The algorithm sorts the keys, preserves the corresponding search frequencies, and uses + * dynamic programming with Knuth's optimization to compute the minimum weighted search cost. + * + *
Reference:
+ * https://en.wikipedia.org/wiki/Optimal_binary_search_tree
+ */
+public final class OptimalBinarySearchTree {
+ private OptimalBinarySearchTree() {
+ }
+
+ /**
+ * Computes the minimum weighted search cost for the given keys and search frequencies.
+ *
+ * @param keys the BST keys
+ * @param frequencies the search frequencies associated with the keys
+ * @return the minimum search cost
+ * @throws IllegalArgumentException if the input is invalid
+ */
+ public static long findOptimalCost(int[] keys, int[] frequencies) {
+ validateInput(keys, frequencies);
+ if (keys.length == 0) {
+ return 0L;
+ }
+
+ int[][] sortedNodes = sortNodes(keys, frequencies);
+ int nodeCount = sortedNodes.length;
+ long[] prefixSums = buildPrefixSums(sortedNodes);
+ long[][] optimalCost = new long[nodeCount][nodeCount];
+ int[][] root = new int[nodeCount][nodeCount];
+
+ for (int index = 0; index < nodeCount; index++) {
+ optimalCost[index][index] = sortedNodes[index][1];
+ root[index][index] = index;
+ }
+
+ for (int length = 2; length <= nodeCount; length++) {
+ for (int start = 0; start <= nodeCount - length; start++) {
+ int end = start + length - 1;
+ long frequencySum = prefixSums[end + 1] - prefixSums[start];
+ optimalCost[start][end] = Long.MAX_VALUE;
+
+ int leftBoundary = root[start][end - 1];
+ int rightBoundary = root[start + 1][end];
+ for (int currentRoot = leftBoundary; currentRoot <= rightBoundary; currentRoot++) {
+ long leftCost = currentRoot > start ? optimalCost[start][currentRoot - 1] : 0L;
+ long rightCost = currentRoot < end ? optimalCost[currentRoot + 1][end] : 0L;
+ long currentCost = frequencySum + leftCost + rightCost;
+
+ if (currentCost < optimalCost[start][end]) {
+ optimalCost[start][end] = currentCost;
+ root[start][end] = currentRoot;
+ }
+ }
+ }
+ }
+
+ return optimalCost[0][nodeCount - 1];
+ }
+
+ private static void validateInput(int[] keys, int[] frequencies) {
+ if (keys == null || frequencies == null) {
+ throw new IllegalArgumentException("Keys and frequencies cannot be null");
+ }
+ if (keys.length != frequencies.length) {
+ throw new IllegalArgumentException("Keys and frequencies must have the same length");
+ }
+
+ for (int frequency : frequencies) {
+ if (frequency < 0) {
+ throw new IllegalArgumentException("Frequencies cannot be negative");
+ }
+ }
+ }
+
+ private static int[][] sortNodes(int[] keys, int[] frequencies) {
+ int[][] sortedNodes = new int[keys.length][2];
+ for (int index = 0; index < keys.length; index++) {
+ sortedNodes[index][0] = keys[index];
+ sortedNodes[index][1] = frequencies[index];
+ }
+
+ Arrays.sort(sortedNodes, Comparator.comparingInt(node -> node[0]));
+
+ for (int index = 1; index < sortedNodes.length; index++) {
+ if (sortedNodes[index - 1][0] == sortedNodes[index][0]) {
+ throw new IllegalArgumentException("Keys must be distinct");
+ }
+ }
+
+ return sortedNodes;
+ }
+
+ private static long[] buildPrefixSums(int[][] sortedNodes) {
+ long[] prefixSums = new long[sortedNodes.length + 1];
+ for (int index = 0; index < sortedNodes.length; index++) {
+ prefixSums[index + 1] = prefixSums[index] + sortedNodes[index][1];
+ }
+ return prefixSums;
+ }
+}
diff --git a/src/test/java/com/thealgorithms/dynamicprogramming/OptimalBinarySearchTreeTest.java b/src/test/java/com/thealgorithms/dynamicprogramming/OptimalBinarySearchTreeTest.java
new file mode 100644
index 000000000000..17ff3ec728dc
--- /dev/null
+++ b/src/test/java/com/thealgorithms/dynamicprogramming/OptimalBinarySearchTreeTest.java
@@ -0,0 +1,73 @@
+package com.thealgorithms.dynamicprogramming;
+
+import static org.junit.jupiter.api.Assertions.assertEquals;
+import static org.junit.jupiter.api.Assertions.assertThrows;
+
+import java.util.Arrays;
+import java.util.stream.Stream;
+import org.junit.jupiter.params.ParameterizedTest;
+import org.junit.jupiter.params.provider.Arguments;
+import org.junit.jupiter.params.provider.MethodSource;
+
+class OptimalBinarySearchTreeTest {
+
+ @ParameterizedTest
+ @MethodSource("validTestCases")
+ void testFindOptimalCost(int[] keys, int[] frequencies, long expectedCost) {
+ assertEquals(expectedCost, OptimalBinarySearchTree.findOptimalCost(keys, frequencies));
+ }
+
+ private static Stream