[Paper Review] Finding monotone patterns in sublinear time

Omri Ben‐Eliezer, Clément L. Canonne+2 more|arXiv (Cornell University)|Oct 3, 2019

Complexity and Algorithms in Graphs38 references4 citations

TL;DR

This paper establishes the optimal non-adaptive query complexity for finding a length-$k$ monotone subsequence in a function $f:[n] \to \mathbb{R}$ that is $\varepsilon$-far from being $(12\dots k)$-free. It introduces a novel structural decomposition using binary profiles and tree descriptors, proving tight $\Theta((\log n)^{\lfloor \log_2 k \rfloor})$ query complexity, significantly improving prior bounds and resolving the complexity for this sublinear-time pattern-finding problem.

ABSTRACT

We study the problem of finding monotone subsequences in an array from the viewpoint of sublinear algorithms. For fixed $k \in \mathbb{N}$ and $\varepsilon > 0$, we show that the non-adaptive query complexity of finding a length-$k$ monotone subsequence of $f \colon [n] o \mathbb{R}$, assuming that $f$ is $\varepsilon$-far from free of such subsequences, is $Θ((\log n)^{\lfloor \log_2 k floor})$. Prior to our work, the best algorithm for this problem, due to Newman, Rabinovich, Rajendraprasad, and Sohler (2017), made $(\log n)^{O(k^2)}$ non-adaptive queries; and the only lower bound known, of $Ω(\log n)$ queries for the case $k = 2$, followed from that on testing monotonicity due to Ergün, Kannan, Kumar, Rubinfeld, and Viswanathan (2000) and Fischer (2004).

Motivation & Objective

• To resolve the query complexity of non-adaptive algorithms for finding a length-$k$ monotone subsequence in functions $\varepsilon$-far from $(12\dots k)$-free.
• To close the gap between prior upper bounds of $(\log n)^{O(k^2)}$ and the only known lower bound of $\Omega(\log n)$ for $k=2$, by establishing tight bounds for all fixed $k\in\mathbb{N}$.
• To develop a new structural characterization of functions containing or avoiding monotone subsequences, using binary profiles and tree descriptors.
• To prove a matching lower bound matching the upper bound, establishing optimality of the query complexity.

Proposed method

• Introduce a hierarchical binary profile system to classify intervals based on their monotonicity structure and the positions of increasing pairs.
• Define a structural dichotomy theorem that partitions intervals into 'growing suffixes' and 'splittable intervals' based on their monotonicity behavior.
• Construct a recursive algorithm that uses non-adaptive queries to identify a length-$k$ monotone subsequence by traversing a tree of descriptors derived from binary profiles.
• Use a matching lower bound argument via Yao’s minimax principle, analyzing the probability that a random query set captures a monotone copy in a carefully constructed distribution of functions.
• Leverage the uniqueness of maximal matching values $M(x,y)$ to ensure that no triple $x<y<z$ has $M(x,y)=M(y,z)$, which prevents spurious monotonicity in substructures.
• Prove that each constructed function $f_{i_1,\dots,i_h}$ contains $n/k$ disjoint $(12\dots k)$-copies, ensuring $\varepsilon$-far distance from $(12\dots k)$-free functions.

Experimental results

Research questions

• RQ1 What is the optimal non-adaptive query complexity for finding a length-$k$ monotone subsequence in a function $f:[n] \to \mathbb{R}$ that is $\varepsilon$-far from being $(12\dots k)$-free?
• RQ2 Can the query complexity be improved beyond the $O((\log n)^{k^2})$ bound of Newman et al. (2017) for non-adaptive algorithms with one-sided error?
• RQ3 Is the dependence on $n$ in the query complexity tight, and if so, what is the exact asymptotic form?
• RQ4 How can the structure of monotone subsequences be characterized using hierarchical binary profiles and tree descriptors?
• RQ5 What is the minimal number of queries required to guarantee finding a monotone subsequence with constant success probability under the $\varepsilon$-far promise?

Key findings

• The non-adaptive query complexity for finding a length-$k$ monotone subsequence in a function $\varepsilon$-far from $(12\dots k)$-free is $\Theta\left((\log n)^{\lfloor \log_2 k \rfloor}\right)$, which is optimal.
• The upper bound is achieved by a non-adaptive algorithm making $O\left((\log n)^{\lfloor \log_2 k \rfloor} \cdot \mathrm{poly}(1/\varepsilon)\right)$ queries.
• The lower bound matches the upper bound, proving that no non-adaptive algorithm can achieve better dependence on $n$ for fixed $k$.
• Each function $f_{i_1,\dots,i_h}$ in the hard instance distribution contains exactly $n/k$ disjoint $(12\dots k)$-copies, ensuring $\varepsilon$-far distance with $\varepsilon = 1/k$.
• The structural characterization via binary profiles and tree descriptors enables a recursive construction that isolates monotone subsequences through hierarchical decomposition.
• The proof relies on a novel inductive argument showing that any increasing subsequence of length $k$ in the constructed functions must have a unique maximal matching position, which prevents overlapping or spurious copies.