This paper revisits the fundamental problem of monitoring the linearizability of concurrent stacks, queues, sets, and multisets. Given a history of a library implementing one of these abstract data types, the monitoring problem is to answer whether the given history is linearizable. For stacks, queues, and (multi)sets, we present monitoring algorithms with complexities 𝓞(𝑛 2 ), 𝓞(𝑛 𝑙𝑜𝑔 𝑛), and 𝓞(𝑛), respectively, where 𝑛 is the number of operations in the input history. For stacks and queues, our results hold under the standard assumption of data-independence, i.e., the behavior of the library is not sensitive to the actual values stored in the data structure. Past works to solve the same problems have cubic time complexity and (more seriously) have correctness issues: they either (i) lack correctness proofs or (ii) the suggested correctness proofs are erroneous (we present counter-examples), or (iii) have incorrect algorithms. Our improved complexity results rely on substantially different algorithms for which we provide detailed proofs of correctness. We have implemented our stack and queue algorithms in 𝐿𝑖𝑀𝑜 (Linearizability Monitor). We evaluate 𝐿𝑖𝑀𝑜 and compare it with the state-of-the-art tool 𝑉𝑖𝑜𝑙𝑖𝑛 – whose correctness proofs we have found errors in – which checks for linearizability violations. Our experimental evaluation confirms that 𝐿𝑖𝑀𝑜 outperforms 𝑉𝑖𝑜𝑙𝑖𝑛 regarding both efficiency and scalability.