This paper presents an algorithm for sorting by using of LIT (left inversions table). The algorithm is named LR. The time complexity of the proposed algorithm analytically evaluated. Two approaches for acceleration of LR are presented. The proposed algorithm and its two improvements are implemented in C++. Experimental comparisons are done between LR and some known algorithms, and between LR and its two modifications. The experiments show that LR is faster than “bubble sort” and “LtoRA” algorithms but it is slower than the algorithms “insertion sort” and “selection sort”. The experiments also show that for rows in which there is a large number of the repetitions, the modification “LR – repeat” is faster than the original algorithm, “Bubble sort”, “Selection sort” and the modification “LRA – minimax”. The algorithm “LR minimax” is faster than algorithm LR in all cases (when the row has large or small number of repetitions).