Approximation Schemes for Edit Distance and LCS in Quasi-Strongly Subquadratic Time

Abstract

We present novel randomized approximation schemes for the Edit Distance (ED) problem and the Longest Common Subsequence (LCS) problem that, for any constant $ε>0$, compute a $(1+ε)$-approximation for ED and a $(1-ε)$-approximation for LCS in time $n^2 / 2^{\log^{Ω(1)}(n)}$ for two strings of total length at most $n$. This running time improves upon the classical quadratic-time dynamic programming algorithms by a quasi-polynomial factor. Our results yield significant insights into fine-grained complexity: Firstly, for ED, prior work indicates that any exact algorithm cannot be improved beyond a few logarithmic factors without refuting established complexity assumptions [Abboud, Hansen, Vassilevska Williams, Williams, 2016]; our quasi-polynomial speed-up shows a separation the complexity of approximate ED from that of exact ED, even for approximation factor arbitrarily close to $1$. Secondly, for LCS, obtaining similar approximation-time tradeoffs via deterministic algorithms would imply breakthrough circuit lower bounds [Chen, Goldwasser, Lyu, Rothblum, Rubinstein, 2019]; our randomized algorithm demonstrates derandomization hardness for LCS approximation.

Figures (6)

• Figure 1: Edit distance as a shortest path on 2D grid (standard)
• Figure 2: Convenient coordinate system
• Figure 3: Sub-sampling
• Figure 4: Overfitting Issue With Naïve Sub-sampling Approach for ED
• Figure 5: In this example, the branching factor is $M = 4$. Naïvely, all three paths are candidate paths. However, in our approach we only consider as a valid candidate the middle path, which is rounded to the straight line from A to E on "anchors" B, C, and D.

Theorems & Definitions (107)

• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 4.0
• Claim 4.1
• proof
• proof : Proof of \ref{['lemma:totalmeandeviation']}
• Lemma 4.2
• proof