From 34135a80152a8a48201704244811c5d55abdc3ea Mon Sep 17 00:00:00 2001
From: Bruno Freitas Tissei <bft15@inf.ufpr.br>
Date: Mon, 30 Apr 2018 14:54:14 +0200
Subject: [PATCH] Add fast matrix pow
Signed-off-by: Bruno Freitas Tissei <bft15@inf.ufpr.br>
---
contests/Cadernaveis/DESCULPA.cpp | 46 ++++++++++++++
contests/Cadernaveis/TOPOLAND.cpp | 102 ++++++++++++++++++++++++++++++
math/fast_matrix_pow.cpp | 63 ++++++++++++++++++
3 files changed, 211 insertions(+)
create mode 100644 contests/Cadernaveis/DESCULPA.cpp
create mode 100644 contests/Cadernaveis/TOPOLAND.cpp
create mode 100644 math/fast_matrix_pow.cpp
diff --git a/contests/Cadernaveis/DESCULPA.cpp b/contests/Cadernaveis/DESCULPA.cpp
new file mode 100644
index 0000000..4b05311
--- /dev/null
+++ b/contests/Cadernaveis/DESCULPA.cpp
@@ -0,0 +1,46 @@
+#include <bits/stdc++.h>
+
+#define MAX 1010
+#define MOD 1000000007
+#define inf 0x3f3f3f3f
+
+#define fi first
+#define se second
+#define sz size()
+#define pb push_back
+#define ende '\n'
+#define all(x) (x).begin(), (x).end()
+#define rall(x) (x).rbegin(), (x).rend()
+
+using namespace std;
+
+typedef long long ll;
+typedef pair<int,int> ii;
+
+int dp[60][MAX];
+int v[MAX], w[MAX];
+
+int main() {
+ ios::sync_with_stdio(0);
+ cin.tie(0);
+
+ int c, f, cas = 1;
+ while (cin >> c >> f && (c || f)) {
+ memset(dp, 0, sizeof dp);
+ for (int i = 1; i <= f; ++i)
+ cin >> w[i] >> v[i];
+
+ for (int i = 1; i <= f; ++i)
+ for (int j = 0; j <= c; ++j)
+ if (j >= w[i])
+ dp[i][j] = max(dp[i-1][j], dp[i-1][j - w[i]] + v[i]);
+ else
+ dp[i][j] = dp[i-1][j];
+
+ cout << "Teste " << cas << ende;
+ cout << dp[f][c] << ende << ende;
+ cas++;
+ }
+
+ return 0;
+}
diff --git a/contests/Cadernaveis/TOPOLAND.cpp b/contests/Cadernaveis/TOPOLAND.cpp
new file mode 100644
index 0000000..3db4470
--- /dev/null
+++ b/contests/Cadernaveis/TOPOLAND.cpp
@@ -0,0 +1,102 @@
+#include <bits/stdc++.h>
+
+#define MAX 101010
+#define MOD 1000000007
+#define inf 0x3f3f3f3f
+
+#define fi first
+#define se second
+#define sz size()
+#define pb push_back
+#define all(x) (x).begin(), (x).end()
+#define rall(x) (x).rbegin(), (x).rend()
+
+using namespace std;
+
+typedef long long ll;
+typedef pair<int,int> ii;
+
+ll N, tree[4 * MAX], v[MAX];
+
+void build_tree(int node = 1, int a = 0, int b = N) {
+ if (a > b)
+ return;
+
+ if (a == b) {
+ tree[node] = v[a];
+ return;
+ }
+
+ build_tree(node * 2, a, (a + b) / 2);
+ build_tree(node * 2 + 1, 1 + (a + b) / 2, b);
+
+ tree[node] = max(tree[node * 2], tree[node * 2 + 1]);
+}
+
+
+void update_tree(int idx, ll val, int node = 1, int a = 0, int b = N) {
+ if (a > b || a > idx || b < idx)
+ return;
+
+ if (a == b) {
+ tree[node] = val;
+ return;
+ }
+
+ update_tree(idx, val, node * 2, a, (a + b) / 2);
+ update_tree(idx, val, node * 2 + 1, 1 + (a + b) / 2, b);
+
+ tree[node] = max(tree[node * 2], tree[node * 2 + 1]);
+}
+
+
+int query_tree(int i, int j, int node = 1, int a = 0, int b = N) {
+ if (a > b || a > j || b < i)
+ return 0;
+
+ if (a >= i && b <= j)
+ return tree[node];
+
+ int res1 = query_tree(i, j, node * 2, a, (a + b) / 2);
+ int res2 = query_tree(i, j, node * 2 + 1, 1 + (a + b) / 2, b);
+
+ return max(res1, res2);
+}
+
+
+int main() {
+ ios::sync_with_stdio(0);
+ cin.tie(0);
+
+ ll t, n, m;
+ cin >> t;
+ for (int cas = 0; cas < t; ++cas) {
+ cout << "Testcase " << cas << ":\n";
+ cin >> n >> m;
+ N = n;
+
+ for (int i = 0; i < n; ++i)
+ cin >> v[i];
+ build_tree();
+
+ ll q, a, b;
+ string op;
+ cin >> q;
+ for (int i = 0; i < q; ++i) {
+ cin >> op;
+ if (op[0] == 'A') {
+ cin >> a;
+ m += a;
+ } else if (op[0] == 'B') {
+ cin >> a >> b;
+ update_tree(a, b);
+ } else {
+ cin >> a >> b;
+ cout << abs(m - query_tree(a, b)) << '\n';
+ }
+ }
+ cout << '\n';
+ }
+
+ return 0;
+}
diff --git a/math/fast_matrix_pow.cpp b/math/fast_matrix_pow.cpp
new file mode 100644
index 0000000..c8c02ef
--- /dev/null
+++ b/math/fast_matrix_pow.cpp
@@ -0,0 +1,63 @@
+/**
+ * Fast Matrix Exponentiation
+ *
+ * Complexity (Time): O(K^3 log n)
+ * Complexity (Space): O(K^2)
+ */
+
+// This algorithm is used to solve recurrences such as:
+// f(n) = x1 * f(n - 1) + x2 * f(n - 1) + ... + xk * f(n - k)
+//
+// It works by defining this recurrence as a linear combination,
+// for example (k = 2):
+// f(n) = [x1 x2] [f(n - 1)]
+// [f(n - 2)]
+// It can be rewriten as:
+// [ f(n) ] = [x1 x2] [f(n - 1)]
+// [f(n - 1)] [ 1 0] [f(n - 2)]
+//
+// And that is solved by calculating the following matrix power:
+// [x1 x2]^n
+// [ 1 0]
+
+#define K 2
+
+struct mat {
+ ll m[K][K];
+
+ // Matrix multiplication - O(k^3)
+ mat operator*(mat a) {
+ mat aux;
+
+ for(int i = 0; i < K; i++)
+ for(int j = 0; j < K; j++) {
+ ll sum = 0;
+ for(int k = 0; k < K; k++)
+ sum += (m[i][k] * a.m[k][j]) % MOD;
+ aux.m[i][j] = sum % MOD;
+ }
+
+ return aux;
+ }
+};
+
+
+// Fast exponentiation (can be used with integers as well) - O(log n)
+ll mat_pow(mat in, ll n) {
+ mat ans, b = in;
+
+ // Set ans as identity matrix
+ memset(ans.m, 0, sizeof ans.m);
+ for (int i = 0; i < K; ++i)
+ ans.m[i][i] = 1;
+
+ while (n) {
+ if (n & 1)
+ ans = ans * b;
+
+ n >>= 1;
+ b = b * b;
+ }
+
+ return ans;
+}
--
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