[Paper Review] Sketching, Streaming, and Fine-Grained Complexity of (Weighted) LCS

Karl Bringmann, Bhaskar Ray Chaudhury|arXiv (Cornell University)|Oct 2, 2018

Algorithms and Data Compression14 references4 citations

TL;DR

This paper presents a deterministic sketching and streaming algorithm for the Longest Common Subsequence (LCS) problem on strings over a fixed-size alphabet, achieving optimal space complexity of $\mathcal{O}(L^{|Σ|-1}\log L)$ bits. It proves tight lower bounds under the Strong Exponential Time Hypothesis (SETH), showing that the algorithm's running time of $\mathcal{O}(\min\{nm, n + m^{|Σ|}\})$ for the weighted LCS variant is optimal up to lower-order factors.

ABSTRACT

We study sketching and streaming algorithms for the Longest Common Subsequence problem (LCS) on strings of small alphabet size $|Σ|$. For the problem of deciding whether the LCS of strings $x,y$ has length at least $L$, we obtain a sketch size and streaming space usage of $\mathcal{O}(L^{|Σ| - 1} \log L)$. We also prove matching unconditional lower bounds. As an application, we study a variant of LCS where each alphabet symbol is equipped with a weight that is given as input, and the task is to compute a common subsequence of maximum total weight. Using our sketching algorithm, we obtain an $\mathcal{O}( extrm{min}\{nm, n + m^{\lvert Σ vert}\})$-time algorithm for this problem, on strings $x,y$ of length $n,m$, with $n \ge m$. We prove optimality of this running time up to lower order factors, assuming the Strong Exponential Time Hypothesis.

Motivation & Objective

• To design efficient sketching and streaming algorithms for LCS on strings over a fixed-size alphabet $\Sigma$, where prior work assumed large alphabets.
• To close the gap in understanding LCS complexity under space and time constraints for small alphabets, where strong lower bounds previously only applied to large alphabets.
• To develop a sketching scheme that captures all common subsequences of length at most $L$, enabling efficient decision and streaming computation.
• To prove optimality of the proposed algorithm's time and space complexity for the weighted LCS problem under the Strong Exponential Time Hypothesis (SETH).
• To establish tight lower bounds for the weighted LCS problem by reducing from the Orthogonal Vectors problem under SETH.

Proposed method

• Construct a subsequence $C_L(x)$ of $x$ that preserves all common subsequences of length $\leq L$, using $\mathcal{O}(L^{|Σ|-1})$ runs of length at most $L$.
• Use a one-pass streaming algorithm to compute $C_L(x)$ in $\mathcal{O}(L^{|Σ|-1}\log L)$ space and $\mathcal{O}(1)$ time per symbol.
• Encode each run using $\mathcal{O}(\log L)$ bits, resulting in a total sketch size of $\mathcal{O}(L^{|Σ|-1}\log L)$ bits.
• Reduce the Orthogonal Vectors (OV) problem to the weighted LCS problem by constructing strings $x$ and $y$ such that $\text{WLCS}(x,y) \geq \tau$ iff an orthogonal pair exists.
• Use a parameter $\alpha$ to control the length ratio $n/m$ and construct strings with $n = m^{\alpha \pm o(1)}$ to achieve tight hardness bounds.
• Prove that solving the weighted LCS problem requires time $(nm)^{1-o(1)}$ under SETH, matching the algorithm's running time up to lower-order factors.

Experimental results

Research questions

• RQ1 Can we design a sketching or streaming algorithm for LCS that achieves sublinear space for strings over a fixed-size alphabet?
• RQ2 Is the sketch size $\mathcal{O}(L^{|Σ|-1}\log L)$ optimal for the LCS decision problem under small alphabet assumptions?
• RQ3 Can we achieve an optimal $\mathcal{O}(\min\{nm, n + m^{|Σ|}\})$-time algorithm for the weighted LCS problem on small alphabets?
• RQ4 Does the Strong Exponential Time Hypothesis (SETH) imply that no faster algorithm exists for weighted LCS on small alphabets?
• RQ5 Can we reduce the Orthogonal Vectors problem to weighted LCS in a way that preserves the time complexity under SETH?

Key findings

• The proposed sketching algorithm achieves a sketch size of $\mathcal{O}(L^{|Σ|-1}\log L)$ bits, which is optimal up to constant factors.
• The streaming algorithm uses $\mathcal{O}(L^{|Σ|-1}\log L)$ space and processes each symbol in $\mathcal{O}(1)$ time, with a total decoding time of $\mathcal{O}(L^{2|Σ|})$.
• An $\mathcal{O}(\min\{nm, n + m^{|Σ|}\})$-time algorithm is obtained for the weighted LCS problem on strings of length $n, m$ with $n \geq m$.
• The running time of this algorithm is optimal up to lower-order factors under the Strong Exponential Time Hypothesis (SETH).
• The paper proves that solving weighted LCS requires time $(nm)^{1-o(1)}$ under SETH, establishing tight hardness for all $\alpha < \sigma$.
• The construction of strings $x$ and $y$ ensures that $n = m^{\alpha \pm o(1)}$ and $|\Sigma| = \sigma$, enabling tight reductions from OV to weighted LCS.