Motivation
---------

When compressing sequences of symbols, it is useful to have many equal symbols follow each other, because then they can be encoded with a run length encoding. For example, RLE encoding of `"aaaabbbbbbbbbbbcccccc"` would give something like `4a 11b 6c`.

(Look [here](http://www.codewars.com/kata/run-length-encoding/) for learning more about the run-length-encoding.)

Of course, RLE is interesting only if the string contains many identical consecutive characters. But what bout human readable text? Here comes the Burrows-Wheeler-Transformation.

Transformation
-------------
There even exists a transformation, which brings equal symbols closer together, it is called the **Burrows-Wheeler-Transformation**. The forward transformation works as follows: Let's say we have a sequence with length n, first write every shift of that string into a *n x n* matrix:
```
Input: "bananabar"

b a n a n a b a r
r b a n a n a b a
a r b a n a n a b
b a r b a n a n a
a b a r b a n a n
n a b a r b a n a
a n a b a r b a n
n a n a b a r b a
a n a n a b a r b
```
Then we sort that matrix by its rows. The output of the transformation then is the **last column** and the **row index** in which the original string is in:
```
               .-.
a b a r b a n a n
a n a b a r b a n
a n a n a b a r b
a r b a n a n a b
b a n a n a b a r <- 4
b a r b a n a n a
n a b a r b a n a
n a n a b a r b a
r b a n a n a b a
               '-'

Output: ("nnbbraaaa", 4)
```

```if:java
To handle the two kinds of output data, we will use the preloaded class `BWT`, whose contract is the following:

    public class BWT {
        
        public String s;
        public int n;
        
        public BWT(String s, int n)
        
        @Override public String  toString()
        @Override public boolean equals(Object o)
        @Override public int     hashCode()
    }

```

Of course we want to restore the original input, therefore you get the following hints:

1. The output contains the last matrix column.
2. The first column can be acquired by sorting the last column.
3. **For every row of the table:** Symbols in the first column follow on symbols in the last column, in the same way they do in the input string.
4. You don't need to reconstruct the whole table to get the input back.

Goal
----

The goal of this Kata is to write both, the `encode` and `decode` functions. Together they should work as the identity function on lists. (*Note:* For the empty input, the row number is ignored.)

Further studies
--------------
You may have noticed that symbols are not always consecutive, but just in proximity, after the transformation. If you're interested in how to deal with that, you should have a look at [this Kata](http://www.codewars.com/kata/move-to-front-encoding/).

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = 'from bisect import bisect_left\n\n\ndef encode(s):\n    if s == "":\n        return (\'\', 0)\n    \n    # Pretend that I am a linked list with pointers to head and tail\n    string_array = list(s)\n    \n    # Create the matrix\n    matrix = []\n    for _ in range(len(s)):\n        # Shift the string array\n        head = string_array.pop(0)\n        string_array.append(head)\n        \n        # Add shifted array to the matrix\n        matrix.append(list(string_array))\n\n    # Sort the rows\n    matrix.sort()\n    \n    # Get the last col string\n    lost_col_array = []\n    for row in matrix:\n        lost_col_array.append(row[-1])\n    \n    # Now find the original string in the matrix\n    row_num = bisect_left(matrix, string_array)\n    return ("".join(lost_col_array), row_num)\n    \n\ndef decode(s, n):\n    if s == "":\n        return ""\n    \n    matrix = []\n    for index, char in enumerate(s):\n        matrix.append([char])\n    \n    matrix.sort()\n    \n    # Do the following s - 1 times\n    for _ in range(len(s) - 1):\n        \n        # Write out the encoded message and then sort\n        for index, char in enumerate(s):\n            matrix[index].insert(0, char)\n        \n        matrix.sort()\n        \n    return "".join(matrix[n])'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'encode'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 6986