = h->c7;
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if(!do_draw(c, V)) continue;
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if(grigorchuk::view_lines) queueline(V * ddspin(c, 2) * xpush0(cgi.tessf/2), V * ddspin(c, 2) * xpush0(-cgi.tessf), 0xFF00FFFF, 2);
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if(grigorchuk::view_labels) queuestr(V, 0.3, grigorchuk::deform(dec[c->master]), 0xFFFFFF);
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if(ccolor::which == &grigorchuk_coloring && c->landparam == 0)
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c->landparam = grigorchuk_coloring(c);
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drawcell(c, V * currentmap->master_relative(c, false));
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for(int i=0; i<3; i++) if(c->move(i))
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dq::enqueue_by_matrix(h->cmove(i), optimized_shift(V * adj(h, i)));
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}
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}
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transmatrix relative_matrixh(heptagon *h2, heptagon *h1, const hyperpoint& hint) override {
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if(gmatrix0.count(h2->c7) && gmatrix0.count(h1->c7))
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return inverse_shift(gmatrix0[h1->c7], gmatrix0[h2->c7]);
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return Id;
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}
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transmatrix relative_matrixc(cell *c2, cell *c1, const struct hyperpoint& hint) override {
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if(gmatrix0.count(c2) && gmatrix0.count(c1))
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return inverse_shift(gmatrix0[c1], gmatrix0[c2]);
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return Id;
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}
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};
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eGeometry gGrigorchuk(eGeometry(-1));
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ccolor::data grigorchuk_coloring = ccolor::data("Grigorchuk", [] { return geometry == gGrigorchuk; }, [] (cell *c, ccolor::data& cco) {
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return 0x102008 * (1 + ((hrmap_grigorchuk*)currentmap)->dec[c->master]->len);
|
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}, {});
|
|
|
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void create_grigorchuk_geometry() {
|
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if(gGrigorchuk != eGeometry(-1)) return;
|
|
ginf.push_back(ginf[gNormal]);
|
|
gGrigorchuk = eGeometry(isize(ginf) - 1);
|
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auto& gi = ginf[gGrigorchuk];
|
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gi.sides = 3;
|
|
gi.vertex = 8;
|
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gi.flags = qANYQ | qEXPERIMENTAL;
|
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gi.tiling_name = "{3,8}";
|
|
gi.quotient_name = "Grigorchuk";
|
|
gi.menu_displayed_name = "Grigorchuk group";
|
|
gi.shortname = "Grig";
|
|
gi.default_variation = eVariation::pure;
|
|
ccolor::all.push_back(&grigorchuk_coloring);
|
|
}
|
|
|
|
int readArgsG() {
|
|
using namespace arg;
|
|
|
|
if(0) ;
|
|
else if(argis("-grig-limit")) {
|
|
shift(); grigorchuk::grig_limit = argi();
|
|
}
|
|
|
|
else if(argis("-grigorchuk")) {
|
|
|
|
PHASEFROM(3);
|
|
|
|
stop_game();
|
|
create_grigorchuk_geometry();
|
|
set_geometry(gGrigorchuk);
|
|
set_variation(eVariation::pure);
|
|
}
|
|
|
|
else if(argis("-grig-nolines")) {
|
|
grigorchuk::view_lines = false;
|
|
}
|
|
|
|
else if(argis("-grig-nolabels")) {
|
|
grigorchuk::view_labels = false;
|
|
}
|
|
|
|
else return 1;
|
|
return 0;
|
|
}
|
|
|
|
auto hook = addHook(hooks_args, 100, readArgsG)
|
|
+ addHook(hooks_newmap, 100, [] { return geometry == gGrigorchuk ? new hrmap_grigorchuk : nullptr; })
|
|
+ addHook(dialog::hooks_display_dialog, 100, [] () {
|
|
if(current_screen_cfunction() == showEuclideanMenu && geometry == gGrigorchuk) {
|
|
dialog::addBoolItem_action(XLAT("Grigorchuk lines"), grigorchuk::view_lines, 'L');
|
|
dialog::addBoolItem_action(XLAT("Grigorchuk labels"), grigorchuk::view_labels, 'M');
|
|
}
|
|
})
|
|
+ addHook(hooks_initialize, 100, create_grigorchuk_geometry)
|
|
+ addHook_rvslides(140, [] (string s, vector<tour::slide>& v) {
|
|
if(s != "mixed") return;
|
|
using namespace rogueviz::pres;
|
|
v.push_back(tour::slide{
|
|
"Grigorchuk group", 10, tour::LEGAL::NONE,
|
|
|
|
"This is a visualization of the Grigorchuk group. It is the first known group with "
|
|
"intermediate growth (i.e., superpolynomial and subexponential).\n\n"
|
|
"Each tile corresponds to two elements of the Grigorchuk group.\n\n"
|
|
"Every element of the Grigorchuk group has finite order. Therefore, if you choose a specific "
|
|
"way of travelling (e.g. turn left, go, turn right, go) you will always eventually reach the "
|
|
"starting point.\n\n"
|
|
"Cells are color-coded by the distance to the origin. Distance is only known for a given number of cells "
|
|
"(initially 10000); if you want to compute more distances, press '5'. Press 'o' to enable/disable lines.\n\n"
|
|
"See grigorchuk.cpp for more comments.",
|
|
|
|
[] (tour::presmode mode) {
|
|
slide_url(mode, 'p', "a paper about Grigorchuk group", "https://arxiv.org/pdf/math/0607384.pdf");
|
|
if(mode == pmStart) {
|
|
grigorchuk::grig_limit = 10000;
|
|
gamestack::push();
|
|
slide_backup(ccolor::which, &grigorchuk_coloring);
|
|
slide_backup(firstland, laCanvas);
|
|
slide_backup(specialland, laCanvas);
|
|
set_geometry(gGrigorchuk);
|
|
start_game();
|
|
resetview();
|
|
}
|
|
if(mode == pmKey) {
|
|
grigorchuk::prepare_to_next(true);
|
|
}
|
|
if(mode == pmStop) {
|
|
gamestack::pop();
|
|
slide_restore_all();
|
|
}
|
|
}}
|
|
);});
|
|
}
|
|
|