diff --git a/algorithms/graph/kruskal.cpp b/algorithms/graph/kruskal.cpp
index 15218e7feb68f422a665e0b2ac852c77f050d4eb..2f86011a94ce12f60d1404de51dafb67426db106 100644
--- a/algorithms/graph/kruskal.cpp
+++ b/algorithms/graph/kruskal.cpp
@@ -1,7 +1,5 @@
/// Kruskal
///
-///
-///
/// Time: O(E log V)
/// Space: O(E)
///
@@ -17,11 +15,7 @@ struct Kruskal {
Kruskal(int N) : N(N), ds(N) {}
- // Returns value of MST and fills mst vector with
- // the edges from the MST.
int run(vector<iii> &mst) {
-
- // Sort by weight of the edges
sort(all(edges), [&](const iii &a, const iii &b) {
// ('>' for maximum spanning tree)
return a.se < b.se;
@@ -32,8 +26,6 @@ struct Kruskal {
int pu = ds.find_set(edges[i].fi.fi);
int pv = ds.find_set(edges[i].fi.se);
- // If the sets are different, then the edge i does
- // not close a cycle
if (pu != pv) {
mst.pb(edges[i]);
size += edges[i].se;
diff --git a/algorithms/math/big_integer.cpp b/algorithms/math/big_integer.cpp
index f88c09f0d43e6a46947257b07523f79d4f69de14..bc69d5eaec850bdecec2cb4022ae0b3dd1603cf9 100644
--- a/algorithms/math/big_integer.cpp
+++ b/algorithms/math/big_integer.cpp
@@ -4,6 +4,8 @@
///
/// Status: Tested (KRAKOVIA)
/// Caution: Just use python.
+/// Include:
+/// - math/karatsuba
const int base = 1000000000;
const int base_d = 9;
@@ -31,7 +33,6 @@ struct BigInt {
BigInt operator+(const BigInt &x) const {
if (sign != x.sign) return *this - (-x);
-
int carry = 0;
BigInt res = x;
@@ -85,8 +86,8 @@ struct BigInt {
return res;
}
- friend pair<BigInt, BigInt> divmod(const BigInt &a1,
- const BigInt &b1)
+ friend pair<BigInt, BigInt> divmod(const BigInt &a1,
+ const BigInt &b1)
{
int norm = base / (b1.back() + 1);
BigInt a = a1.abs() * norm;
@@ -150,23 +151,15 @@ struct BigInt {
return m * sign;
}
- void operator+=(const BigInt &x) {
- *this = *this + x;
- }
- void operator-=(const BigInt &x) {
- *this = *this - x;
- }
- void operator*=(const BigInt &x) {
- *this = *this * x;
- }
- void operator/=(const BigInt &x) {
- *this = *this / x;
- }
+ void operator+=(const BigInt &x) { *this = *this + x; }
+ void operator-=(const BigInt &x) { *this = *this - x; }
+ void operator*=(const BigInt &x) { *this = *this * x; }
+ void operator/=(const BigInt &x) { *this = *this / x; }
bool operator<(const BigInt &x) const {
if (sign != x.sign)
return sign < x.sign;
-
+
if (size() != x.size())
return size() * sign < x.size() * x.sign;
@@ -224,17 +217,17 @@ struct BigInt {
friend BigInt gcd(const BigInt &a, const BigInt &b) {
return b.is_zero() ? a : gcd(b, a % b);
}
+
friend BigInt lcm(const BigInt &a, const BigInt &b) {
return a / gcd(a, b) * b;
}
- // Transforms string into BigInt.
void read(const string &s) {
sign = 1;
num.clear();
int pos = 0;
- while (pos < s.size() &&
+ while (pos < s.size() &&
(s[pos] == '-' || s[pos] == '+')) {
if (s[pos] == '-')
sign = -sign;
@@ -268,9 +261,9 @@ struct BigInt {
return stream;
}
- static vector<int> convert_base(const vector<int> &a,
- int oldd, int newd)
- {
+ static vector<int> convert_base(
+ const vector<int> &a,
+ int oldd, int newd) {
vector<ll> p(max(oldd, newd) + 1);
p[0] = 1;
for (int i = 1; i < p.size(); i++)
@@ -297,44 +290,6 @@ struct BigInt {
return res;
}
- static vector<ll> karatsuba(const vector<ll> &a,
- const vector<ll> &b)
- {
- int n = a.size();
- vector<ll> res(n + n);
-
- if (n <= 32) {
- for (int i = 0; i < n; i++)
- for (int j = 0; j < n; j++)
- res[i + j] += a[i] * b[j];
- return res;
- }
-
- int k = n >> 1;
- vector<ll> a1(a.begin(), a.begin() + k);
- vector<ll> a2(a.begin() + k, a.end());
- vector<ll> b1(b.begin(), b.begin() + k);
- vector<ll> b2(b.begin() + k, b.end());
-
- vector<ll> a1b1 = karatsuba(a1, b1);
- vector<ll> a2b2 = karatsuba(a2, b2);
-
- for (int i = 0; i < k; i++)
- a2[i] += a1[i];
- for (int i = 0; i < k; i++)
- b2[i] += b1[i];
-
- vector<ll> r = karatsuba(a2, b2);
- for (int i = 0; i < a1b1.size(); i++) r[i] -= a1b1[i];
- for (int i = 0; i < a2b2.size(); i++) r[i] -= a2b2[i];
-
- for (int i = 0; i < r.size(); i++) res[i + k] += r[i];
- for (int i = 0; i < a1b1.size(); i++) res[i] += a1b1[i];
- for (int i = 0; i < a2b2.size(); i++) res[i + n] += a2b2[i];
-
- return res;
- }
-
BigInt operator*(const BigInt &x) const {
vector<int> a6 = convert_base(this->num, base_d, 6);
vector<int> b6 = convert_base(x.num, base_d, 6);
@@ -344,10 +299,11 @@ struct BigInt {
while (a.size() < b.size()) a.pb(0);
while (b.size() < a.size()) b.pb(0);
- while (a.size() & (a.size() - 1))
+ while (a.size() & (a.size() - 1))
a.pb(0), b.pb(0);
- vector<ll> c = karatsuba(a, b);
+ Karatsuba karatsuba;
+ vector<ll> c = karatsuba.run(a, b);
BigInt res;
int carry = 0;
diff --git a/algorithms/math/karatsuba.cpp b/algorithms/math/karatsuba.cpp
new file mode 100644
index 0000000000000000000000000000000000000000..a3c5d10fe9f5ef85d0f794e6ae6d837a991e4801
--- /dev/null
+++ b/algorithms/math/karatsuba.cpp
@@ -0,0 +1,48 @@
+/// Karatsuba
+///
+/// Time: O(n^{log(3)})
+/// Space: O(n)
+
+struct Karatsuba {
+ vector<ll> run(const vector<ll> &a, const vector<ll> &b) {
+ int n = a.size();
+ vector<ll> res(n + n);
+
+ if (n <= 32) {
+ for (int i = 0; i < n; i++)
+ for (int j = 0; j < n; j++)
+ res[i + j] += a[i] * b[j];
+
+ return res;
+ }
+
+ int k = n >> 1;
+ vector<ll> a1(a.begin(), a.begin() + k);
+ vector<ll> a2(a.begin() + k, a.end());
+ vector<ll> b1(b.begin(), b.begin() + k);
+ vector<ll> b2(b.begin() + k, b.end());
+
+ vector<ll> a1b1 = run(a1, b1);
+ vector<ll> a2b2 = run(a2, b2);
+
+ for (int i = 0; i < k; i++)
+ a2[i] += a1[i];
+ for (int i = 0; i < k; i++)
+ b2[i] += b1[i];
+
+ vector<ll> r = run(a2, b2);
+ for (int i = 0; i < a1b1.size(); i++)
+ r[i] -= a1b1[i];
+ for (int i = 0; i < a2b2.size(); i++)
+ r[i] -= a2b2[i];
+
+ for (int i = 0; i < r.size(); i++)
+ res[i + k] += r[i];
+ for (int i = 0; i < a1b1.size(); i++)
+ res[i] += a1b1[i];
+ for (int i = 0; i < a2b2.size(); i++)
+ res[i + n] += a2b2[i];
+
+ return res;
+ }
+};
diff --git a/caderno.pdf b/caderno.pdf
index 3ae532569df79b62bfa711e4c399d821fab6ba0d..f9ce52fea430d4dc1a24f210c59ff867c4b99e5e 100644
Binary files a/caderno.pdf and b/caderno.pdf differ