Added two new Algorithms in Bit Manipulation and Number Theory

Author: zee-bit

06-Bit Manipulation/Unique Bitwise OR of subarrays.cpp

@@ -0,0 +1,35 @@
+// This algorithm finds unique Bitwise OR of
+// all the subarrays of an array.
+// The Time Complexity might look like
+// its O(n^2), but upon taking a closer look 
+// at algorithm, it comes out to be of the 
+// order of O(n)
+#include <bits/stdc++.h>
+using namespace std;
+
+int subarrayBitwiseORs(vector<int>& A) {
+    unordered_set<int> st;
+    int n = A.size(), chk = 0, curr;
+    for(int i = n - 1; i >= 0; i--) {
+        curr = A[i];
+        st.insert(curr);
+        chk = 0;
+        for(int j = i + 1; j < n && chk != curr; j++) {
+            curr |= A[j];
+            chk |= A[j];
+            st.insert(curr);
+        }
+    }
+    return st.size();
+}
+
+int main() {
+    int n;
+    cin >> n;
+    vector<int> arr(n);
+    for(int i = 0; i < n; i++) {
+        cin >> arr[i];
+    }
+    cout << subarrayBitwiseORs(arr) << endl;
+    return 0;
+}

09- Number Theory/Binary Multiplication.cpp

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+// This code will have a complexity of O(log b), but
+// we can use __int128 in C++ to avoid any potential
+// overflows and work in O(1).
+#include <bits/stdc++.h>
+using namespace std;
+typedef long long ll;
+
+ll binaryMultiplication(ll a, ll b) {
+	ll x = 0, y = a;
+	while (b > 0) {
+		if (b & 1)
+			x = (x + y);
+		y = (y + y);
+		b >>= 1;
+	}
+	return x;
+}
+
+int main() {
+    ll a, b;
+    cin >> a >> b;
+    // Finding a^b
+    cout << binaryMultiplication(a, b) << endl;
+    return 0;
+}