{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T18:52:34Z","timestamp":1782240754353,"version":"3.54.5"},"reference-count":44,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2017,9,1]],"date-time":"2017-09-01T00:00:00Z","timestamp":1504224000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007835","name":"Politechnika \u015al\u0105ska","doi-asserted-by":"publisher","award":["RGH 2017"],"award-info":[{"award-number":["RGH 2017"]}],"id":[{"id":"10.13039\/501100007835","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Modern architectures make possible development in new algorithms for large data sets and distributed computing. The newly proposed versions can benefit both from faster computing on the multi core architectures, and intelligent programming techniques that use efficient procedures available in the latest programming studios. Frequently used algorithms to sort arrays of data in NoSQL databases is merge sort, where as NoSQL we understand any database without typical SQL programming interpreter. The author describes how to use the parallelization of the sorting processes for the modified method of sorting by merging for large data sets. The subject of this research is the claim that the parallelization of the sorting method is faster and beneficial for multi-core systems. Presented results show how the number of processors influences the sorting performance. The results are presented in theoretical assumptions and confirmed in practical benchmark tests. The method is compared to other sorting methods like quick sort, heap sort, and merge sort to show potential efficiency.<\/jats:p>","DOI":"10.3390\/sym9090176","type":"journal-article","created":{"date-parts":[[2017,9,1]],"date-time":"2017-09-01T11:05:24Z","timestamp":1504263924000},"page":"176","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":28,"title":["Parallelization of Modified Merge Sort Algorithm"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6593-7158","authenticated-orcid":false,"given":"Zbigniew","family":"Marsza\u0142ek","sequence":"first","affiliation":[{"name":"Institute of Mathematics, Silesian University of Technology, ul. Kaszubska 23, 44-100 Gliwice, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,9,1]]},"reference":[{"key":"ref_1","unstructured":"Aho, I., Hopcroft, J., and Ullman, J. (1975). The Design and Analysis of Computer Algorithms, Addison-Wesley Publishing Company."},{"key":"ref_2","unstructured":"Knuth, D. (1998). The Art of Computer Programming Vol.3: Sorting and Searching, Addison-Wesley."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/BF01939369","article-title":"A one-way, stack less quick sort algorithm","volume":"26","author":"Knuth","year":"1986","journal-title":"BIT"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1016\/0167-8191(92)90089-P","article-title":"A parallel partition for enhanced parallel quick sort","volume":"18","author":"Francis","year":"1992","journal-title":"Parallel Comput."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Rauh, A., and Arce, G. (2010, January 26\u201329). A fast weighted median algorithm based on quick select. Proceedings of the IEEE 17th International Conference on Image Processing, Hong Kong, China.","DOI":"10.1109\/ICIP.2010.5651855"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Tsigas, P., and Zhang, Y. (2003, January 5\u20137). A simple, fast parallel implementation of quick sort and its performance evaluation on SUN enterprise 10,000. Proceedings of the Euromicro Workshop on Parallel, Distributed and Network-Based Processing, Genova, Italy.","DOI":"10.1109\/EMPDP.2003.1183613"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Daoud, A., Abdel-Jaber, H., and Ababneh, J. (2011, January 20\u201322). Efficient non-quadratic quick sort (NQQuickSort). Proceedings of the International Conference on Digital Enterprise and Information Systems, London, UK.","DOI":"10.1007\/978-3-642-22603-8_58"},{"key":"ref_8","unstructured":"Edmondson, J. (2005, January 20\u201323). Pivot sort\u2014Replacing quick sort. Proceedings of the 2005 International Conference on Algorithmic Mathematics and Computer Science, Las Vegas, NV, USA."},{"key":"ref_9","unstructured":"Kushagra, S., L\u00f3pez-Ortiz, A., Munro, J., and Qiao, A. (2016, January 10\u201312). Multi-pivot quicksort: Theory and experiments. Proceedings of the Meeting on Algorithm Engineering & Experiments. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Ben-Or, M. (1983, January 25\u201327). Lower bounds for algebraic computation trees. Proceedings of the 15th ACM Symposium on Theory of Computing (STOC), Boston, MA, USA.","DOI":"10.1145\/800061.808735"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/BF01941462","article-title":"Inserting a new element into a heap","volume":"21","author":"Doberkat","year":"1983","journal-title":"BIT Numer. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"917","DOI":"10.1109\/32.29490","article-title":"The external heap sort","volume":"15","author":"Lutz","year":"1989","journal-title":"IEEE Trans. Softw. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Sumathi, S., Prasad, A., and Suma, V. (2014, January 14\u201315). Optimized heap sort technique (OHS) to enhance the performance of the heap sort by using two-swap method. Proceedings of the 3rd International Conference on Frontiers of Intelligent Computing: Theory and Applications (FICTA-2014), Bhubaneswar, India.","DOI":"10.1007\/978-3-319-11933-5_78"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1006\/jagm.2001.1160","article-title":"Digital access to comparison-based tree data structures and algorithms","volume":"40","author":"Roura","year":"2001","journal-title":"J. Algorithms"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/0196-6774(89)90017-5","article-title":"A simple parallel tree construction algorithm","volume":"10","author":"Abrahamson","year":"1987","journal-title":"J. Algorithms"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1007\/3-540-52921-7_74","article-title":"Sublinear merging and natural merge sort","volume":"450","author":"Carlsson","year":"1990","journal-title":"Algorithms"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1137\/0217049","article-title":"Parallel merge sort","volume":"17","author":"Cole","year":"1988","journal-title":"SIAM J. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1016\/S0167-9473(02)00065-8","article-title":"Approximation quality for sorting rules","volume":"40","author":"Gedigaa","year":"2002","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_19","first-page":"1339","article-title":"Sorting unsorted and partially sorted lists using the natural merge sort","volume":"11","author":"Gubias","year":"2006","journal-title":"Softw. Pract. Exp."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1007\/BF00572988","article-title":"Merging sorted runs using main memory","volume":"27","author":"Huang","year":"1989","journal-title":"Acta Inform."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1145\/42392.42403","article-title":"Practical in-place merging","volume":"31","author":"Huang","year":"1989","journal-title":"Commun. ACM"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1109\/69.494169","article-title":"Speeding up external merge sort","volume":"8","author":"Zhang","year":"1996","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"ref_23","unstructured":"Zhang, W., and Larson, P. (1997, January 25\u201329). Dynamic memory adjustment for external merge sort. Proceedings of the Very Large Data Bases Conference, San Francisco, CA, USA."},{"key":"ref_24","unstructured":"Zhang, W., and Larson, P. (1998, January 24\u201327). Buffering and read-ahead strategies for external merge sort. Proceedings of the Very Large Data Bases Conference, New York, NY, USA."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Vignesh, R., and Pradhan, T. (2016, January 25\u201327). Merge sort enhanced in place sorting algorithm. Proceedings of the 2016 International Conference on Advanced Communication Control and Computing Technologies (ICACCCT), Ramanathapuram, India.","DOI":"10.1109\/ICACCCT.2016.7831730"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Cheema, S., Sarwar, N., and Yousaf, F. (2016, January 24\u201326). Contrastive analysis of bubble & merge sort proposing hybrid approach. Proceedings of the 2016 International Conference on Innovative Computing Technology (INTECH), Dublin, Ireland.","DOI":"10.1109\/INTECH.2016.7845075"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"982","DOI":"10.1016\/j.procs.2016.07.292","article-title":"Enhanced merge sort\u2014A new approach to the merging process","volume":"93","author":"Smita","year":"2016","journal-title":"Procedia Comput. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1109\/TSE.1984.5010255","article-title":"Virtual memory behavior of some sorting algorithm","volume":"10","author":"Alanko","year":"1984","journal-title":"IEEE Trans. Softw. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Larson, P.-\u00c5., and Graefe, G. (1998, January 2\u20134). Memory management during run generation in External Sorting. Proceedings of the SIGMOD, Seattle, WA, USA.","DOI":"10.1145\/276305.276346"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1006\/jagm.1998.0985","article-title":"The influence of caches on the performance of sorting","volume":"31","author":"LaMarca","year":"1999","journal-title":"J. Algorithms"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/3-540-44867-5_7","article-title":"Search data structures for skewed strings","volume":"Volume 2647","author":"Crescenzi","year":"2003","journal-title":"Lecture Notes in Computer Science"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1145\/146370.146381","article-title":"A survey of adaptive sorting algorithms","volume":"24","author":"Wood","year":"1992","journal-title":"Comput. Surv."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"11269","DOI":"10.1109\/ACCESS.2016.2642979","article-title":"Chrological big data curation: A study on the enhanced information retrieval system","volume":"5","author":"Choi","year":"2017","journal-title":"IEEE Access"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Axtmann, M., Bigmann, T., Schulz, C., and Sanders, P. (2015, January 13\u201315). Practical massively parallel sorting. Proceedings of the 27th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA\u201915), Portland, OR, USA.","DOI":"10.1145\/2755573.2755595"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"7350","DOI":"10.1109\/ACCESS.2017.2692254","article-title":"Self-sorting-based MAC protocol for high-density vehicular Ad Hot networks","volume":"5","author":"Shen","year":"2017","journal-title":"IEEE Access"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1930","DOI":"10.1109\/TVLSI.2017.2661746","article-title":"An efficient O(N) comparison-free sorting algorithm","volume":"25","year":"2017","journal-title":"IEEE Trans. Very Large Scale Integr. Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1016\/j.future.2017.01.034","article-title":"Bidirectional conditional insertion sort algorithm; An efficient progress on the classical insertion sort","volume":"71","author":"Saher","year":"2017","journal-title":"Future Gener. Comput. Syst. Int. J. ESci."},{"key":"ref_38","first-page":"111","article-title":"Preprocessing large data sets by the use of quick sort algorithm","volume":"364","author":"Gabryel","year":"2015","journal-title":"Adv. Intell. Syst. Comput. KICSS\u20192013"},{"key":"ref_39","unstructured":"Skulimowski, A.M.J. (2015). Triple heap sort algorithm for large data sets. Looking into the Future of Creativity and Decision Support Systems, Progress & Business Publishers."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Wo\u017aniak, M., Marsza\u0142ek, Z., Gabryel, M., and Nowicki, R. (2013, January 9\u201313). Modified merge sort algorithm for large scale data sets. Proceedings of the International Conference on Artificial Intelligence and Soft Computing, Zakopane, Poland.","DOI":"10.1007\/978-3-642-38610-7_56"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Marsza\u0142ek, Z., Wo\u017aniak, M., Borowik, G., Wazirali, R., Napoli, C., Pappalardo, G., and Tramontana, E. (2015, January 14\u201316). Benchmark tests on improved merge for big data processing. Proceedings of the Asia-Pacific Conference on Computer Aided System Engineering (APCASE\u20192015), Quito, Ecuador.","DOI":"10.1109\/APCASE.2015.24"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Czerwi\u0144ski, D. (2015, January 16\u201319). Digital filter implementation in Hadoop data mining system. Proceedings of the International Conference on Computer Networks, Brunow, Poland.","DOI":"10.1007\/978-3-319-19419-6_39"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Marsza\u0142ek, Z. (2016, January 13\u201315). Novel recursive fast sort algorithm. Proceedings of the 22nd International Conference on Information and Software Technologies (ICIST), Druskininkai, Lithuania.","DOI":"10.1007\/978-3-319-46254-7_27"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Uyar, A. (2014, January 15\u201317). Parallel merge sort with double merging. Proceedings of the 2014 IEEE 8th International Conference on Application of Information and Communication Technologies (AICT), Astana, Kazakhstan. Book Series: International Conference on Application of Information and Communication Technologies.","DOI":"10.1109\/ICAICT.2014.7036012"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/9\/176\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:43:57Z","timestamp":1760208237000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/9\/176"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,9,1]]},"references-count":44,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2017,9]]}},"alternative-id":["sym9090176"],"URL":"https:\/\/doi.org\/10.3390\/sym9090176","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,9,1]]}}}