Lipschitz bijections between boolean functions

We answer four questions from a recent paper of Rao and Shinkar [17] on Lipschitz bijections between functions from {0, 1} n to {0, 1}. (1) We show that there is no O(1)-bi-Lipschitz bijection from Dictator to XOR such that each output bit depends on O(1) input bits. (2) We give a construction for a mapping from XOR to Majority which has average stretch $O(\sqrt{n})$ , matching a previously known lower bound. (3) We give a 3-Lipschitz embedding $\phi \colon \{0,1\}^n \to \{0,1\}^{2n+1}$ such that $${\rm{XOR }}(x) = {\rm{ Majority }}(\phi (x))$$ for all $x \in \{0,1\}^n$ . (4) We show that with high probability there is an O(1)-bi-Lipschitz mapping from Dictator to a uniformly random balanced function.

Identifikasi Akuifer Air Tanah Dalam Menggununakan Metode Geolistrik Schlumberger di Desa Wedomartani, Kabupaten Sleman

Water is the main problem in every region that has a rapid population increase, one of which is Wedomartani village, Sleman Regency. This study aims to determine the water reserves in Wedomartani village and find out the direction of groundwater flow in the area. The research carried out by measuring geolectricity method (Schlumberger configuration) survey of 15 very point acquisition with an average stretch of 250 m. The tool used in geoelectric measurement is Resistivity Meter Syscal Jr. The software used in processing is Progress 3.0. The results of data processing is resistivity variations function of depth per point acquisition that will interpreted into rock types and water content. The average resistivity of groundwater aquifers in Wedomartani village is 19.49 Ωm and the average depth of groundwater aquifers in Wedomartani village is 101 m. The direction of deep groundwater flow in Wedomartani village generally leads from north to south and west (in the southern part of the research area.

On-the-fly scheduling versus reservation-based scheduling for unpredictable workflows

Scientific insights in the coming decade will clearly depend on the effective processing of large data sets generated by dynamic heterogeneous applications typical of workflows in large data centers or of emerging fields like neuroscience. In this article, we show how these big data workflows have a unique set of characteristics that pose challenges for leveraging HPC methodologies, particularly in scheduling. Our findings indicate that execution times for these workflows are highly unpredictable and are not correlated with the size of the data set involved or the precise functions used in the analysis. We characterize this inherent variability and sketch the need for new scheduling approaches by quantifying significant gaps in achievable performance. Through simulations, we show how on-the-fly scheduling approaches can deliver benefits in both system-level and user-level performance measures. On average, we find improvements of up to 35% in system utilization and up to 45% in average stretch of the applications, illustrating the potential of increasing performance through new scheduling approaches.

On Lipschitz Bijections Between Boolean Functions

Given two functions f,g : {0,1}n → {0,1}, a mapping ψ : {0,1}n → {0,1}n is said to be a mapping from f to g if it is a bijection and f(z) = g(ψ(z)) for every z ∈ {0,1}n. In this paper we study Lipschitz mappings between Boolean functions.Our first result gives a construction of a C-Lipschitz mapping from the Majority function to the Dictator function for some universal constant C. On the other hand, there is no n/2-Lipschitz mapping in the other direction, namely from the Dictator function to the Majority function. This answers an open problem posed by Daniel Varga in the paper of Benjamini, Cohen and Shinkar (FOCS 2014 [1]).We also show a mapping from Dictator to XOR that is 3-local, 2-Lipschitz, and its inverse is O(log(n))-Lipschitz, where by L-local mapping we mean that each output bit of the mapping depends on at most L input bits.Next, we consider the problem of finding functions such that any mapping between them must have large average stretch, where the average stretch of a mapping φ is defined as $${\sf avgstretch}(\phi) = {\mathbb E}_{x,i}[{\sf dist}(\phi(x),\phi(x+e_i)].$$ We show that any mapping φ from XOR to Majority must satisfy avgStretch(φ) ≥ c$\sqrt{n}$ for some absolute constant c > 0. In some sense, this gives a ‘function analogue’ to the question of Benjamini, Cohen and Shinkar (FOCS 2014 [1]), who asked whether there exists a set A ⊆ {0,1}n of density 0.5 such that any bijection from {0,1}n−1 to A has large average stretch.Finally, we show that for a random balanced function f: {0,1}n → {0,1}n, with high probability there is a mapping φ from Dictator to f such that both φ and φ−1 have constant average stretch. In particular, this implies that one cannot obtain lower bounds on average stretch by taking uniformly random functions.