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6

This is a very good question, which I don't think I can completely answer. What I can try to do is give some history of how the idea emerged (although I haven't been able to discover as much of the story as I'd like). provide an explanation of why FSM+datapath is a valuable abstraction technique. (Note: an explanation, not the explanation) I don't think I'...

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If a problem $A$ is verifiable in polynomial-time by a non-deterministic Turing machine $T$ (given an instance of the problem and certificate of length polynomially bounded in the size of the instance) then it is also solvable in polynomial-time by a non-deterministic Turing machine $T'$. The Turing machine $T'$ simply (non-deterministically) guesses the ...

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A gentle but rigorous introduction would be Algorithm Design by Tardos and Kleinberg. It covers all the necessary topics, and discusses the ideas and intuitions behind an algorithm before introducing it. Unlike CLRS, you don't need to "pick and choose" chapters to read as a beginner; instead, you can just follow through the chapters since they are ...

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A (tough) introduction is Jeff Erickson's "Algorithms". It is available as PDF for free. If you want to learn about how to program (how to put together the above to create useful applications), consider Downey's "Think Python" (be sure to get the second edition, as it is written for Python 3, the current version). Also a free PDF. Much of ...

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There certainly are division algorithms around, check e.g. Knuth's "Seminumerical Algorithms" or the book describing how libtommath works (this library was written more to explain how to do things than for utmost speed). With today's machines, where the speed is more limited by memory access than computation as such, brilliant hacks like the fast ...

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Newton's method is pretty good for this. Specifically, a good strategy is to first calculate an approximation $x \approx \dfrac{1}{b}$ so that $ax$ is then a good approximation for $\dfrac{a}{b}$. To do this, we will use Newton's method to approximate a root of the function $f(x) = \dfrac{1}{x} - b$. Newton's method asks us to recursively calculate ...

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More generally, this area is known as geometric deep learning, i.e., it encompasses learning not only on graph but on other non-Euclidean domains. A good broad start is the survey of Bronstein et al. [1], after which I can recommend basically any material by Bronstein like his many excellent presentations (also available on Youtube - just do a search). For ...

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FSM+Datapath, in this context, is a design technique of logic design. Any real-world sequential circuit can be modeled as an FSM, but in practice only sequential control logic is modeled as an FSM. For example, a D Flip-Flop can be modeled as an FSM, i.e. in terms of explicit states and their transitions, or it can be modeled in terms of logic gates, or a ...

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Although it is not an introduction for beginners, knowledge of the existence Knuth's work on fundamental algorithms is important for a beginning Computer Scientist. One day in the future they will need to consult a copy. So for completeness, and for your future as a computer scientist put this on your "wish list": The Art of Computer Programming, ...

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The class $\mathsf{NP}$ can be defined either as the class of languages that can be verified by a deterministic polynomial time algorithm; the class of languages that can be solved by a nondeterministic polynomial time algorithm. Regarding 1), a verification algorithm is a two-argument algorithm $A$, where one argument is an ordinary input string $x$, and ...

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