# CUDA and OpenMp Implementation of Boolean Matrix Product with Applications in Visual SLAM
## 5 Ultrametric-PAM and Ultrametric-FABMAP
## 7 Conclusions

In this paper, we demonstrated that the degree of ultrametricity has a direct relationship with the performance of the PAM clustering algorithm. Ultrametric PAM is a thousand times faster than regular PAM. The performance was also evaluated using the clustering Donn index, compactness, sum of squared errors, and normalised mutual information. Clusters with a better Donn index, higher normalised mutual information, and a lower sum of squared errors are obtained by using the PAM algorithm in spaces with higher ultrametric configuration. Furthermore, we demonstrated that our new measure and [PERSON]'s measure are pointing in the same direction, with a Pearson's correlation close to 1. It has been demonstrated that adding a hierarchical structure to the dataset distance matrices improves the accuracy and recall of loop closure detection by at least 6%. In future works, we will attempt to define Boolean Strassen matrix multiplication in order to reduce transformation time calculation. Not applicable. ## References

* (1) [PERSON]; [PERSON] Data Ultrametricity and Clusterability. _J. Phys. Conf. Ser._**2020**, _1334_, 012002. [CrossRef]
* (2) [PERSON] On the powers of matrices over a distributive lattice. _Linear Algebra Its Appl._**2001**, _336_, 1-14. [CrossRef]
* (3) [PERSON] On the transitive matrices over distributive lattices. _Linear Algebra Its Appl._**2005**, _400_, 169-191. [CrossRef]
* (4) [PERSON] Sparse p-adic data coding for computationally efficient and effective big data analytics. _P-Adic Numbers Ultrametric Anal. Appl._**2016**, \(8\), 27-42. [CrossRef]
* (5) [PERSON]; [PERSON]; [PERSON] Unsupervised Feature Selection Based on Ultrametricity and Sparse Training Data: A Case Study for the Classification of High-Dimensional Hyperspectral Data. _Remote Sens._**2018**, _10_, 1564. [CrossRef]
* (6) [PERSON] Identifying and Exploiting Ultrametricity. In _Proceedings of the Advances in Data Analysis, Proceedings of the 30 th Annual Conference of the Gesellschaft fur Klassifikation e.V., Freie Universitat Berlin, 8-10 March 2006_; [PERSON], [PERSON]; Springer: New York, NY, USA, 2006; pp. 263-272. [CrossRef]
* (7) [PERSON]; [PERSON] The Future of Search and Discovery in Big Data Analytics: Ultrametric Information Spaces. _arXiv_**2012**, arXiv:1202.3451. [CrossRef]
* (8) [PERSON]; [PERSON] _Finding Groups in Data: An Introduction to Cluster Analysis_; [PERSON]: New York, NY, USA, 2009. * (9) K-Medoids. Available online: [[https://en.wikipedia.org/wiki/K-medoids](https://en.wikipedia.org/wiki/K-medoids)]([https://en.wikipedia.org/wiki/K-medoids](https://en.wikipedia.org/wiki/K-medoids)) (accessed on 10 December 2022). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dataset-Size** & **PAM** & **Ultrametric-PAM** & **Speedup** \\ \hline
6736 \(\times\) 6736 & 3.5 days & 18 min & \(\times\)280 \\
11,712 \(\times\) 11,712 & 14 days & 2 h and 23 min & \(\times\)1254. \\ \hline \hline \end{tabular}
\end{table} 
Table 9: PAM vs. Ultrametric-PAM speed of clustering. * (10) K-Medoids Clustering with Solved Example. Available online: [[https://www.geeksforgeeks.org/ml-k-medoids-clustering-with-example/](https://www.geeksforgeeks.org/ml-k-medoids-clustering-with-example/)]([https://www.geeksforgeeks.org/ml-k-medoids-clustering-with-example/](https://www.geeksforgeeks.org/ml-k-medoids-clustering-with-example/)) (accessed on 10 December 2022). * (11) [PERSON]; [PERSON] FAB-MAP: Probabilistic Localization and Mapping in the Space of Appearance. _Int. J. Robot. Res._**2008**, _27_, 647-665. [CrossRef]
* (12) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] OpenFABMAP: An Open Source Toolbox for Appearance-based Loop Closure Detection. In Proceedings of the International Conference on Robotics and Automation, Saint Paul, MN, USA, 14-18 May 2012; IEEE: St. Paul, MN, USA, 2012. * (13) [PERSON]; [PERSON]; [PERSON] On the degree of ultrametricity. _J. Phys. Lett._**1985**, _46_, 945-952.