Revision #3 Authors: Paul Beame, Widad Machmouchi

Accepted on: 21st April 2011 03:08

Downloads: 1960

Keywords:

We prove a time-space tradeoff lower bound of $T =

\Omega\left(n\log(\frac{n}{S}) \log \log(\frac{n}{S})\right) $ for

randomized oblivious branching programs to compute $1GAP$, also

known as the pointer jumping problem, a problem for which there is a

simple deterministic time $n$ and space $O(\log n)$ RAM (random

access machine) algorithm.

We give a similar time-space tradeoff of $T =

\Omega\left(n\log(\frac{n}{S}) \log \log(\frac{n}{S})\right)$ for

Boolean randomized oblivious branching programs computing

$\GIPshuffle$, a variation of the generalized inner product problem that

can be computed in time $n$ and space $O(\log^2 n)$ by a

deterministic Boolean branching program.

These are also the first lower bounds for randomized oblivious

branching programs computing explicit functions that apply for

$T=\omega(n\log n)$.

They also show that any simulation of

general branching programs by randomized oblivious ones requires either

a superlogarithmic increase in time

or an exponential increase in space.

Revision #2 Authors: Paul Beame, Widad Machmouchi

Accepted on: 16th December 2010 07:01

Downloads: 1583

Keywords:

We prove a time-space tradeoff lower bound of $T =

\Omega\left(n\log(\frac{n}{S}) \log \log(\frac{n}{S})\right) $ for

randomized oblivious branching programs to compute $1GAP$, also

known as the pointer jumping problem, a problem for which there is a

simple deterministic time $n$ and space $O(\log n)$ RAM (random

access machine) algorithm.

We give a similar time-space tradeoff of $T =

\Omega\left(n\log(\frac{n}{S}) \log \log(\frac{n}{S})\right)$ for

Boolean randomized oblivious branching programs computing

$\GIPshuffle$, a variation of the generalized inner product problem that

can be computed in time $n$ and space $O(\log^2 n)$ by a

deterministic Boolean branching program.

These are also the first lower bounds for randomized oblivious

branching programs computing explicit functions that apply for

$T=\omega(n\log n)$.

They also show that any simulation of

general branching programs by randomized oblivious ones requires either

a superlogarithmic increase in time

or a very substantial increase in space.

Revision #1 Authors: Paul Beame, Widad Machmouchi

Accepted on: 16th December 2010 06:56

Downloads: 1139

Keywords:

We prove a time-space tradeoff lower bound of $T =

\Omega\left(n\log(\frac{n}{S}) \log \log(\frac{n}{S})\right) $ for

randomized oblivious branching programs to compute $1GAP$, also

known as the pointer jumping problem, a problem for which there is a

simple deterministic time $n$ and space $O(\log n)$ RAM (random

access machine) algorithm.

We give a similar time-space tradeoff of $T =

\Omega\left(n\log(\frac{n}{S}) \log \log(\frac{n}{S})\right)$ for

Boolean randomized oblivious branching programs computing

$\GIPshuffle$, a variation of the generalized inner product problem that

can be computed in time $n$ and space $O(\log^2 n)$ by a

deterministic Boolean branching program.

These are also the first lower bounds for randomized oblivious

branching programs computing explicit functions that apply for

$T=\omega(n\log n)$.

They also show that any simulation of

general branching programs by randomized oblivious ones requires either

a superlogarithmic increase in time

or a very substantial increase in space.

TR10-104 Authors: Paul Beame, Widad Machmouchi

Publication: 29th June 2010 03:04

Downloads: 1868

Keywords:

We prove a time-space tradeoff lower bound of $T =

\Omega\left(n\log(\frac{n}{S}) \log \log(\frac{n}{S})\right) $ for

randomized oblivious branching programs to compute $1GAP$, also

known as the pointer jumping problem, a problem for which there is a

simple deterministic time $n$ and space $O(\log n)$ RAM (random

access machine) algorithm.

In a STOC 2010 paper, Ajtai (and independently, Damg{\aa}rd,

Meldgaard, and Nielsen) derived simulations of general RAMs by

randomized oblivious RAMs with only a polylogarithmic factor

increase in time and space. Our lower bound implies that a

superlogarithmic factor increase is indeed necessary in any such

simulation.

In contrast to the claim in the original version of our paper, the

model considered by Ajtai is incomparable to the one that we

consider here. (The same applies to Damgaard and Nielsen's

simulation to the extent that we understand it.)

Russell Impagliazzo brought to our attention that the randomized

oblivious branching programs we are considering are less general

than the oblivious randomized algorithms that Ajtai's simulation

uses. The main difference is that, in our model, for each fixing of

the random choices, the sequence of locations accessed must be

independent of the input though that sequence may depend on the

random choices, while in Ajtai's simulation, it is only necessary

that the {\em probability distribution} of the sequence of locations

accessed is input independent. Upon further investigation, we found

that Ajtai's simulation assumes that the original RAM algorithm only

has sequential access to its input, in which case the small space

upper bound for $1GAP_n$ does not apply (or alternatively the

original RAM algorithm has random access but at least linear space

which a deterministic oblivious branching program can use to

compute any function in linear time). Therefore, the upper bound

derived by Ajtai no longer applies in our model.

In our revised version of the paper, we derive the same

quantitative separation between general and randomized oblivious

algorithms in the case of Boolean inputs, a result that does not

follow from the separation for $1GAP_n$. We obtain this by

considering a variation of the $GIP$ problem, $GIP$-$MAP$ which

includes a pseudorandom permutation on the input bits of $GIP$ as

part of its input. To more accurately represent our results and to

avoid any confusion about their applicability we have changed the

title of our paper slightly.