Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2312.13178

$\mathcal{O}(\log\log{n})$ Passes is Optimal for Semi-Streaming Maximal Independent Set

Sepehr Assadi·Christian Konrad·Kheeran K. Naidu·Janani Sundaresan
View original source ↗Read source document ↗Export metadata ↓

Abstract

In the semi-streaming model for processing massive graphs, an algorithm makes multiple passes over the edges of a given $n$-vertex graph and is tasked with computing the solution to a problem using $O(n \cdot \text{polylog}(n))$ space. Semi-streaming algorithms for Maximal Independent Set (MIS) that run in $O(\log\log{n})$ passes have been known for almost a decade, however, the best lower bounds can only rule out single-pass algorithms. We close this large gap by proving that the current algorithms are optimal: Any semi-streaming algorithm for finding an MIS with constant probability of success requires $Ω(\log\log{n})$ passes. This settles the complexity of this fundamental problem in the semi-streaming model, and constitutes one of the first optimal multi-pass lower bounds in this model. We establish our result by proving an optimal round vs communication tradeoff for the (multi-party) communication complexity of MIS. The key ingredient of this result is a new technique, called hierarchical embedding, for performing round elimination: we show how to pack many but small hard $(r-1)$-round instances of the problem into a single $r$-round instance, in a way that enforces any $r$-round protocol to effectively solve all these $(r-1)$-round instances also. These embeddings are obtained via a novel application of results from extremal graph theory -- in particular dense graphs with many disjoint unique shortest paths -- together with a newly designed graph product, and are analyzed via information-theoretic tools such as direct-sum and message compression arguments.

Explore related subjects

cs.DScs.CCcs.DC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sepehr Assadi, Christian Konrad, Kheeran K. Naidu, Janani Sundaresan. 2023-12-20. $\mathcal{O}(\log\log{n})$ Passes is Optimal for Semi-Streaming Maximal Independent Set. https://arxiv.org/abs/2312.13178

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

KEEP EXPLORING

Related papers

arXiv2026

Path Enumeration by Position-Visit Counts in Recombining Trinomial Trees

Recombining trinomial trees are a workhorse for modeling discrete-event systems in option pricing, logistics, and feedback control. Because each node stores a state-dependent quantity, a depth-$D$ tree contains $3^D$ raw trajectories, making exhaustive enumeration rapidly infeasible. However, when each node's value depends only on its position, a raw trajectory's aggregate is determined by its position-visit counts. We call these count vectors cardinality tuples and decompose the admissible tuples into weak-composition mass layers. Leveraging these structures, we introduce a mass-shifting enumeration algorithm that slides integer ``masses'' through cardinality tuples to generate exactly one representative of each path-equivalence class, while the accompanying weak-composition bijections yield exact counting formulas for the generated families. This suppresses redundant raw-path orderings a priori rather than enumerating and deduplicating them afterward. For the full-tuple implementation, we prove an output-sensitive running-time bound at each fixed endpoint, together with a uniform worst-case upper bound $\mathscr{O}(D2^D)$ and an exact worst-case exponential growth base of $2$, compared with base $3$ for exhaustive raw-path enumeration. Thus the construction achieves a provable exponential reduction in the enumeration space, up to polynomial factors. The same framework also recovers the information compressed by the equivalence classes: we derive an exact degeneracy formula for the number of raw paths represented by every cardinality tuple. We further prove that the nonnegative return specialization is exactly the classical Motzkin family, recover its recursive and generating-function structure and the Dyck specialization, and derive a multivariate occupation-profile $J$-fraction whose coefficients recover the corresponding cardinality-tuple degeneracies.

cs.DS↗
arXiv2026

The Longest Common Bitonic Subsequence: Match-Sensitive Algorithms and Conditional Hardness

The longest common bitonic subsequence problem asks for a longest common subsequence of two ordered sequences whose values strictly increase and then strictly decrease; either phase may be empty. We formulate the problem through increasing and decreasing endpoint values at matching position pairs. This gives a constructive quadratic baseline and a matchsensitive algorithm based on two standard dominance-maximum passes. Its time is the sum of an input-sorting term and the number of matches times a squared logarithmic factor. We state the endpoint interface that permits reuse of increasing subsequence algorithms, and distinguish this specialization from new range searching machinery. A linear-size padding reduction transfers the conditional strongly subquadratic lower bound for longest common increasing subsequence to the bitonic problem. Reproducible implementations, exhaustive small-instance checks, and newly measured synthetic experiments document correctness and the practical tradeoff between sparse and dense processing.

cs.DS↗
arXiv2026

Locally Approximating the Top Eigenvector of Bounded Entry Matrices

We provide a local computation algorithm to approximate the top eigenvector $x \in \mathbb{R}^n$ of a symmetric matrix $A \in \mathbb{R}^{n \times n}$ with entries between $-1$ and $1$, building on the work of Swartworth and Woodruff [SODA 25] who show how to approximate the eigenvalues up to additive-$\varepsilon n$ error using $\tilde{O}(1/\varepsilon^4)$ queries. Our local computation algorithm has a preprocessing complexity of $\tilde{O}(1/\varepsilon^4)$ and per-coordinate query complexity of $\tilde{O}(1/\varepsilon^2)$ for an additive-$\varepsilon n$ approximation whenever {$|λ_{\min}(A)| = O(λ_{\max}(A))$. When $λ_{\min}(A)$ greatly exceeds $λ_{\max}(A)$, our complexity degrades to at most $\tilde{O}(1/\varepsilon^{6.\overline{6}})$ in preprocessing and $\tilde{O}(1/\varepsilon^{3.\overline{3}})$ per query. Furthermore, we show a lower bound of $Ω(n/\varepsilon^2)$ on the total number of queries needed to output an approximately top eigenvector (implying that the per-coordinate query complexity of $Ω(1/\varepsilon^2)$ is necessary). As an application, we use our algorithm to provide local computation algorithms for the sparsest-cut and max-cut problems in the dense graph model of Goldreich, Goldwasser, Ron [JACM 98]. By accessing the top eigenvectors (of an approximate normalized adjacency), we implement local versions of Cheeger's inequality and Trevisan's algorithm [SICOMP 12] to obtain "square-root-opt" approximations in polynomial time (as opposed to exponential-in-$\text{poly}(1/\varepsilon)$ time which is incurred in Goldreich, Goldwasser, Ron.

cs.DS↗