Pairwise sequence alignment remains a fundamental problem in computational biology and bioinformatics. Recent advances in genomics and sequencing technologies demand faster and scalable algorithms that can cope with the ever-increasing sequence lengths. Classical pairwise alignment algorithms based on dynamic programming are strongly limited by quadratic requirements in time and memory. The recently proposed wavefront alignment algorithm (WFA) introduced an efficient algorithm to perform exact gap-affine alignment in O(ns) time, where s is the optimal score and n is the sequence length. Notwithstanding these bounds, WFA's O(s2) memory requirements become computationally impractical for genome-scale alignments, leading to a need for further improvement. In this article, we present the bidirectional WFA algorithm, the first gap-affine algorithm capable of computing optimal alignments in O(s) memory while retaining WFA's time complexity of O(ns). As a result, this work improves the lowest known memory bound O(n) to compute gap-affine alignments. In practice, our implementation never requires more than a few hundred MBs aligning noisy Oxford Nanopore Technologies reads up to 1 Mbp long while maintaining competitive execution times. All code is publicly available at https://github.com/smarco/BiWFA-paper. Supplementary data are available at Bioinformatics online.