You are working as an analyst in a company working on a new system for big data storage. This system will store $n$ different objects. Each object should have a unique ID.
To create the system, you choose the parameters of the system — integers $m \ge 1$ and $b_{1}, b_{2}, \ldots, b_{m}$. With these parameters an ID of some object in the system is an array of integers $[a_{1}, a_{2}, \ldots, a_{m}]$ where $1 \le a_{i} \le b_{i}$ holds for every $1 \le i \le m$.
Developers say that production costs are proportional to $\sum_{i=1}^{m} b_{i}$. You are asked to choose parameters $m$ and $b_{i}$ so that the system will be able to assign unique IDs to $n$ different objects and production costs are minimized. Note that you don’t have to use all available IDs.
Input
In the only line of input there is one positive integer $n$. The length of the decimal representation of $n$ is no greater than $1.5 \cdot 10^{6}$. The integer does not contain leading zeros.
Output
Print one number — minimal value of $\sum_{i=1}^{m} b_{i}$.
Examples
input
36
output
10
input
37
output
11
input
12345678901234567890123456789
output
177
Solution:
//#undef _GLIBCXX_DEBUG #include <bits/stdc++.h> using namespace std; string to_string(string s) { return '"' + s + '"'; } string to_string(const char* s) { return to_string((string) s); } string to_string(bool b) { return (b ? "true" : "false"); } template <typename A, typename B> string to_string(pair<A, B> p) { return "(" + to_string(p.first) + ", " + to_string(p.second) + ")"; } template <typename A> string to_string(A v) { bool first = true; string res = "{"; for (const auto &x : v) { if (!first) { res += ", "; } first = false; res += to_string(x); } res += "}"; return res; } void debug_out() { cerr << endl; } template <typename Head, typename... Tail> void debug_out(Head H, Tail... T) { cerr << " " << to_string(H); debug_out(T...); } #ifdef LOCAL #define debug(...) cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__) #else #define debug(...) 42 #endif namespace fft { typedef double dbl; struct num { dbl x, y; num() { x = y = 0; } num(dbl x, dbl y) : x(x), y(y) { } }; inline num operator+(num a, num b) { return num(a.x + b.x, a.y + b.y); } inline num operator-(num a, num b) { return num(a.x - b.x, a.y - b.y); } inline num operator*(num a, num b) { return num(a.x * b.x - a.y * b.y, a.x * b.y + a.y * b.x); } inline num conj(num a) { return num(a.x, -a.y); } int base = 1; vector<num> roots = {{0, 0}, {1, 0}}; vector<int> rev = {0, 1}; const dbl PI = acosl(-1.0); void ensure_base(int nbase) { if (nbase <= base) { return; } rev.resize(1 << nbase); for (int i = 0; i < (1 << nbase); i++) { rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1)); } roots.resize(1 << nbase); while (base < nbase) { dbl angle = 2 * PI / (1 << (base + 1)); // num z(cos(angle), sin(angle)); for (int i = 1 << (base - 1); i < (1 << base); i++) { roots[i << 1] = roots[i]; // roots[(i << 1) + 1] = roots[i] * z; dbl angle_i = angle * (2 * i + 1 - (1 << base)); roots[(i << 1) + 1] = num(cos(angle_i), sin(angle_i)); } base++; } } void fft(vector<num> &a, int n = -1) { if (n == -1) { n = a.size(); } assert((n & (n - 1)) == 0); int zeros = __builtin_ctz(n); ensure_base(zeros); int shift = base - zeros; for (int i = 0; i < n; i++) { if (i < (rev[i] >> shift)) { swap(a[i], a[rev[i] >> shift]); } } for (int k = 1; k < n; k <<= 1) { for (int i = 0; i < n; i += 2 * k) { for (int j = 0; j < k; j++) { num z = a[i + j + k] * roots[j + k]; a[i + j + k] = a[i + j] - z; a[i + j] = a[i + j] + z; } } } /* for (int len = 1; len < n; len <<= 1) { for (int i = 0; i < n; i += 2 * len) { for (int j = i, k = i + len; j < i + len; j++, k++) { num z = a[k] * roots[k - i]; a[k] = a[j] - z; a[j] = a[j] + z; } } }*/ } vector<num> fa, fb; vector<int> multiply(vector<int> &a, vector<int> &b) { int need = a.size() + b.size() - 1; int nbase = 1; while ((1 << nbase) < need) nbase++; ensure_base(nbase); int sz = 1 << nbase; if (sz > (int) fa.size()) { fa.resize(sz); } for (int i = 0; i < sz; i++) { int x = (i < (int) a.size() ? a[i] : 0); int y = (i < (int) b.size() ? b[i] : 0); fa[i] = num(x, y); } fft(fa, sz); num r(0, -0.25 / (sz >> 1)); for (int i = 0; i <= (sz >> 1); i++) { int j = (sz - i) & (sz - 1); num z = (fa[j] * fa[j] - conj(fa[i] * fa[i])) * r; if (i != j) { fa[j] = (fa[i] * fa[i] - conj(fa[j] * fa[j])) * r; } fa[i] = z; } for (int i = 0; i < (sz >> 1); i++) { num A0 = (fa[i] + fa[i + (sz >> 1)]) * num(0.5, 0); num A1 = (fa[i] - fa[i + (sz >> 1)]) * num(0.5, 0) * roots[(sz >> 1) + i]; fa[i] = A0 + A1 * num(0, 1); } fft(fa, sz >> 1); vector<int> res(need); for (int i = 0; i < need; i++) { if (i % 2 == 0) { res[i] = fa[i >> 1].x + 0.5; } else { res[i] = fa[i >> 1].y + 0.5; } } return res; } vector<long long> square(const vector<int> &a) { int need = a.size() + a.size() - 1; int nbase = 1; while ((1 << nbase) < need) nbase++; ensure_base(nbase); int sz = 1 << nbase; if ((sz >> 1) > (int) fa.size()) { fa.resize(sz >> 1); } for (int i = 0; i < (sz >> 1); i++) { int x = (2 * i < (int) a.size() ? a[2 * i] : 0); int y = (2 * i + 1 < (int) a.size() ? a[2 * i + 1] : 0); fa[i] = num(x, y); } fft(fa, sz >> 1); num r(1.0 / (sz >> 1), 0.0); for (int i = 0; i <= (sz >> 2); i++) { int j = ((sz >> 1) - i) & ((sz >> 1) - 1); num fe = (fa[i] + conj(fa[j])) * num(0.5, 0); num fo = (fa[i] - conj(fa[j])) * num(0, -0.5); num aux = fe * fe + fo * fo * roots[(sz >> 1) + i] * roots[(sz >> 1) + i]; num tmp = fe * fo; fa[i] = r * (conj(aux) + num(0, 2) * conj(tmp)); fa[j] = r * (aux + num(0, 2) * tmp); } fft(fa, sz >> 1); vector<long long> res(need); for (int i = 0; i < need; i++) { if (i % 2 == 0) { res[i] = fa[i >> 1].x + 0.5; } else { res[i] = fa[i >> 1].y + 0.5; } } return res; } vector<int> multiply_mod(vector<int> &a, vector<int> &b, int m, int eq = 0) { int need = a.size() + b.size() - 1; int nbase = 0; while ((1 << nbase) < need) nbase++; ensure_base(nbase); int sz = 1 << nbase; if (sz > (int) fa.size()) { fa.resize(sz); } for (int i = 0; i < (int) a.size(); i++) { int x = (a[i] % m + m) % m; fa[i] = num(x & ((1 << 15) - 1), x >> 15); } fill(fa.begin() + a.size(), fa.begin() + sz, num {0, 0}); fft(fa, sz); if (sz > (int) fb.size()) { fb.resize(sz); } if (eq) { copy(fa.begin(), fa.begin() + sz, fb.begin()); } else { for (int i = 0; i < (int) b.size(); i++) { int x = (b[i] % m + m) % m; fb[i] = num(x & ((1 << 15) - 1), x >> 15); } fill(fb.begin() + b.size(), fb.begin() + sz, num {0, 0}); fft(fb, sz); } dbl ratio = 0.25 / sz; num r2(0, -1); num r3(ratio, 0); num r4(0, -ratio); num r5(0, 1); for (int i = 0; i <= (sz >> 1); i++) { int j = (sz - i) & (sz - 1); num a1 = (fa[i] + conj(fa[j])); num a2 = (fa[i] - conj(fa[j])) * r2; num b1 = (fb[i] + conj(fb[j])) * r3; num b2 = (fb[i] - conj(fb[j])) * r4; if (i != j) { num c1 = (fa[j] + conj(fa[i])); num c2 = (fa[j] - conj(fa[i])) * r2; num d1 = (fb[j] + conj(fb[i])) * r3; num d2 = (fb[j] - conj(fb[i])) * r4; fa[i] = c1 * d1 + c2 * d2 * r5; fb[i] = c1 * d2 + c2 * d1; } fa[j] = a1 * b1 + a2 * b2 * r5; fb[j] = a1 * b2 + a2 * b1; } fft(fa, sz); fft(fb, sz); vector<int> res(need); for (int i = 0; i < need; i++) { long long aa = fa[i].x + 0.5; long long bb = fb[i].x + 0.5; long long cc = fa[i].y + 0.5; res[i] = (aa + ((bb % m) << 15) + ((cc % m) << 30)) % m; } return res; } vector<int> square_mod(vector<int> &a, int m) { return multiply_mod(a, a, m, 1); } }; void multiply(vector<int> &a, int b) { int carry = 0; for (int i = 0; i < (int) a.size(); i++) { carry += a[i] * b; a[i] = carry % 100000; carry /= 100000; } while (carry > 0) { a.push_back(carry % 100000); carry /= 100000; } } void divide(vector<int> &a, int b) { int md = 0; for (int i = (int) a.size() - 1; i >= 0; i--) { md = 100000 * md + a[i]; a[i] = md / b; md %= b; } assert(md == 0); while (!a.empty() && a.back() == 0) { a.pop_back(); } } bool is_less(const vector<int> &a, const vector<int> &b) { if (a.size() != b.size()) { return (a.size() < b.size()); } for (int i = (int) a.size() - 1; i >= 0; i--) { if (a[i] != b[i]) { return (a[i] < b[i]); } } return false; } vector<int> sp3; vector<long long> sp3_aux; void get_p3(int p3) { if (p3 <= 0) { return; } if (p3 & 1) { get_p3(p3 - 1); multiply(sp3, 3); } else { get_p3(p3 >> 1); sp3_aux = fft::square(sp3); sp3.resize(sp3_aux.size()); long long carry = 0; for (int i = 0; i < (int) sp3_aux.size(); i++) { carry += sp3_aux[i]; sp3[i] = carry % 100000; carry /= 100000; } while (carry > 0) { sp3.push_back(carry % 100000); carry /= 100000; } } } int main() { ios::sync_with_stdio(false); cin.tie(0); // debug(fft::square({100, 200, 300, 400, 500})); // return 0; string foo; cin >> foo; if (foo == "1") { cout << 1 << '\n'; return 0; } int len = (int) foo.size(); vector<int> s_10(len); for (int i = 0; i < len; i++) { s_10[i] = (int) (foo[len - 1 - i] - '0'); } vector<int> s((len + 4) / 5); for (int i = 0; i < (int) s.size(); i++) { s[i] = s_10[5 * i]; if (5 * i + 1 < (int) s_10.size()) s[i] += 10 * s_10[5 * i + 1]; if (5 * i + 2 < (int) s_10.size()) s[i] += 100 * s_10[5 * i + 2]; if (5 * i + 3 < (int) s_10.size()) s[i] += 1000 * s_10[5 * i + 3]; if (5 * i + 4 < (int) s_10.size()) s[i] += 10000 * s_10[5 * i + 4]; } int p3 = (int) (logl(10) * (len - 1) / logl(3)); sp3.assign(1, 1); get_p3(p3); int ans = 3 * p3; while (is_less(sp3, s)) { if (ans <= 3) { sp3[0] = ans + 1; } else { if (ans % 3 == 0) { divide(sp3, 3); multiply(sp3, 4); } else { divide(sp3, 2); multiply(sp3, 3); } } ans++; } cout << ans << '\n'; return 0; }