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2011 26th Annual IEEE Conference on Computational Complexity
Making Branching Programs Oblivious Requires Superlogarithmic Overhead
San Jose, California USA
June 08-June 11
ISBN: 978-0-7695-4411-3
We prove a time-space tradeoff lower bound of $T =\Omega\left(n\log(\frac{n}{S}) \log \log(\frac{n}{S})\right) $ forrandomized oblivious branching programs to compute $1GAP$, alsoknown as the pointer jumping problem, a problem for which there is asimple deterministic time $n$ and space $O(\log n)$ RAM (randomaccess machine) algorithm.%Although no simulations of general%computation on randomized oblivious algorithms requiring only%polylogarithmic increase in time and space are yet known, our lower%bound implies that a superlogarithmic factor increase in time and%space is in fact necessary in any such simulation.We give a similar time-space tradeoff of $T =\Omega\left(n\log(\frac{n}{S}) \log \log(\frac{n}{S})\right)$ forBoolean randomized oblivious branching programs computing$GIP$-$MAP$, a variation of the generalized inner product problem thatcan be computed in time $n$ and space $O(\log^2 n)$ by adeterministic Boolean branching program.These are also the first lower bounds for randomized obliviousbranching programs computing explicit functions that apply for$T=\omega(n\log n)$. They also show that any simulation ofgeneral branching programs by randomized oblivious ones requires eithera superlogarithmic increase in time or an exponential increase in space.
Index Terms:
time-space tradeoffs, lower bounds, branching programs, oblivious computation, randomization
Paul Beame, Widad Machmouchi, "Making Branching Programs Oblivious Requires Superlogarithmic Overhead," ccc, pp.12-22, 2011 26th Annual IEEE Conference on Computational Complexity, 2011
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