Brief Announcement: Upper and Lower Bounds for Edit Distance in Space-Efficient MPC

In the Massively Parallel Computation (MPC) model, data is distributed across multiple processors, and we call an algorithm space-efficient if each machine has n^1-ε + o(1) memory with a machine count of Ømega(n^ε).

In this paper, we study the string edit distance problem in the MPC model, presenting both a new algorithm and lower-bound results. A space-efficient MPC algorithm computing the exact edit distance using O(n^ε) communication rounds is known by updating the algorithm of Chowdhury and Ramachandran (SPAA 2008). A key contribution of our work is the introduction of the first space-efficient MPC algorithm, which uses subpolynomial number of rounds and provides an n^o(1) -approximation of edit distance, where n denotes the length of the input strings.

Further, we complement this algorithm with new lower-bound results. The Orthogonal Vector (O.V.) conjecture states that no truly subquadratic time algorithm exists for the Orthogonal Vector problem, and it follows directly from the Strong Exponential Time Hypothesis (SETH). Drawing inspiration from this, we propose the Strong O.V. Conjecture that posits that there is no space-efficient MPC algorithm capable of solving O.V. using n^ε - Ømega(1) communication rounds. The Strong O.V. conjecture has far-reaching consequences, yielding the first (or strengthened) lower bounds for a myriad of problems in the MPC model including graph diameter estimation, computing Fréchet distance, longest common subsequence, and dynamic time warping. Via an MPC reduction from O.V. to edit distance, we give the first conditional lower bound for string edit distance in the MPC model showing that there does not exist any space-efficient, n^ε - Ømega(1) -round MPC exact edit distance algorithm.