Constant-round O(1)-approximate Euclidean MST in MPC

Develop a fully-scalable Massively Parallel Computation algorithm that, in O(1) rounds, computes an O(1)-approximate solution for the Euclidean Minimum Spanning Tree problem.

Background

The authors survey progress on Euclidean MST in MPC, noting that value estimation admits O(1)-approximations in O(1) rounds, but existing solution algorithms either require O(log log n) rounds for O(1)-approximation or achieve only polylogarithmic approximations in O(1) rounds.

They explicitly point out that attaining an O(1)-approximate MST in O(1) rounds remains unresolved, identifying it as a notable open direction in geometric graph optimization under MPC.