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How would you hash a string in such a way that it can be un-hashed?

Suppose that you hash a string in python using a custom-made hash function named sash().
sash("hello world") returns something like 2769834847158000631.
What code (in python) would implement sash() function and a unsash() function such that unsash(sash("hello world")) returns "hello world"?
If you like, assume that the string contains ASCII characters only.
There are 128 ASCII characters.
Thus, each python string is like a natural number written in base 128.

A hash is fixed in size, whereas a string is not. Therefore there will be more more possible strings than hash values, making it impossible to reverse.
In your example, you have an 11-character string containing 77 bits. Your corresponding integer would fit in 64 bits (actually 62 bits, but I will take 64 bits as what you might have been imagining). If we consider only 11-character strings (obviously there are far more), we have 277 possible strings. Assuming a 64-bit hash, there are only 264 hash values. Each hash value would have, on average, 8192 strings that map to it. So given just the hash value, you would have no idea which of those 8192 strings to decode it to.
If you don't mind a hash of unbounded size, then sure, you can simply consider the string itself to be the hash. Then no decoding required. You can get a little fancier, since you are limiting the characters to 0..127, and pack seven bits for each character into a string of bytes, reducing the size by 1/8th. This is effectively the base-128 number you are referring to. You may be able to get it smaller with compression if your 0..127 characters do not have the same probability. Then on average, the string can be compressed, with some possible strings necessarily getting larger instead of smaller.

Python: Time complexity of .isAlpha

So I can't find the official documentation on how the isalpha method was written (of the string module), but I imagine the algorithm used would be: 1). Convert char in question to int.2). Compare it to the larger alpha-ascii values (i.e. 90 or 122) to see if it less than or equal to these values.3). Compare it to the larger alpha-ascii values, i.e. 55 or 97 depending on the upper bound used (if only less than 90 use 55...), to see if it greater than or equal to these values.Am I correct in this assessment of the isalpha method or is it something different altogether? If so does it have a complexity of O(3)?

Python handles text as unicode. As such, what character is alphabetic and what is not depends on the character unicode category, enconpassing all characters defined on the unicode version compiled along with Python. That is tens of hundreds of characters, and hundreds of scripts, etc... each with their own alphabetic ranges. Although it all boils down to numeric ranges of the codepoints that could be compared using other algorithms, it is almost certain all characters are iterated, and the character unicode category is checked. If you want the complexity, it is then O(n) .
(Actually, it would have been O(n) on your example as well, since all characters have to be checked. For a single character, Python uses a dict, or dict-like table to get from the character to its category infomation, and that is O(1))

Probability with collision on uuid slice

I want to give my user a unique ID that is ten digits long and randomized. I was previously using:
user.guid = str(uuid.uuid4()).replace('-','')
'5d2f251a32ed437689e7d66575aee09f'
However, I also want to make it a bit easier to 'type in', as there some places where that is required. I was thinking about doing:
>>> str(uuid.uuid4()).replace('-','')[:10].upper()
'AA6560AB32'
What are the chances that there is a collision on this? And is there a better way to guarantee uniqueness (without storing previously added IDs)?

The letters abcdef in a UUID string are hex digits. So you can change them to uppercase without problems.
A UUID is a guaranteed-unique 128-bit number. If you truncate it to 40 bits (ten hex digits) it is no longer guaranteed unique. If it's vital to tell your users apart, you probably should collision-test these 40-bit numbers after generating them before assigning them to users.

Fast hash for strings

I have a set of ASCII strings, let's say they are file paths. They could be both short and quite long.
I'm looking for an algorithm that could calculate hash of such a strings and this hash will be also a string, but will have a fixed length, like youtube video ids:
https://www.youtube.com/watch?v=-F-3E8pyjFo
^^^^^^^^^^^
MD5 seems to be what I need, but it is critical for me to have a short hash strings.
Is there a shell command or python library which can do that?

As of Python 3 this method does not work:
Python has a built-in hash() function that's very fast and perfect for most uses:
>>> hash("dfds")
3591916071403198536
You can then make it unsigned:
>>> hashu=lambda word: ctypes.c_uint64(hash(word)).value
You can then turn it into a 16 byte hex string:
>>> hashu("dfds").to_bytes(8,"big").hex()
Or an N*2 byte string, where N is <= 8:
>>> hashn=lambda word, N : (hashu(word)%(2**(N*8))).to_bytes(N,"big").hex()
..etc. And if you want N to be larger than 8 bytes, you can just hash twice. Python's built-in is so vastly faster, it's never worth using hashlib for anything unless you need security... not just collision resistance.
>>> hashnbig=lambda word, N : ((hashu(word)+2**64*hashu(word+"2"))%(2**(N*8))).to_bytes(N,"big").hex()
And finally, use the urlsafe base64 encoding to make a much better string than "hex" gives you
>>> hashnbigu=lambda word, N : urlsafe_b64encode(((hashu(word)+2**64*hash(word+"2"))%(2**(N*8))).to_bytes(N,"big")).decode("utf8").rstrip("=")
>>> hashnbigu("foo",16)
'ZblnvrRqHwAy2lnvrR4HrA'
Caveats:
Be warned that in Python 3.3 and up, this function is
randomized and won't work for some use cases. You can disable this with PYTHONHASHSEED=0
See https://github.com/flier/pyfasthash for fast, stable hashes that
that similarly won't overload your CPU for non-cryptographic applications.
Don't use this lambda style in real code... write it out! And
stuffing things like 2**32 in your code, instead of making them
constants is bad form.
In the end 8 bytes of collision resistance is OK for a smaller
applications.... with less than a million entries, you've got
collision odds of < 0.0000001%. That's a 12 byte b64 encoded
string. But it might not be enough for larger apps.
16 bytes is enough for a UUID/OID in a cache, etc.
Speed comparison for producing 300k 16 byte hashes from a bytes-input.
builtin: 0.188
md5: 0.359
fnvhash_c: 0.113
For a complex input (tuple of 3 integers, for example), you have to convert to bytes to use the non-builtin hashes, this adds a lot of conversion overhead, making the builtin shine.
builtin: 0.197
md5: 0.603
fnvhash_c: 0.284

I guess this question is off-topic, because opinion based, but at least one hint for you, I know the FNV hash because it is used by The Sims 3 to find resources based on their names between the different content packages. They use the 64 bits version, so I guess it is enough to avoid collisions in a relatively large set of reference strings. The hash is easy to implement, if no module satisfies you (pyfasthash has an implementation of it for example).
To get a short string out of it, I would suggest you use base64 encoding. For example, this is the size of a base64-encoded 64 bits hash: nsTYVQUag88= (and you can get rid or the padding =).
Edit: I had finally the same problem as you, so I implemented the above idea: https://gist.github.com/Cilyan/9424144

Another option: hashids is designed to solve exactly this problem and has been ported to many languages, including Python. It's not really a hash in the sense of MD5 or SHA1, which are one-way; hashids "hashes" are reversable.
You are responsible for seeding the library with a secret value and selecting a minimum hash length.
Once that is done, the library can do two-way mapping between integers (single integers, like a simple primary key, or lists of integers, to support things like composite keys and sharding) and strings of the configured length (or slightly more). The alphabet used for generating "hashes" is fully configurable.
I have provided more details in this other answer.

You could use the sum program (assuming you're on linux) but keep in mind that the shorter the hash the more collisions you might have. You can always truncate MD5/SHA hashes as well.
EDIT: Here's a list of hash functions: List of hash functions

Something to keep in mind is that hash codes are one way functions - you cannot use them for "video ids" as you cannot go back from the hash to the original path. Quite apart from anything else hash collisions are quite likely and you end up with two hashes both pointing to the same video instead of different ones.
To create an Id like the youtube one the easiest way is to create a unique id however you normally do that (for example an auto key column in a database) and then map that to a unique string in a reversible way.
For example you could take an integer id and map it to 0-9a-z in base 36...or even 0-9a-zA-Z in base 62, padding the generated string out to the desired length if the id on its own does not give enough characters.

Shortest hash in python to name cache files

What is the shortest hash (in filename-usable form, like a hexdigest) available in python? My application wants to save cache files for some objects. The objects must have unique repr() so they are used to 'seed' the filename. I want to produce a possibly unique filename for each object (not that many). They should not collide, but if they do my app will simply lack cache for that object (and will have to reindex that object's data, a minor cost for the application).
So, if there is one collision we lose one cache file, but it is the collected savings of caching all objects makes the application startup much faster, so it does not matter much.
Right now I'm actually using abs(hash(repr(obj))); that's right, the string hash! Haven't found any collisions yet, but I would like to have a better hash function. hashlib.md5 is available in the python library, but the hexdigest is really long if put in a filename. Alternatives, with reasonable collision resistance?
Edit:
Use case is like this:
The data loader gets a new instance of a data-carrying object. Unique types have unique repr. so if a cache file for hash(repr(obj)) exists, I unpickle that cache file and replace obj with the unpickled object. If there was a collision and the cache was a false match I notice. So if we don't have cache or have a false match, I instead init obj (reloading its data).
Conclusions (?)
The str hash in python may be good enough, I was only worried about its collision resistance. But if I can hash 2**16 objects with it, it's going to be more than good enough.
I found out how to take a hex hash (from any hash source) and store it compactly with base64:
# 'h' is a string of hex digits
bytes = "".join(chr(int(h[i:i+2], 16)) for i in xrange(0, len(h), 2))
hashstr = base64.urlsafe_b64encode(bytes).rstrip("=")

The birthday paradox applies: given a good hash function, the expected number of hashes before a collision occurs is about sqrt(N), where N is the number of different values that the hash function can take. (The wikipedia entry I've pointed to gives the exact formula). So, for example, if you want to use no more than 32 bits, your collision worries are serious for around 64K objects (i.e., 2**16 objects -- the square root of the 2**32 different values your hash function can take). How many objects do you expect to have, as an order of magnitude?
Since you mention that a collision is a minor annoyance, I recommend you aim for a hash length that's roughly the square of the number of objects you'll have, or a bit less but not MUCH less than that.
You want to make a filename - is that on a case-sensitive filesystem, as typical on Unix, or do you have to cater for case-insensitive systems too? This matters because you aim for short filenames, but the number of bits per character you can use to represent your hash as a filename changes dramatically on case-sensive vs insensitive systems.
On a case-sensitive system, you can use the standard library's base64 module (I recommend the "urlsafe" version of the encoding, i.e. this function, as avoiding '/' characters that could be present in plain base64 is important in Unix filenames). This gives you 6 usable bits per character, much better than the 4 bits/char in hex.
Even on a case-insensitive system, you can still do better than hex -- use base64.b32encode and get 5 bits per character.
These functions take and return strings; use the struct module to turn numbers into strings if your chosen hash function generates numbers.
If you do have a few tens of thousands of objects I think you'll be fine with builtin hash (32 bits, so 6-7 characters depending on your chosen encoding). For a million objects you'd want 40 bits or so (7 or 8 characters) -- you can fold (xor, don't truncate;-) a sha256 down to a long with a reasonable number of bits, say 128 or so, and use the % operator to cut it further to your desired length before encoding.

The builtin hash function of strings is fairly collision free, and also fairly short. It has 2**32 values, so it is fairly unlikely that you encounter collisions (if you use its abs value, it will have only 2**31 values).
You have been asking for the shortest hash function. That would certainly be
def hash(s):
return 0
but I guess you didn't really mean it that way...

You can make any hash you like shorter by simply truncating it. md5 is always 32 hex digits, but an arbitrary substring of it (or any other hash) has the proper qualities of a hash: equal values produce equal hashes, and the values are spread around a bunch.

I'm sure that there's a CRC32 implementation in Python, but that may be too short (8 hex digits). On the upside, it's very quick.
Found it, binascii.crc32

If you do have a collision, how are you going to tell that it actually happened?
If I were you, I would use hashlib to sha1() the repr(), and then just get a limited substring of it (first 16 characters, for example).
Unless you are talking about huge numbers of these objects, I would suggest that you just use the full hash. Then the opportunity for collision is so, so, so, so small, that you will never live to see it happen (likely).
Also, if you are dealing with that many files, I'm guessing that your caching technique should be adjusted to accommodate it.

We use hashlib.sha1.hexdigest(), which produces even longer strings, for cache objects with good success. Nobody is actually looking at cache files anyway.

Condsidering your use case, if you don't have your heart set on using separate cache files and you are not too far down that development path, you might consider using the shelve module.
This will give you a persistent dictionary (stored in a single dbm file) in which you store your objects. Pickling/unpickling is performed transparently, and you don't have to concern yourself with hashing, collisions, file I/O, etc.
For the shelve dictionary keys, you would just use repr(obj) and let shelve deal with stashing your objects for you. A simple example:
import shelve
cache = shelve.open('cache')
t = (1,2,3)
i = 10
cache[repr(t)] = t
cache[repr(i)] = i
print cache
# {'(1, 2, 3)': (1, 2, 3), '10': 10}
cache.close()
cache = shelve.open('cache')
print cache
#>>> {'(1, 2, 3)': (1, 2, 3), '10': 10}
print cache[repr(10)]
#>>> 10

Short hashes mean you may have same hash for two different files. Same may happen for big hashes too, but its way more rare.
Maybe these file names should vary based on other references, like microtime (unless these files may be created too quickly).