From 98a632c944843f47be7ef4869f9d68ebec03780f Mon Sep 17 00:00:00 2001
From: Milan Pokharna <MilanPokharna@users.noreply.github.com>
Date: Sat, 24 Oct 2020 08:15:15 +0530
Subject: [PATCH] conceptual algorithms

---
 Fast_Exponentiation.cpp   | 22 ++++++++++++
 Modular Exponentation.cpp | 31 ++++++++++++++++
 Segmented Sieve.cpp       | 76 +++++++++++++++++++++++++++++++++++++++
 linearRecurssion.cpp      | 47 ++++++++++++++++++++++++
 4 files changed, 176 insertions(+)
 create mode 100644 Fast_Exponentiation.cpp
 create mode 100644 Modular Exponentation.cpp
 create mode 100644 Segmented Sieve.cpp
 create mode 100644 linearRecurssion.cpp

diff --git a/Fast_Exponentiation.cpp b/Fast_Exponentiation.cpp
new file mode 100644
index 00000000..9ed8e9b6
--- /dev/null
+++ b/Fast_Exponentiation.cpp
@@ -0,0 +1,22 @@
+#include<bits/stdc++.h>
+using namespace std;
+
+typedef long long int ll;
+
+ll fastExpo(int x,int n)
+{	
+	if (n==0) return 1;
+	if (n & 1)
+		return fastExpo(x,n-1) *  x;
+	int y = fastExpo(x,n/2);
+	return y*y;
+}
+
+int main()
+{
+	int base,power;
+	cin>>base>>power;
+	cout<<fastExpo(base,power)<<endl;
+
+	return 0;
+}
\ No newline at end of file
diff --git a/Modular Exponentation.cpp b/Modular Exponentation.cpp
new file mode 100644
index 00000000..202dbb44
--- /dev/null
+++ b/Modular Exponentation.cpp	
@@ -0,0 +1,31 @@
+#include <iostream>
+using namespace std ;
+
+int power(int x, unsigned int y, int p)
+{
+   int res = 1;      // Initialize result
+
+   x = x % p;  // Update x if it is more than or 
+               // equal to p
+
+   while (y > 0)
+   {
+       // If y is odd, multiply x with result
+       if (y & 1)
+           res = (res*x) % p;
+
+       // y must be even now
+       y = y>>1; // y = y/2
+       x = (x*x) % p;  
+   }
+   return res;
+}
+
+int main()
+{
+  int x = 2;
+  int y = 5;
+  int p = 13;
+  cout << "Power is " << power(x, y, p);
+  return 0;
+}
\ No newline at end of file
diff --git a/Segmented Sieve.cpp b/Segmented Sieve.cpp
new file mode 100644
index 00000000..c2001fe8
--- /dev/null
+++ b/Segmented Sieve.cpp	
@@ -0,0 +1,76 @@
+#include <bits/stdc++.h>
+using namespace std;
+
+
+void simpleSieve(int limit, vector<int> &prime)
+{
+    bool mark[limit+1];
+    memset(mark, true, sizeof(mark));
+ 
+    for (int p=2; p*p<limit; p++)
+    {
+        // If p is not changed, then it is a prime
+        if (mark[p] == true)
+        {
+            // Update all multiples of p
+            for (int i=p*2; i<limit; i+=p)
+                mark[i] = false;
+        }
+    }
+ 
+    // Print all prime numbers and store them in prime
+    for (int p=2; p<limit; p++)
+    {
+        if (mark[p] == true)
+        {
+            prime.push_back(p);
+            cout << p << "  ";
+        }
+    }
+}
+
+
+void segmentedSieve(int n)
+{
+    int limit = floor(sqrt(n))+1;
+    vector<int> prime;  
+    simpleSieve(limit, prime); 
+ 
+    int low  = limit;
+    int high = 2*limit;
+ 
+    while (low < n)
+    {
+        bool mark[limit+1];
+        memset(mark, true, sizeof(mark));
+ 
+        for (int i = 0; i < prime.size(); i++)
+        {
+            int loLim = floor(low/prime[i]) * prime[i];
+            if (loLim < low)
+                loLim += prime[i];
+ 
+            for (int j=loLim; j<high; j+=prime[i])
+                mark[j-low] = false;
+        }
+ 
+        // Numbers which are not marked as false are prime
+        for (int i = low; i<high; i++)
+            if (mark[i - low] == true)
+                cout << i << "  ";
+ 
+        // Update low and high for next segment
+        low  = low + limit;
+        high = high + limit;
+        if (high >= n) high = n;
+    }
+}
+ 
+// Driver program to test above function
+int main()
+{
+    int n = 100;
+    cout << "Primes smaller than " << n << ":n";
+    segmentedSieve(n);
+    return 0;
+}
diff --git a/linearRecurssion.cpp b/linearRecurssion.cpp
new file mode 100644
index 00000000..8864e828
--- /dev/null
+++ b/linearRecurssion.cpp
@@ -0,0 +1,47 @@
+//Template for solving Linear Recurence Relations in LogN time by using fast exponentiation
+#include <bits/stdc++.h>
+using namespace std;
+
+typedef long long ll;
+typedef unsigned long long ull;
+typedef unsigned long long int ulli;
+#define flash ios_base::sync_with_stdio(false); cin.tie(NULL);
+#define mat(x, y, name) vector< vector<ll> > name (x, vector<ll>(y));
+#define printMat(name) for (int i = 0; i < name.size(); i++) {for (int j = 0; j < res[i].size(); j++) cout << res[i][j] << " "; cout << endl;}
+
+#define MOD 98765431
+
+vector< vector<ll> > matMul(vector< vector<ll> > A, vector< vector<ll> > B)
+{
+    vector< vector<ll> > C(A.size(), vector<ll>(B[0].size()));
+    for (int i = 0; i < A.size(); i++)
+    {
+        for (int j = 0; j < B[0].size(); j++)
+        {
+            C[i][j] = 0;
+            for (int k = 0; k < B.size(); k++)
+                C[i][j] = (C[i][j] + ((A[i][k] * B[k][j]) % MOD)) % MOD;
+        }
+    }
+    return C;
+}
+
+vector< vector<ll> > matPow(vector< vector<ll> > A, int p)
+{
+    if (p == 1)
+        return A;
+    if (p&1)
+        return matMul(A, matPow(A, p-1));
+    else
+    {
+        vector< vector<ll> > C = matPow(A, p/2);
+        return matMul(C, C);
+    }
+}
+
+int main()
+{
+    flash;
+
+    return 0;
+}