File size: 7,942 Bytes
1fd0050 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 | #include <bits/stdc++.h>
using namespace std;
static const long long P = 998244353;
// Count distinct T values for the generalized chain structure
// where push_val[i] can be any value in 1..d, not just i+1
int main() {
for (int d = 3; d <= 10; d++) {
// For each instruction i (0..d-1): push_val[i] in 1..d, push_goto[i] in 0..i
// HALT: halt_push in 1..d, halt_goto in 0..d-1
// Total configs per instruction: d * (i+1)
// Total: product(d * (i+1), i=0..d-1) * d * d = d^(d+1) * d! * d ... too many
// Let's just count for standard chain (push_val=i+1) vs generalized
// Standard: push_goto[i] in 0..i, halt fixed
// Generalized: push_val[i] in 1..d, push_goto[i] in 0..i, halt fixed
// For speed, fix halt_push=1, halt_goto=d-1 (last POP)
// and use recursive DP
unordered_set<long long> std_vals, gen_vals;
// Standard chain
{
vector<int> gy(d, 0);
while (true) {
long long Sv[15]; Sv[0] = 0;
for (int j = 0; j < d; j++) {
long long c = ((Sv[j] - Sv[gy[j]] + (j - gy[j]) + 1) % P + P) % P;
Sv[j + 1] = (Sv[j] + c) % P;
}
long long T = (Sv[d] + d + 1) % P;
std_vals.insert(T);
int pos = d - 1;
while (pos >= 0) { gy[pos]++; if (gy[pos] <= pos) break; gy[pos] = 0; pos--; }
if (pos < 0) break;
}
}
// Generalized: vary push_val and push_goto
// Use the actual checker DP
{
// Too many configs for large d, sample
int maxVal = d;
mt19937 rng(42);
int samples = min((long long)10000000, (long long)1);
// Actually, let's enumerate for small d
// push_val[i] in 1..d: d choices
// push_goto[i] in 0..i: (i+1) choices
// Total: product(d*(i+1), i=0..d-1) = d^d * d!
long long total = 1;
for (int i = 0; i < d; i++) {
total *= (long long)d * (i + 1);
if (total > 100000000) break;
}
if (total <= 100000000) {
// Enumerate
// Config: array of (push_val, push_goto) for each instruction
vector<int> pv(d, 1), pg(d, 0); // push_val 1..d, push_goto 0..i
long long count = 0;
while (true) {
// Evaluate using recursive DP
// States: (0..d, 0..maxVal)
int nstates = (d + 1) * (maxVal + 1);
vector<int> dest(nstates, -2);
vector<long long> cost(nstates, 0);
vector<int8_t> st(nstates, 0);
bool cycle = false;
function<pair<int,long long>(int,int)> solve = [&](int i, int x) -> pair<int,long long> {
int id = i * (maxVal + 1) + x;
if (st[id] == 2) return {dest[id], cost[id]};
if (st[id] == 1) { cycle = true; return {-2, 0}; }
st[id] = 1;
int dd; long long cc;
if (i < d) {
if (x == i + 1) { dd = i + 1; cc = 1; }
else {
auto [j, u] = solve(pg[i], pv[i]);
if (cycle) return {-2, 0};
if (j < 0 || j > d) { cycle = true; return {-2, 0}; }
auto [k, v] = solve(j, x);
if (cycle) return {-2, 0};
dd = k; cc = (u + v + 1) % P;
}
} else {
if (x == 0) { dd = -1; cc = 1; }
else {
// HALT pushes 1, goto d-1
auto [j, u] = solve(d - 1, 1);
if (cycle) return {-2, 0};
if (j < 0 || j > d) { cycle = true; return {-2, 0}; }
auto [k, v] = solve(j, x);
if (cycle) return {-2, 0};
dd = k; cc = (u + v + 1) % P;
}
}
st[id] = 2; dest[id] = dd; cost[id] = cc;
return {dd, cc};
};
auto [_, T] = solve(0, 0);
if (!cycle) gen_vals.insert(T);
count++;
if (count % 5000000 == 0) cerr << "d=" << d << " count=" << count << "/" << total << endl;
// Increment config
int pos = d - 1;
while (pos >= 0) {
// Try incrementing push_goto first, then push_val
pg[pos]++;
if (pg[pos] <= pos) break;
pg[pos] = 0;
pv[pos]++;
if (pv[pos] <= d) break;
pv[pos] = 1;
pos--;
}
if (pos < 0) break;
}
} else {
cerr << "d=" << d << ": too many configs (" << total << "), sampling" << endl;
for (int s = 0; s < 10000000; s++) {
vector<int> pv(d), pg(d);
for (int i = 0; i < d; i++) {
pv[i] = 1 + rng() % d;
pg[i] = rng() % (i + 1);
}
int maxVal2 = d;
int nstates = (d + 1) * (maxVal2 + 1);
vector<int> dest(nstates, -2);
vector<long long> cost(nstates, 0);
vector<int8_t> st(nstates, 0);
bool cycle = false;
function<pair<int,long long>(int,int)> solve = [&](int i, int x) -> pair<int,long long> {
int id = i * (maxVal2 + 1) + x;
if (st[id] == 2) return {dest[id], cost[id]};
if (st[id] == 1) { cycle = true; return {-2, 0}; }
st[id] = 1;
int dd; long long cc;
if (i < d) {
if (x == i + 1) { dd = i + 1; cc = 1; }
else {
auto [j, u] = solve(pg[i], pv[i]);
if (cycle) return {-2, 0};
if (j < 0 || j > d) { cycle = true; return {-2, 0}; }
auto [k, v] = solve(j, x);
if (cycle) return {-2, 0};
dd = k; cc = (u + v + 1) % P;
}
} else {
if (x == 0) { dd = -1; cc = 1; }
else {
auto [j, u] = solve(d - 1, 1);
if (cycle) return {-2, 0};
if (j < 0 || j > d) { cycle = true; return {-2, 0}; }
auto [k, v] = solve(j, x);
if (cycle) return {-2, 0};
dd = k; cc = (u + v + 1) % P;
}
}
st[id] = 2; dest[id] = dd; cost[id] = cc;
return {dd, cc};
};
auto [_, T] = solve(0, 0);
if (!cycle) gen_vals.insert(T);
}
}
}
cout << "d=" << d << ": std=" << std_vals.size() << " gen=" << gen_vals.size() << endl;
}
return 0;
}
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