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On utilizing the transitivity pursuit-enhanced object partitioning to optimize self-organizing lists-on-lists

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In this paper, the Transitivity Pursuit-Enhanced Object Migration Automata (TPEOMA) is used to capture the dependence of elements in a hierarchical Singly-Linked-Lists on Singly-Linked-Lists (SLLs-on-SLLs) “adaptive” data structure. In doing so, the TPEOMA-enhanced hierarchical SLLs-on-SLLs learns the probability distribution of elements in a Non-stationary Environment. In this framework, we divide the hierarchical Singly-Linked-Lists on Singly-Linked-Lists (SLLs-on-SLLs) into an outer and inner list context. The inner-list context is itself a SLLs containing sub-elements of the list, while the outer-list context contains these sublist partitions as its primitive elements. The elements belonging to a particular sublist partition are determined using the TPEOMA reinforcement learning scheme from the theory of Learning Automata. The idea of Transitivity builds on the Pursuit concept that injects a noise filter into the EOMA to filter divergent queries from the Environment, thereby increasing the likelihood of training the Automaton to approximate the “true” distribution of the Environment. The Transitivity phenomenon can infer “dependent” query pairs from non-accessed elements in the transitivity relation based on the statistical distribution of the queried elements. The TPEOMA-enhanced hierarchical SLLs-on-SLLs schemes results in superior performances to the MTF and TR schemes as well as to the EOMA-enhanced hierarchical SLLs-on-SLLs schemes in NSEs.

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  1. Albeit referred to as a “deadlock” in the literature, it could more appropriately be described as a “livelock”.

  2. Due to space limitations, the background material is only briefly surveyed. The seminal work by Narendra and Thathachar (2012) and the MCS thesis of the first author (Bisong 2018) contain exhaustive details of the theory and applications of LA. This thesis can be made available to the reader.

  3. The converse, i.e., that there might be some relations from \(O_j\) that can be introduced to \(O_i\) as well, is taken care of by the symmetry of the Pursuit matrix, and by the fact that \(O_i\) and \(O_j\) are “dummy” indices.

  4. The Appendix is included in the interest of submitting a comprehensive set of experimental results. It can be deleted if the Referees recommend it.

  5. We are grateful to the Anonymous Referee who requested this.

  6. The Appendix is included in the interest of submitting a comprehensive set of experimental results. It has been included as per the request of the Referees.


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Correspondence to B. John Oommen.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We are very grateful for the feedback from the anonymous Referees of the original submission. Their input significantly improved the quality of this final version.

In this paper, the primitive data structure is the SLL. However, these concepts can be extended to more complicated data structures.

A preliminary version of these results was presented at IFIP’2020 and published in its proceedings. Since the latter paper was reckoned as one of the better papers of the conference, we were invited to present the results, having minimal overlap with the above paper, for possible publication in a special issue of this present journal.



This AppendixFootnote 6 contains the experimental results for the OMA Hierarchical SLL on SLLs. The results contrast the performance of the MTF and TR with the hierarchical MTF-MTF-TPEOMA, TR-MTF-TPEOMA, MTF-TR-TPEOMA, and TR-TR-TPEOMA in dependent non-stationary Environments (Tables 5, 6, 7, 8).

For a list containing 128 elements, the results are from an ensemble of 300,000 queries over 10 experiments. The ensemble asymptotic and amortized cost are used as evaluation metrics for the experiments. The asymptotic cost averages the access cost of the last 20% of the simulations, while the amortized cost is the average of the entire simulations access costs. The results are separated by the size of the sublist, \(k = \{2, 4, 8, 16, 32, 64\}\), with the MSE utilizing values of \(\alpha \in \{0.2, 0.9\}\), and for the PSE \(T = \{30\}\).

Table 5 TPEOMA-Augmented Hierarchical SLLs Results for MSE with \(\alpha = 0.9\) and \(k=2,4,8,16,32,64\)
Table 6 TPEOMA-Augmented Hierarchical SLLs Results for MSE with \(\alpha = 0.2\) and \(k=2,4,8,16,32,64\)
Table 7 TPEOMA-Augmented Hierarchical SLLs Results for PSE with \(T = 30\) and \(k=2,4,8\)
Table 8 TPEOMA-Augmented Hierarchical SLLs Results for PSE with \(T = 30\) and \(k=16,32,64\)

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Bisong, O.E., Oommen, B.J. On utilizing the transitivity pursuit-enhanced object partitioning to optimize self-organizing lists-on-lists. Evolving Systems 12, 655–686 (2021).

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