Dujmović et al. (FOCS2019) recently proved that every planar graph G is a subgraph of \(H\boxtimes P\), where \(\boxtimes\) denotes the strong graph product, H is a graph of treewidth 8 and P is a path. This result has found numerous applications to linear graph layouts, graph colouring, and graph labelling. The proof given by Dujmović et al. is based on a similar decomposition of Pilipczuk and Siebertz (SODA2019) which is constructive and leads to an \(O(n^2)\) time algorithm for finding H and the mapping from V(G) onto \(V(H\boxtimes P)\). In this note, we show that this algorithm can be made to run in \(O(n\log n)\) time.