We are all familiar with the sorting algorithms. The algorithm will take an input array A$[]$ with size n$,$ and reorder the elements to put them in order. In many cases, a sorting algorithm will find two elements in the input and swap them. By performing a series of $$swaps$,$ all elements are put in order. For simplicity, let$$s assume all input elements are distinct. Now let$$s consider a special sorting algorithm: this algorithm can only move elements by swapping adjacent elements. Let$$s call such a swap an adjacent swap. In other words, to reorder the elements, the only operation that is allowed is to swap A$[$i$]$ with A$[$i $+1],$ for some i$.$ It can read and compare elements arbitrarily $($but only move elements by adjacent swaps$).$ We are interested in studying that, how $$powerful$$ such an algorithm is$,$ and how does that affect the upper and lower bounds of sorting algorithms. In general, we will consider a computational model, such that we only count the number of adjacent swaps as the cost of the algorithm.