Random generation of combinatorial structures from a uniform
Theoretical Computer Science
Algorithms for random generation and counting: a Markov chain approach
Algorithms for random generation and counting: a Markov chain approach
Random generation of embedded graphs and an extension to Dobrushin uniqueness (extended abstract)
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Lecture notes on approximation algorithms: Volume I
Lecture notes on approximation algorithms: Volume I
Algorithms for Almost-uniform Generation with an Unbiased Binary Source
Algorithms for Almost-uniform Generation with an Unbiased Binary Source
Random Cayley graphs and expanders
Random Structures & Algorithms
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We consider the problem of uniform generation of random integers in the range [1, n] given only a binary source of randomness. Standard models of randomized algorithms (e.g. probabilistic Turing machines) assume the availability of a random binary source that can generate independent random bits in unit time withu niform probability. This makes the task trivial if n is a power of 2. However, exact uniform generation algorithms with bounded run time do not exist if n is not a power of 2. We analyze several almost-uniform generation algorithms and discuss the tradeoff between the distance of the generated distribution from the uniform distribution, and the number of operations required per random number generated. In particular, we present a new algorithm which is based on a circulant, symmetric, rapidly mixing Markov chain. For a given positive integer N, the algorithm produces an integer i in the range [1, n] withp robability pi = pi(N) using O(N log n) bit operations such that | pi - 1/n | cβN, for some constant c, where β = 21/4/π (√2√2 -√ 5 - √ 5) ≅ 0.4087. This rate of convergence is superior to the estimates obtainable by commonly used methods of bounding the mixing rate of Markov chains such as conductance, direct canonical paths, and couplings.