653 research outputs found
Balanced Boolean functions that can be evaluated so that every input bit is unlikely to be read
A Boolean function of n bits is balanced if it takes the value 1 with
probability 1/2. We exhibit a balanced Boolean function with a randomized
evaluation procedure (with probability 0 of making a mistake) so that on
uniformly random inputs, no input bit is read with probability more than
Theta(n^{-1/2} sqrt{log n}). We give a balanced monotone Boolean function for
which the corresponding probability is Theta(n^{-1/3} log n). We then show that
for any randomized algorithm for evaluating a balanced Boolean function, when
the input bits are uniformly random, there is some input bit that is read with
probability at least Theta(n^{-1/2}). For balanced monotone Boolean functions,
there is some input bit that is read with probability at least Theta(n^{-1/3}).Comment: 11 page
Tug-of-war and the infinity Laplacian
We prove that every bounded Lipschitz function F on a subset Y of a length
space X admits a tautest extension to X, i.e., a unique Lipschitz extension u
for which Lip_U u = Lip_{boundary of U} u for all open subsets U of X that do
not intersect Y.
This was previously known only for bounded domains R^n, in which case u is
infinity harmonic, that is, a viscosity solution to Delta_infty u = 0. We also
prove the first general uniqueness results for Delta_infty u = g on bounded
subsets of R^n (when g is uniformly continuous and bounded away from zero), and
analogous results for bounded length spaces.
The proofs rely on a new game-theoretic description of u. Let u^epsilon(x) be
the value of the following two-player zero-sum game, called tug-of-war: fix
x_0=x \in X minus Y. At the kth turn, the players toss a coin and the winner
chooses an x_k with d(x_k, x_{k-1})< epsilon. The game ends when x_k is in Y,
and player one's payoff is
F(x_k) - (epsilon^2/2) sum_{i=0}^{k-1} g(x_i)
We show that the u^\epsilon converge uniformly to u as epsilon tends to zero.
Even for bounded domains in R^n, the game theoretic description of
infinity-harmonic functions yields new intuition and estimates; for instance,
we prove power law bounds for infinity-harmonic functions in the unit disk with
boundary values supported in a delta-neighborhood of a Cantor set on the unit
circle.Comment: 44 pages, 4 figure
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