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A search algorithm is a recursive procedure which accepts an instance and a partial solution and attempts to extend it to a complete solution bit by bit. For example, consider a search algorithm trying to solve the eight queens puzzle. The search is initiated with an empty board. On any given position, the search algorithm attempts to place one more queen, ...

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The choice of function is done in nondeterministic polynomial time which is something you shouldn't try to "make sense of" in the real world. It is a mathematical model of computing where we assume that the machine can perform many simultaneous operations, kind of in parallel, and if any of these operations "succeed", you get one of those ...

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Let $f(n) = \begin{cases} 1 & \mbox{if$n$is odd}\\ n & \mbox{if$n$even} \end{cases}$, and $g(n) = \begin{cases} n & \mbox{if$n$is odd}\\ 1 & \mbox{if$n$even} \end{cases}$. Then, $h(n)=f(n)+g(n) = n+1 \not\in O(f(n))$. Indeed, for any $n_0$ and $c>0$, there is some $n \ge n_0$ such that $h(n) > c f(n)$. Simply pick $n$ as an ...

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In general, if NP turns out to be the subset of any tractable class then that will be a black swan event for those who depend on cryptography as it exists today. $NP \subseteq BQP$ is one such example. But it need not be a fatal blow to use of cryptography. Quantum computers with a large number of qubits have yet to be built, and it is unknown whether they ...

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This paper: Linearly-growing Reductions of Karp's 21 NP-complete Problems does not show any linear reductions, but references my quasi-linear reduction of Hamiltonian Cycle to SAT, and SAT to 3-SAT has a linear reduction. My reduction reduces a Hamiltonian Cycle problem (HCP) with n vertices and m directed edges to a Boolean Satisfiability problem (SAT) ...

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P, you may recall, is the class of problems for which there exists a deterministic Turing machine which solves the problem and the number of steps taken by that Turing machine is polynomial in the size of the input on the tape. NP is defined similarly, only the Turing machine is nondeterministic. That the "complexity zoo" is defined in terms of ...

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