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Accumulate the grouped sum of values across trillions of values

I have a data reduction issue that is proving to be very difficult to solve.
Essentially, I have a program that calculates incremental values (floating point) for pairs of keys from a set of about 60 million keys total. The program will generate values for about 53 trillion pairs 'relatively' quickly (simply iterating through the values would take about three days 🤣). Not every pair of keys will occur, and many pairs will come up many times. There is no reasonable way to have the pairs come up in a particular order. What I need is a way to find the sum of the values generated for each pair of keys.
For data that would fit in memory, this is a very simple problem. In python it would look something like:
from collections import Counter
res = Counter()
for key1,key2,val in data_generator():
res[(key1,key2)] += val
The problem, of course, is that a mapping like that won't fit in memory. So I'm looking for a way to do this efficiently with a mix of on-disk and in-memory processing.
So far I've tried:
A postgresql table with upserts (ON CONFLICT UPDATE). This turned out to be far, far to slow.
A hybrid of in-memory dictionaries in python that write to a RocksDB or LMDB key value store when they get too big. Though these DBs are much faster than postgresql for this kind of task, the time to complete is still on the order of months.
At this point, I'm hoping someone has a better approach that I could try. Is there a way to break this problem up into smaller parts? Is there a standard MapReduce approach to this kind of problem?
Any tips or pointers would be greatly appreciated. Thanks!
Edit:
The computer I'm using has 64GB of RAM, 96 cores (most of my work is very parallelizable), and terabytes of HDD (and some SSD) storage.
It's hard to estimate the total number of key pairs that will be in the reduced result, but it will certainly be at least in the hundreds of billions.

As Frank Yellin observes, there's a one-round MapReduce algorithm. The mapper produces key-value pairs with key key1,key2 and value val. The MapReduce framework groups these pairs by key (the shuffle). The reducer sums the values.
In order to control the memory usage, MapReduce writes the intermediate data to disk. Traditionally there are n files, and all of the pairs with key key1,key2 go to file hash((key1,key2)) mod n. There is a tension here: n should be large enough that each file can be handled by an in-memory map, but if n is too large, then the file system falls over. Back of the envelope math suggests that n might be between 1e4 and 1e5 for you. Hopefully the OS will use RAM to buffer the file writes for you, but make sure that you're maxing out your disk throughput or else you may have to implement buffering yourself. (There also might be a suitable framework, but you don't have to write much code for a single machine.)
I agree with user3386109 that you're going to need a Really Big Disk. If you can regenerate the input multiple times, you can trade time for space by making k passes that each save only a 1/k fraction of the files.
I'm concerned that the running time of this MapReduce will be too large relative to the mean time between failures. MapReduce is traditionally distributed for fault tolerance as much as parallelism.
If there's anything you can tell us about how the input arises, and what you're planning to do with the output, we might be able to give you better advice.

what the fastest method to loop through a very large number of account in a log in page

I was wonder what if i had 1 million users, how much time will take it to loop every account to check for the username and email if the registration page to check if they are already used or not, or check if the username and password and correct in the log in page, won't it take ages if i do it in a traditional for loop?

Rather than give a detailed technical answer, I will try to give a theoretical illustration of how to address your concerns. What you seem to be saying is this:
Linear search might be too slow when logging users in.
I imagine what you have in mind is this: a user types in a username and password, clicks a button, and then you loop through a list of username/password combinations and see if there is a matching combination. This takes time that is linear in terms of the number of users in the system; if you have a million users in the system, the loop will take about a thousand times as long as when you just had a thousand users... and if you get a billion users, it will take a thousand times longer over again.
Whether this is a performance problem in practice can only be determined through testing and requirements analysis. However, if it is determined to be a problem, then there is a place for theory to come to the rescue.
Imagine one small improvement to our original scheme: rather than storing the username/password combinations in arbitrary order and looking through the whole list each time, imagine storing these combinations in alphabetic order by username. This enables us to use binary search, rather than linear search, to determine whether there exists a matching username:
check the middle element in the list
if the target element is equal to the middle element, you found a match
otherwise, if the target element comes before the middle element, repeat binary search on the left half of the list
otherwise, if the target element comes after the middle element, repeat binary search on the right half of the list
if you run out of list without finding the target, it's not in the list
The time complexity of this is logarithmic in terms of the number of users in the system: if you go from a thousand users to a million users, the time taken goes up by a factor of roughly ten, rather than one thousand as was the case for linear search. This is already a vast improvement over linear search and for any realistic number of users is probably going to be efficient enough. However, if additional performance testing and requirements analysis determine that it's still too slow, there are other possibilities.
Imagine now creating a large array of username/password pairs and whenever a pair is added to the collection, a function is used to transform the username into a numeric index. The pair is then inserted at that index in the array. Later, when you want to find whether that entry exists, you use the same function to calculate the index, and then check just that index to see if your element is there. If the function that maps the username to indices (called a hash function; the index is called a hash) is perfect - different strings don't map to the same index - then this unambiguously tells you whether your element exists. Notably, under somewhat reasonable assumptions, the time to make this determination is mostly independent from the number of users currently in the system: you can get (amortized) constant time behavior from this scheme, or something reasonably close to it. That means the performance hit from going from a thousand to a million users might be negligible.
This answer does not delve into the ugly real-world minutia of implementing these ideas in a production system. However, real world systems to implement these ideas (and many more) for precisely the kind of situation presented.
EDIT: comments asked for some pointers on actually implementing a hash table in Python. Here are some thoughts on that.
So there is a built-in hash() function that can be made to work if you disable the security feature that causes it to produce different hashes for different executions of the program. Otherwise, you can import hashlib and use some hash function there and convert the output to an integer using e.g. int.from_bytes. Once you get your number, you can take the modulus (or remainder after division, using the % operator) w.r.t. the capacity of your hash table. This gives you the index in the hash table where the item gets put. If you find there's already an item there - i.e. the assumption we made in theory that the hash function is perfect turns out to be incorrect - then you need a strategy. Two strategies for handling collisions like this are:
Instead of putting items at each index in the table, put a linked list of items. Add items to the linked list at the index computed by the hash function, and look for them there when doing the search.
Modify the index using some deterministic method (e.g., squaring and taking the modulus) up to some fixed number of times, to see if a backup spot can easily be found. Then, when searching, if you do not find the value you expected at the index computed by the hash method, check the next backup, and so on. Ultimately, you must fall back to something like method 1 in the worst case, though, since this process could fail indefinitely.
As for how large to make the capacity of the table: I'd recommend studying recommendations but intuitively it seems like creating it larger than necessary by some constant multiplicative factor is the best bet generally speaking. Once the hash table begins to fill up, this can be detected and the capacity expanded (all hashes will have to be recomputed and items re-inserted at their new positions - a costly operation, but if you increase capacity in a multiplicative fashion then I imagine this will not be too frequent an issue).

LevelDB for 100s of millions entries

What are the top factors to consider when tuning inserts for a LevelDB store?
I'm inserting 500M+ records in the form:
key="rs1234576543" very predictable structure. rs<1+ digits>
value="1,20000,A,C" string can be much longer but usually ~ 40 chars
keys are unique
key insert order is random
into a LevelDB store using the python plyvel, and see dramatic drop in speed as the number of records grows. I guess this is expected but are there tuning measures I could look at to make it scale better?
Example code:
import plyvel
BATCHSIZE = 1000000
db = plyvel.DB('/tmp/lvldbSNP151/', create_if_missing=True)
wb = db.write_batch()
# items not in any key order
for key, value in DBSNPfile:
wb.put(key,value)
if i%BATCHSIZE==0:
wb.write()
wb.write()
I've tried various batch sizes, which helps bit, but am hoping there's something else I've missed. For example, can knowing the max length of a key (or value) be leveraged?

(Plyvel author here.)
LevelDB keeps all database items in sorted order. Since you are writing in a random order, this basically means that all parts of the database get rewritten all the time since LevelDB has to merge SSTs (this happens in the background). Once your database gets larger, and you keep adding more items to it, this results in a reduced write throughput.
I suspect that performance will not degrade as badly if you have better locality of your writes.
Other ideas that may be worth trying out are:
increase the write_buffer_size
increase the max_file_size
experiment with a larger block_size
use .write_batch(sync=False)
The above can all be used from Python using extra keyword arguments to plyvel.DB and to the .write_batch() method. See the api docs for details.

How many times string appears in another string

I have a large static binary (10GB) that doesn't change.
I want to be able to take as input small strings (15 bytes or lower each) and then to determine which string is the least frequent.
I understand that without actually searching the whole binary I wont be able to determine this exactly, so I know it will be an approximation.
Building a tree/hash table isn't feasible since it will require about 256^15 bytes which is ALOT.
I have about 100GB of disk space and 8GB RAM which will be dedicated into this task, but I can't seem to find any way to accomplish this task without actually going over the file.
I have as much time as I want to prepare the big binary, and after that I'll need to decide which is the least frequent string many many times.
Any ideas?
Thanks!
Daniel.
(BTW: if it matters, I'm using Python)

Maybe build a hashtable with the counts for as many n-tuples as you can afford storage for? You can prune the trees that don't appear anymore. I wouldn't call it "approximation", but could be "upper bounds", with assurance to detect strings that don't appear.
So, say you can build all 4-tuples.
Then to count occurrences for "ABCDEF" you'd have the minimum of count(ABCD), count(BCDE), count(CDEF). If that is zero for any of those, the string is guaranteed to not appear. If it is one, it will appear at most once (but maybe not at all).

Because you have a large static string that does not change you could distinguish one-time work preprocessing this string which never has to be repeated from the work of answering queries. It might be convenient to do the one-time work on a more powerful machine.
If you can find a machine with an order of magnitude or so more internal storage you could build a suffix array - an array of offsets into the stream in sorted order of the suffixes starting at the offset. This could be stored in external storage for queries, and you could use this with binary search to find the first and last positions in sorted order where your query string appears. Obviously the distance between the two will give you the number of occurrences, and a binary search will need about 34 binary chops to do 16 Gbyte assuming 16Gbytes is 2^34 bytes so each query should cost about 68 disk seeks.
It may not be reasonable to expect you to find that amount of internal storage, but I just bought a 1TB USB hard drive for about 50 pounds, so I think you could increase external storage for one time work. There are algorithms for suffix array construction in external memory, but because your query strings are limited to 15 bytes you don't need anything that complicated. Just create 200GB of data by writing out the 15-byte string found at every offset followed by an 5-byte offset number, then sort these 20-byte records with an external sort. This will give you 50Gbytes of indexes into the string in sorted order for you to put into external storage to answer queries with.

If you know all of the queries in advance, or are prepared to batch them up, another approach would be to build an http://en.wikipedia.org/wiki/Aho%E2%80%93Corasick_string_matching_algorithm tree from them. This takes time linear in the total size of the queries. Then you can stream the 10GB data past them in time proportional to the sum of the size of that data and the number of times any string finds a match.

Since you are looking for which is least frequent, and are willing to accept approximate solution. You could use a series of Bloom filters instead of a hash table. If you use sufficiently large ones, you shouldn't need to worry about the query size, as you can probably keep the false positive rate low.
The idea would be to go through all of the possible query sizes and make sub-strings out of them. For example, if the queries will be between 3 and 100, then it would cost (N * (sum of (i) from i = 3 to i = 100)). Then one by one add the subsets to one of the bloom filters, such that the query doesn't exist within the filter, creating a new one Bloom filter with the same hash functions if needed. You obtain the count by going through each filter and checking if the query exists within it. Each query then simply goes through each of the filter and checks if it's there, if it is, it adds 1 to a count.
You'll need to try to balance the false positive rate as well as the number of filters. If the false positive rate gets too high on one of the filters it isn't useful, likewise it's bad if you have trillions of bloom filters (quite possible if you one filter per sub-string). There are a couple of ways these issues can be dealt with.
To reduce the number of filters:
Randomly delete filters until there are only so many left. This will likely increase the false negative rate, which probably means it's better to simply delete the filters with the highest expected false positive rates.
Randomly merge filters until there are only so many left. Ideally avoiding merging a filter too often as it increases the false positive rate. Practically speaking, you probably have too many to do this without making use of the scalable version (see below), as it'll probably be hard enough to manage the false positive rate.
It also may not be a bad to avoid a greedy approach when adding to a bloom filter. Be rather selective in which filter something is added to.
You might end up having to implement scalable bloom filters to keep things manageable, which sounds similar to what I'm suggesting anyway, so should work well.

BST or Hash Table?

I have large text files upon which all kinds of operations need to be performed, mostly involving row by row validations. The data are generally of a sales / transaction nature, and thus tend to contain a huge amount of redundant information across rows, such as customer names. Iterating and manipulating this data has become such a common task that I'm writing a library in C that I hope to make available as a Python module.
In one test, I found that out of 1.3 million column values, only ~300,000 were unique. Memory overhead is a concern, as our Python based web application could be handling simultaneous requests for large data sets.
My first attempt was to read in the file and insert each column value into a binary search tree. If the value has never been seen before, memory is allocated to store the string, otherwise a pointer to the existing storage for that value is returned. This works well for data sets of ~100,000 rows. Much larger and everything grinds to a halt, and memory consumption skyrockets. I assume the overhead of all those node pointers in the tree isn't helping, and using strcmp for the binary search becomes very painful.
This unsatisfactory performance leads me to believe I should invest in using a hash table instead. This, however, raises another point -- I have no idea ahead of time how many records there are. It could be 10, or ten million. How do I strike the right balance of time / space to prevent resizing my hash table again and again?
What are the best data structure candidates in a situation like this?
Thank you for your time.

Hash table resizing isn't a concern unless you have a requirement that each insert into the table should take the same amount of time. As long as you always expand the hash table size by a constant factor (e.g. always increasing the size by 50%), the computational cost of adding an extra element is amortized O(1). This means that n insertion operations (when n is large) will take an amount of time that is proportionate to n - however, the actual time per insertion may vary wildly (in practice, one of the insertions will be very slow while the others will be very fast, but the average of all operations is small). The reason for this is that when you insert an extra element that forces the table to expand from e.g. 1000000 to 1500000 elements, that insert will take a lot of time, but now you've bought yourself 500000 extremely fast future inserts before you need to resize again. In short, I'd definitely go for a hash table.

You need to use incremental resizing of your hash table. In my current project, I keep track of the hash key size used in every bucket, and if that size is below the current key size of the table, then I rehash that bucket on an insert or lookup. On a resizing of the hash table, the key size doubles (add an extra bit to the key) and in all the new buckets, I just add a pointer back to the appropriate bucket in the existing table. So if n is the number of hash buckets, the hash expand code looks like:
n=n*2;
bucket=realloc(bucket, sizeof(bucket)*n);
for (i=0,j=n/2; j<n; i++,j++) {
bucket[j]=bucket[i];
}

library in C that I hope to make
available as a Python module
Python already has very efficient finely-tuned hash tables built in. I'd strongly suggest that you get your library/module working in Python first. Then check the speed. If that's not fast enough, profile it and remove any speed-humps that you find, perhaps by using Cython.
setup code:
shared_table = {}
string_sharer = shared_table.setdefault
scrunching each input row:
for i, field in enumerate(fields):
fields[i] = string_sharer(field, field)
You may of course find after examining each column that some columns don't compress well and should be excluded from "scrunching".