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Problem 1919 K-way Merging sort

## Problem Description

As we all known, merge sort is an O(nlogn) comparison-based sorting algorithm. The merge sort achieves its good runtime by a divide-and-conquer strategy, namely, that of halving the list being sorted: the front and back halves of the list are recursively sorted separately, then the two results are merged into the answer list. An implementation is shown as follows:

The procedure Merge(L1,L2:in List_type;L:out List_type) that we have in mind for sorting two lists is described as follows. Initialize pointers to the first item in each list L1,L2, and then

repeat

compare the two items pointed at;

move the smaller into L;

move the pointer which originally points at the smaller one to the next number;

until one of L1,L2 exhausts;

drain the remainder of the unexhausted list into L;

Now let us come to the situation when there are k pointers, here k≥2. Let L be a list of n elements. Divide L into k disjoint contiguous sublists L1,L2,…,Lk of nearly equal length. Some Li’s (namely, n reminder k of them, so possibly none) will have length , let these have the low indices: L1,L2,…,Ln%k Other Li’s will have length , and high indices are assigned: Ln%k+1,…,Lk-1,Lk. We intend to recursively sort the Li’s and merge the k results into an answer list.

We use Linear-Search-Merge here to merge k sorted lists. We find the smallest of k items (one from each of the k sorted source lists), at a cost of k-1 comparisons. Move the smallest into the answer list and advances its corresponding pointer (the next smallest element) in the source list from which it came. Again there are k items, from among which the smallest is to be selected. (When i (1 ≤ i < k) lists are empty, k-way merging sort becomes to (k-i)-way merging sort, and the draining process will start when the total order of all the elements have been found)

Given a list containing n elements, your task is to find out the maximum number of comparisons in k-way merging sort.

## Input

The first line of the input contains an integer T (T <= 100), indicating the number of cases. Each case begins with a line containing two integer n (1 ≤ n ≤ 10100) and k (2 ≤ k ≤ 20), the number of elements in the list, and it is k-way merging sort.

## Output

For each test case, print a line containing the test case number (beginning with 1) and the maximum number of comparisons in k-way merging sort.

## Sample Input

4 2 2 3 2 100 7 1000 10

## Sample Output

Case 1: 1 Case 2: 3 Case 3: 1085 Case 4: 22005

## Source

2010年全国大学生程序设计邀请赛（福州）

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