Question

Suppose we are given array L of sorted lists of integers L[1], ..., [k]; for convenience, I will use L, to refer to L[i].

Further, assume that n is a multiple of k, k is a power of 2, and that each list has integers so that the total size of the lists is n. In this problem, we are interested in merging all the k sorted lists to produce one sorted list.
By "C Merge (A, B)", I mean sorted lists A and B are merged to produce the sorted list C as studied in the class. Note that the lists need not be distinct. Assume that lists are implemented in such a way that any addition (deletion) to (from) front/back of the list can be done in constant time.
Below are algorithms for the problem. Analyze each algorithm to find an expression for the complexity of the algorithm; provide details. Give a theta bound for the complexity, but you do not have to prove the bound.
The complexity will clearly depend on n and k.
(a) L = empty list;
for i = 1 to k /* List L is the sorted list of all the items */
L = Merge (L, Li);
(b) Consider the following approach using divide and conquer: if there is only one list to deal with, return that as answer. Otherwise, recursively (using the same algorithm) merge lists L1, L to get list Left. Then, recursively merge lists L+1, L to get list Right. Merge Left with Right and return the result. Analyze the algorithm to determine its complexity.

Answer 1

The complexity for the sequential merge algorithm is O(kn), where k is the number of lists and n is the total size of all lists. The divide and conquer algorithm has a complexity of O(nlogk), combining efficiency when merging each pair of lists and the depth of recursive calls represented by log2(k).

Step-by-step explanation:

Analysis of Merging k Sorted Lists Algorithms

When analyzing the complexity of given algorithms for merging k sorted lists of integers, where the total size of the lists is n and k is a power of 2, we can approach each algorithm individually.

(a) Sequential Merge Algorithm

For the first algorithm, where a new list L is formed by merging each list sequentially, we will perform a merge for each of the k lists. Since each list has n/k elements and merging cost is proportional to the sum of the lengths of both lists being merged, the complexity can be expressed as O(n + 2n + ... + kn/k) = O(kn). Thus, the complexity for this algorithm is O(kn).

(b) Divide and Conquer Merge Algorithm

The second algorithm employs a divide and conquer strategy to merge the lists. At each recursive step, the problem is divided in half, with each half taking O(n) time to merge. The depth of recursion is log2(k), leading to a total complexity of O(nlogk), hence the complexity is O(nlogk).

Answer 2

Algorithm (a) has a time complexity of O(nk) as it performs a sequential merge of k lists. Algorithm (b) uses a divide and conquer strategy and has a time complexity of O(n log k), which is a result of merging n elements over log k recursion levels.

Step-by-step explanation:

Analyzing the complexity of the given algorithms involves understanding how the operations scale with the size of the input, denoted by n for the total number of elements and k for the number of lists.

(a) Sequential Merge

Algorithm (a) merges each list Li one by one with an accumulating list L. The time complexity of merging two lists of size i and j is O(i + j). Initially, L is empty, and we add k lists one after the other. The complexity of this can be described as O(n) for each merge, repeated k times, resulting in O(nk).

(b) Divide and Conquer Merge

The divide and conquer approach splits the problem into merging two halves recursively. Each recursive call halves the number of lists until there is only one list left, which takes O(log k) splits. The final merge of each half has a complexity of O(n) since it merges two lists with a combined total of n elements. Since there are O(log k) levels of recursion, and the merge at each level costs O(n), the overall complexity is O(n log k).

Accordingly, the theta bound for the complexity of algorithm (a) is Θ(nk), and for algorithm (b) is Θ(n log k).