We show that any linear program (LP) in $n$ nonnegative variables and
$m$ equality constraints defines in a
natural way a \emph{unique sink orientation} of the $n$-dimensional
cube. From the sink of the cube, we can either read off an optimal
solution to the LP, or we obtain certificates for infeasibility or
unboundedness.
This reduction complements the implicit local neighborhoods
induced by the vertex-edge structure of the feasible region with an
explicit neighborhood structure that allows random access to all $2^n$
candidate solutions. Using the currently best sink-finding algorithm
for general unique sink orientations, we obtain the fastest
deterministic LP algorithm in the RAM model, for the central
case $n=2m$.