RL (complexity)

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Randomized Logarithmic-space (RL), [1] sometimes called RLP (Randomized Logarithmic-space Polynomial-time), [2] is the complexity class of computational complexity theory problems solvable in logarithmic space and polynomial time with probabilistic Turing machines with one-sided error. It is named in analogy with RP, which is similar but has no logarithmic space restriction.

Contents

Definition

The probabilistic Turing machines in the definition of RL never accept incorrectly but are allowed to reject incorrectly less than 1/3 of the time; this is called one-sided error. The constant 1/3 is arbitrary; any x with 0 ≤ x ≤ 1 would suffice. This error can be made 2−p(x) times smaller for any polynomial p(x) without using more than polynomial time or logarithmic space by running the algorithm repeatedly.

Relation to other complexity classes

Sometimes the name RL is reserved for the class of problems solvable by logarithmic-space probabilistic machines in unbounded time. However, this class can be shown to be equal to NL using a probabilistic counter, and so is usually referred to as NL instead; this also shows that RL is contained in NL. RL is contained in BPL , which is similar but allows two-sided error (incorrect accepts). RL contains L , the problems solvable by deterministic Turing machines in log space, since its definition is just more general.

Noam Nisan showed in 1992 the weak derandomization result that RL is contained in SC , [3] the class of problems solvable in polynomial time and polylogarithmic space on a deterministic Turing machine; in other words, given polylogarithmic space, a deterministic machine can simulate logarithmic space probabilistic algorithms.

It is believed that RL is equal to L, that is, that polynomial-time logspace computation can be completely derandomized; major evidence for this was presented by Reingold et al. in 2005. [4] A proof of this is the holy grail of the efforts in the field of unconditional derandomization of complexity classes. A major step forward was Omer Reingold's proof that SL is equal to L.

Related Research Articles

In computational complexity theory, a branch of computer science, bounded-error probabilistic polynomial time (BPP) is the class of decision problems solvable by a probabilistic Turing machine in polynomial time with an error probability bounded by 1/3 for all instances. BPP is one of the largest practical classes of problems, meaning most problems of interest in BPP have efficient probabilistic algorithms that can be run quickly on real modern machines. BPP also contains P, the class of problems solvable in polynomial time with a deterministic machine, since a deterministic machine is a special case of a probabilistic machine.

In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm.

<span class="mw-page-title-main">ZPP (complexity)</span> Concept in computer science

In complexity theory, ZPP is the complexity class of problems for which a probabilistic Turing machine exists with these properties:

In theoretical computer science, a probabilistic Turing machine is a non-deterministic Turing machine that chooses between the available transitions at each point according to some probability distribution. As a consequence, a probabilistic Turing machine can—unlike a deterministic Turing Machine—have stochastic results; that is, on a given input and instruction state machine, it may have different run times, or it may not halt at all; furthermore, it may accept an input in one execution and reject the same input in another execution.

The space complexity of an algorithm or a data structure is the amount of memory space required to solve an instance of the computational problem as a function of characteristics of the input. It is the memory required by an algorithm until it executes completely. This includes the memory space used by its inputs, called input space, and any other (auxiliary) memory it uses during execution, which is called auxiliary space.

<span class="mw-page-title-main">Time complexity</span> Estimate of time taken for running an algorithm

In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor.

A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output are random variables.

<span class="mw-page-title-main">Complexity class</span> Set of problems in computational complexity theory

In computational complexity theory, a complexity class is a set of computational problems "of related resource-based complexity". The two most commonly analyzed resources are time and memory.

<span class="mw-page-title-main">PP (complexity)</span> Class of problems in computer science

In complexity theory, PP, or PPT is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of less than 1/2 for all instances. The abbreviation PP refers to probabilistic polynomial time. The complexity class was defined by Gill in 1977.

In computational complexity theory, an Arthur–Merlin protocol, introduced by Babai (1985), is an interactive proof system in which the verifier's coin tosses are constrained to be public. Goldwasser & Sipser (1986) proved that all (formal) languages with interactive proofs of arbitrary length with private coins also have interactive proofs with public coins.

In computational complexity theory, an advice string is an extra input to a Turing machine that is allowed to depend on the length n of the input, but not on the input itself. A decision problem is in the complexity class P/f(n) if there is a polynomial time Turing machine M with the following property: for any n, there is an advice string A of length f(n) such that, for any input x of length n, the machine M correctly decides the problem on the input x, given x and A.

In computational complexity theory, NL is the complexity class containing decision problems that can be solved by a nondeterministic Turing machine using a logarithmic amount of memory space.

In computational complexity theory, L is the complexity class containing decision problems that can be solved by a deterministic Turing machine using a logarithmic amount of writable memory space. Formally, the Turing machine has two tapes, one of which encodes the input and can only be read, whereas the other tape has logarithmic size but can be read as well as written. Logarithmic space is sufficient to hold a constant number of pointers into the input and a logarithmic number of boolean flags, and many basic logspace algorithms use the memory in this way.

In computational complexity theory, SL is the complexity class of problems log-space reducible to USTCON, which is the problem of determining whether there exists a path between two vertices in an undirected graph, otherwise described as the problem of determining whether two vertices are in the same connected component. This problem is also called the undirected reachability problem. It does not matter whether many-one reducibility or Turing reducibility is used. Although originally described in terms of symmetric Turing machines, that equivalent formulation is very complex, and the reducibility definition is what is used in practice.

In computer science, st-connectivity or STCON is a decision problem asking, for vertices s and t in a directed graph, if t is reachable from s.

In computational complexity theory, NL-complete is a complexity class containing the languages that are complete for NL, the class of decision problems that can be solved by a nondeterministic Turing machine using a logarithmic amount of memory space. The NL-complete languages are the most "difficult" or "expressive" problems in NL. If a deterministic algorithm exists for solving any one of the NL-complete problems in logarithmic memory space, then NL = L.

In computational complexity theory, L/poly is the complexity class of logarithmic space machines with a polynomial amount of advice. L/poly is a non-uniform logarithmic space class, analogous to the non-uniform polynomial time class P/poly.

In computational complexity theory, BPL, sometimes called BPLP, is the complexity class of problems solvable in logarithmic space and polynomial time with probabilistic Turing machines with two-sided error. It is named in analogy with BPP, which is similar but has no logarithmic space restriction.

In computational complexity theory, SC is the complexity class of problems solvable by a deterministic Turing machine in polynomial time and polylogarithmic space. It may also be called DTISP(poly, polylog), where DTISP stands for deterministic time and space. Note that the definition of SC differs from P ∩ PolyL, since for the former, it is required that a single algorithm runs in both polynomial time and polylogarithmic space; while for the latter, two separate algorithms will suffice: one that runs in polynomial time, and another that runs in polylogarithmic space..

<span class="mw-page-title-main">Structural complexity theory</span>

In computational complexity theory of computer science, the structural complexity theory or simply structural complexity is the study of complexity classes, rather than computational complexity of individual problems and algorithms. It involves the research of both internal structures of various complexity classes and the relations between different complexity classes.

References

  1. ↑ Complexity Zoo : RL
  2. ↑ A. Borodin, S.A. Cook, P.W. Dymond, W.L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM Journal on Computing, 18(3):559–578. 1989.
  3. ↑ Nisan, Noam (1992), "RL ⊆ SC", Proceedings of the 24th ACM Symposium on Theory of computing (STOC '92), Victoria, British Columbia, Canada, pp. 619–623, doi:10.1145/129712.129772 {{citation}}: CS1 maint: location missing publisher (link).
  4. ↑ O. Reingold and L. Trevisan and S. Vadhan. Pseudorandom walks in biregular graphs and the RL vs. L problem, ECCC   TR05-022, 2004.
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