Detailed algorithm description (corrected).

README.md

@@ -184,8 +184,8 @@ The SA-IS algorithm is quite elegant, yet implementing it efficiently presents m
         SA[buckets[v3]++] = (p3 - 1) | ((sa_sint_t)(buckets[2 + v3] != d) << (SAINT_BIT - 1)); buckets[2 + v3] = d;
     }
 ```
-* In the SA-IS algorithm, after induced sorting, the ranks of LMS substrings are computed in suffix order. These ranks then need to be scattered to reorder them in string order before being gathered again to form the compacted string for recursion. At this point, some LMS substrings may be unique, meaning they don't share their rank with any other LMS substring. Being unique, these substrings are essentially already sorted, and their position relative to other LMS substrings is already determined. However, these unique LMS substrings may still be necessary for sorting other, non-unique LMS substrings during recursion—unless a unique LMS substring is immediately followed by another unique LMS substring in the string. In such cases, the rank of any subsequent unique LMS substrings becomes redundant in the compacted string, as it will not be utilized. Leveraging this insight, libsais employs a strategy to further reduce the size of the compacted string by omitting such redundant LMS substring ranks. This process involves a few steps. First, unique LMS substrings are identified by looking ahead while scanning LMS-positions in the suffix array during the ranking and scattering phase. When scattering LMS substring ranks to form the compacted string, the most significant bit (MSB) of the rank is used to mark that this rank is unique. Next, as the library scans the ranks in string order and detects tandems of unique ranks using the MSB, it then  recalculates the MSB for ranks which are redundant, thus markign them for removal from the compacted string. Subsequently, the libsais rescans the LMS-positions in suffix order to recompute the ranks, now focusing only on the ranks of the remaining LMS substrings. The library also uses MSB of first symbol of LMS substrings to mark that LMS substring is removed from the compacted string. Finally, the library builds the compacted string based on the newly recalculated ranks for the remaining LMS substrings, while also saving the final positions for the removed LMS substrings before proceeding with recursion. This reduction process not only further decreases the size of the compacted string but also reduces the alphabet size of the reduced string and creates additional free space in the suffix array, which can be utilized during recursion.
-* The SA-IS algorithm, while robust for suffix array construction, is not considered lightweight due to its need for additional memory for tasks such as position classification, induced sorting, the creation of compacted string representations, and recursive decomposition. To mitigate this, libsais optimizes memory usage by not storing position classifications and striving to reuse the memory space allocated for the suffix array for induced sorting, compacted string representations, and recursive decomposition processes. Since position classifications are not stored, the library recalculates them as needed, typically involving checks of adjacent symbols for a given position. Although this approach may seem straightforward, it introduces the challenge of random memory access. Nevertheless, libsais manages these accesses in a manner that either avoids unnecessary memory fetches or minimizes cache penalties. In situations where avoiding cache penalties is unfeasible, the library leverages the most significant bit (MSB) bits for computations, as branch mispredictions on modern microprocessors generally incur lower penalties than cache misses. Memory reuse for the suffix array, despite appearing straightforward, also presents hidden challenges related to implementation complexity. In certain cases, the available space in the suffix array may not suffice for the most optimal algorithm implementation mentioned above. Although such instances are rare, the library aims to deliver optimal performance without additional memory allocation by resorting to a less efficient variant of induced sorting. To accommodate various scenarios, libsais includes four distinct implementations tailored to different breakpoints based on alphabet size (denoted by 'k'): 6k, 4k, 2k, and 1k, with each implementation optimized to ensure performance efficiency. Extensive efforts have been dedicated to refining these implementations, including significant time invested in using various sanitizers to confirm the correctness of the algorithms. Ultimately, while there are specific inputs under which libsais might require additional memory—most of which tend to be synthetic tests designed specifically to challenge the SA-IS algorithm—such instances are relatively rare. In these exceptional cases, the library is designed to allocate only the minimum necessary amount of memory while still delivering the best possible performance.
+* In the SA-IS algorithm, after induced sorting, the ranks of LMS substrings are computed in suffix order. These ranks then need to be scattered to reorder them in string order before being gathered again to form the compacted string for recursion. At this point, some LMS substrings may be unique, meaning they don't share their rank with any other LMS substring. Being unique, these substrings are essentially already sorted, and their position relative to other LMS substrings is already determined. However, these unique LMS substrings may still be necessary for sorting other, non-unique LMS substrings during recursion-unless a unique LMS substring is immediately followed by another unique LMS substring in the string. In such cases, the rank of any subsequent unique LMS substrings becomes redundant in the compacted string, as it will not be utilized. Leveraging this insight, libsais employs a strategy to further reduce the size of the compacted string by omitting such redundant LMS substring ranks. This process involves a few steps. First, unique LMS substrings are identified by looking ahead while scanning LMS-positions in the suffix array during the ranking and scattering phase. When scattering LMS substring ranks to form the compacted string, the most significant bit (MSB) of the rank is used to mark that this rank is unique. Next, as the library scans the ranks in string order and detects tandems of unique ranks using the MSB, it then  recalculates the MSB for ranks which are redundant, thus markign them for removal from the compacted string. Subsequently, the libsais rescans the LMS-positions in suffix order to recompute the ranks, now focusing only on the ranks of the remaining LMS substrings. The library also uses MSB of first symbol of LMS substrings to mark that LMS substring is removed from the compacted string. Finally, the library builds the compacted string based on the newly recalculated ranks for the remaining LMS substrings, while also saving the final positions for the removed LMS substrings before proceeding with recursion. This reduction process not only further decreases the size of the compacted string but also reduces the alphabet size of the reduced string and creates additional free space in the suffix array, which can be utilized during recursion.
+* The SA-IS algorithm, while robust for suffix array construction, is not considered lightweight due to its need for additional memory for tasks such as position classification, induced sorting, the creation of compacted string representations, and recursive decomposition. To mitigate this, libsais optimizes memory usage by not storing position classifications and striving to reuse the memory space allocated for the suffix array for induced sorting, compacted string representations, and recursive decomposition processes. Since position classifications are not stored, the library recalculates them as needed, typically involving checks of adjacent symbols for a given position. Although this approach may seem straightforward, it introduces the challenge of random memory access. Nevertheless, libsais manages these accesses in a manner that either avoids unnecessary memory fetches or minimizes cache penalties. In situations where avoiding cache penalties is unfeasible, the library leverages the most significant bit (MSB) bits for computations, as branch mispredictions on modern microprocessors generally incur lower penalties than cache misses. Memory reuse for the suffix array, despite appearing straightforward, also presents hidden challenges related to implementation complexity. In certain cases, the available space in the suffix array may not suffice for the most optimal algorithm implementation mentioned above. Although such instances are rare, the library aims to deliver optimal performance without additional memory allocation by resorting to a less efficient variant of induced sorting. To accommodate various scenarios, libsais includes four distinct implementations tailored to different breakpoints based on alphabet size (denoted by 'k'): 6k, 4k, 2k, and 1k, with each implementation optimized to ensure performance efficiency. Extensive efforts have been dedicated to refining these implementations, including significant time invested in using various sanitizers to confirm the correctness of the algorithms. Ultimately, while there are specific inputs under which libsais might require additional memory-most of which tend to be synthetic tests designed specifically to challenge the SA-IS algorithm-such instances are relatively rare. In these exceptional cases, the library is designed to allocate only the minimum necessary amount of memory while still delivering the best possible performance.
 * The libsais library, initially was developed for constructing suffix arrays, but has broadened its scope to include the calculation of the longest common prefix (LCP) and both the forward and inverse Burrows-Wheeler Transform (BWT) with considerable efforts has been dedicated to refining these algorithms to ensure they deliver maximum performance and maintain the correctness. An illustrative example is the forward BWT, which performance is nearly identical to that of its suffix array construction which is achieved by integrating a modified version of the induced sorting implementation within the final stage of the SA-IS algorithm. Rather than inducing suffix positions at this stage, the library induces the Burrows-Wheeler Transform directly. This approach also supports in-place transformation, maintaining a memory usage of 5n, making it an sutable for data compression applications. Similarly, the inverse BWT is fine-tuned to operate in-place, adhering to the same memory efficiency of 5n with an additional optimization of a bi-gram LF-mapping technique, which allows for the decoding of two symbols simultaneously effectively reduces the number of cache misses during the inversion of the Burrows-Wheeler Transform.
 
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