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| 1 | +package com.thealgorithms.compression; |
| 2 | + |
| 3 | +import java.util.Arrays; |
| 4 | +import java.util.HashMap; |
| 5 | +import java.util.Map; |
| 6 | + |
| 7 | +/** |
| 8 | + * Implementation of the Burrows-Wheeler Transform (BWT) and its inverse. |
| 9 | + * <p> |
| 10 | + * BWT is a reversible data transformation algorithm that rearranges a string into runs of |
| 11 | + * similar characters. While not a compression algorithm itself, it significantly improves |
| 12 | + * the compressibility of data for subsequent algorithms like Move-to-Front encoding and |
| 13 | + * Run-Length Encoding. |
| 14 | + * </p> |
| 15 | + * |
| 16 | + * <p>The transform works by: |
| 17 | + * <ol> |
| 18 | + * <li>Generating all rotations of the input string</li> |
| 19 | + * <li>Sorting these rotations lexicographically</li> |
| 20 | + * <li>Taking the last column of the sorted matrix as output</li> |
| 21 | + * <li>Recording the index of the original string in the sorted matrix</li> |
| 22 | + * </ol> |
| 23 | + * </p> |
| 24 | + * |
| 25 | + * <p><b>Important:</b> The input string should end with a unique end-of-string marker |
| 26 | + * (typically '$') that: |
| 27 | + * <ul> |
| 28 | + * <li>Does not appear anywhere else in the text</li> |
| 29 | + * <li>Is lexicographically smaller than all other characters</li> |
| 30 | + * <li>Ensures unique rotations and enables correct inverse transformation</li> |
| 31 | + * </ul> |
| 32 | + * Without this marker, the inverse transform may not correctly reconstruct the original string. |
| 33 | + * </p> |
| 34 | + * |
| 35 | + * <p><b>Time Complexity:</b> |
| 36 | + * <ul> |
| 37 | + * <li>Forward transform: O(n² log n) where n is the string length</li> |
| 38 | + * <li>Inverse transform: O(n) using the LF-mapping technique</li> |
| 39 | + * </ul> |
| 40 | + * </p> |
| 41 | + * |
| 42 | + * <p><b>Example:</b></p> |
| 43 | + * <pre> |
| 44 | + * Input: "banana$" |
| 45 | + * Output: BWTResult("annb$aa", 4) |
| 46 | + * - "annb$aa" is the transformed string (groups similar characters) |
| 47 | + * - 4 is the index of the original string in the sorted rotations |
| 48 | + * </pre> |
| 49 | + * |
| 50 | + * @see <a href="https://en.wikipedia.org/wiki/Burrows%E2%80%93Wheeler_transform">Burrows–Wheeler transform (Wikipedia)</a> |
| 51 | + */ |
| 52 | +public final class BurrowsWheelerTransform { |
| 53 | + |
| 54 | + private BurrowsWheelerTransform() { |
| 55 | + } |
| 56 | + |
| 57 | + /** |
| 58 | + * A container for the result of the forward BWT. |
| 59 | + * <p> |
| 60 | + * Contains the transformed string and the index of the original string |
| 61 | + * in the sorted rotations matrix, both of which are required for the |
| 62 | + * inverse transformation. |
| 63 | + * </p> |
| 64 | + */ |
| 65 | + public static class BWTResult { |
| 66 | + /** The transformed string (last column of the sorted rotation matrix) */ |
| 67 | + public final String transformed; |
| 68 | + |
| 69 | + /** The index of the original string in the sorted rotations matrix */ |
| 70 | + public final int originalIndex; |
| 71 | + |
| 72 | + /** |
| 73 | + * Constructs a BWTResult with the transformed string and original index. |
| 74 | + * |
| 75 | + * @param transformed the transformed string (L-column) |
| 76 | + * @param originalIndex the index of the original string in sorted rotations |
| 77 | + */ |
| 78 | + public BWTResult(String transformed, int originalIndex) { |
| 79 | + this.transformed = transformed; |
| 80 | + this.originalIndex = originalIndex; |
| 81 | + } |
| 82 | + |
| 83 | + @Override |
| 84 | + public boolean equals(Object obj) { |
| 85 | + if (this == obj) { |
| 86 | + return true; |
| 87 | + } |
| 88 | + if (obj == null || getClass() != obj.getClass()) { |
| 89 | + return false; |
| 90 | + } |
| 91 | + BWTResult bwtResult = (BWTResult) obj; |
| 92 | + return originalIndex == bwtResult.originalIndex && transformed.equals(bwtResult.transformed); |
| 93 | + } |
| 94 | + |
| 95 | + @Override |
| 96 | + public int hashCode() { |
| 97 | + return 31 * transformed.hashCode() + originalIndex; |
| 98 | + } |
| 99 | + |
| 100 | + @Override |
| 101 | + public String toString() { |
| 102 | + return "BWTResult[transformed=" + transformed + ", originalIndex=" + originalIndex + "]"; |
| 103 | + } |
| 104 | + } |
| 105 | + |
| 106 | + /** |
| 107 | + * Performs the forward Burrows-Wheeler Transform on the input string. |
| 108 | + * <p> |
| 109 | + * The algorithm generates all cyclic rotations of the input, sorts them |
| 110 | + * lexicographically, and returns the last column of this sorted matrix |
| 111 | + * along with the position of the original string. |
| 112 | + * </p> |
| 113 | + * |
| 114 | + * <p><b>Note:</b> It is strongly recommended that the input string ends with |
| 115 | + * a unique end-of-string marker (e.g., '$') that is lexicographically smaller |
| 116 | + * than any other character in the string. This ensures correct inversion.</p> |
| 117 | + * |
| 118 | + * @param text the input string to transform; must not be {@code null} |
| 119 | + * @return a {@link BWTResult} object containing the transformed string (L-column) |
| 120 | + * and the index of the original string in the sorted rotations matrix; |
| 121 | + * returns {@code BWTResult("", -1)} for empty input |
| 122 | + * @throws NullPointerException if {@code text} is {@code null} |
| 123 | + */ |
| 124 | + public static BWTResult transform(String text) { |
| 125 | + if (text == null || text.isEmpty()) { |
| 126 | + return new BWTResult("", -1); |
| 127 | + } |
| 128 | + |
| 129 | + int n = text.length(); |
| 130 | + |
| 131 | + // Generate all rotations of the input string |
| 132 | + String[] rotations = new String[n]; |
| 133 | + for (int i = 0; i < n; i++) { |
| 134 | + rotations[i] = text.substring(i) + text.substring(0, i); |
| 135 | + } |
| 136 | + |
| 137 | + // Sort rotations lexicographically |
| 138 | + Arrays.sort(rotations); |
| 139 | + int originalIndex = Arrays.binarySearch(rotations, text); |
| 140 | + StringBuilder lastColumn = new StringBuilder(n); |
| 141 | + for (int i = 0; i < n; i++) { |
| 142 | + lastColumn.append(rotations[i].charAt(n - 1)); |
| 143 | + } |
| 144 | + |
| 145 | + return new BWTResult(lastColumn.toString(), originalIndex); |
| 146 | + } |
| 147 | + |
| 148 | + /** |
| 149 | + * Performs the inverse Burrows-Wheeler Transform using the LF-mapping technique. |
| 150 | + * <p> |
| 151 | + * The LF-mapping (Last-First mapping) is an efficient method to reconstruct |
| 152 | + * the original string from the BWT output without explicitly reconstructing |
| 153 | + * the entire sorted rotations matrix. |
| 154 | + * </p> |
| 155 | + * |
| 156 | + * <p>The algorithm works by: |
| 157 | + * <ol> |
| 158 | + * <li>Creating the first column by sorting the BWT string</li> |
| 159 | + * <li>Building a mapping from first column indices to last column indices</li> |
| 160 | + * <li>Following this mapping starting from the original index to reconstruct the string</li> |
| 161 | + * </ol> |
| 162 | + * </p> |
| 163 | + * |
| 164 | + * @param bwtString the transformed string (L-column) from the forward transform; must not be {@code null} |
| 165 | + * @param originalIndex the index of the original string row from the forward transform; |
| 166 | + * use -1 for empty strings |
| 167 | + * @return the original, untransformed string; returns empty string if input is empty or {@code originalIndex} is -1 |
| 168 | + * @throws NullPointerException if {@code bwtString} is {@code null} |
| 169 | + * @throws IllegalArgumentException if {@code originalIndex} is out of valid range (except -1) |
| 170 | + */ |
| 171 | + public static String inverseTransform(String bwtString, int originalIndex) { |
| 172 | + if (bwtString == null || bwtString.isEmpty() || originalIndex == -1) { |
| 173 | + return ""; |
| 174 | + } |
| 175 | + |
| 176 | + int n = bwtString.length(); |
| 177 | + if (originalIndex < 0 || originalIndex >= n) { |
| 178 | + throw new IllegalArgumentException("Original index must be between 0 and " + (n - 1) + ", got: " + originalIndex); |
| 179 | + } |
| 180 | + |
| 181 | + char[] lastColumn = bwtString.toCharArray(); |
| 182 | + char[] firstColumn = bwtString.toCharArray(); |
| 183 | + Arrays.sort(firstColumn); |
| 184 | + |
| 185 | + // Create the "next" array for LF-mapping. |
| 186 | + // next[i] stores the row index in the last column that corresponds to firstColumn[i] |
| 187 | + int[] next = new int[n]; |
| 188 | + |
| 189 | + // Track the count of each character seen so far in the last column |
| 190 | + Map<Character, Integer> countMap = new HashMap<>(); |
| 191 | + |
| 192 | + // Store the first occurrence index of each character in the first column |
| 193 | + Map<Character, Integer> firstOccurrence = new HashMap<>(); |
| 194 | + |
| 195 | + for (int i = 0; i < n; i++) { |
| 196 | + if (!firstOccurrence.containsKey(firstColumn[i])) { |
| 197 | + firstOccurrence.put(firstColumn[i], i); |
| 198 | + } |
| 199 | + } |
| 200 | + |
| 201 | + // Build the LF-mapping |
| 202 | + for (int i = 0; i < n; i++) { |
| 203 | + char c = lastColumn[i]; |
| 204 | + int count = countMap.getOrDefault(c, 0); |
| 205 | + int firstIndex = firstOccurrence.get(c); |
| 206 | + next[firstIndex + count] = i; |
| 207 | + countMap.put(c, count + 1); |
| 208 | + } |
| 209 | + |
| 210 | + // Reconstruct the original string by following the LF-mapping |
| 211 | + StringBuilder originalString = new StringBuilder(n); |
| 212 | + int currentRow = originalIndex; |
| 213 | + for (int i = 0; i < n; i++) { |
| 214 | + originalString.append(firstColumn[currentRow]); |
| 215 | + currentRow = next[currentRow]; |
| 216 | + } |
| 217 | + |
| 218 | + return originalString.toString(); |
| 219 | + } |
| 220 | +} |
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