approximation – How are matchings a lower bound for an approximate vertex cover?

I am reading Algorithms by Dasgupta et al and they mention maximal matchings as approximations for vertex cover.

They mention that the 2-approximation ratio is a lower bound. How is a maximal matching a lower bound? Does it mean that it is the lowest approximation value we can get compared to $log n$ of the greedy algorithm that is polynomial?

The images below show the optimal vertex cover = 8 but and a maximal matching of 12.

Maximal matching in Dasgupta

I decided to find another matching which I later discovered is called a maximum matching from Wikipedia. Since a maximum matching is also a maximal matching, I assume that this the worst that we can get, a maximum matching of 16 vertices.

Maximum matching

How can one improve on the edges picked then? My 16 vertices versus the book’s 12?

A closed form or a good approximation of an infinite series related to the negative binomial distribution

Does anyone know a closed form for this expression:
$$sum_{r =1}^{infty}{{alpha + 2r – 1}choose{ r – 1}}(1 – p)^{alpha + r}p^{r},$$
where $alpha geq 1$ and $0<p<1$. A good approximation of this expression works too. Thanks!

nt.number theory – Question related to diophantine approximation involving $|n_1 alpha_1 + … + n_k alpha_k|$ for $n_i in mathbb{Z}$

Let $alpha_1, …, alpha_k$ be real numbers with following properties.
For all $-N leq n_1, …, n_k leq N$, not all $0$,
$$
|n_1 alpha_1 + … + n_k alpha_k| > 1/delta(N)
$$

for all $N > 1$.

I am interested in explicit examples of such $alpha_i$ and function $delta(N)$. I was able to find some articles dealing with upper bounds but I couldn’t find anything for lower bounds as in this question. Any comments or references are appreciated!

stochastic processes – Differentiable approximation of Brownian diffusion with bounded volatility

Yes. Let $X_t := int^t_0 sigma_s mathrm dW_s$. Due to Theorem V.6 from the book Stochastic Integration and Differential Equations (second edition) by P.E. Protter, there is a continuous and adapted process ${tilde X^n_t}_{tin(0;T)}$ such that
$$
tilde X^n_t
=
int^t_0 n cdot big( tilde X^n_s – X_s big) mathrm ds.
$$

Hence we define $tildesigma^n_s := n cdot ( tilde X^n_s – X_s )$, which is adapted and even continuous.

To prove the limit property, we first prove the following:

Lemma 1. For all $beta,delta in (0;infty)$, there exists $nu in (0;infty)$ such that
$$
mathbb P bigg( sup_{0 le s le t le min(s+nu,T)} bigg| int^t_s sigma_u mathrm dW_u bigg| le beta bigg)
ge 1 – delta.
$$

Proof: We discretize the interval $(0;T)$ and consider events that its increments stay bounded. For all integer $0 le k < N$ and all $alpha in (0;infty)$, we define
$$
A^alpha_{k,N}
:= bigg{ max_{frac{T}{N} k le t le frac{T}{N} (k+1)} bigg| int_{frac{T}{N} k}^t sigma_u mathrm dW_u bigg| ge alpha bigg}.
$$

Due to the Burkholder–Davis–Gundy inequality (since $sigma$ is bounded, say $vertsigmavert le overlinesigma$),
$$
mathbb Ebigg( max_{T/Ncdot k le t le T/Ncdot (k+1)} bigg| int_{T/Ncdot k}^t sigma_u mathrm dW_u bigg|^4 bigg)
le C_4 mathbb Ebigg( bigglangle int_{T/Ncdot k}^cdot sigma_u mathrm dW_u biggrangle_{T/Ncdot (k+1)}^2 bigg)
\=
C_4 mathbb Ebigg( bigg( int_{T/Ncdot k}^{T/Ncdot (k+1)} big(sigma_ubig)^2 mathrm du bigg)^2 bigg)
le C_4 bigg( frac{T} N cdot overlinesigma^2 bigg)^2
= N^{-2} C
$$

and due to the Markov inequality, we obtain
$$
mathbb P big( A^alpha_{k,N} big)
le alpha^{-4} mathbb Ebigg( max_{T/Ncdot k le t le T/Ncdot (k+1)} bigg| int_{T/Ncdot k}^t sigma_u mathrm dW_u bigg|^4 bigg)
le alpha^{-4} N^{-2} C
$$

Now we assume that $omega in Omega backslash bigcup_{k=0}^{N-1} A^alpha_{k,N}$ and assume $s, t in (0;T)$ with $s le t le s + T/N$. Then we can find a $k in {0,ldots,N-1}$ such that $T/Ncdot k le s le T/Ncdot (k+1) le t le T/Ncdot (k+2)$ or $T/Ncdot k le s le t le T/Ncdot (k+1)$. In the first case, we obtain
$$
bigg| bigg(int^t_s sigma_u mathrm dW_ubigg)(omega) bigg|
le bigg| bigg(int_{T/Ncdot (k+1)}^t sigma_u mathrm dW_ubigg)(omega) bigg|
+ bigg| bigg(int_{T/Ncdot k}^{T/Ncdot (k+1)} sigma_u mathrm dW_ubigg)(omega) bigg|
\
quad + bigg| bigg(int_{T/Ncdot k}^s sigma_u mathrm dW_ubigg)(omega) bigg|
le 3 alpha.
$$

In the second case, we get the same result analogously.

Let $omega in Omega backslash bigcup_{k=0}^{N-1} A^alpha_{k,N}$ and $s, t in (0;T)$ with $|s – t| le frac{T}{N}$. Then, $bigg| bigg(int^t_s sigma_u mathrm dW_ubigg)(omega) bigg| leq 3 alpha$ and so
$$
Omega backslash bigcup_{k=0}^{N-1} A^alpha_{k,N}
subseteq bigg{ max_{s,tin (0;T), |s-t| le frac T N} bigg| int^t_s sigma_u mathrm dW_u bigg| le 3 alpha bigg}.
$$

As a result, if $N$ is large enough,
$$
mathbb P bigg( max_{s,tin (0;T), |s-t| le frac T N} bigg| int^t_s tildesigma_u mathrm dW_u bigg| le 3 alpha bigg)
\ge
1 – sum_{k=0}^{N-1} mathbb P big( A^alpha_{k,N} big)
ge 1 – frac{C}{alpha^{4} N^{1}}
ge 1 – delta,
$$

which proves the statement.

Since $tilde X^n$ always moves into the direction of $X$, we also have the following:

Lemma 2.
$$
sup_{t in (0;T)} vert tilde X^n_t vert
le
sup_{t in (0;T)} vert X_t vert
$$

Now since the increments of $X$ are bounded on an event of large probability due to Lemma 1, it is also straightforward to prove this:

Lemma 3. Let
$$
M^{beta,nu}:=bigg{sup_{0 le s le t le min(s+nu,T)} bigg| int^t_s sigma_u mathrm dW_u bigg| le betabigg}.
$$

Then for all $omegain M$, we have
$$
sup_{tin (0;T)} bigvert tilde X^{beta/nu}_t – X_t bigvert
le 3 beta.
$$

Now we prove the main statement. Let $n:=beta/nu$. Due to the Minkovski inequality,
$$
sqrt{ mathbb Ebigg( int^T_0 big( tilde X^n_s – X_s big)^2 mathrm dt bigg) }
\le
sqrt{ mathbb Ebigg( mathbb 1_{M^{beta,nu}} int^T_0 big( tilde X^n_s – X_s big)^2 mathrm dt bigg) }
+ sqrt{ mathbb Ebigg( mathbb 1_{Omegabackslash M^{beta,nu}} int^T_0 big( tilde X^n_s – X_s big)^2 mathrm dt bigg) }
$$

The first summand can be bound directly by $3 beta sqrt T$ using Lemma 2. The second summand can be bound using Hölder inequality by
$$
mathbb Ebigg( int^T_0 mathbb 1_{Omegabackslash M^{beta,nu}} big( tilde X^n_s – X_s big)^2 mathrm dt bigg)
\le
sqrt{ mathbb Ebigg( int^T_0 mathbb 1_{Omegabackslash M^{beta,nu}} mathrm dt bigg) }
sqrt{ mathbb Ebigg( int^T_0 big( tilde X^n_s – X_s big)^4 mathrm dt bigg) }
=
sqrt T sqrt{1 – mathbb Pbig(M^{beta,nu}big) }
sqrt{ mathbb Ebigg( int^T_0 big( tilde X^n_s – X_s big)^4 mathrm dt bigg) }
$$

The first factor can be made arbitrarily small if choosing $nu$ small enough depending on $beta$ due to Lemma 1, and the second factor is bounded due to the boundedness of $sigma$ and Lemma 2.

Best approximation theorem

enter image description hereBest Approximation Theorem Let W is a finite dimensional subspace of an inner product space V and any vector in V. The best approximation to is not Proj ) in W. we have y from W is then Pr phi w ^ * for every w (that ||v-Pr rho x ^ prime ||<||v-w|| y is

diophantine approximation – More about Roth’s theorem: bound for the constant and multidimensional case

For a real number $x$, we denote
$$ |x|=inf_{min {Bbb Z}}|x+m|.$$

Problem 1:

Roth’s theorem states that given any irrational algebraic number $alpha$ and for any $epsilon>0$, there exists a constant $C(alpha,epsilon)$ such that

$$|qalpha|>frac{C(alpha,epsilon)}{q^{1+epsilon}}.$$

Is there an explicit bound for $C(alpha,epsilon)$$ ?$

Problem 2:

Let $t=(t_1,cdots, t_k)$ be a vector whose components are all integers and do not vanish simultaneously. Suppose that $1,alpha_1,cdots, alpha_k$ are real algebraic numbers and linear independent over ${Bbb Z}$.
Does there exist a constant $C$ independent of $t$ such that
$$|tcdotalpha|=|t_1alpha_1+cdots+t_kalpha_k|geq frac{C(alpha,epsilon)}{Pi_{t_jnot=0}|t_j|^{1+epsilon}} ?$$

For both of the problems, if the general case looks a little bit difficult, what about the special one that $epsilon=1 ?$

approximation – An example of bad estimate in the CountMin sketch algorithm

Suppose the CountMin sketch algorithm use exactly one hash function that maps elements of the stream to ${0,…,m-1}$, where $m=10$. Assume elements of the stream are from set $(1..n)$ the hash function $h:(1..n)rightarrow(0..(m-1))$ is chosen uniformly at random from a $2$-universal hash family $mathcal H$. That is for any $x,yin(1..n)$, if $xneq y$ then $Pr_{h sim mathcal H}(h(x)=h(y))=frac{1}{m}$.
Additionally, (if needed) assume that $mathcal H$ is $3$-uniform as well.

Is there an example of an input stream $sigma$ of length $t$, such that the probability is very high that at least one of the items, say $k in sigma$, the estimate of its frequency is much larger than its actual frequency

network – There is no official Mining Node counts….Is there an easy approximation of it?

I would like to have your opinions. To me, the number of Mining Nodes on the Network appears a fundamental and crucial information, yet I appear to be one of the few asking the question “How Many?”. I developed my own technique to form an approximation but am open to suggestions, or to being directed to the data which would help me assess.
My approximation only approximate the possible Number of Mining Nodes on the Network, which is not the number of Miners. Further, understand that I have no idea how the Network Difficulty assessment is presented, that is, is there statistical smoothing involved (probably), and to what degree…but my exercise is just to get a rough idea….
I round numbers on their theoretical basis just for demonstration purposes(i.e.: 55,000 blocks per year).
First, you take the Reward estimate earned by the Fastest Hashing rate Machine. Currently, to my knowledge, it appears to be the Antminer S19 Pro at 110Th/seconds (The WhatsMiner M30S++ clocks at 112Th/s, but is not gaining market momentum). So, you chose the best mining performance calculator you can find. I use a calculator from https://www.buybitcoinworldwide.com/mining/hardware/ because it is the only one ajusted for Network Difficulty when producing an estimate.
With this Hash Rate (110Th/s) this calculator estimates I get 0,2911 Bitcoin in rewards per year with the device running 24/7/365. This means that at 6.25 Bitcoin rewards per block, I contributed to solving 0.0465 of 1 block in 1 year. If all the Network used only these machines, then I divide the 55,000 Block per year by 0.0465 = 1,182,795 Mining Nodes. This is the Lower Bound, the minimum number of Mining Nodes on the system.
Now, my other assumption is that Network Difficulty is equivalent to the average weighted Hash rate of the sum of all Mining Nodes on the network (time frame is probably smoothed here). If the current, March 18 2021, Network Difficulty is 21.6 Th/seconds (from https://www.blockchain.com/charts/difficulty, then the same calculator calculates that if my device ran at this speed I would earn 0.0572 Bitcoin in 1 year. Using the same calculation, 55,000/(1/(6.25/.0572)) = 6,009,615 Mining Nodes. This would be the estimated number of total Mining Nodes on the Network.
Blockchain.com explains that it gets the Network Difficulty information from Full Nodes which have the number of Hash used to solve a block in a set time (around 600 seconds) once it has been authenticated. So in fact this is described as the Weighted average Hash rate of the Miners, which is used to either increase the difficulty if times falls measurably under 600 seconds per block, or loosen difficulty if time rises measurably over 600 seconds. But in the long term, the difficulty has risen at a rate of 400% per year. The calculator mentioned above does make an adjustment in calculating rewards, by using a discounting of performance of 0,45% per day.

optimization – Online approximation algorithm for median?

Is there a well-known or (relatively) easily-implementable streaming algorithm for approximating the median of the last, say, $k$ elements of a stream $c_1,c_2,c_3,dots$?

The scenario is: I have a stream of numbers, and at any point in the streaming process I should be able to query for the (approximate) median in the last $k$ elements.

Thanks!

How to derive the multivariate Pade Approximation for $ln left( {1 + frac{x}{y}} right)$?

How to derive the multivariate Pade Approximation for $ln left( {1 + frac{x}{y}} right)$ ?