Summary: We give an \(n^{O(\log n)}\)-time (\(n\) is the input size) blackbox polynomial identity testing algorithm for unknown-order read-once oblivious arithmetic branching programs (ROABPs). The best time complexity known for blackbox polynomial identity testing (PIT) for this class was \(n^{O(\log^2 n)}\) due to M. A. Forbes et al. [in: Proceedings of the 46th annual ACM symposium on theory of computing, STOC ’14. New York, NY: Association for Computing Machinery (ACM). 867–875 (2014; Zbl 1315.68127)]. Moreover, their result holds only when the individual degree is small, while we do not need any such assumption. With this, we match the time complexity for the unknown-order ROABP with the known-order ROABP (due to M. A. Forbes and A. Shpilka [“Quasipolynomial-time identity testing of non-commutative and read-once oblivious algebraic branching programs”, in: Proceedings of the IEEE 54th annual symposium on foundations of computer science, FOCS’13. Los Alamitos, CA: IEEE Computer Society. 243–252 (2013; doi:10.1109/FOCS.2013.34)]) and also with the depth-3 set-multilinear circuits (due to M. Agrawal et al. [in: Proceedings of the 45th annual ACM symposium on theory of computing, STOC ’13. New York, NY: Association for Computing Machinery (ACM). 321–330 (2013; Zbl 1293.94140)]). Our proof is simpler and involves a new technique called basis isolation. The depth-3 model has recently gained much importance, as it has become a stepping stone to understanding general arithmetic circuits. Multilinear depth-3 circuits are known to have exponential lower bounds but no polynomial time blackbox identity tests. In this paper, we take a step toward designing such hitting-sets. We give the first subexponential whitebox PIT for the sum of constantly many set-multilinear depth-3 circuits. To achieve this, we define the notions of distance and base sets. Distance, for a multilinear depth-3 circuit (say, in \(n\) variables and \(k\) product gates), measures how far the variable partitions corresponding to the product gates are from being a mere refinement of each other. The 1-distance circuits strictly contain the set-multilinear model, while \(n\)-distance captures general multilinear depth-3. We design a hitting-set in time \((nk)^{O(\Delta \log n)}\) for \(\Delta\)-distance. Further, we give an extension of our result to models where the distance is large (close to \(n\)) but is small when restricted to certain base sets (of variables). We also explore a new model of ROABPs where the factor matrices are invertible (called invertible-factor ROABPs). We design a hitting-set in time \(\mathrm{poly}(n^{w^2})\) for width-\(w\) invertible-factor ROABPs. Further, we could do without the invertibility restriction when \(w=2\). Previously, the best result for width-2 ROABPs was quasi-polynomial time [Forbes et al., loc. cit.].