In small-to-large merging, when we merge two components of size $$$a$$$ and $$$b$$$, we only iterate over the nodes in the smaller of the two components, so its $$$\mathcal O(min(a, b))$$$ work. The divide-and-conquer technique is the basis of efficient algorithms. Combine: count inversions where a i and a j are in different halves, and return sum of three quantities. The solutions to the sub-problems are then combined to give a solution to the original problem. Conquer: recursively count inversions in each half. Each state that splits into two child state can be thought of in reverse: two child states merge to become one state. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. Now based on this analogy, I refer to the recursive tree. Otherwise, ais divided into two (almost) equalhalvesaleftandaright. If so, thentheais sorted in place by making at most one swap. The algorithm begins by checking if input arrayahas two or fewer elements. And since we can only delete at most n−1 edges, we only have at most 2(n−1)+1 states, or O(n) states! A naive bound on the amount of work we do at each state is O(n), so the complexity is O(n2). TheMergesortalgorithm is a divide-and-conquer algorithm for sorting an array of comparableelements. What does a split of a state into two other states represent in this analogy? It represents deleting some edge (m,m+1) and splitting the component into two separate ones. And initially, we have representing a line graph with edges connecting (i,i+1) for all 1≤i
0 kommentar(er)
