I am looking for a shuffle algorithm to shuffle a set of sequential numbers without buffering. Another way to state this is that I’m looking for a random sequence of unique numbers that have a given period.
Your typical Fisher–Yates shuffle needs to have each element all of the elements it is going to shuffle, so that isn’t going to work.
A Linear-Feedback Shift Register (LFSR) does what I want, but only works for periods that are powers-of-two less two. Here is an example of using a 4-bit LFSR to shuffle the numbers 1-14:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
8
12
14
7
4
10
5
11
6
3
2
1
9
13
The first two is the input, and the second row the output. What’s nice is that the state is very small—just the current index. You can start of any index and get a difference set of numbers (starting at 1 yields: 8, 12, 14; starting at 9: 6, 3, 2), although the sequence is always the same (5 is always followed by 11). If I want a different sequence, I can pick a different generator polynomial.
The limitations to the LFSR are that the periods are always power-of-two less two (the min and max are always the same, thus unshuffled) and there not enough enough generator polynomials to allow every possible random sequence.
A block cipher algorithm would work. Every key produces a uniquely shuffled set of numbers. However all block ciphers (that I know about) have power-of-two block sizes, and usually a fixed or limited number of block sizes. A block cipher with a arbitrary non-binary block size would be perfect if such a thing exists.
There are a couple of projects I have that could benefit from such an algorithm. One is for small embedded micros that need to produce a shuffled sequence of numbers with a period larger than the memory they have available (think Arduino Uno needing to shuffle 1 to 100,000).
Does such an algorithm exist? If not, what things might I search for to help me develop such an algorithm? Or is this simply not possible?
Edit 2022-01-30
I have received a lot of good feedback and I need to better explain what I am searching for.
In addition to the Arduino example, where memory is an issue, there is also the shuffle of a large number of records (billions to trillions). The desire is to have a shuffle applied to these records without needing a buffer to hold the shuffle order array, or the time needed to build that array.
I do not need an algorithm that could produce every possible permutation, but a large number of permutations. Something like a typical block cipher in counter mode where each key produces a unique sequence of values.
A Linear Congruential Generator using coefficients to produce the desired sequence period will only produce a single sequence. This is the same problem for a Linear Feedback Shift Register.
Format-Preserving Encryption (FPE), such as AES FFX, shows promise and is where I am currently focusing my attention. Additional feedback welcome.
It is certainly not possible to produce an algorithm which could potentially generate every possible sequence of length N with less than N (log2N - 1.45) bits of state, because there are N! possible sequence and each state can generate exactly one sequence. If your hypothetical Arduino application could produce every possible sequence of 100,000 numbers, it would require at least 1,516,705 bits of state, a bit more than 185Kib, which is probably more memory than you want to devote to the problem [Note 1].
That's also a lot more memory than you would need for the shuffle buffer; that's because the PRNG driving the shuffle algorithm also doesn't have enough state to come close to being able to generate every possible sequence. It can't generate more different sequences than the number of different possible states that it has.
So you have to make some compromise :-)
One simple algorithm is to start with some parametrisable generator which can produce non-repeating sequences for a large variety of block sizes. Then you just choose a block size which is as least as large as your target range but not "too much larger"; say, less than twice as large. Then you just select a subrange of the block size and start generating numbers. If the generated number is inside the subrange, you return its offset; if not, you throw it away and generate another number. If the generator's range is less than twice the desired range, then you will throw away less than half of the generated values and producing the next element in the sequence will be amortised O(1). In theory, it might take a long time to generate an individual value, but that's not very likely, and if you use a not-very-good PRNG like a linear congruential generator, you can make it very unlikely indeed by restricting the possible generator parameters.
For LCGs you have a couple of possibilities. You could use a power-of-two modulus, with an odd offset and a multiplier which is 5 mod 8 (and not too far from the square root of the block size), or you could use a prime modulus with almost arbitrary offset and multiplier. Using a prime modulus is computationally more expensive but the deficiencies of LCG are less apparent. Since you don't need to handle arbitrary primes, you can preselect a geometrically-spaced sample and compute the efficient division-by-multiplication algorithm for each one.
Since you're free to use any subrange of the generator's range, you have an additional potential parameter: the offset of the start of the subrange. (Or even offsets, since the subrange doesn't need to be contiguous.) You can also increase the apparent randomness by doing any bijective transformation (XOR/rotates are good, if you're using a power-of-two block size.)
Depending on your application, there are known algorithms to produce block ciphers for subword bit lengths [Note 2], which gives you another possible way to increase randomness and/or add some more bits to the generator state.
Notes
The approximation for the minimum number of states comes directly from Stirling's approximation for N!, but I computed the number of bits by using the commonly available lgamma function.
With about 30 seconds of googling, I found this paper on researchgate.net; I'm far from knowledgable enough in crypto to offer an opinion, but it looks credible; also, there are references to other algorithms in its footnotes.
Related
I want a simple (non-cryptographic) random number generation algorithm where I can freely choose the period.
One candidate would be a special instance of LCG:
X(n+1) = (aX(n)+c) mod m (m,c relatively prime; (a-1) divisible by all prime factors of m and also divisible by 4 if m is).
This has period m and does not restrict possible values of m.
I intend to use this RNG to create a permutation of an array by generating indices into it. I tried the LCG and it might be OK. However, it may not be "random enough" in that distances between adjacent outputs have very few possible values (i.e, plotting x(n) vs n gives a wrapped line). The arrays I want to index into have some structure that has to do with this distance and I want to avoid potential issues with this.
Of course, I could use any good PRNG to shuffle (using e.g. Fisher–Yates) an array [1,..., m]. But I don't want to have to store this array of indices. Is there some way to capture the permuted indices directly in an algorithm?
I don't really mind the method ending up biased w.r.t choice of RNG seed. Only the period matters and the permuted sequence (for a given seed) being reasonably random.
Encryption is a one-to-one operation. If you encrypt a range of numbers, you will get the same count of apparently random numbers back. In this case the period will be the size of the chosen range. So for a period of 20, encrypt the numbers 0..19.
If you want the output numbers to be in a specific range, then pick a block cipher with an appropriately sized block and use Format Preserving Encryption if needed, as #David Eisenstat suggests.
It is not difficult to set up a cipher with almost any reasonable block size, so long as it is an even number of bits, using the Feistel structure. If you don't require cryptographic security then four or six Feistel rounds should give you enough randomness.
Changing the encryption key will give you a different ordering of the numbers.
I have a set of 64-bit unsigned integers with length >= 2. I pick 2 random integers, a, b from that set. I apply a deterministic operation to combine a and b into different 64-bit unsigned integers, c_1, c_2, c_3, etc. I add those c_ns to the set. I repeat that process.
What procedure can I use to guarantee that c will practically never collide with an existing bitstring on the set, even after millions of steps?
Since you're generating multiple 64-bit values from a pair of 64-bit numbers, I would suggest that you select two numbers at random, and use them to initialize a 64 bit xorshift random number generator with 128 bits of state. See https://en.wikipedia.org/wiki/Xorshift#xorshift.2B for an example.
However, it's rather difficult to predict the collision probability when you're using multiple random number generators. With a single PRNG, the rule of thumb is that you'll have a 50% chance of a collision after generating the square root of the range. For example, if you were generating 32-bit random numbers, your collision probability reaches 50% after about 70,000 numbers generated. Square root of 2^32 is 65,536.
With a single 64-bit PRNG, you could generate more than a billion random numbers without too much worry about collisions. In your case, you're picking two numbers from a potentially small pool, then initializing a PRNG and generating a relatively small number of values that you add back to the pool. I don't know how to calculate the collision probability in that case.
Note, however, that whatever the probability of collision, the possibility of collision always exists. That "one in a billion" chance does in fact occur: on average once every billion times you run the program. You're much better off saving your output numbers in a hash set or other data structure that won't allow you to store duplicates.
I think the best you can do without any other given constraints is to use a pseudo-random function that maps two 64-bit integers to a 64-bit integer. Depending on whether the order of a and b matter for your problem or not (i.e. (3, 5) should map to something else than (5, 3)) you shouldn't or should sort them before.
The natural choice for a pseudo-random function that maps a larger input to a smaller input is a hash function. You can select any hash function that produces an output of at least 64-bit and truncate it. (My favorite in this case would be SipHash with an arbitrary fixed key, it is fast and has public domain implementations in many languages, but you might just use whatever is available.)
The expected amount of numbers you can generate before you get a collision is determined by the birthday bound, as you are essentially selecting values at random. The linked article contains a table for the probabilities for 64-bit values. As an example, if you generate about 6 million entries, you have a collision probability of one in a million.
I don't think it is possible to beat this approach in the general case, as you could encode an arbitrary amount of information in the sequence of elements you combine while the amount of information in the output value is fixed to 64-bit. Thus you have to consider collisions, and a random function spreads out the probability evenly among all possible sequences.
I need pseudo random numbers generated for hardware (either in VHDL or Verilog) that meet the following criteria.
- Each number is 1-bit (doesn't have to be, but that would complicate things more)
- The N pseudo random numbers cannot be correlated with each other.
- The N pseudo random numbers need to be generated at the same time (every clock edge).
I understand that the following will not work :
- Using N different seeds for a given polynomial - they will simply be shifted versions of each other
- Using N different polynomials for a given length LFSR - not practical since N can be as large as 64, and I don't know what length LSFR would give 64 different tap combinations, too huge if possible at all.
If using LFSR, the lengths do not need to be identical. For a small N, say 4, I thought about using 4 different prime number lengths (to minimize repeatability), e.g., 15, 17, 19, 23, but again, for a large N, it gets very messy. Let's say, something on the order of 2^16 gives sufficient length for an LFSR.
Is there an elegant way of handling this problem? By elegant, I mean not having to code N different unique modules (15, 17, 19, 23 above as an example). Using N different instances of Mersenne Twister, with different seeds? I do not have unlimited amount of hardware resources (FF, LUT, BRAM), but for the sake of this discussion it's probably best to ignore resource issues.
Than you in advance.
One option is to use a cryptographic hash, these are typically wide (64-256 bits), and good hashes have the property that a single bit input change will propagate to all output bits in unpredictable fashion. Run an incrementing counter into the hash and start the counter at a random value.
The GHASH used in AES-GCM is hardware-friendly and can generate new output values every clock.
I need to generate around 9-100 million non-repeating random numbers, ranging from zero to the amount of numbers generated, and I need them to be generated very quickly. Several answers to similar questions proposed simply shuffling an array in order to get the random numbers, and others proposed using a bloom filter. The question is, which one is more efficient, and in case of it being the bloom filter, how do I use it?
You don't want random numbers at all. You want exactly the numbers 0 to N-1, in random order.
Simply filling the array and shuffling should be very quick. A proper Fisher-Yates shuffle is O(n), so an array of 100 million should take well under a second in C or even Java, slightly slower in a higher-level language like Python.
You only have to generate N-1 random numbers to do the shuffle (maybe up to 1.3N if you use rejection sampling to get perfect uniformity), so the speed will depend largely on how fast your RNG is.
You'll never need to look up whether a number has already be generated; that will deadly be slow no matter which algorithm you use, especially toward the end of the run.
If you need slightly fewer than N total numbers, fill the array from 0 to N-1, then just abort the shuffle early and take the partial result. Only if the amount of numbers you need is very small compared to their range should you consider the generate-and-check-for-dups approach. In that case Bob Floyd's algorithm might be good.
As an alternative you could use an appropriately sized block cypher. Use the block cypher to encrypt the numbers 0, 1, 2, ... and you will get a series of non-repeating random numbers out. Exactly what series will depend on the key you use. They are guaranteed not to repeat, because a block cypher is a reversible permutation.
For 64 bit numbers use DES, for 32 bit use Hasty Pudding (which allows a large range of block sizes) or write your own simple Feistel cypher. Assuming that security is not a big issue for this, then writing your own is possible.
For sure its better create an algorithm to shuffle the numbers, if you use a seed, as for example, the server microtime or timestamp, you can have one different random string for each milisecond .
Start creating an array using range function, set number of numbers as you like .
Than, you need to use a seed to make the pseudo-randomness better .
So, instead of rand, you gotta use SHUFFLE,
so you set array on range as 1 to 90, set the seed, than use shuffle to shuffle the array.. than you got all numbers in a random order (corresponding to the seed) .
You gotta change the seed to have another result .
The order of the numbers is the result .
as .. ball 1 : 42 ... ball 2: 10.... ball 3: 50.... ball 1 is 0 in the array. ;)
You can also use slice function and create a for / each loop, incrementing the slice factor, so you loop
slice array 0,1 the the result .. ball 1...
slice array 0.2 ball 2...
slice array 0.3
Thats the logic, i hope you understand, if so .. it ill help you a lot .
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Possible Duplicate:
Create Random Number Sequence with No Repeats
I'd like to write an URL shortener that only uses numbers as short string.
I don't want to count up, I want the next new number to be random (or pseudo random).
At first thought algorithm would then look like this (pseudo code):
do
{
number = random(0,10000)
}
while (datastore.contains(number))
datastore.store(number, url)
The problem with this implementation is: As the datastore contains more numbers, the more likely it is that the loop will be executed multiple times. The performance will decrease over time.
Isn't there a better way to get a random number that is not already in use?
1) fill an array with sequential values
2) shuffle the array
Use an encryption. Since encryption is reversible, unique inputs generate unique outputs. For 64 bit numbers use a cypher with a 64 bit blocksize. For smaller block sizes, such as 32 bit or 16 bit, have a look at the Hasty Pudding Cypher.
Whatever block size you need, just encrypt the numbers 0, 1, 2, ... (in the appropriate block size) to generate as many unique non-sequential numbers as you need.
Some related questions: # 2394246, # 54059, # 158716, # 196017, and # 1608181.
The proper approach depends on how many numbers you will generate and on if realtime performance is required. If you draw no more than a small fraction of the numbers available in a range, average time per number for your code snippet is O(1), with slight increase of time per later number but still O(1). See, for example, question #1608181 answer in which I show that getting k numbers from a range of more than 2*k numbers with such code is O(k). (That answer also has C code to generate M numbers from a range of N numbers, in O(M) time when M<N/2, and explains how to use it for O(M) time when M>=N/2.)
If you want O(1) performance with a hard time limit, you can use the program just mentioned to pre-load an array, or can shuffle the whole range of integers, as mentioned by Justin. After that preprocessing, each access is O(1). Buf if you know you won't draw more than say 3000 numbers from your 1...10000 range, and don't have a hard time limit, the code you have will run in O(1) time on average, with probability of k passes decreasing like 0.3 ^ k; i.e., at worst about 70% chance of 1 pass, 21% for 2, 6% for 3, 2% for 4, 0.6% for 5, and so forth.