Merge pull request #403 from rgkbitw/PR2

Added Binary Lifting O(nlogn) method to find Lowest Common Ancestor

Dynamic Programming/Longest Increasing Subsequence (nlogn).cpp

@@ -0,0 +1,46 @@
+//Program to calculate length of longest increasing subsequence in an array
+// in O(n log n)
+// tested on : https://cses.fi/problemset/task/1145/
+
+#include <bits/stdc++.h>
+
+using namespace std;
+int LIS(int arr[], int n)
+{
+    set < int > active; // The current built LIS.
+    active.insert(arr[0]);
+    // Loop through every element.
+    for (int i = 1; i < n; ++i)
+    {
+        auto get = active.lower_bound(arr[i]);
+        if (get == active.end())
+	{
+            active.insert(arr[i]);
+        } // current element is the greatest so LIS increases by 1.
+        else
+	{
+            int val = * get; // we find the position where arr[i] will be in the LIS. If it is in the LIS already we do nothing
+            if (val > arr[i])
+	    {
+                // else we remove the bigger element and add a smaller element (which is arr[i]) and continue;
+                active.erase(get);
+                active.insert(arr[i]);
+            }
+        }
+    }
+    return active.size(); // size of the LIS.
+}
+int main(int argc, char const * argv[])
+{
+    int n;
+    cout << "Enter size of array: ";
+    cin >> n;
+    int a[n];
+    cout << "Enter array elements: ";
+    for (int i = 0; i < n; ++i)
+    {
+        cin >> a[i];
+    }
+    cout << LIS(a, n) << endl;
+    return 0;
+}

Graph/lca.cpp

@@ -0,0 +1,120 @@
+#include<bits/stdc++.h>
+
+using namespace std;
+// Find the lowest common ancestor using binary lifting in O(nlogn)
+// Zero based indexing
+// Resource : https://cp-algorithms.com/graph/lca_binary_lifting.html
+const int N = 1005;
+const int LG = log2(N) + 1;
+struct lca
+{
+    int n;
+    vector<int> adj[N]; // Graph
+    int up[LG][N]; // build this table
+    int level[N]; // get the levels of all of them
+
+    lca(int n_): n(n_)
+    {
+        memset(up, -1, sizeof(up));
+        memset(level, 0, sizeof(level));
+        for (int i = 0; i < n - 1; ++i)
+	{
+            int a, b;
+            cin >> a >> b;
+            a--;
+            b--;
+            adj[a].push_back(b);
+            adj[b].push_back(a);
+        }
+        level[0] = 0;
+        dfs(0, -1);
+        build();
+    }
+    void verify()
+    {
+        for (int i = 0; i < n; ++i)
+	{
+            cout << i << " : level: " << level[i] << endl;
+        }
+        cout << endl;
+        for (int i = 0; i < LG; ++i)
+	{
+            cout << "Power:" << i << ": ";
+            for (int j = 0; j < n; ++j)
+	    {
+                cout << up[i][j] << " ";
+            }
+            cout << endl;
+        }
+    }
+
+    void build()
+    {
+        for (int i = 1; i < LG; ++i)
+	{
+            for (int j = 0; j < n; ++j)
+	    {
+                if (up[i - 1][j] != -1)
+		{
+                    up[i][j] = up[i - 1][up[i - 1][j]];
+                }
+            }
+        }
+    }
+
+    void dfs(int node, int par)
+    {
+        up[0][node] = par;
+        for (auto i: adj[node])
+	{
+            if (i != par)
+	    {
+                level[i] = level[node] + 1;
+                dfs(i, node);
+            }
+        }
+    }
+    int query(int u, int v)
+    {
+        u--;
+        v--;
+        if (level[v] > level[u])
+	{
+            swap(u, v);
+        }
+        // u is at the bottom.
+        int dist = level[u] - level[v];
+        // Go up this much distance
+        for (int i = LG - 1; i >= 0; --i)
+	{
+            if (dist & (1 << i))
+	    {
+                u = up[i][u];
+            }
+        }
+        if (u == v)
+	{
+            return u;
+        }
+        assert(level[u] == level[v]);
+        for (int i = LG - 1; i >= 0; --i)
+	{
+            if (up[i][u] != up[i][v])
+	    {
+                u = up[i][u];
+                v = up[i][v];
+            }
+        }
+        assert(up[0][u] == up[0][v]);
+        return up[0][u];
+    }
+};
+
+int main()
+{
+    int n; // number of nodes in the tree.
+    lca l(n); // will take the input in the format given
+    // n-1 edges of the form
+    // a b
+    // Use verify function to see.
+}

Range queries/bit.cpp

@@ -0,0 +1,76 @@
+// Binary Indexed Tree.
+#include<bits/stdc++.h>
+
+using namespace std;
+
+class Bit
+{
+    int n;
+    vector<int> bit;
+    inline int offset(int x)
+    {
+        return (x & (-x));
+    }
+
+    public:
+
+        Bit(vector<int>& arr)
+	{
+            n = arr.size();
+            bit.assign(n + 1, 0);
+            for (int i = 0; i < n; ++i)
+	    {
+                update(i, arr[i]);
+            }
+        }
+    Bit(int x)
+    {
+        n = x;
+        bit.assign(n + 1, 0);
+    }
+
+    void update(int id, int val)
+    {
+        // Add val at id
+        id++;
+        while (id <= n)
+	{
+            bit[id] += val;
+            id += offset(id);
+        }
+    }
+
+    int sum(int id)
+    {
+        // Get prefix sum upto id.
+        id++;
+        int res = 0;
+        while (id > 0)
+	{
+            res += bit[id];
+            id -= offset(id);
+        }
+        return res;
+    }
+
+    int sum_range(int l, int r)
+    {
+        return sum(r) - sum(l - 1);
+    }
+};
+
+int main()
+{
+    int n = 5;
+    vector<int> arr = { 1, 2, 3, 4, 5 };
+    Bit x(arr);
+
+    assert(x.sum_range(0, 0) == 1);
+    assert(x.sum_range(0, 1) == 3);
+    assert(x.sum_range(0, 2) == 6);
+    x.update(0, 6);
+    assert(x.sum_range(0, 0) == 6);
+    assert(x.sum_range(0, 1) == 8);
+    assert(x.sum_range(0, 2) == 11);
+	return 0;
+}