11,277 research outputs found
A linear lower bound for incrementing a space-optimal integer representation in the bit-probe model
We present the first linear lower bound for the number of bits required to be
accessed in the worst case to increment an integer in an arbitrary space-
optimal binary representation. The best previously known lower bound was
logarithmic. It is known that a logarithmic number of read bits in the worst
case is enough to increment some of the integer representations that use one
bit of redundancy, therefore we show an exponential gap between space-optimal
and redundant counters.
Our proof is based on considering the increment procedure for a space optimal
counter as a permutation and calculating its parity. For every space optimal
counter, the permutation must be odd, and implementing an odd permutation
requires reading at least half the bits in the worst case. The combination of
these two observations explains why the worst-case space-optimal problem is
substantially different from both average-case approach with constant expected
number of reads and almost space optimal representations with logarithmic
number of reads in the worst case.Comment: 12 pages, 4 figure
Multiple-Environment Markov Decision Processes
We introduce Multi-Environment Markov Decision Processes (MEMDPs) which are
MDPs with a set of probabilistic transition functions. The goal in a MEMDP is
to synthesize a single controller with guaranteed performances against all
environments even though the environment is unknown a priori. While MEMDPs can
be seen as a special class of partially observable MDPs, we show that several
verification problems that are undecidable for partially observable MDPs, are
decidable for MEMDPs and sometimes have even efficient solutions
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