feat: add spare table data structure

Added implementation of sparse table for a 'duplicate invariant function'. Implementation of min(a1,a2...,aN) being the duplicate invariant function is done.

data_structures/sparse_table.cpp

@@ -0,0 +1,139 @@
+#include<bits/stdc++.h>
+#include <iostream>
+#include <cassert>
+/**
+ *
+ * @file
+ * @brief Implementation of Sparse Table
+ *
+ * @author [manncodes](https://github.com/manncodes)
+ *
+ * @details
+ * Sparse Table is a data structure, that allows answering range queries. 
+ * It can answer most range queries in O(logn), but its true power is answering
+ * range minimum queries (or equivalent range maximum queries). For those queries
+ * it can compute the answer in O(1) time. The only drawback of this data structure is,
+ * that it can only be used on immutable arrays. This means, that the array cannot be 
+ * changed between two queries. 
+ * 
+ * If any element in the array changes, the complete data structure has to be recomputed.
+ *
+ * @warning this sparse table is made for min(a1,a2,...an) duplicate invariant function.
+ * This implementation can be changed to other functions like gcd(),lcm(),max() by changing 
+ * few lines of code.
+ */
+
+
+/**
+ * @namespace data_structures
+ * @brief Data Structures algorithms
+ */
+namespace data_structures {
+    
+/**
+ * @namespace sparse_table
+ * @brief Implementation of Sparse Table for function : min()
+ * 
+ */
+namespace sparse_table {
+
+    /**
+     * @brief A struct to represent sparse table
+     * 
+     */
+    struct Sparse_table{
+        // N : the maximum size of the array.
+        // M : ceil(log2(N)).
+        const static int N = 12345, M = 14;
+
+        // The array to compute its sparse table.
+        int n, a[N];
+
+        //
+        // Sparse table related variables.
+        //
+        // ST[j][i] : the sparse table value (min, max, ...etc) in the interval [i, i + (2^j) - 1].
+        // LOG[i]   : floor(log2(i)).
+        int ST[M][N], LOG[N];
+
+        /**
+         * 
+         * Builds the sparse table for computing min/max/gcd/lcm/...etc
+         * for any contiguous sub-segment of the array.
+         * 
+         * This is an example of computing the index of the minimum value.
+         * 
+         * Complexity: O(n.log(n))
+         */
+        void buildST() {
+            LOG[0] = -1;
+
+            for (int i = 0; i < n; ++i) {
+                ST[0][i] = i;
+                LOG[i + 1] = LOG[i] + !(i & (i + 1));
+            }
+
+            for (int j = 1; (1 << j) <= n; ++j) {
+                for (int i = 0; (i + (1 << j)) <= n; ++i) {
+                    int x = ST[j - 1][i];
+                    int y = ST[j - 1][i + (1 << (j - 1))];
+
+                    ST[j][i] = (a[x] <= a[y] ? x : y);
+                }
+            }
+        }
+
+        /**
+         * Queries the sparse table for the value of the interval [l, r]
+         * (i.e. from l to r inclusive).
+         * 
+         * Complexity: O(1)
+         * 
+         * @param l the left index of the range (inclusive).
+         * @param r the right index of the range (inclusive).
+         * 
+         * @return the computed value of the given interval.
+         */
+        int query(int l, int r) {
+            int g = LOG[r - l + 1];
+            int x = ST[g][l];
+            int y = ST[g][r - (1 << g) + 1];
+            return (a[x] <= a[y] ? x : y);
+        }
+    };
+} // namespace sparse_table
+} // namespace data_structures
+
+/**
+ * @brief testcase for sparse_table
+ */
+static void test(){
+    int testcase[] = {1,2,3,4,5,6,7,8,9,10};
+    int testcase_size = sizeof(testcase)/sizeof(testcase[0]);
+    
+    data_structures::sparse_table::Sparse_table st;
+    
+    // copying testcase to the struct
+    std::copy(std::begin(testcase),std::end(testcase),std::begin(st.a));
+    st.n = testcase_size;
+
+    //precomputing sparse tree
+    st.buildST();
+    // pass queries of form [l,r]
+    assert(st.query(1,9) == 1); // as 1 is smallest from 1..9
+    assert(st.query(2,6) == 2); // as 2 is smallest from 2..6
+    assert(st.query(3,8) == 3); // as 3 is smallest from 3..8
+
+    std::cout << "Testcase passed!" << std::endl;
+}
+
+/**
+ * @brief Main function
+ * @param argc commandline argument count (ignored)
+ * @param argv commandline array of arguments (ignored)
+ * @returns 0 on exit
+ */
+int main(int argc, char *argv[]){
+    test(); // execute the tests
+    return 0;
+}