Add Booth's algorithm for minimal string rotation (#7644)

* Add Booth's algorithm for minimal string rotation

* Add tests for Booth's minimal rotation algorithm

* Use constant-space comparisons in Booth's algorithm

Author: vedant0517

src/main/java/com/thealgorithms/strings/BoothsAlgorithm.java

@@ -0,0 +1,69 @@
+package com.thealgorithms.strings;
+
+/**
+ * Finds the lexicographically smallest cyclic rotation of a string using
+ * Booth's algorithm.
+ *
+ * <p>The algorithm compares candidate rotations without materializing them,
+ * running in O(n) time and O(1) auxiliary space.</p>
+ *
+ * @see <a href="https://en.wikipedia.org/wiki/Lexicographically_minimal_string_rotation">
+ *     Lexicographically minimal string rotation</a>
+ */
+public final class BoothsAlgorithm {
+    private BoothsAlgorithm() {
+    }
+
+    /**
+     * Returns the starting index of the lexicographically smallest cyclic
+     * rotation of {@code input}. If several rotations are equal and minimal,
+     * the algorithm returns the smallest starting index among them.
+     *
+     * <p>For example, the smallest rotation of {@code "baca"} is
+     * {@code "abac"}, which starts at index {@code 3}.</p>
+     *
+     * @param input the string whose minimal cyclic rotation is sought
+     * @return the starting index of the minimal rotation, or {@code 0} for an
+     *         empty string
+     * @throws IllegalArgumentException if {@code input} is {@code null}
+     */
+    public static int minimalRotationIndex(String input) {
+        if (input == null) {
+            throw new IllegalArgumentException("Input string must not be null");
+        }
+
+        int length = input.length();
+        if (length == 0) {
+            return 0;
+        }
+
+        int first = 0;
+        int second = 1;
+        int offset = 0;
+
+        while (first < length && second < length && offset < length) {
+            char firstChar = input.charAt((first + offset) % length);
+            char secondChar = input.charAt((second + offset) % length);
+
+            if (firstChar == secondChar) {
+                offset++;
+                continue;
+            }
+
+            if (firstChar > secondChar) {
+                first += offset + 1;
+                if (first <= second) {
+                    first = second + 1;
+                }
+            } else {
+                second += offset + 1;
+                if (second <= first) {
+                    second = first + 1;
+                }
+            }
+            offset = 0;
+        }
+
+        return Math.min(first, second);
+    }
+}

src/test/java/com/thealgorithms/strings/BoothsAlgorithmTest.java

@@ -0,0 +1,45 @@
+package com.thealgorithms.strings;
+
+import static org.junit.jupiter.api.Assertions.assertEquals;
+import static org.junit.jupiter.api.Assertions.assertThrows;
+
+import org.junit.jupiter.api.Test;
+
+public class BoothsAlgorithmTest {
+
+    @Test
+    public void testMinimalRotationIndexForExample() {
+        assertEquals(3, BoothsAlgorithm.minimalRotationIndex("baca"));
+    }
+
+    @Test
+    public void testEmptyString() {
+        assertEquals(0, BoothsAlgorithm.minimalRotationIndex(""));
+    }
+
+    @Test
+    public void testSingleCharacter() {
+        assertEquals(0, BoothsAlgorithm.minimalRotationIndex("x"));
+    }
+
+    @Test
+    public void testRepeatedCharacters() {
+        assertEquals(0, BoothsAlgorithm.minimalRotationIndex("aaaaa"));
+    }
+
+    @Test
+    public void testPeriodicString() {
+        assertEquals(1, BoothsAlgorithm.minimalRotationIndex("baba"));
+        assertEquals(1, BoothsAlgorithm.minimalRotationIndex("cabcab"));
+    }
+
+    @Test
+    public void testStringAlreadyHasMinimalRotationAtStart() {
+        assertEquals(0, BoothsAlgorithm.minimalRotationIndex("abc"));
+    }
+
+    @Test
+    public void testNullInput() {
+        assertThrows(IllegalArgumentException.class, () -> BoothsAlgorithm.minimalRotationIndex(null));
+    }
+}