Uniform sets with few progressions via colorings

Ruzsa asked whether there exist Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and 4-term arithmetic progression (4-APs) density at most $\alpha^C$, for arbitrarily large $C$. Gowers constructed Fourier uniform sets with density $\alpha$ and 4-AP density at most $\alpha^{4+c}$ for some small constant $c>0$. We show that an affirmative answer to Ruzsa's question would follow from the existence of an $N^{o(1)}$-coloring of $[N]$ without symmetrically colored 4-APs. For a broad and natural class of constructions of Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$, we show that Ruzsa's question is equivalent to our arithmetic Ramsey question. We prove analogous results for all even-length APs. For each odd $k\geq 5$, we show that there exist $U^{k-2}$-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and $k$-AP density at most $\alpha^{c_k \log(1/\alpha)}$. We also prove generalizations to arbitrary one-dimensional patterns.Comment: 20 page

Space-Query Tradeoffs in Range Subgraph Counting and Listing

This paper initializes the study of range subgraph counting and range subgraph listing, both of which are motivated by the significant demands in practice to perform graph analytics on subgraphs pertinent to only selected, as opposed to all, vertices. In the first problem, there is an undirected graph G where each vertex carries a real-valued attribute. Given an interval q and a pattern Q, a query counts the number of occurrences of Q in the subgraph of G induced by the vertices whose attributes fall in q. The second problem has the same setup except that a query needs to enumerate (rather than count) those occurrences with a small delay. In both problems, our goal is to understand the tradeoff between space usage and query cost, or more specifically: (i) given a target on query efficiency, how much pre-computed information about G must we store? (ii) Or conversely, given a budget on space usage, what is the best query time we can hope for? We establish a suite of upper- and lower-bound results on such tradeoffs for various query patterns

Enumerating Subgraphs of Constant Sizes in External Memory

We present an indivisible I/O-efficient algorithm for subgraph enumeration, where the objective is to list all the subgraphs of a massive graph G : = (V, E) that are isomorphic to a pattern graph Q having k = O(1) vertices. Our algorithm performs O((|E|^{k/2})/(M^{{k/2}-1} B) log_{M/B}(|E|/B) + (|E|^?)/(M^{?-1} B) I/Os with high probability, where ? is the fractional edge covering number of Q (it always holds ? ? k/2, regardless of Q), M is the number of words in (internal) memory, and B is the number of words in a disk block. Our solution is optimal in the class of indivisible algorithms for all pattern graphs with ? > k/2. When ? = k/2, our algorithm is still optimal as long as M/B ? (|E|/B)^? for any constant ? > 0

Recurrent Contour-based Instance Segmentation with Progressive Learning

Contour-based instance segmentation has been actively studied, thanks to its flexibility and elegance in processing visual objects within complex backgrounds. In this work, we propose a novel deep network architecture, i.e., PolySnake, for contour-based instance segmentation. Motivated by the classic Snake algorithm, the proposed PolySnake achieves superior and robust segmentation performance with an iterative and progressive contour refinement strategy. Technically, PolySnake introduces a recurrent update operator to estimate the object contour iteratively. It maintains a single estimate of the contour that is progressively deformed toward the object boundary. At each iteration, PolySnake builds a semantic-rich representation for the current contour and feeds it to the recurrent operator for further contour adjustment. Through the iterative refinements, the contour finally progressively converges to a stable status that tightly encloses the object instance. Moreover, with a compact design of the recurrent architecture, we ensure the running efficiency under multiple iterations. Extensive experiments are conducted to validate the merits of our method, and the results demonstrate that the proposed PolySnake outperforms the existing contour-based instance segmentation methods on several prevalent instance segmentation benchmarks. The codes and models are available at https://github.com/fh2019ustc/PolySnake