We provide a fast witness-sensitive algorithm for detecting an induced diamond (a K₄ minus an edge) in an n-vertex graph containing t induced diamonds. Our algorithm runs in time Õ(min(n^2.425/t^0.25 + n², n^ω)) with high probability, improving upon the prior state of the art (witness-oblivious) algorithm that runs in time O(n^ω log n) [Vassilevska Williams, Wang, Williams, Yu, SODA 2014] whenever t ≥ n^{(3-ω)/3}, where ω < 2.372 is the matrix multiplication exponent. Our key insight is that the size of a clique containing one of the triangles of an induced diamond plays a crucial role in detecting such a diamond. We say that a diamond is r-heavy if this size is at least r, and we provide a fast detection algorithm for r-heavy diamonds in Õ(r⋅(n/r)^ω + (n/r)³+ nr) time. When there are no r-heavy diamonds, we provide a different fast detection algorithm in Õ(MM(n,n,n√{r/t})) time, where MM(a,b,c) denotes the time to multiply an a × b matrix by a b × c matrix, which is conditionally optimal for r = Õ(1). Our main technical contribution is in designing a refinement framework for sampling vectors, which allows sampling vertices for detecting diamonds in a manner that is adaptive to the structure of graphs with no r-heavy diamonds. We establish that our technique is of a wide applicability, by showing how it also allows for faster witness-sensitive algorithms for 4-SUM and for a special case of 4-cycles.

We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, Tětek [ICALP'22] gave an algorithm that returns a (1±ε)-approximation in Õ(n^ω/t^{ω-2}) time, where t is the unknown number of triangles in the given n node graph and ω < 2.372 is the matrix multiplication exponent. We obtain an improved algorithm whose running time is, within polylogarithmic factors the same as that for multiplying an n× n/t matrix by an n/t × n matrix. We then extend our framework to obtain the first nontrivial (1± ε)-approximation algorithms for the number of h-cycles in a graph, for any constant h ≥ 3. Our running time is Õ(MM(n,n/t^{1/(h-2)},n)), the time to multiply n × n/(t^{1/(h-2)}) by n/(t^{1/(h-2)) × n matrices. Finally, we show that under popular fine-grained hypotheses, this running time is optimal.

Many applications of approximate membership query data structures, or filters , require only an incremental filter that supports insertions but not deletions. However, the design space of incremental filters is missing a "sweet spot" filter that combines space efficiency, fast queries, and fast insertions. Incremental filters, such as the Bloom and blocked Bloom filter, are not space efficient. Dynamic filters (i.e., supporting deletions), such as the cuckoo or vector quotient filter, are space efficient but do not exhibit consistently fast insertions and queries. In this paper, we propose the prefix filter , an incremental filter that addresses the above challenge: (1) its space (in bits) is similar to state-of-the-art dynamic filters; (2) query throughput is high and is comparable to that of the cuckoo filter; and (3) insert throughput is high with overall build times faster than those of the vector quotient filter and cuckoo filter by 1.39X--1.46X and 3.2X--3.5X, respectively. We present a rigorous analysis of the prefix filter that holds also for practical set sizes (i.e., n = 2 25 ). The analysis deals with the probability of failure, false positive rate, and probability that an operation requires accessing more than a single cache line.