Added Burrows-Wheeler transform algorithm. (#1029)

* Added doctest and more explanation about Dijkstra execution.

* tests were not passing with python2 due to missing __init__.py file at number_theory folder

* Removed the dot at the beginning of the imported modules names because 'python3 -m doctest -v data_structures/hashing/*.py' and 'python3 -m doctest -v data_structures/stacks/*.py' were failing not finding hash_table.py and stack.py modules.

* Moved global code to main scope and added doctest for project euler problems 1 to 14.

* Added test case for negative input.

* Changed N variable to do not use end of line scape because in case there is a space after it the script will break making it much more error prone.

* Added problems description and doctests to the ones that were missing. Limited line length to 79 and executed python black over all scripts.

* Changed the way files are loaded to support pytest call.

* Added __init__.py to problems to make them modules and allow pytest execution.

* Added project_euler folder to test units execution

* Changed 'os.path.split(os.path.realpath(__file__))' to 'os.path.dirname()'

* Added Burrows-Wheeler transform algorithm.

* Added changes suggested by cclauss

Author: brunohadlich

Diff for: compression/burrows_wheeler.py

@@ -0,0 +1,176 @@
+"""
+https://en.wikipedia.org/wiki/Burrows%E2%80%93Wheeler_transform
+
+The Burrows–Wheeler transform (BWT, also called block-sorting compression)
+rearranges a character string into runs of similar characters. This is useful
+for compression, since it tends to be easy to compress a string that has runs
+of repeated characters by techniques such as move-to-front transform and
+run-length encoding. More importantly, the transformation is reversible,
+without needing to store any additional data except the position of the first
+original character. The BWT is thus a "free" method of improving the efficiency
+of text compression algorithms, costing only some extra computation.
+"""
+from typing import List, Dict
+
+
+def all_rotations(s: str) -> List[str]:
+    """
+    :param s: The string that will be rotated len(s) times.
+    :return: A list with the rotations.
+    :raises TypeError: If s is not an instance of str.
+    Examples:
+
+    >>> all_rotations("^BANANA|") # doctest: +NORMALIZE_WHITESPACE
+    ['^BANANA|', 'BANANA|^', 'ANANA|^B', 'NANA|^BA', 'ANA|^BAN', 'NA|^BANA',
+    'A|^BANAN', '|^BANANA']
+    >>> all_rotations("a_asa_da_casa") # doctest: +NORMALIZE_WHITESPACE
+    ['a_asa_da_casa', '_asa_da_casaa', 'asa_da_casaa_', 'sa_da_casaa_a',
+    'a_da_casaa_as', '_da_casaa_asa', 'da_casaa_asa_', 'a_casaa_asa_d',
+    '_casaa_asa_da', 'casaa_asa_da_', 'asaa_asa_da_c', 'saa_asa_da_ca',
+    'aa_asa_da_cas']
+    >>> all_rotations("panamabanana") # doctest: +NORMALIZE_WHITESPACE
+    ['panamabanana', 'anamabananap', 'namabananapa', 'amabananapan',
+    'mabananapana', 'abananapanam', 'bananapanama', 'ananapanamab',
+    'nanapanamaba', 'anapanamaban', 'napanamabana', 'apanamabanan']
+    >>> all_rotations(5)
+    Traceback (most recent call last):
+        ...
+    TypeError: The parameter s type must be str.
+    """
+    if not isinstance(s, str):
+        raise TypeError("The parameter s type must be str.")
+
+    return [s[i:] + s[:i] for i in range(len(s))]
+
+
+def bwt_transform(s: str) -> Dict:
+    """
+    :param s: The string that will be used at bwt algorithm
+    :return: the string composed of the last char of each row of the ordered
+    rotations and the index of the original string at ordered rotations list
+    :raises TypeError: If the s parameter type is not str
+    :raises ValueError: If the s parameter is empty
+    Examples:
+
+    >>> bwt_transform("^BANANA")
+    {'bwt_string': 'BNN^AAA', 'idx_original_string': 6}
+    >>> bwt_transform("a_asa_da_casa")
+    {'bwt_string': 'aaaadss_c__aa', 'idx_original_string': 3}
+    >>> bwt_transform("panamabanana")
+    {'bwt_string': 'mnpbnnaaaaaa', 'idx_original_string': 11}
+    >>> bwt_transform(4)
+    Traceback (most recent call last):
+        ...
+    TypeError: The parameter s type must be str.
+    >>> bwt_transform('')
+    Traceback (most recent call last):
+        ...
+    ValueError: The parameter s must not be empty.
+    """
+    if not isinstance(s, str):
+        raise TypeError("The parameter s type must be str.")
+    if not s:
+        raise ValueError("The parameter s must not be empty.")
+
+    rotations = all_rotations(s)
+    rotations.sort()  # sort the list of rotations in alphabetically order
+    # make a string composed of the last char of each rotation
+    return {
+        "bwt_string": "".join([word[-1] for word in rotations]),
+        "idx_original_string": rotations.index(s),
+    }
+
+
+def reverse_bwt(bwt_string: str, idx_original_string: int) -> str:
+    """
+    :param bwt_string: The string returned from bwt algorithm execution
+    :param idx_original_string: A 0-based index of the string that was used to
+    generate bwt_string at ordered rotations list
+    :return: The string used to generate bwt_string when bwt was executed
+    :raises TypeError: If the bwt_string parameter type is not str
+    :raises ValueError: If the bwt_string parameter is empty
+    :raises TypeError: If the idx_original_string type is not int or if not
+    possible to cast it to int
+    :raises ValueError: If the idx_original_string value is lower than 0 or
+    greater than len(bwt_string) - 1
+
+    >>> reverse_bwt("BNN^AAA", 6)
+    '^BANANA'
+    >>> reverse_bwt("aaaadss_c__aa", 3)
+    'a_asa_da_casa'
+    >>> reverse_bwt("mnpbnnaaaaaa", 11)
+    'panamabanana'
+    >>> reverse_bwt(4, 11)
+    Traceback (most recent call last):
+        ...
+    TypeError: The parameter bwt_string type must be str.
+    >>> reverse_bwt("", 11)
+    Traceback (most recent call last):
+        ...
+    ValueError: The parameter bwt_string must not be empty.
+    >>> reverse_bwt("mnpbnnaaaaaa", "asd") # doctest: +NORMALIZE_WHITESPACE
+    Traceback (most recent call last):
+        ...
+    TypeError: The parameter idx_original_string type must be int or passive
+    of cast to int.
+    >>> reverse_bwt("mnpbnnaaaaaa", -1)
+    Traceback (most recent call last):
+        ...
+    ValueError: The parameter idx_original_string must not be lower than 0.
+    >>> reverse_bwt("mnpbnnaaaaaa", 12) # doctest: +NORMALIZE_WHITESPACE
+    Traceback (most recent call last):
+        ...
+    ValueError: The parameter idx_original_string must be lower than
+    len(bwt_string).
+    >>> reverse_bwt("mnpbnnaaaaaa", 11.0)
+    'panamabanana'
+    >>> reverse_bwt("mnpbnnaaaaaa", 11.4)
+    'panamabanana'
+    """
+    if not isinstance(bwt_string, str):
+        raise TypeError("The parameter bwt_string type must be str.")
+    if not bwt_string:
+        raise ValueError("The parameter bwt_string must not be empty.")
+    try:
+        idx_original_string = int(idx_original_string)
+    except ValueError:
+        raise TypeError(
+            (
+                "The parameter idx_original_string type must be int or passive"
+                " of cast to int."
+            )
+        )
+    if idx_original_string < 0:
+        raise ValueError(
+            "The parameter idx_original_string must not be lower than 0."
+        )
+    if idx_original_string >= len(bwt_string):
+        raise ValueError(
+            (
+                "The parameter idx_original_string must be lower than"
+                " len(bwt_string)."
+            )
+        )
+
+    ordered_rotations = [""] * len(bwt_string)
+    for x in range(len(bwt_string)):
+        for i in range(len(bwt_string)):
+            ordered_rotations[i] = bwt_string[i] + ordered_rotations[i]
+        ordered_rotations.sort()
+    return ordered_rotations[idx_original_string]
+
+
+if __name__ == "__main__":
+    entry_msg = "Provide a string that I will generate its BWT transform: "
+    s = input(entry_msg).strip()
+    result = bwt_transform(s)
+    bwt_output_msg = "Burrows Wheeler tranform for string '{}' results in '{}'"
+    print(bwt_output_msg.format(s, result["bwt_string"]))
+    original_string = reverse_bwt(
+        result["bwt_string"], result["idx_original_string"]
+    )
+    fmt = (
+        "Reversing Burrows Wheeler tranform for entry '{}' we get original"
+        " string '{}'"
+    )
+    print(fmt.format(result["bwt_string"], original_string))