optimization – Clustering sets by set difference

Suppose you have $n$ nonequal sets $S_1, ldots, S_n$ and some constant $0 le k < n$. The goal of set clustering is to find a partition of the set $mathbf{S} = {S_1, ldots, S_n}$ such that the sum of the total distance for each subset of $mathbf{S}$ is minimized and such that $mathrm{cardinality}(mathbf{S}) = k$ (in reality, this latter constraint is not quite so tight, but the size of $mathbf{S}$ must be less than $k$, and hopefully near it). The total distance $d$ of a set $X in mathbf{S}$ is $d(X) = sum_{A, B in X} mathrm{cardinality}(A ominus B)$ where $ominus$ is symmetric difference. Assume for the purposes of the problem that the sets consume $O(1)$ space (so symmetric difference can be computed in constant time).

Is there a good greedy linear(ithmic) heuristic for this problem? Is there any literature on this problem, or similar ones?


All I’ve come up with so far is an $O(n^2 log n)$ heuristic that looks like:

  1. Set $I = {1, ldots, n}$
  2. Choose some $i in I$
  3. Emit a cluster $C subset I$ containing all $m in I$ such that $mathrm{cardinality}(S_i ominus S_m) le d_mathrm{max}$.
  4. Set $I = I – C$
  5. Go to step 2.

where this process occurs in each step of a binary search that finds the best value of $d_mathrm{max}$ for a given $k$.

I was thinking that if there were some way to sort the list of sets such that nearby sets in the list have small symmetric difference, then a linearithmic solution might be easy to write.

Can I set up a Google Analytics view that filters out anyone with a user-id tagged on them?

If you are capturing the user-id and sending it as a user scoped custom dimension, then you will be able to apply a filter to a View to exclude the users assigned a value for that particular custom dimension.

In doing this, any data before they are assigned a user-id and have it applied as a custom dimension will remain in the reports.

postgresql – wal_level set to replica at database level but i dont see that in configuration file

wal_level is commented in postgresql.conf and i dont see any entry in auto.conf either. but at the databsae level i see its set to REPLICA. Is there any other place whee it could have been set?

postgres@postgresqlmaster:/etc/postgresql/12/data$ psql -p 5433
psql (12.3 (Ubuntu 12.3-1.pgdg18.04+1))
Type "help" for help.

postgres=# select name, setting, sourcefile, sourceline from pg_settings where name = 'wal_level';
   name    | setting | sourcefile | sourceline
-----------+---------+------------+------------
 wal_level | replica |            |
(1 row)

postgres@postgresqlmaster:/etc/postgresql/12/data$ cat /var/lib/postgresql/12/data/postgresql.auto.conf
# Do not edit this file manually!
# It will be overwritten by the ALTER SYSTEM command.
listen_addresses = '*'
shared_buffers = '200MB'
synchronous_standby_names = 'pgstandby_synch1'

How to generate a set of orthogonal polynomials with a special weight function

Regarding to $e^{-ax-bx^2}$ as the weight function , how can I generate a set of orthogonal polynomials? I just know the Orthogonalize command in Mathematica which seems that uses 1 as weight function by Gram-Schmidt method, other commands such as HermiteH and LaguerreL generate special polynomials. But I don’t know how to do my case.
Will it be reasonable to produce a basis as $$A={x^n e^{-ax-bx^2} :n in N}$$ and then orthogonalize it?

lightning network – How to set up LndHub on top of RaspiBolt ir RaspiBlitz?

The RaspiBlitz is running LND. So, when you look at https://github.com/BlueWallet/LndHub, you will find that you only need to install Redis and then you can set up LndHub.io as in their documentation:

git clone git@github.com:BlueWallet/LndHub.git
cd LndHub
npm i

Install bitcoind, lnd and redis. Edit config.js and set it up correctly. Copy admin.macaroon and tls.cert in root folder of LndHub.

You can do all of that via ssh on your RaspiBlitz.

What’s an example of a set that’s not closed,open, or bounded?

Can’t think of an example of a set within the usual plane that isn’t closed,open, or bounded.

Magento 2 product edit page featured product option value No set

I am a beginner in magento 2 , i would like to add featured product value "No" in product edit page through programmically

How to set the screen size in Ubuntu to correctly fit your monitor so that there are no scroll bars on the bottom & right & no windows go out of view?

Pretty much what’s there in the question.
Following are the specs.
OS: Ubuntu 20.04 LTS
Monitor specs : 15.6"
Running on Oracle Virtual box

Thank you

dataset – Type of data in a data set columns

How can I check the type of data in the columns of the dataset and count how many are of that data type?

Here is the data set I am working with:

Dataset[{
  <|"speed" -> 4, "distance" -> 2.01|>,
  <|"speed" -> 4, "distance" -> 10.03|>,
  <|"speed" -> 7, "distance" -> 4.04|>,
  <|"speed" -> 7, "distance" -> 22.8|>,
  <|"speed" -> 8, "distance" -> 16.2|>,
  <|"speed" -> 16, "distance" -> 10.1|>,
  <|"speed" -> 8, "distance" -> 12.1|>,
  <|"speed" -> 9, "distance" -> 22.1|>,
  <|"speed" -> 11, "distance" -> 32.3|>,
  <|"speed" -> 6, "distance" -> 11.12|>}
 ]

real analysis, in a metric space, what does a set inside both open and closed mean?

in a metric space, what does a set inside both open and closed mean? could you explain specifically? Just like the discrete metric.