The Transformer has been on a lot of people’s minds over the last year five years. This post presents an

annotated version of the paper in the form of a line-by-line implementation. It reorders and deletes

some sections from the original paper and adds comments throughout. This document itself is a working

notebook, and should be a completely usable implementation. Code is available here here here here here here here here here here here here here.

This document presents a deterministic algorithm for the single-source shortest path (SSSP) problem on directed graphs with non-negative edge weights, achieving a time complexity of O(m log^(2/3) n). This groundbreaking result surpasses the long-standing O(m + n log n) barrier of Dijkstra's algorithm, demonstrating that Dijkstra's is not optimal for SSSP on sparse graphs when the vertex ordering by distance is not strictly required. The approach ingeniously merges concepts from Dijkstra's and Bellman-Ford algorithms using a recursive partitioning technique to manage the "frontier" of uncertain distances more efficiently, avoiding the sorting bottleneck inherent in traditional methods. It introduces a "FindPivots" procedure and a specialized data structure to limit the size of the set of vertices that need active consideration, thereby reducing computational overhead and improving performance.

• We first introduce a new class of alphas with intriguingstrengths: like formulaic alphas, these alphas can modelscalar features and thus are simple to mine into a weaklycorrelated set, but, like machine learning alphas, they arehigh-dimensional data-driven models utilizing long-termfeatures. We then propose a novel alpha mining framework,AlphaEvolve, to generate the new alphas. To the best of ourknowledge, we are the first to solve the stock prediction problem based on AutoML and the first to tackle the problem ofmining weakly correlated alphas.• We enable AlphaEvolve to selectively inject relational domain knowledge without any strong structural assumptionin an alpha.• We propose an optimization technique to accelerate alphamining by pruning redundant alphas.• We conduct extensive experimental study on AlphaEvolveusing the stock price data of NASDAQ. The results show thatAlphaEvolve generates alphas with weakly correlated highreturns.

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