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We give a brief statement and proof of Yao's technique for lowering bounding the randomized complexity of a problem. Yao's technique is essentially an application of VonNeumann's mini-max principle.

Notation: Let A be a generic deterministic algorithm, or equivalently a pure strategy in game theoretic terms. Let $\cal A$ be a generic randomized algorithm algorithm, or a equivalently a generic distribution over deterministic algorithms, or equivalently a mixed strategy in game theoretic terms. Let I be a generic input. Let $\cal I$ and $\cal J$ be a generic distribution over the inputs. We use E[A, I] to denote the time used by algorithm A on input I, $E[{\cal A}, I]$ to denoted the expected time used by algorithm $\cal A$ on input I, $E[{\cal A}, {\cal I}]$ to denoted the expected time used by algorithm $\cal A$ on input distribution $\cal I$, and $E[A, {\cal I}]$ to denote the expected time of algorithm A on input distribution $\cal I$.

Statement:

\begin{displaymath}\min_{\cal A} \max_I E[{\cal A}, I] \ge
\max_{\cal I} \min_{A} E[A, {\cal I}]\end{displaymath}

That is, you can lower bounded the randomized time complexity by the average case time on any distribution.

Proof: Consider any input distribution $\cal I$. Since the optimal distribution $\cal A$ puts all the probability on the best algorithm, it follows that:

\begin{displaymath}\min_{A} E[A, {\cal I}] = \min_{\cal A} E[A, {\cal I}] \end{displaymath}

Now

\begin{displaymath}
\min_{\cal A} E[A, {\cal I}] \le
\min_{\cal A} \max_{\cal J} E[{\cal A}, {\cal J}]\end{displaymath}

since the maximizer may choose $\cal J$ to be $\cal I$. Then

\begin{displaymath}
\min_{\cal A} \max_{\cal J} E[{\cal A}, {\cal J}] =
\min_{\cal A} \max_{I} E[{\cal A}, I]\end{displaymath}

since the optimal choice for the maximizer is to put all the probability on the input that maximizes the expected running time for $\cal A$

Example Application: Since the expected running time of every deterministic comparison-based algorithm for sorting is $\Omega(n \log n)$ it follows by Yao's technique that the expected running time of every randomized comparison-based algorithm is $\Omega(n \log n)$.




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Kirk Pruhs
1999-04-01