Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1402.3444

Subgraph Enumeration in Massive Graphs

Francesco Silvestri
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the problem of enumerating all instances of a given pattern graph in a large data graph. Our focus is on determining the input/output (I/O) complexity of this problem. Let $E$ be the number of edges in the data graph, $k=O(1)$ be the number of vertices in the pattern graph, $B$ be the block length, and $M$ be the main memory size. The main results of the paper are two algorithms that enumerate all instances of the pattern graph. The first one is a deterministic algorithm that exploits a suitable independent set of the pattern graph of size $1\leq s \leq k/2$ and requires $O\left(E^{k-s}/\left(BM^{k-s-1}\right)\right)$ I/Os. The second algorithm is a randomized algorithm that enumerates all instances in $O\left(E^{k/2}/\left(BM^{k/2-1}\right)\right)$ expected I/Os; the same bound also applies with high probability under some assumptions. A lower bound shows that the deterministic algorithm is optimal for some pattern graphs with $s=k/2$ (e.g., paths and cycles of even length, meshes of even side), while the randomized algorithm is optimal for a wide class of pattern graphs, called Alon class (e.g., cliques, cycles and every graph with a perfect matching).

Explore related subjects

cs.DS

Keep this discovery

BibTeXRIS

Francesco Silvestri. 2014-02-14. Subgraph Enumeration in Massive Graphs. https://arxiv.org/abs/1402.3444

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗