How Can I Efficiently Generate Unique Random Numbers Without Repetition?

Generating Unique Random Numbers without Repetition

Creating a sequence of random numbers without repeats is a common problem in programming. Wanting to generate a sequence of numbers without repeats within a specific range is a popular variant of this problem.

Approaching the Problem

One straightforward method for generating a sequence of unique numbers is to create a list of all numbers in the desired range, shuffle the list, and then iterate through the shuffled list. However, this approach requires substantial memory allocation for large ranges.

Mathematical Algorithms

A more efficient approach involves using a linear feedback shift register (LFSR). LFSRs are hardware or software constructs that generate a sequence of pseudo-random numbers by shifting and XORing the bits within a register. By carefully selecting the tap points (bits used for feedback), LFSRs can produce sequences with very long periods, even for a short register size.

For example, a 16-bit LFSR can generate a sequence of up to 65,535 numbers without repeats. LFSRs are deterministic, producing the same sequence each time they are initialized with the same seed. However, they are statistically random and meet many of the requirements for cryptographic applications.

Implementing LFSRs

Implementing LFSRs requires careful selection of the tap points to achieve maximal length sequences. There are established methods for constructing LFSRs with desired periods. Many programming languages provide libraries or functions that implement LFSRs for convenience.

By using LFSRs, programmers can generate unique random number sequences without the need for large memory allocations or excessive shuffling operations. LFSRs are particularly useful when generating large random numbers or when non-repeating sequences are crucial for applications such as cryptography or simulation.

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