Questions tagged [randomized-algorithms]

An algorithm whose behaviour is determined by its input and a generator producing uniformly random numbers.

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41 votes
7 answers
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When does randomization speed up algorithms and it "shouldn't"?

Adleman's proof that $BPP$ is contained in $P/poly$ shows that if there is a randomized algorithm for a problem that runs in time $t(n)$ on inputs of size $n$, then there also is a deterministic ...
Dana Moshkovitz's user avatar
39 votes
9 answers
4k views

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 ...
adrianN's user avatar
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36 votes
1 answer
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Consequences of $\mathsf{NP}$ containing $\mathsf{BPP}$

Many believe that $\mathsf{BPP} = \mathsf{P} \subseteq \mathsf{NP}$. However we only know that $\mathsf{BPP}$ is in the second level of polynomial hierarchy, i.e. $\mathsf{BPP}\subseteq \Sigma^ \...
Kaveh's user avatar
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33 votes
9 answers
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Randomized algorithm that "looks" deterministic?

Is there an interesting example of a randomized algorithm for a search problem that always outputs the same (correct) answer, regardless of its internal randomness, but which exploits the randomness ...
arnab's user avatar
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28 votes
10 answers
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Probabilistic (randomized) algorithms before "modern" computer science appeared

Edit: I choice the answer with highest score by December 06, 2012. This is a soft question. The concept of (deterministic) algorithms dates back to BC. What about the probabilistic algorithms? In ...
28 votes
1 answer
1k views

Is uniform RNC contained in polylog space?

Log-space-uniform NC is contained in deterministic polylog space (sometimes written PolyL). Is log-space-uniform RNC also in this class? The standard randomized version of PolyL should be in PolyL, ...
Ryan O'Donnell's user avatar
27 votes
4 answers
2k views

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 ...
András Salamon's user avatar
27 votes
1 answer
915 views

Other applications of Karger-Stein branching amplification?

I just taught the Karger-Stein randomized mincut algorithm in my graduate algorithms class. This is a real algorithmic gem, so I can't not teach it, but it always leaves me frustrated, because I don'...
26 votes
1 answer
2k views

Who first proposed using $x^2+y^2 < 1$ Monte Carlo algorithm to calculate Pi?

I'm sure everybody knows of Buffon's needle experiment in the 18th century, that is one of the first probabilistic algorithms to calculate $\pi$. The implementation of the algorithm in computers ...
Jérémie's user avatar
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24 votes
1 answer
624 views

The randomized query complexity of the conjoined trees problem

An important 2003 paper by Childs et al. introduced the "conjoined trees problem": a problem admitting an exponential quantum speedup that's unlike just about any other such problem that we know of. ...
Scott Aaronson's user avatar
23 votes
3 answers
3k views

Generalizing the "median trick" to higher dimensions?

For randomized algorithms $\mathcal{A}$ taking real values, the "median trick" is a simple way to reduce the probability of failure to any threshold $\delta > 0$, at the cost of only a ...
Clement C.'s user avatar
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22 votes
2 answers
2k views

Bounds on $E[f(x)]$ in terms of $f(E[x])$ other than Jensen's inequality?

If $f$ is a convex function then Jensen's inequality states that $f(\textbf{E}[x]) \le \textbf{E}[f(x)]$, and mutatis mutandis when $f$ is concave. Clearly in the worst case you cannot upper bound $\...
Ian's user avatar
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22 votes
2 answers
3k views

Yao's Minimax Principle on Monte Carlo Algorithms

The celebrated Yao's Minimax Principle states the relation between distributional complexity and randomized complexity. Let $P$ be a problem with a finite set $\mathcal{X}$ of inputs and a finite set $...
Federico Magallanez's user avatar
21 votes
1 answer
723 views

A flowchart for concentration bounds

When I teach tail bounds, I use the usual progression: If your r.v is positive, you can apply Markov's inequality If you have independence and also bounded variance, you can apply Chebyshev's ...
Suresh Venkat's user avatar
20 votes
2 answers
652 views

Estimating Average in Polynomial Time

Let $f \colon \lbrace 0,1 \rbrace ^ n \to (2^{-n},1]$ be a function. We want to estimate the average of $f$; that is: $\mathbb{E}[f(n)]=2^{-n}\sum_{x\in \lbrace 0,1 \rbrace ^ n}f(x)$. ...
Sadeq Dousti's user avatar
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19 votes
3 answers
694 views

Running a BPP algorithm with a half-random, half-adversarial string

Consider the following model: an n-bit string r=r1...rn is chosen uniformly at random. Next, each index i∈{1,...,n} is put into a set A with independent probability 1/2. Finally, an adversary ...
Scott Aaronson's user avatar
18 votes
3 answers
816 views

Does randomness buy us anything inside P?

Let $\mathsf{BPTIME}(f(n))$ be the class of the decision problems having a bounded two-sided error randomized algorithm running in time $O(f(n))$. Do we know of any problem $Q \in \mathsf{P}$ such ...
aelguindy's user avatar
  • 951
18 votes
2 answers
2k views

Beginner's Guide to Derandomization

I found the book Pairwise Independence and Derandomization on the subject, but it's more research-oriented than tutorial oriented. I'm new to the subject of "Derandomization," and as such, I wanted ...
Sadeq Dousti's user avatar
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18 votes
2 answers
976 views

Are theoretically sound pseudorandom generators used in practice?

As far as I'm aware, most implementations of pseudorandom number generation in practice use methods such as linear shift feedback registers (LSFRs), or these "Mersenne Twister" algorithms. While they ...
Henry Yuen's user avatar
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18 votes
3 answers
809 views

In what class are randomized algorithms that err with exactly 25% chance?

Suppose I consider the following variant of BPP, which let us call E(xact)BPP: A language is in EBPP if there is a polynomial time randomized TG that accepts every word of the language with exactly 3/...
domotorp's user avatar
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17 votes
3 answers
725 views

Randomize or Not?

This question is inspired by the Georgia Tech Algorithms and Randomness Center's t-shirt, which asks "Randomize or not?!" There are many examples where randomizing helps, especially when operating in ...
Lev Reyzin's user avatar
  • 12k
17 votes
1 answer
1k views

The complexity of sampling (approximately) the Fourier transform of a Boolean function

One thing that quantum computers can do (possibly even with just BPP + log-depth quantum circuits) is to approximate-sample the Fourier transform of a Boolean $\pm 1$-valued function in P. Here and ...
Gil Kalai's user avatar
  • 6,023
16 votes
5 answers
2k views

Examples of successful derandomization from BPP to P

What are some major examples of successful derandomization or at least progress in showing concrete evidence towards $P=BPP$ goal (not the hardness randomness connection)? The only example that comes ...
user avatar
16 votes
2 answers
960 views

Faster join of treap-like data structures with approximately the same size

Given two AVL trees $T_1$ and $T_2$ and a value $t_r$ such that $\forall x \in T_1, \forall y \in T_2, x < t_r < y$, it is easy to construct a new AVL tree containing $t_r$ and the values in $...
jbapple's user avatar
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15 votes
2 answers
1k views

Which randomized algorithms have exponentially small error probability?

Suppose that a randomized algorithm uses $r$ random bits. The lowest error probability one can expect (falling short of a deterministic algorithm with 0 error) is $2^{-\Omega(r)}$. Which randomized ...
Dana Moshkovitz's user avatar
15 votes
1 answer
468 views

Natural theorems proven only "to high probability"?

There are plenty of situations where a randomized "proof" is much easier than a deterministic proof, the canonical example being polynomial identity testing. Question: Are there any natural ...
Geoffrey Irving's user avatar
15 votes
2 answers
575 views

Reusing 5-independent hash functions for linear probing

In hash tables that resolve collisions by linear probing, in order to ensure $O(1)$ expected performance, it is both necessary and sufficient that the hash function be from a 5-independent family. (...
jbapple's user avatar
  • 11.2k
14 votes
1 answer
481 views

Is it enough for linear program constraints to be satisfied in expectation?

In the paper Randomized Primal-Dual analysis of RANKING for Online Bipartite Matching, while proving that the RANKING algorithm is $\left(1 - \frac{1}{e}\right)$-competitive, the authors show that the ...
Arindam Pal's user avatar
  • 1,591
14 votes
1 answer
355 views

Generating Graphs with Trivial Automorphisms

I'm revising some cryptographic model. To show its inadequacy, I've devised a contrived protocol based on graph isomorphism. It is "commonplace" (yet controversial!) to assume the existence ...
Sadeq Dousti's user avatar
  • 16.5k
14 votes
1 answer
454 views

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 ...
Stasys's user avatar
  • 6,765
13 votes
4 answers
644 views

Can we fast generate perfectly uniformly mod 3 or solve NP problem?

To be honest, I don't know that much about how random number are generated (comments are welcome!) but let's assume the following theoretical model: We can get integers uniformly random from $[1,2^n]$ ...
domotorp's user avatar
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13 votes
2 answers
960 views

What is the proof of this nonstandard version of Azuma's inequality?

In Appendix B of Boosting and Differential Privacy by Dwork et al., the authors state the following result without proof and refer to it as Azuma's inequality: Let $C_1, \dots, C_k$ be real-valued ...
William Hoza's user avatar
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13 votes
2 answers
671 views

What is a very simple pseudodeterministic algorithm (for educational purposes)?

Definition. A randomized algorithm for a search problem is pseudodeterministic if it produces a fixed canonical solution to the search problem with high probability. Question. The notion of a ...
user69687's user avatar
  • 133
12 votes
5 answers
3k views

List of quantum-inspired algorithms

Advances in quantum computing have led to the development of new classical algorithms. Notable recent examples are quantum-inspired algorithms for linear algebra: A quantum-inspired classical ...
Juan Miguel Arrazola's user avatar
12 votes
1 answer
3k views

Count $k$-hop neighborhood for every vertex

For a node $v$ of a directed unweighted graph $G$, I define the $k$-hop neighborhood of $v$ as the set of vertices that are reachable from $v$ in $k$ hops or fewer (that is following a path with $k$ ...
Lipoff's user avatar
  • 121
12 votes
0 answers
223 views

Hitting edges in graphs at random and let them die with honor

Let $G=(V,E)$ be a finite simple 2-connected graph. Let $B\subseteq E$ be a set of bad edges of size $k:=|B|$. For each edge $e\in E$ we toss a fair coin ($p=1/2$) and if the outcome is head, we hit ...
XORwell's user avatar
  • 650
11 votes
2 answers
874 views

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 ...
Shaull's user avatar
  • 5,571
11 votes
1 answer
336 views

Randomness and small circuits complexity classes

Let $\mathcal{C}$ be a complexity class and $\textrm{BP-}\mathcal{C}$ be the randomized counterpart of $\mathcal{C}$ defined as $\textrm{BPP}$ with respect to $\textrm{P}$. More formally we provide ...
C.P.'s user avatar
  • 992
11 votes
2 answers
623 views

Lower bound on estimating $\sum_{k=1}^n a_k$ for non-increasing $(a_k)_k$

I'd like to know (related to this other question) if lower bounds were known for the following testing problem: one is given query access to a sequence of non-negative numbers $a_n \geq \dots\geq a_1$ ...
Clement C.'s user avatar
  • 4,461
11 votes
1 answer
312 views

Randomized Polynomial Hierarchy?

I wonder, what would happen, if in the definition of $PH$ (Polynomial Hierarchy, see, e.g., here), the role of $NP$ would be replaced by $RP$? It seems, we could still build a hierarchy, the same ...
Andras Farago's user avatar
11 votes
0 answers
201 views

On preprocessing a convex polyhedron prior to sampling

Most of the algorithms for estimating the volume of a convex polyhedron $K \subset R^d$ assume the existence of an affine transform $T$ with the property that $$ B \subset TK \tilde{\subset}\ \...
Suresh Venkat's user avatar
10 votes
3 answers
645 views

What is the fastest known simulation of BPP using Las Vegas algorithms?

$\mathsf{BPP}$ and $\mathsf{ZPP}$ are two of basic probabilistic complexity classes. $\mathsf{BPP}$ is the class of languages decided by probabilistic polynomial-time Turing algorithms where the ...
echuly's user avatar
  • 549
10 votes
2 answers
847 views

A converse to Fano's inequality ?

Fano's inequality can be stated in many forms, and one particularly useful one is due (with a minor modification) to Oded Regev: Let $X$ be a random variable, and let $Y = g(X)$ where $g(\cdot)$ is ...
Suresh Venkat's user avatar
10 votes
1 answer
457 views

Can true randomness (provably) be replaced with Kolmogorov randomness for RP?

Have there been any attempts to show that Kolmogorov randomness would be sufficient for RP? Would the probability used in the statement "If the correct answer is YES, then it (the probabilistic Turing ...
Thomas Klimpel's user avatar
10 votes
2 answers
318 views

What is the minimum over all distributions of unit vectors of the variance of the dot product of the vectors?

I am trying to find a distribution over $n$ random vectors, say $x_1,\ldots, x_n$, on the $k$-dimensional unit sphere (where $n > k$) that minimizes $\max_{i\neq j} \mathrm{Var}(x_i^T x_j)$ subject ...
peng's user avatar
  • 101
10 votes
1 answer
573 views

What is the advantage of designing deterministic distributed algorithms?

Distributed algorithms that are resilient to failures can either be deterministic or probabilistic. Take for example the consensus problem. Paxos is deterministic in the sense that given the ...
danyhow's user avatar
  • 201
10 votes
2 answers
544 views

Exact formula for the number of spanning trees of a rectangle

This blog talks about generating "twisty little mazes" using a computer an enumerating them. The enumeration can be done using Wilson's algorithm to get the UST, but I don't remember the formula for ...
john mangual's user avatar
10 votes
1 answer
554 views

Determine the minimum number of coin-weighings

In the paper On two problems of information theory, Erdõs and Rényi give lower bounds on the minimum number of weighings one must do to determine the number of false coins in a set of $n$ coins. ...
Nicholas Mancuso's user avatar
10 votes
2 answers
797 views

How to shuffle colour balls?

I have 400 balls, in which 100 are red, 40 are yellow, 50 are green, 60 are blue, 70 are purple, 80 are black. (balls of the same colour are identical) i need an efficient shuffling algorithm, so ...
colinfang's user avatar
  • 271
10 votes
1 answer
277 views

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 $\textrm{P}$. ...
C.P.'s user avatar
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