backtracking: add unique permutation algorithm with test cases (#7078)

* Add unique permutation algorithm and test cases

* Fix: add braces for checkstyle

* Apply clang-format

src/main/java/com/thealgorithms/backtracking/UniquePermutation.java

@@ -0,0 +1,62 @@
+package com.thealgorithms.backtracking;
+
+import java.util.ArrayList;
+import java.util.Arrays;
+import java.util.List;
+
+/**
+ * Generates all UNIQUE permutations of a string, even when duplicate characters exist.
+ *
+ * Example:
+ *   Input: "AAB"
+ *   Output: ["AAB", "ABA", "BAA"]
+ *
+ * Time Complexity: O(n! * n)
+ */
+public final class UniquePermutation {
+
+    private UniquePermutation() {
+        // Prevent instantiation
+        throw new UnsupportedOperationException("Utility class");
+    }
+
+    public static List<String> generateUniquePermutations(String input) {
+        List<String> result = new ArrayList<>();
+        if (input == null) {
+            return result;
+        }
+
+        char[] chars = input.toCharArray();
+        Arrays.sort(chars); // important: sort to detect duplicates
+
+        backtrack(chars, new boolean[chars.length], new StringBuilder(), result);
+        return result;
+    }
+
+    private static void backtrack(char[] chars, boolean[] used, StringBuilder current, List<String> result) {
+
+        if (current.length() == chars.length) {
+            result.add(current.toString());
+            return;
+        }
+
+        for (int i = 0; i < chars.length; i++) {
+
+            // skip duplicates
+            if (i > 0 && chars[i] == chars[i - 1] && !used[i - 1]) {
+                continue;
+            }
+
+            if (!used[i]) {
+                used[i] = true;
+                current.append(chars[i]);
+
+                backtrack(chars, used, current, result);
+
+                // undo changes
+                used[i] = false;
+                current.deleteCharAt(current.length() - 1);
+            }
+        }
+    }
+}

src/test/java/com/thealgorithms/backtracking/UniquePermutationTest.java

@@ -0,0 +1,31 @@
+package com.thealgorithms.backtracking;
+
+import static org.junit.jupiter.api.Assertions.assertEquals;
+
+import java.util.Arrays;
+import java.util.List;
+import org.junit.jupiter.api.Test;
+
+public class UniquePermutationTest {
+
+    @Test
+    void testUniquePermutationsAab() {
+        List<String> expected = Arrays.asList("AAB", "ABA", "BAA");
+        List<String> result = UniquePermutation.generateUniquePermutations("AAB");
+        assertEquals(expected, result);
+    }
+
+    @Test
+    void testUniquePermutationsAbc() {
+        List<String> expected = Arrays.asList("ABC", "ACB", "BAC", "BCA", "CAB", "CBA");
+        List<String> result = UniquePermutation.generateUniquePermutations("ABC");
+        assertEquals(expected, result);
+    }
+
+    @Test
+    void testEmptyString() {
+        List<String> expected = Arrays.asList("");
+        List<String> result = UniquePermutation.generateUniquePermutations("");
+        assertEquals(expected, result);
+    }
+}