refactor: bwt encoding optimized, comments updated in bwtDecode.ts

src/algorithms/bwt/bwtDecode.ts

@@ -4,12 +4,11 @@ import { END_OF_STRING } from "./utils/constants";
 /**
  * Decodes a string that was encoded using the Burrows-Wheeler Transform (BWT).
  *
-
  * 1. Create a createFrequencyMap for chars from input string.
  *
  * 2. Construct a mapping (`firstOccurrenceTable`) that indicates the starting position of each character in the sorted BWT string.
  *
- * 3. Compute the index of each character in the BWT string as it appears in the sorted BWT string. 
+ * 3. Compute the index of each character in the BWT string as it appears in the sorted BWT string.
  *
  * 4. Using the `charsIndexesInSortedInput` array, trace back from the given index to reconstruct the original string.
  *

src/algorithms/bwt/bwtEncode.ts

@@ -1,39 +1,41 @@
 import { END_OF_STRING } from "./utils/constants";
+import { buildSuffixArray } from "./utils/suffix-array";
 
 /**
  * Encodes a given string using the Burrows-Wheeler Transform (BWT).
  *
- * The function virtualizes all cyclic permutations of the input string, sorts them
- * lexicographically, and constructs the BWT string from the last characters of each
- * permutation. It also returns the index of the original string in the sorted list of permutations.
+ * This function constructs the suffix array for the input string using a
+ * prefix doubling algorithm with radix sort, which operates in O(n log n) time.
+ * The BWT string is then derived from the suffix array by taking the character
+ * preceding each suffix.
  *
- * @param {string} input - The string to be encoded using the Burrows-Wheeler Transform.
+ * @param {string} input - The string to be encoded using the BWT.
  * @returns {{ bwt: string; index: number }} An object containing the encoded BWT string and the index of the original string.
  */
 export function bwtEncode(input: string): { bwt: string; index: number } {
   input += END_OF_STRING;
   const length = input.length;
 
-  const permutationsIndexes = Array.from({ length }, (_, i) => i);
-
-  // sorting all permutations
-  // using indexes
-  // without creating actual strings
-  permutationsIndexes.sort((a, b) => {
-    for (let i = 0; i < length; i++) {
-      const charA = input[(a + i) % length];
-      const charB = input[(b + i) % length];
-      if (charA < charB) return -1;
-      if (charA > charB) return 1;
-    }
-    return 0;
-  });
+  // To construct the BWT string, we utilize the suffix array.
+  // The suffix array contains the starting indixes of all suffixes of the string, sorted in lexicographical order.
+  // This sorted order helps us build the BWT string by accessing the characters just before the start of each suffix
+  // in the cyclic permutation of the string.
+  const suffixArray = buildSuffixArray(input);
 
   const bwtResult: string[] = [];
   let originalIndex = -1;
 
+  // For each suffix indexed in the suffix array, we find the character that precedes it in this cyclic view.
+  // The formula (index + length - 1) % length computes the position of this preceding character:
+  // - `index` is the start of the current suffix in the sorted suffix array.
+  // - Add length - 1 to the index to move to the preceding position in the cyclic permutation.
+  // - Use modulo length to wrap around to the start of the string if needed.
+  // Example: For a suffix starting at index 5 in a string of length 6:
+  // - Calculate the preceding character position: (5 + 6 - 1) % 6 = 10 % 6 = 4
+  // - The character at index 4 in the string is the one just before the suffix starting at index 5.
+  // Collecting these preceding characters for all suffixes in the sorted order gives us the BWT string.
   for (let i = 0; i < length; i++) {
-    const index = permutationsIndexes[i];
+    const index = suffixArray[i];
     bwtResult.push(input[(index + length - 1) % length]);
     if (index === 0) {
       originalIndex = i;

src/algorithms/bwt/utils/suffix-array.ts

@@ -0,0 +1,122 @@
+export function buildSuffixArray(input: string): number[] {
+  const length = input.length;
+  // The `ranks` array holds the `rank` of each suffix.
+  // As comparisonRange increases, these ranks begin to represent the order of suffixes based
+  // on multiple characters.
+  //
+  // Example for the string "banana":
+  // Initially, `ranks` contains ASCII codes of the characters:
+  // String:  "banana"
+  // Indexes:  0  1   2   3   4   5
+  // ranks:  [98, 97, 110, 97, 110, 97]
+  //
+  // After the first iteration (`comparisonSpan = 1`), ranks get updated:
+  // Now `ranks` represents the relative order of suffixes based on the first character
+  // (and the second character, if available): [1, 0, 2, 0, 2, 0]
+  let ranks = new Array(length);
+  // `nextRanks` is a temp array where new ranks are stored after recalculating them during
+  // each iteration. After updating, these ranks will be copied back to ranks for the next iteration.
+  let nextRanks = new Array(length);
+  // `comparisonRange` controls how many characters we compare in each iteration. Initially, it’s set to 1
+  // (compare only the first character), then it doubles (compare first two characters,
+  // then four, and so on) until all characters in the suffix are considered.
+  let comparisonRange = 1;
+
+  // indexes of sorted suffixes
+  const sortedSuffixes = new Array(length);
+
+  // set initial values for `ranks` and `orderedSuffixes`
+  for (let i = 0; i < length; i++) {
+    ranks[i] = input.charCodeAt(i);
+    sortedSuffixes[i] = i;
+  }
+
+  while (comparisonRange < length) {
+    // Step 1: Sort suffixes based on ranks starting `comparisonRange` positions ahead.
+    // Example: Suppose `comparisonRange` = 1.
+    // - `ranks` = [98, 97, 110, 97, 110, 97] (character codes for suffixes).
+    // - Suffixes are sorted based on ranks of positions one ahead:
+    //   - For suffix starting at index 0, rank is ranks[(0 + 1) % 6] = ranks[1] = 97.
+    //   - For suffix starting at index 1, rank is ranks[(1 + 1) % 6] = ranks[2] = 110.
+    //   - Continue similarly for all suffixes.
+    // - Sorting the suffix array based on these "ahead" ranks gives us a new order.
+    radixSort(sortedSuffixes, (i) => ranks[(i + comparisonRange) % length]);
+    // Step 2: Sort suffixes based on current ranks.
+    // Example: Suppose after the previous sorting, the ranks are as follows:
+    // - `ranks` = [1, 0, 2, 0, 2, 0] (ranks of suffixes based on initial comparison).
+    // - Sorting `suffixArray` directly by these ranks would order suffixes based on current rank values.
+    radixSort(sortedSuffixes, (i) => ranks[i]);
+    recalculateRanks(ranks, nextRanks, sortedSuffixes, comparisonRange, length);
+
+    [ranks, nextRanks] = [nextRanks, ranks];
+    comparisonRange *= 2;
+  }
+
+  return sortedSuffixes;
+}
+
+/**
+ * Sorts the `suffixIndexes` array using radix sort based on the ranks of suffixes.
+ * @param suffixIndexes - An array of suffix indices to be sorted.
+ * @param getRank - A function that returns the rank of a given suffix index.
+ */
+const radixSort = (
+  sortedSuffixes: number[],
+  getRank: (index: number) => number
+) => {
+  const buckets: number[][] = Array.from({ length: 16 }, () => []);
+
+  let bitPointer = 0;
+  let maxValue = Math.max(...sortedSuffixes.map(getRank));
+  const bitsCount = Math.ceil(Math.log2(maxValue + 1));
+
+  // We sort suffix indexes using radix sort, processing the rank bits in chunks of 4 bits.
+  // 1. Loop over `suffixIndexes` and determine the rank for each suffix.
+  // 2. Extract 4 bits from the rank, starting from `bitPointer`.
+  // 3. Use these 4 bits to decide the bucket where the suffix index will go.
+  // 4. After distributing all suffixes into buckets based on the current 4-bit segment,
+  //    move to the next 4-bit segment by incrementing `bitPointer` by 4.
+  while (bitPointer < bitsCount) {
+    for (let i = 0; i < sortedSuffixes.length; i++) {
+      const rank = getRank(sortedSuffixes[i]);
+      const bucketIndex = (rank >> bitPointer) & 0b1111;
+      buckets[bucketIndex].push(sortedSuffixes[i]);
+    }
+
+    sortedSuffixes.length = 0;
+    for (const bucket of buckets) {
+      sortedSuffixes.push(...bucket);
+      bucket.length = 0;
+    }
+    bitPointer += 4;
+  }
+
+  return sortedSuffixes;
+};
+
+function recalculateRanks(
+  currentRanks: number[],
+  nextRanks: number[],
+  suffixArray: number[],
+  comparisonSpan: number,
+  length: number
+): void {
+  let rank = 1;
+  nextRanks[suffixArray[0]] = rank;
+
+  for (let i = 1; i < length; i++) {
+    const currentSuffix = suffixArray[i];
+    const previousSuffix = suffixArray[i - 1];
+
+    const differentRanks =
+      currentRanks[currentSuffix] !== currentRanks[previousSuffix];
+    const differentNextRanks =
+      currentRanks[(currentSuffix + comparisonSpan) % length] !==
+      currentRanks[(previousSuffix + comparisonSpan) % length];
+
+    if (differentRanks || differentNextRanks) {
+      rank++;
+    }
+    nextRanks[currentSuffix] = rank;
+  }
+}