Faster Algorithms for All Pairs Non-Decreasing Paths Problem

In this paper, we present an improved algorithm for the All Pairs Non-decreasing Paths (APNP) problem on weighted simple digraphs, which has running time O~(n^{{3 + omega}/{2}}) = O~(n^{2.686}). Here n is the number of vertices, and omega < 2.373 is the exponent of time complexity of fast matrix multiplication [Williams 2012, Le Gall 2014]. This matches the current best upper bound for (max, min)-matrix product [Duan, Pettie 2009] which is reducible to APNP. Thus, further improvement for APNP will imply a faster algorithm for (max, min)-matrix product. The previous best upper bound for APNP on weighted digraphs was O~(n^{1/2(3 + {3 - omega}/{omega + 1} + omega)}) = O~(n^{2.78}) [Duan, Gu, Zhang 2018]. We also show an O~(n^2) time algorithm for APNP in undirected simple graphs which also reaches optimal within logarithmic factors

Faster Matrix Multiplication via Asymmetric Hashing

Fast matrix multiplication is one of the most fundamental problems in algorithm research. The exponent of the optimal time complexity of matrix multiplication is usually denoted by $\omega$. This paper discusses new ideas for improving the laser method for fast matrix multiplication. We observe that the analysis of higher powers of the Coppersmith-Winograd tensor [Coppersmith & Winograd 1990] incurs a "combination loss", and we partially compensate for it using an asymmetric version of CW's hashing method. By analyzing the eighth power of the CW tensor, we give a new bound of $\omega<2.37188$, which improves the previous best bound of $\omega<2.37286$ [Alman & Vassilevska Williams 2020]. Our result breaks the lower bound of $2.3725$ in [Ambainis, Filmus & Le Gall 2015] because of the new method for analyzing component (constituent) tensors.Comment: 67 page

Self-supervised Video Representation Learning with Motion-Aware Masked Autoencoders

Masked autoencoders (MAEs) have emerged recently as art self-supervised spatiotemporal representation learners. Inheriting from the image counterparts, however, existing video MAEs still focus largely on static appearance learning whilst are limited in learning dynamic temporal information hence less effective for video downstream tasks. To resolve this drawback, in this work we present a motion-aware variant -- MotionMAE. Apart from learning to reconstruct individual masked patches of video frames, our model is designed to additionally predict the corresponding motion structure information over time. This motion information is available at the temporal difference of nearby frames. As a result, our model can extract effectively both static appearance and dynamic motion spontaneously, leading to superior spatiotemporal representation learning capability. Extensive experiments show that our MotionMAE outperforms significantly both supervised learning baseline and state-of-the-art MAE alternatives, under both domain-specific and domain-generic pretraining-then-finetuning settings. In particular, when using ViT-B as the backbone our MotionMAE surpasses the prior art model by a margin of 1.2% on Something-Something V2 and 3.2% on UCF101 in domain-specific pretraining setting. Encouragingly, it also surpasses the competing MAEs by a large margin of over 3% on the challenging video object segmentation task. The code is available at https://github.com/happy-hsy/MotionMAE.Comment: 17 pages, 6 figure

A Simple Near-Linear Pseudopolynomial Time Randomized Algorithm for Subset Sum

Given a multiset S of n positive integers and a target integer t, the Subset Sum problem asks to determine whether there exists a subset of S that sums up to t. The current best deterministic algorithm, by Koiliaris and Xu [SODA\u2717], runs in O~(sqrt{n}t) time, where O~ hides poly-logarithm factors. Bringmann [SODA\u2717] later gave a randomized O~(n + t) time algorithm using two-stage color-coding. The O~(n+t) running time is believed to be near-optimal. In this paper, we present a simple and elegant randomized algorithm for Subset Sum in O~(n + t) time. Our new algorithm actually solves its counting version modulo prime p>t, by manipulating generating functions using FFT