A New Study of Applying Complexity Theoretical Tools in Algorithm Design

Given n vectors with dimension m in Boolean domain, how to find two vectors whose pairwise Hamming distance is minimum? This problem is known as the Closest Pair Problem. If these vectors are generated uniformly at random except two of them are correlated with Pearson-correlation coefficient, then the problem is called the Light Bulb Problem. In this work, we propose a novel coding-based scheme for the Closest Pair Problem. We design both randomized and deterministic algorithms, which achieve the best-known running time when the length of input vectors m is small and the minimum distance is very small compared to m. When applied to the Light Bulb Problem, our result yields state-of-the-art deterministic running time when the Pearson-correlation coefficient is very large. Specifically, when it is greater than 0.9933, our deterministic algorithm runs faster than the previously best deterministic algorithm (Alman, SOSA 2019)

Efficient Algorithms for the Closest Pair Problem and Applications

The closest pair problem (CPP) is one of the well studied and fundamental problems in computing. Given a set of points in a metric space, the problem is to identify the pair of closest points. Another closely related problem is the fixed radius nearest neighbors problem (FRNNP). Given a set of points and a radius $R$, the problem is, for every input point $p$, to identify all the other input points that are within a distance of $R$ from $p$. A naive deterministic algorithm can solve these problems in quadratic time. CPP as well as FRNNP play a vital role in computational biology, computational finance, share market analysis, weather prediction, entomology, electro cardiograph, N-body simulations, molecular simulations, etc. As a result, any improvements made in solving CPP and FRNNP will have immediate implications for the solution of numerous problems in these domains. We live in an era of big data and processing these data take large amounts of time. Speeding up data processing algorithms is thus much more essential now than ever before. In this paper we present algorithms for CPP and FRNNP that improve (in theory and/or practice) the best-known algorithms reported in the literature for CPP and FRNNP. These algorithms also improve the best-known algorithms for related applications including time series motif mining and the two locus problem in Genome Wide Association Studies (GWAS)

Fast Heavy Inner Product Identification Between Weights and Inputs in Neural Network Training

In this paper, we consider a heavy inner product identification problem, which generalizes the Light Bulb problem~(\cite{prr89}): Given two sets $A \subset \{-1,+1\}^d$ and $B \subset \{-1,+1\}^d$ with $|A|=|B| = n$, if there are exact $k$ pairs whose inner product passes a certain threshold, i.e., $\{(a_1, b_1), \cdots, (a_k, b_k)\} \subset A \times B$ such that $\forall i \in [k], \langle a_i,b_i \rangle \geq \rho \cdot d$, for a threshold $\rho \in (0,1)$, the goal is to identify those $k$ heavy inner products. We provide an algorithm that runs in $O(n^{2 \omega / 3+ o(1)})$ time to find the $k$ inner product pairs that surpass $\rho \cdot d$ threshold with high probability, where $\omega$ is the current matrix multiplication exponent. By solving this problem, our method speed up the training of neural networks with ReLU activation function.Comment: IEEE BigData 202

Strategic Transfer in Logical Abilities in Children Playing Mastermind and an Analogue

Strategy development and the use of strategy as a mechanism of transfer was examined in sixty elementary students while playing the logical deduction game Mastermind and a familiar analogue. In the first couple of two-way ANOVAs subjects showed that they are in fact learning or developing a task-specific strategy that can be applied across the two types of games regardless of which game was in the target position of a transfer paradigm. This suggests that subjects were able to focus on structural similarities rather than surface features and apply what was learned between the game isomorphs. Both the third and fourth ANOVAs indicate that strategic transfer did occur between the Family Dinner Table game and Mastermind game, when mastermind was in the target position of a transfer paradigm. This suggests that strategies can be used as a mechanism in transfer

Strategic Transfer in Logical Abilities in Children Playing Mastermind and an Analogue

Strategy development and the use of strategy as a mechanism of transfer was examined in sixty elementary students while playing the logical deduction game Mastermind and a familiar analogue. In the first couple of two-way ANOVAs subjects showed that they are in fact learning or developing a task-specific strategy that can be applied across the two types of games regardless of which game was in the target position of a transfer paradigm. This suggests that subjects were able to focus on structural similarities rather than surface features and apply what was learned between the game isomorphs. Both the third and fourth ANOVAs indicate that strategic transfer did occur between the Family Dinner Table game and Mastermind game, when mastermind was in the target position of a transfer paradigm. This suggests that strategies can be used as a mechanism in transfer

Strategic Transfer in Logical Abilities in Children Playing Mastermind and an Analogue

Strategy development and the use of strategy as a mechanism of transfer was examined in sixty elementary students while playing the logical deduction game Mastermind and a familiar analogue. In the first couple of two-way ANOVAs subjects showed that they are in fact learning or developing a task-specific strategy that can be applied across the two types of games regardless of which game was in the target position of a transfer paradigm. This suggests that subjects were able to focus on structural similarities rather than surface features and apply what was learned between the game isomorphs. Both the third and fourth ANOVAs indicate that strategic transfer did occur between the Family Dinner Table game and Mastermind game, when mastermind was in the target position of a transfer paradigm. This suggests that strategies can be used as a mechanism in transfer