OT (Operational Transformation) was invented for supporting real-time co-editors in the late 1980s and has evolved to become core techniques widely used in today's working co-editors and adopted in industrial products. CRDT (Commutative Replicated Data Type) for co-editors was first proposed around 2006, under the name of WOOT (WithOut Operational Transformation). Follow-up CRDT variations are commonly labeled as "post-OT" techniques capable of making concurrent operations natively commutative in co-editors. On top of that, CRDT solutions have made broad claims of superiority over OT solutions, and often portrayed OT as an incorrect and inefficient technique. Over one decade later, however, CRDT is rarely found in working co-editors; OT remains the choice for building the vast majority of today's co-editors. Contradictions between the reality and CRDT's purported advantages have been the source of much confusion and debate in co-editing researcher and developer communities. To seek truth from facts, we set out to conduct a comprehensive and critical review on representative OT and CRDT solutions and working co-editors based on them. From this work, we have made important discoveries about OT and CRDT, and revealed facts and evidences that refute CRDT claims over OT on all accounts. These discoveries help explain the underlying reasons for the choice between OT and CRDT in the real world. We report these results in a series of three articles. In this article (the second in the series), we reveal the differences between OT and CRDT in their basic approaches to realizing the same general transformation and how such differences had resulted in different technical challenges and consequential correctness and complexity issues. Moreover, we reveal hidden complexity and algorithmic flaws with representative CRDT solutions, and discuss common myths and facts related to correctness and complexity of OT and CRDT. We hope the discoveries from this work help clear up common myths and confusions surrounding OT and CRDT, and accelerate progress in co-editing technology for real world applications.

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OT (Operational Transformation) was invented for supporting real-time co-editors in the late 1980s and has evolved to become a core technique used in today's working co-editors and adopted in major industrial products. CRDT (Commutative Replicated Data Type) for co-editors was first proposed around 2006, under the name of WOOT (WithOut Operational Transformation). Follow-up CRDT variations are commonly labeled as "post-OT" techniques and have made broad claims of superiority over OT solutions, in terms of correctness, time and space complexity, simplicity, etc. Over one decade later, however, OT remains the choice for building the vast majority of co-editors, whereas CRDT is rarely found in working co-editors. Why? Based on comprehensive review and comparison on representative OT and CRDT solutions and co-editors based on them, we present our discoveries in relation to this question and beyond in a series of three articles.In prior work, we have revealed that CRDT is like OT in following the same general transformation approach and CRDT is not natively commutative for concurrent operations in co-editors, which helps to clarify what CRDT really is and is not for co-editors. In this article, we reveal OT and CRDT differences in correctness and complexity by dissecting and examining representative OT and CRDT solutions. We explore how different basic approaches ̶ the concurrency-centric approach taken by OT and the content-centric approach taken by CRDT ̶ had resulted in different technical challenges, correctness and complexity issues, and solutions. Moreover, we reveal hidden algorithmic flaws with representative CRDT solutions, and discuss common myths and facts related to correctness, time and space complexity, and simplicity of OT and CRDT. We present facts and evidences that refute CRDT claimed advantages over OT. We hope the discoveries from this work help clear up common myths and confusions surrounding OT and CRDT, and accelerate progress in co-editing technology for real world applications. ; URL: https://www.ntu.edu.sg/home/CZSun 20,38,43,54,55,58,61,70,71,77], and support real-time collaboration in rich-text co-editing in word processors [60,65,68,82], HTML/XML Web document co-editing [10], spreadsheet co-editing [69], 3D model co-editing in digital media design tools [1,2], and file synchronization in cloud storage systems [3]. Recent years have seen OT being widely adopted in industry products as the core technique for consistency maintenance, ranging from battle-tested online collaborative richtext editors like Google Docs 1 [11], to emerging start-up products, such as Codox Apps 2 .A variety of alternative techniques for consistency maintenance in co-editors had also been explored in the past decades [14,16,18,41,42,72]. One notable class of techniques is CRDT 3 (Commutative Replicated Data Type) for co-editors [4,5,8,25,32,37,39,40,41,45,47,48,79,80,81]. The first CRDT solution for plain-text co-editing appeared around 2006 [40,41], under the name of WOOT (WithOut Operational Transformation). One mo...

Multi-user collaborative editors are useful computer-aided tools to support human-to-human collaboration. For multi-user collaborative editors, selective undo is an essential utility enabling users to undo any editing operations at any time. Collaborative editors usually adopt operational transformation (OT) to address concurrency and consistency issues. However, it is still a great challenge to design an efficient and correct OT algorithm capable of handling both normal do operations and user-initiated undo operations because these two kinds of operations can interfere with each other in various forms. In this paper, we propose a semi-transparent selective undo algorithm that handles both do and undo in a unified framework, which separates the processing part of do operations from the processing part of undo operations. Formal proofs are provided to prove the proposed algorithm under the well-established criteria. Theoretical analysis and experimental evaluation are conducted to show that the proposed algorithm outperforms the prior OT-based selective undo algorithms.