Merge branch 'master' into add/number_of_paths

dynamic_programming/Unbounded_0_1_Knapsack.cpp

@@ -0,0 +1,151 @@
+/**
+ * @file
+ * @brief Implementation of the Unbounded 0/1 Knapsack Problem
+ * 
+ * @details 
+ * The Unbounded 0/1 Knapsack problem allows taking unlimited quantities of each item. 
+ * The goal is to maximize the total value without exceeding the given knapsack capacity. 
+ * Unlike the 0/1 knapsack, where each item can be taken only once, in this variation, 
+ * any item can be picked any number of times as long as the total weight stays within 
+ * the knapsack's capacity.
+ * 
+ * Given a set of N items, each with a weight and a value, represented by the arrays 
+ * `wt` and `val` respectively, and a knapsack with a weight limit W, the task is to 
+ * fill the knapsack to maximize the total value.
+ *
+ * @note weight and value of items is greater than zero 
+ *
+ * ### Algorithm
+ * The approach uses dynamic programming to build a solution iteratively. 
+ * A 2D array is used for memoization to store intermediate results, allowing 
+ * the function to avoid redundant calculations.
+ * 
+ * @author [Sanskruti Yeole](https://github.com/yeolesanskruti)
+ * @see dynamic_programming/0_1_knapsack.cpp
+ */
+
+#include <iostream>  // Standard input-output stream
+#include <vector>   // Standard library for using dynamic arrays (vectors)
+#include <cassert>  // For using assert function to validate test cases
+#include <cstdint>  // For fixed-width integer types like std::uint16_t
+
+/**
+ * @namespace dynamic_programming
+ * @brief Namespace for dynamic programming algorithms
+ */
+namespace dynamic_programming {
+
+/**
+ * @namespace Knapsack
+ * @brief Implementation of unbounded 0-1 knapsack problem
+ */
+namespace unbounded_knapsack {
+
+/**
+ * @brief Recursive function to calculate the maximum value obtainable using 
+ *        an unbounded knapsack approach.
+ *
+ * @param i Current index in the value and weight vectors.
+ * @param W Remaining capacity of the knapsack.
+ * @param val Vector of values corresponding to the items.
+ * @note "val" data type can be changed according to the size of the input.
+ * @param wt Vector of weights corresponding to the items.
+ * @note "wt" data type can be changed according to the size of the input.
+ * @param dp 2D vector for memoization to avoid redundant calculations.
+ * @return The maximum value that can be obtained for the given index and capacity.
+ */
+std::uint16_t KnapSackFilling(std::uint16_t i, std::uint16_t W, 
+                    const std::vector<std::uint16_t>& val, 
+                    const std::vector<std::uint16_t>& wt, 
+                    std::vector<std::vector<int>>& dp) {
+    if (i == 0) {
+        if (wt[0] <= W) {
+            return (W / wt[0]) * val[0]; // Take as many of the first item as possible
+        } else {
+            return 0; // Can't take the first item
+        }
+    }
+    if (dp[i][W] != -1) return dp[i][W]; // Return result if available
+
+    int nottake = KnapSackFilling(i - 1, W, val, wt, dp); // Value without taking item i
+    int take = 0;
+    if (W >= wt[i]) {
+        take = val[i] + KnapSackFilling(i, W - wt[i], val, wt, dp); // Value taking item i
+    }
+    return dp[i][W] = std::max(take, nottake); // Store and return the maximum value
+}
+
+/**
+ * @brief Wrapper function to initiate the unbounded knapsack calculation.
+ *
+ * @param N Number of items.
+ * @param W Maximum weight capacity of the knapsack.
+ * @param val Vector of values corresponding to the items.
+ * @param wt Vector of weights corresponding to the items.
+ * @return The maximum value that can be obtained for the given capacity.
+ */
+std::uint16_t unboundedKnapsack(std::uint16_t N, std::uint16_t W, 
+                      const std::vector<std::uint16_t>& val, 
+                      const std::vector<std::uint16_t>& wt) {
+    if(N==0)return 0; // Expect 0 since no items
+    std::vector<std::vector<int>> dp(N, std::vector<int>(W + 1, -1)); // Initialize memoization table
+    return KnapSackFilling(N - 1, W, val, wt, dp); // Start the calculation
+}
+
+} // unbounded_knapsack
+
+} // dynamic_programming
+
+/**
+ * @brief self test implementation
+ * @return void
+ */
+static void tests() {
+    // Test Case 1
+    std::uint16_t N1 = 4;  // Number of items
+    std::vector<std::uint16_t> wt1 = {1, 3, 4, 5}; // Weights of the items
+    std::vector<std::uint16_t> val1 = {6, 1, 7, 7}; // Values of the items
+    std::uint16_t W1 = 8; // Maximum capacity of the knapsack
+    // Test the function and assert the expected output
+    assert(unboundedKnapsack(N1, W1, val1, wt1) == 48);
+    std::cout << "Maximum Knapsack value " << unboundedKnapsack(N1, W1, val1, wt1) << std::endl;
+
+    // Test Case 2
+    std::uint16_t N2 = 3; // Number of items
+    std::vector<std::uint16_t> wt2 = {10, 20, 30}; // Weights of the items
+    std::vector<std::uint16_t> val2 = {60, 100, 120}; // Values of the items
+    std::uint16_t W2 = 5; // Maximum capacity of the knapsack
+    // Test the function and assert the expected output
+    assert(unboundedKnapsack(N2, W2, val2, wt2) == 0);
+    std::cout << "Maximum Knapsack value " << unboundedKnapsack(N2, W2, val2, wt2) << std::endl;
+
+    // Test Case 3
+    std::uint16_t N3 = 3; // Number of items
+    std::vector<std::uint16_t> wt3 = {2, 4, 6}; // Weights of the items
+    std::vector<std::uint16_t> val3 = {5, 11, 13};// Values of the items 
+    std::uint16_t W3 = 27;// Maximum capacity of the knapsack 
+    // Test the function and assert the expected output
+    assert(unboundedKnapsack(N3, W3, val3, wt3) == 27);
+    std::cout << "Maximum Knapsack value " << unboundedKnapsack(N3, W3, val3, wt3) << std::endl;
+
+    // Test Case 4
+    std::uint16_t N4 = 0; // Number of items
+    std::vector<std::uint16_t> wt4 = {}; // Weights of the items
+    std::vector<std::uint16_t> val4 = {}; // Values of the items 
+    std::uint16_t W4 = 10; // Maximum capacity of the knapsack
+    assert(unboundedKnapsack(N4, W4, val4, wt4) == 0); 
+    std::cout << "Maximum Knapsack value for empty arrays: " << unboundedKnapsack(N4, W4, val4, wt4) << std::endl;
+  
+    std::cout << "All test cases passed!" << std::endl;
+
+}
+
+/**
+ * @brief main function
+ * @return 0 on successful exit
+ */
+int main() {
+    tests(); // Run self test implementation 
+    return 0;
+}
+

math/sieve_of_eratosthenes.cpp

@@ -1,6 +1,7 @@
 /**
  * @file
- * @brief Get list of prime numbers using Sieve of Eratosthenes
+ * @brief Prime Numbers using [Sieve of
+ * Eratosthenes](https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes)
  * @details
  * Sieve of Eratosthenes is an algorithm that finds all the primes
  * between 2 and N.
@@ -11,21 +12,39 @@
  * @see primes_up_to_billion.cpp prime_numbers.cpp
  */
 
-#include <cassert>
+#include <cassert>   /// for assert
-#include <iostream>
+#include <iostream>  /// for IO operations
-#include <vector>
+#include <vector>    /// for std::vector
 
 /**
- * This is the function that finds the primes and eliminates the multiples.
+ * @namespace math
+ * @brief Mathematical algorithms
+ */
+namespace math {
+/**
+ * @namespace sieve_of_eratosthenes
+ * @brief Functions for finding Prime Numbers using Sieve of Eratosthenes
+ */
+namespace sieve_of_eratosthenes {
+/**
+ * @brief Function to sieve out the primes
+ * @details
+ * This function finds all the primes between 2 and N using the Sieve of
+ * Eratosthenes algorithm. It starts by assuming all numbers (except zero and
+ * one) are prime and then iteratively marks the multiples of each prime as
+ * non-prime.
+ *
  * Contains a common optimization to start eliminating multiples of
  * a prime p starting from p * p since all of the lower multiples
  * have been already eliminated.
- * @param N number of primes to check
+ * @param N number till which primes are to be found
- * @return is_prime a vector of `N + 1` booleans identifying if `i`^th number is a prime or not
+ * @return is_prime a vector of `N + 1` booleans identifying if `i`^th number is
+ * a prime or not
  */
 std::vector<bool> sieve(uint32_t N) {
-    std::vector<bool> is_prime(N + 1, true);
+    std::vector<bool> is_prime(N + 1, true);  // Initialize all as prime numbers
-    is_prime[0] = is_prime[1] = false;
+    is_prime[0] = is_prime[1] = false;        // 0 and 1 are not prime numbers
+
     for (uint32_t i = 2; i * i <= N; i++) {
         if (is_prime[i]) {
             for (uint32_t j = i * i; j <= N; j += i) {
@@ -37,9 +56,10 @@ std::vector<bool> sieve(uint32_t N) {
 }
 
 /**
- * This function prints out the primes to STDOUT
+ * @brief Function to print the prime numbers
- * @param N number of primes to check
+ * @param N number till which primes are to be found
- * @param is_prime a vector of `N + 1` booleans identifying if `i`^th number is a prime or not
+ * @param is_prime a vector of `N + 1` booleans identifying if `i`^th number is
+ * a prime or not
  */
 void print(uint32_t N, const std::vector<bool> &is_prime) {
     for (uint32_t i = 2; i <= N; i++) {
@@ -50,23 +70,53 @@ void print(uint32_t N, const std::vector<bool> &is_prime) {
     std::cout << std::endl;
 }
 
+}  // namespace sieve_of_eratosthenes
+}  // namespace math
+
 /**
- * Test implementations
+ * @brief Self-test implementations
+ * @return void
  */
-void tests() {
+static void tests() {
-  //                    0      1      2     3     4      5     6      7     8      9      10
+    std::vector<bool> is_prime_1 =
-  std::vector<bool> ans{false, false, true, true, false, true, false, true, false, false, false};
+        math::sieve_of_eratosthenes::sieve(static_cast<uint32_t>(10));
-  assert(sieve(10) == ans);
+    std::vector<bool> is_prime_2 =
+        math::sieve_of_eratosthenes::sieve(static_cast<uint32_t>(20));
+    std::vector<bool> is_prime_3 =
+        math::sieve_of_eratosthenes::sieve(static_cast<uint32_t>(100));
+
+    std::vector<bool> expected_1{false, false, true,  true,  false, true,
+                                 false, true,  false, false, false};
+    assert(is_prime_1 == expected_1);
+
+    std::vector<bool> expected_2{false, false, true,  true,  false, true,
+                                 false, true,  false, false, false, true,
+                                 false, true,  false, false, false, true,
+                                 false, true,  false};
+    assert(is_prime_2 == expected_2);
+
+    std::vector<bool> expected_3{
+        false, false, true,  true,  false, true,  false, true,  false, false,
+        false, true,  false, true,  false, false, false, true,  false, true,
+        false, false, false, true,  false, false, false, false, false, true,
+        false, true,  false, false, false, false, false, true,  false, false,
+        false, true,  false, true,  false, false, false, true,  false, false,
+        false, false, false, true,  false, false, false, false, false, true,
+        false, true,  false, false, false, false, false, true,  false, false,
+        false, true,  false, true,  false, false, false, false, false, true,
+        false, false, false, true,  false, false, false, false, false, true,
+        false, false, false, false, false, false, false, true,  false, false,
+        false};
+    assert(is_prime_3 == expected_3);
+
+    std::cout << "All tests have passed successfully!\n";
 }
 
 /**
- * Main function
+ * @brief Main function
+ * @returns 0 on exit
  */
 int main() {
     tests();
-
-    uint32_t N = 100;
-    std::vector<bool> is_prime = sieve(N);
-    print(N, is_prime);
     return 0;
 }