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Is there only one unique DPDA that accepts a specific language?

Or is it possible to construct more than one DPDA that accepts exactly the same language?

networking – Since UDP multi/broadcast over internet is not possible – how do games that uses UDP over internet send to thousands/millions unique clients?

I am asking from a programming point of view as I struggle to clearly understand the architecture ex online multiplayer games do when they communicate UDP packets over the internet to thousands/millions of players.

I understand that UDP broadcast and multicast is not easily available when its not internal network but across the internet.

The only other way I can see is that you then manually have to loop through each UDP connection and send the same data to each client ( just as you would if you had a lot of TCP clients ) ? is this true ? I have tried to loop through TCP connections this way and at a certain point things go slow and CPU go high.

if no 1 is the only way, then it would mean that client 1 will get things a lot faster than client number 10000 when it gets to this client in the loop ? are there techniques to get around this in any way ?

since we would have to send the same packet of data to each and every client that of course creates A LOT of network traffic but also it would i assume max out the CPU also if we all the time would have to loop through all clients for each little packet we have to send ?

Please note I am specifically talking about UDP in raw form – not interested in Web-socket versions or ‘use TCP instead’ as its the nature of UDP I am trying to understand here and have been reading all weekend here and while I have a fairly good understanding of the packet level of UDP etc. my network technical knowledge is really not good and it does not tend to stick when I try to read non-programming technical articles about networking/routers/switches/ISP support.

So hoping that someone out there could help me understand it with a programming mindset in mind how something like multiplayer games/VOIP/large IOT networks achieve the ‘real-time’ capabilities of UDP if it has to manually process and send packets onto each UDP sockets.

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ag.algebraic geometry – Unique almost complex structure up to diffeomorphism

You were lucky to find the only possible example. If you take any manifold of dimension $ge 4$ you can pick an almost complex structure that is integrable in some closed ball and make it non-integrable outside of it. Since complex $n$-balls have more than one holomorphic structure, we are done. And all surfaces apart from $S^2$ have moduli of complex structures.

algorithm – Number of unique elements only using equality comparisons

Is there a way to find the number of unique elements in a list in faster (worst case) running time than O(n^2) if you can only use equality comparisons between two elements? No deleting, copying or other indexing of elements is allowed (other than looping through them to compare). We basically don’t know what the values of the elements are, we can only tell if two of them are the same. There is no more information about the distribution of elements and you can’t assume integers only.

The best I could do is just brute force it – comparing the current element to all the previous elements which is O(n^2) but I’m not sure how to prove that this is the best possible running time.