added another minimalItems knapcksack subset

Author: JakubSchwenkbeck

src/main/scala/algorithms/dynamic/MinimalItems.scala

@@ -0,0 +1,54 @@
+package algorithms.dynamic
+
+/// Minimal Items Knapsack Problem
+/// Given an array of positive weights and a target weight W,
+/// this function computes the minimum number of items needed to exactly reach weight W using dynamic programming.
+/// Each item may be used at most once.
+///
+/// dp(x) represents the minimal number of items required to achieve weight x.
+///   - Base case: dp(0) = 0 (no item is needed to achieve weight 0)
+///   - Recurrence: For each item with weight w and for each x from W down to w:
+///         if dp(x - w) is not INF then
+///             dp(x) = min(dp(x), dp(x - w) + 1)
+///
+/// Time Complexity: O(n * W), where n is the number of items.
+def minimalItems(weights: Array[Int], W: Int): (Int, Array[Int]) = {
+  // Define INF as a value greater than any possible number of items (here: W + 1)
+  val INF = W + 1
+  // Initialize dp array: dp(x) stores the minimal number of items to reach weight x.
+  val dp: Array[Int] = Array.fill(W + 1)(INF)
+  dp(0) = 0
+
+  // Process each weight (item) exactly once.
+  for (w <- weights) {
+    // Iterate backwards to ensure each item is used only once.
+    for (x <- W to w by -1) {
+      if (dp(x - w) != INF) {
+        dp(x) = math.min(dp(x), dp(x - w) + 1)
+      }
+    }
+  }
+
+  // If dp(W) is still INF, there is no solution.
+  val result = if (dp(W) == INF) -1 else dp(W)
+  (result, dp)
+}
+
+@main
+def mainMinimalItems(): Unit = {
+  // Example: Items with weights 2, 5, 3, 6 and target weight W = 10.
+  val weights: Array[Int] = Array(2, 5, 3, 6)
+  val W = 10
+  val (minItems, dp) = minimalItems(weights, W)
+
+  if (minItems == -1)
+    println(s"Cannot exactly reach weight $W with the given items.")
+  else
+    println(s"Minimum number of items to exactly reach weight $W: $minItems")
+
+  println("DP Array (minimal number of items for each weight):")
+  dp.zipWithIndex.foreach { case (items, weight) =>
+    val info = if (items == W + 1) "not reachable" else items.toString
+    println(s"Weight $weight: $info")
+  }
+}