# Questions tagged [derandomization]

Every randomized algorithm can be simulated by a deterministic algorithm, at the expense of an exponential increase in running time. Derandomization is about converting randomized algorithms into efficient deterministic algorithms.

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### More on PH in PP?

A recent question by Huck Bennett asking whether the class PH was contained in the class PP, received somewhat contradictory answers (all true, it seems). On one hand, several oracle results were ...
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### Efficient and simple randomized algorithms where determinism is difficult

I often hear that for many problems we know very elegant randomized algorithms, but no, or only more complicated, deterministic solutions. However, I only know a few examples for this. Most ...
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### Why does randomness have stronger effect on reductions than on algorithms?

It is conjectured that randomness does not extend the power of polynomial time algorithms, that is, ${\bf P}={\bf BPP}$ is conjectured to hold. On the other hand, randomness seems to have a quite ...
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### Is BPP= P known for ANY uniform model of computation?

Many believe that BPP $=$ P "should" hold for Turing machines. We even have some "witnesses" for this: otherwise some "strange" things would happen; see e.g. this paper by Implagliazzo and Wigderson. ...
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### Derandomizing Valiant-Vazirani?

The Valiant-Vazirani theorem says that if there is a polynomial time algorithm (deterministic or randomized) for distinguishing between a SAT formula that has exactly one satisfying assignment, and an ...
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### Hierarchy for BPP vs derandomization

In one sentence: would the existence of a hierarchy for $\mathsf{BPTIME}$ imply any derandomization results? A related but vaguer question is: does the existence of a hierarchy for $\mathsf{BPTIME}$ ...
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### Problems in $\mathsf{BPP}$ not known to be in $\mathsf P$?

What problems are known to belong to $\mathsf{BPP}$ but not known to belong to $\mathsf P$? More precisely, I am interested in independent problems, that is whose derandomizations are not known to be ...
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### What specific evidence is there for P = RP?

RP is the class of problems decidable by a nondeterministic Turing machine that terminates in polynomial time, but that is also allowed one-sided error. P is the usual class of problems decidable by ...
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### Is BPP vs. P a real problem after we know BPP lies in P/poly?

We know (for now about 40 years, thank Adleman, Bennet and Gill) that the inclusion BPP $\subseteq$ P/poly, and an even stronger BPP/poly $\subseteq$ P/poly hold. The "/poly" means that we ...
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### Space efficient "industrial" unbalanced expanders

I am looking for unbalanced expanders that are "good" and "space-efficient". Specifically, a bipartite left-regular graph $G=(A,B,E)$, $|A|=n$, $|B|=m$, with left degree $d$ is a $(k,\epsilon)$-...
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### Fine-grained complexity of BPP

If E does not have i.o.-$2^{o(n)}$ circuits, then P=BPP, but this does not tell us about the fine-grained containments between $\mathrm{Time}(n^a)$ and $\mathrm{BPTime}(n^b)$. Are there reasonable ...
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### Does Nisan's pseudo-random generator relativize?

Nisan proved in "Psuedorandom Generators for Space-Bounded Computation", that there exists a pseudo-random generator which "fools" space-bounded computations. Does this construction hold for every ...
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### More efficient non-uniform derandomization ?

Adleman, FOCS'78 showed that any randomized circuit for inputs of length $n$ can be non-uniformly derandomized. However, the construction effectively duplicates the original circuit $O(n)$ times, so ...
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### Adleman's theorem over infinite semirings?

Adleman has shown in 1978 that $\mathrm{BPP}\subseteq \mathrm{P/poly}$: if a boolean function $f$ of $n$ variables can be computed by a probabilistic boolean circuit of size $M$, then $f$ can be also ...
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### When does randomization stops helping within PSPACE

It is known that adding bounded-error randomization to PSPACE doesn't add power. That is, BPPSAPCE=PSPACE. It is famously unknown whether P=BPP, but it is known that $BPP\subseteq \Sigma_2\cap \Pi_2$....
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### Pairwise independent gaussians

Given $X_1,\ldots,X_k$ (i.i.d. gaussians with mean $0$ and variance $1$), is it possible (how?) to sample (for $m=k^2$) $Y_1, \ldots, Y_m$ such that $Y_i$'s are pairwise independent gaussians with ...
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### Streaming derandomization

Stream algorithms require randomization for the most part to do anything nontrivial, and because of the small-space constraint, need PRGs that use little space. I know of two methods that have been ...
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### Impagliazzo and Wigderson's famous P=BPP paper

I'm reading Impagliazzo and Wigderson's famous $\mathsf P=\mathsf{BPP}$ paper in 1997. Since I'm new to this field and the paper is a concise conference version, I have difficulty following their ...
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### On fooling $AC^0$

I have a few questions regarding fooling constant depth circuits. It's known that $\log^{O(d)}(n)$-wise independence is necessary to fool $AC^0$ circuits of depth $d$, where $n$ is the size of the ...
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### Randomized algorithms using a stack

I have developed a new derandomization technique which is aimed at recursive randomized algorithms (or) more generally randomized algorithms that use a stack. Unfortunately, I could not find natural ...
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### Randomized algorithms not based on Schwartz-Zippel

Are there any problems that are known to be in a randomized complexity class (e.g. RNC, ZPP, RP, BPP, or even PP), but not in any lower non-randomized class (e.g. NC, P, NP), and whose membership in ...
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### Uniform way of quantifying "branching" in nondeterministic, probabilistic, and quantum computation?

The computation of a nondeterministic Turing machine (NTM) is well known to be representable as a tree of configurations, rooted at the starting configuration. Any transition in the program is ...
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### Consequence of PIT over $\Bbb Z[x_1,\dots,x_n]$ not having efficient algorithm

Given $p(x_1,\dots,x_n),q(x_1,\dots,x_n)\in \Bbb Z[x_1,\dots,x_n]$ such that coefficients of $p,q$ are bounded by $B$, does $p\equiv q$ hold? Schwartz-Zippel lemma applies here since it holds for ...
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### Borel-Cantelli Lemma and Derandomization

I was reading a paper titled Random Oracles with(out) Programmability. The last paragraph of section 2.3 reads: [Using our novel approach] there is no need to apply well-known classical ...
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### Constructions better than a random one.

I am interested in examples of constructions in the complexity theory which are better than a random constructions. The only one example of such construction which I know is in the field of error-...
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### $BPL$ with polylog random bits is in $L$

Consider a $BPL$ machine (namely, a probabilistic algorithm that uses logspace and polynomially many random bits). It is known (Saks-Zhou) that $BPL \subseteq DSPACE(log^{1.5}(n))$. My question is ...
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### On derandomizing polynomial identity testing

In polynomial identity testing we seek a deterministic algorithm to infer equality of two polynomials $g,h\in\Bbb Z[x_1,\dots,x_n]$. Derandomizing known efficient randomized algorithms and producing ...
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### Uniform derandomisation of circuit complexity classes

Let $\mathcal{C}$ be a complexity class and $\textrm{BP-}\mathcal{C}$ be the randomized counterpart of $\mathcal{C}$ defined in the same way as $\textrm{BPP}$ is defined with respect to $\textrm{P}$. ...
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### Is it known whether $BPP\cap NP\subseteq RP$?

The reverse inclusion is obvious, as is the fact that any self-reducible NP language in BPP is also in RP. Is this also known to hold for non-self-reducible NP languages?
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### Has the derandomization of slightly non-uniform classes, e.g BPP/linear, been studied?

By BPP/linear I refer to BPP machines with linear advice, which fulfills the promise when given the "correct" advice, and the derandomization should give us, say, a P/linear or (SUBEXP/linear) ...
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### Can we construct a k-wise independent permutation on [n] using only constant time and space?

Let $k>0$ be a fixed constant. Given an integer $n$, we want to construct a permutation $\sigma \in S_n$ such that: The construction uses constant time and space (i.e. preprocessing takes constant ...
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### What are some results on algorithms that estimate polynomials over a given set of points?

There seem to be many randomized algorithms for polynomial identity testing, checking whether or not a given polynomial is zero. Are there any results of algorithms that do some sort of estimation of ...
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### Extensions of Affine Dispersers

A function $f\colon\{0,1\}^n\to\{0,1\}$ is called an affine disperser for dimension $d$, if for every affine subspace $S\subseteq \{0,1\}^n$ of dimension at least $d$, $f$ is not constant on $S$. This ...
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### Extracting randomness from Santha-Vazirani sources using a seed of constant length

This question is actually an exercise problem from Salil Vadhan's draft survey "Pseudorandomness" marked with a star (*) (see Chapter 6, Problem 6.6). I do not know other references. We say a random ...
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### P vs. NP and Pseudorandom Bit Generators

According to an article on pseudorandom number generators (PRNG) by Jeff Lagarias, he states that trying to prove that a PRNG is unpredictable (secure) is just "as hard" as trying to prove ...
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### Why should we believe that $NEXP \not \subset P/poly$

I am sorry if this is not an advanced question. Most computer scientists believed that $NEXP \not \subset P/poly$ but they are not even close to this assumption. The main evidence that they are used ...
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### Deterministic approximation algorithms for treewidth

As far as I understand, the factor $O(\sqrt{\log OPT})$ approximation algorithm for treewidth of Feige, Hajiaghayi, and Lee is randomized, and no deterministic approximation algorithm with this factor ...
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### On the shortest vector problem (is it $NP$-complete?)
Ajtai has shown that shortest vector problem is $NP$-hard by using randomized reduction from subset sum. Has this been derandomized?