Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1511.05886

Randomization can be as helpful as a glimpse of the future in online computation

Jesper W. Mikkelsen
View original source ↗Read source document ↗Export metadata ↓

Abstract

We provide simple but surprisingly useful direct product theorems for proving lower bounds on online algorithms with a limited amount of advice about the future. As a consequence, we are able to translate decades of research on randomized online algorithms to the advice complexity model. Doing so improves significantly on the previous best advice complexity lower bounds for many online problems, or provides the first known lower bounds. For example, if $n$ is the number of requests, we show that: (1) A paging algorithm needs $Ω(n)$ bits of advice to achieve a competitive ratio better than $H_k=Ω(\log k)$, where $k$ is the cache size. Previously, it was only known that $Ω(n)$ bits of advice were necessary to achieve a constant competitive ratio smaller than $5/4$. (2) Every $O(n^{1-\varepsilon})$-competitive vertex coloring algorithm must use $Ω(n\log n)$ bits of advice. Previously, it was only known that $Ω(n\log n)$ bits of advice were necessary to be optimal. For certain online problems, including the MTS, $k$-server, paging, list update, and dynamic binary search tree problem, our results imply that randomization and sublinear advice are equally powerful (if the underlying metric space or node set is finite). This means that several long-standing open questions regarding randomized online algorithms can be equivalently stated as questions regarding online algorithms with sublinear advice. For example, we show that there exists a deterministic $O(\log k)$-competitive $k$-server algorithm with advice complexity $o(n)$ if and only if there exists a randomized $O(\log k)$-competitive $k$-server algorithm without advice. Technically, our main direct product theorem is obtained by extending an information theoretical lower bound technique due to Emek, Fraigniaud, Korman, and Rosén [ICALP'09].

Explore related subjects

cs.DScs.CC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jesper W. Mikkelsen. 2016-08-19. Randomization can be as helpful as a glimpse of the future in online computation. https://arxiv.org/abs/1511.05886

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↗