[All Your Base] Approaches Doc (#3652)

* [All Your Base] draft approaches doc

* Cleaned up the language a bit and added a couple of examples.

* Added deque ref at end of doc.

---------

Co-authored-by: BethanyG <BethanyG@users.noreply.github.com>

Author: colinleach

exercises/practice/all-your-base/.approaches/config.json

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+{
+  "introduction": {
+    "authors": ["colinleach",
+                "BethanyG"],
+    "contributors": []
+  }
+}

exercises/practice/all-your-base/.approaches/introduction.md

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+# Introduction
+
+The main aim of `All Your Base` is to understand how non-negative integers work in different bases.
+Given that mathematical understanding, implementation can be relatively straightforward.
+
+
+For this approach and its variations, no attempt was made to benchmark performance as this would distract from the main focus of writing clear, correct code for conversion.
+
+
+## General guidance
+
+All successful solutions for base conversion involve three steps:
+
+1. Check that inputs are valid (no non-integer or negative values).
+2. Convert the input list to a Python `int`, per the examples given in the instructions.
+3. Convert the `int` from step 2 into an output list in the new base.
+
+Some programmers prefer to separate the two conversions into separate functions, others put everything in a single function.
+This is largely a matter of taste, and either structure can be made reasonably concise and readable.
+
+
+## 1. Checking the inputs
+
+Solution code should check that the input base is at least 2, and that the output base is 2 or greater.
+Bases outside the range should rase `ValueError`s for input base and output base respectively.
+
+```python
+    if input_base < 2:
+        raise ValueError("input base must be >= 2")
+
+    if not all( 0 <= digit < input_base for digit in digits) :
+        raise ValueError("all digits must satisfy 0 <= d < input base")
+
+    if not output_base >= 2:
+        raise ValueError("output base must be >= 2")
+
+```
+
+Additionally, all input numbers should be positive integers greater or equal to 0 and strictly less than the given number base.
+For the familiar base-10 system, that would mean 0 to 9.
+As implemented, the tests require that invalid inputs raise a `ValueError` with  "all digits must satisfy 0 <= d < input base" as an error message.
+
+
+## 2. Convert the input digits to an `int`
+
+The next step in the conversion process requires that the input list of numbers be converted into a single integer.
+The four code fragments below all show variations of this conversion:
+
+```python
+# Simple loop
+    value = 0
+    for digit in digits:
+        value = input_base * value + digit
+
+# Loop, separating the arithmetic steps
+    value = 0
+    for digit in digits:
+        value *= input_base
+        value += digit
+
+# Sum a generator expression over reversed digits
+    value = sum(digit * input_base ** position for position, digit in enumerate(reversed(digits)))
+
+# Sum a generator expression with alternative reversing
+    value = sum(digit * (input_base ** (len(digits) - 1 - index)) for index, digit in enumerate(digits))
+```
+
+In the first two, the `value *= input_base` step essentially left-shifts all the previous digits, and `value += digit` adds a new digit on the right.
+In the two generator expressions, an exponentation like `input_base ** position` left-shifts the current digit to the appropriate position in the output.
+
+
+````exercism/note
+
+It is important to think about these procedures until they makes sense: these short code fragments are the main point of the exercise. 
+In each code fragment, the Python `int` is called `value`, a deliberately neutral identifier. 
+Surprisingly many students use names like `decimal` or `base10` for the intermediate value, which is misleading.  
+
+A Python `int` is an object with a complicated (but largely hidden) implementation.
+There are methods to convert an `int` to string representations such as octal, binary or hexadecimal, but these do not change the internal representation.
+````
+
+
+## 3. Convert the intermediate `int` to output digits
+
+The `int` created in step 2 can now be reversed, using a different base.
+
+Again, there are multiple code snippets shown below, which all do the same thing (essentially).
+In each case, we need the value and the remainder of integer division.
+The first snippet adds new digits at the start of the `list`, while the next two add them at the end.
+The final snippet uses [`collections.deque()`][deque] to prepend, then converts to a `list` in the `return` statement.
+
+
+These snippets represent choices of where to take the performance hit: appending to the end is a **much** faster and more memory efficient way to grow a `list` (O(1)), but the solution then needs an extra reverse step, incurring O(n) performance for the reversal.
+_Prepending_ to the `list` is very expensive, as every addition needs to move all other elements of the list "over" into new memory.
+The `deque` has O(1) prepends and appends, but then needs to be converted to a `list` before being returned, which is an  O(n) operation.
+
+
+```python
+from collections import deque
+
+
+out = []
+
+# Step forward, insert new digits at index 0 (front of list).
+# Least performant, and not recommended for large amounts of data.
+    while value > 0:
+        out.insert(0, value % output_base)
+        value = value // output_base
+
+# Append values to the end (mor efficient), then reverse the list.
+    while value:
+        out.append(value % output_base)
+        value //= output_base
+    out.reverse()
+
+# Use divmod() and reverse list, same efficiency a above.
+    while value:
+        div, mod = divmod(value, output_base)
+        out.append(mod)
+        value = div
+    out.reverse()
+    
+# Use deque() for effcient appendleft(), convert to list.
+    converted_digits = deque()
+
+    while number > 0:
+        converted_digits.appendleft(number % output_base)
+        number  = number // output_base
+
+    return list(converted_digits) or [0]
+```
+
+
+Finally, we return the digits just calculated.
+
+A minor complication is that a zero value needs to be `[0]`, not `[]` according to the tests.
+Here, we cover this case in the `return` statement, but it could also have been trapped at the beginning of the program, with an early `return`:
+
+
+```python
+# return, with guard for empty list
+    return out or [0]
+```
+
+## Recursion option
+
+An unusual solution to the two conversions is shown below.
+It works, and the problem is small enough to avoid stack overflow (Python has no tail recursion).
+
+
+In practice, few Python programmers would take this approach without carefully thinking about the bounds of the program and any possible memoization/performance optimizations they could take to avoid issues.
+While Python *allows* recursion, it does nothing to *encourage* it, and the default recursion limit is set to only 1000 stack frames.
+
+
+```python
+def base_to_dec(input_base, digits):
+    if not digits:
+        return 0
+    return input_base * base_to_dec(input_base, digits[:-1]) + digits[-1]
+
+
+def dec_to_base(number, output_base):
+    if not number:
+        return []
+    return [number % output_base] + dec_to_base(number // output_base, output_base)
+```
+
+[deque]: https://docs.python.org/3/library/collections.html#collections.deque
+