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## Find the max partition of unique elements where each element corresponds to the set pool containing that element

Given a list of sets:

a b c -> _ c d   -> d b d   -> b a c   -> a a c   -> c 

The objective is to find the max partition of unique elements with each element corresponding to the set containing that element.

I was thinking about ordering the elements in n * log(n) based on occurrence in other groups and then iteratively start with the lowest, and reorder the list each time based on subtracting the occurrences of other elements within the lists containing the removed element. We are able to do so, as each unique element contains a set of pointers to the lists where the element is contained in.

I can store the unique elements with its occurrences in a Min-Heap, where each unique element has handler to the node within the Min-Heap, thus we can remove the min and also decrease the key one for others within the same list as the contained element in log(n) giving that we have it’s handler.

Is the approach feasible at all, if not, what approach I can make use of?

## What is the most unique data identifier for a phone user that cannot be repeated?

I’m currently developing an android (and probably iOS in the future) application for my company.

I was wondering what is the most unique data identifier to authenticate the users. A data that cannot be repeated through users.

For example:

Email? That user can log in with another phone using the email and password

Phone number? Could be the most unique one but it would required to verify the phone and I will have to setup a SMS validation service like WhatsApp

IMEI? It pretty much validates the unique phone but it can be spoofed or replaced. Although I don’t know if the application required permissions for this.

EDIT: Maybe a mix of all this methods?

My main goal is to save this data as a database and make it the primary key of it and with this know exactly who’s the user that it’s really using the company web services.

I hope you guys can help me.

Thank you.

## Make a secure USB (unique identifier)

Background: There is a requirement for the system security that I have to only allow the company approved USB devices (e.g., USB mass storage, keyboard, mouse, Bluetooth,, etc.,) and block all the rest (non-approved).

Even though PID, VID, serial number are unique identifiers to USB devices, but, if somebody knows those information he/she can easily create a USB with the identifiers mentioned above and produce an approved USB.

Problem: Is there any way that I can add unique and secure identifiers to USBs (except VID,PID, S/N) and setup a mechanism to differentiate between company approved USBs and non-approved ones and allow only the approved ones?

Expected result: Secure USB for devices that are left unattended (e.g., kiosk) in the public places.

Thank you!

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## What is the logical reasoning behind Arden’s Theorem proof of unique solution?

Here is the proof for Arden’s Theorem assertion that R=QP* is the unique (only solution) to R=Q+RP. My question is: what is the logical reasoning to prove that any equation is the unique (only solution)? Particularly in this case, how can the procedure

(1) recursively substitute R with R=Q+RP in R=Q+RP, then (2) establish the recursive definition of R, and finally (3) generalize the definition to R=QP*

logically lead to the proof that R=QP* must be the unique (only solution)?

Here is an example of the proof:

## Partitioning bag of sets such that each set in a group has a unique element

Suppose I have a bag (or multiset) of sets $$S = \{s_1, s_2, \dots, s_n\}$$ and $$\emptyset\notin S$$. I wish to partition $$S$$ into groups of sets such that within each group each set has at least one element not found in any other set in that group. Formally, the criterion for a group $$G = \{g_1, g_2, \dots \} \subseteq S$$ is:

$$\forall i: \left(g_i \setminus \bigcup_{j\neq i} g_j\;\neq\;\emptyset\right)$$

The partition $$P = \{\{s_1\}, \{s_2\}, \dots\}$$ always satisfies this requirement, so there is always a valid solution. But what is the smallest number of groups needed? Is this problem feasible or NP-complete?

Another formulation of this problem is to partition a multiset of integers into groups such that each integer has a bit set in its binary expansion that no other integer in its group has set.

## Branch and scope globally unique identifiers

Say we are working with a Prolog-like system where variables are dynamically created in different branch contexts and scopes, yet these variables are also globally viewable by the system regardless of the current context. Is there a way to label each variable with a unique scope ID which can be used to determine if the variable assignment is considered live (i.e. not discarded by backtracking) in the current context.

Naively, you could create a bitstream of left/right branch decisions, but the memory usage would be really excessive and also have bad query time for liveness. Can anything better be done?