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Let $A$ be a randomised algorithm and $F$ be a function such that $A$ returns $F(x)$ on any input $x$. Furthermore suppose that, for input $x$ of size $n$, the $\textbf{expected}$ running time of $A$ is $O(n)$. Give an algorithm that is guaranteed to terminate in time $O(n)$ for every input, and which on input $x$ outputs $F(x)$ with probability at least $1-\varepsilon$ and otherwise returns ${\tt timeout}$.

Is there a way to do it with somehow derandomizing $A$ and using the timeout possibility in a clever way? I have no other ideas. Any help appreciated!

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Write $T_x$ for the random variable representing the running time of $A$ on input $x$. By Markov's inequality,

$$\Pr(T_x \geq 2 \operatorname{E}(T_x)) \leq \frac{1}{2}.$$

See if you can figure out a solution now.

Solution:

Choose $N$ so that $2^{-N} < \varepsilon$. For $1..N$, repeat the following: simulate $A(x)$ for $2 \operatorname{E}(T_x)$ steps with fresh independent random bits; if a value is returned from this simulation, then return that immediately. If we reach the end of the loop and none of the $N$ simulations returned a value, then return timeout.

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  • $\begingroup$ Hence for an integer constant $N$ such that $(\frac{1}{2})^N < \varepsilon$ we have that $N$ trials will suffice, i.e. our new algorithm is "Launch $A$ exactly $N$ times and at the trials at which it does not stop in $O(n)$ time, stop it manually and put timeout". Then we have that with probability $1-\varepsilon$ in at least one of these trials it will stop in $O(n)$ time. Does this work? $\endgroup$ – DesmondMiles Feb 6 at 13:35
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    $\begingroup$ @DesmondMiles That's the general idea, yeah. The way you wrote it is missing a couple details though, such as exactly how long you have to run $A$ for each time for this particular proof to go through. See my edit to the answer. $\endgroup$ – Aaron Rotenberg Feb 6 at 14:02

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