124 research outputs found
The zero-error randomized query complexity of the pointer function
The pointer function of G{\"{o}}{\"{o}}s, Pitassi and Watson
\cite{DBLP:journals/eccc/GoosP015a} and its variants have recently been used to
prove separation results among various measures of complexity such as
deterministic, randomized and quantum query complexities, exact and approximate
polynomial degrees, etc. In particular, the widest possible (quadratic)
separations between deterministic and zero-error randomized query complexity,
as well as between bounded-error and zero-error randomized query complexity,
have been obtained by considering {\em
variants}~\cite{DBLP:journals/corr/AmbainisBBL15} of this pointer function.
However, as was pointed out in \cite{DBLP:journals/corr/AmbainisBBL15}, the
precise zero-error complexity of the original pointer function was not known.
We show a lower bound of on the zero-error
randomized query complexity of the pointer function on bits;
since an upper bound is already known
\cite{DBLP:conf/fsttcs/MukhopadhyayS15}, our lower bound is optimal up to a
factor of \polylog\, n
An entropy based proof of the Moore bound for irregular graphs
We provide proofs of the following theorems by considering the entropy of
random walks: Theorem 1.(Alon, Hoory and Linial) Let G be an undirected simple
graph with n vertices, girth g, minimum degree at least 2 and average degree d:
Odd girth: If g=2r+1,then n \geq 1 + d*(\Sum_{i=0}^{r-1}(d-1)^i) Even girth: If
g=2r,then n \geq 2*(\Sum_{i=0}^{r-1} (d-1)^i) Theorem 2.(Hoory) Let G =
(V_L,V_R,E) be a bipartite graph of girth g = 2r, with n_L = |V_L| and n_R =
|V_R|, minimum degree at least 2 and the left and right average degrees d_L and
d_R. Then, n_L \geq \Sum_{i=0}^{r-1}(d_R-1)^{i/2}(d_L-1)^{i/2} n_R \geq
\Sum_{i=0}^{r-1}(d_L-1)^{i/2}(d_R-1)^{i/2}Comment: 6 page
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