Question

Show that the recurrence equation for the worst-case time complexity for Mergesort (Algorithms \(2.2 \text { and } 2.4)\) is given by \\[ W(n)=W\left(\left[\frac{n}{2}\right\rfloor\right)+W\left(\left\lceil\frac{n}{2}\right]\right)+n-1 \\] when \(n\) is not restricted to being a power of 2

Short answer

The given recurrence equation \(W(n) = W(\left\lfloor\frac{n}{2}\right\rfloor) + W(\left\lceil\frac{n}{2}\right\rceil) + n - 1\) truly reflects the worst-case time complexity of the MergeSort algorithm. It accounts for the complexity of sorting each half and the process of merging them which takes at most \(n - 1\) comparisons.

Step-by-step solution

Understand MergeSort Algorithm

MergeSort is a recursive divide-and-conquer algorithm that splits a list into two halves, sorts them and merges the sorted halves. Its time complexity is given by the time it takes to split the array (which is constant regardless of the array's size since we are just finding the midpoint) plus the time to sort each half (each of which is a subproblem half the size of the original) plus the time to merge the two halves. This gives rise to a recurrence equation that represents the time complexity of MergeSort.

Explicit illustration of recurrence equation

We want to illustrate that the recurrence equation \(W(n) = W(\left\lfloor\frac{n}{2}\right\rfloor) + W(\left\lceil\frac{n}{2}\right\rceil) + n - 1\), where \(\lfloor\frac{n}{2}\rfloor\), denotes the floor of \(n/2\), and \(\lceil\frac{n}{2}\rceil\) denotes the ceiling of \(n/2\), represents the worst-case time complexity of MergeSort. MergeSort will divide the array into 2 parts of size \(\lfloor\frac{n}{2}\rfloor\) and \(\lceil\frac{n}{2}\rceil\). The worst-case time complexity for sorting each of these halves is denoted by \(W(\left\lfloor\frac{n}{2}\right\rfloor)\), and \(W(\left\lceil\frac{n}{2}\rceil)\) respectively. The process of merging these halves requires at most \(n - 1\) comparisons.

Validating the Recurrence Equation and Wrap Up

Each recursive call to MergeSort on a list of size \(n\) will end up in base cases of lists of size 1. Once those base cases are reached, the algorithm will begin merging each pair of sorted lists, using a maximum of \(n - 1\) comparisons per merge, and proceed until it reaches the top of the recursion tree where the initially divided list is now sorted. This is why we see the \(n - 1\) operation in our recurrence equation. This reflects the worst-case number of comparisons needed to merge the sorted lists at each level of recursion.

Key concepts

Worst-Case Time Complexity

When analyzing the efficiency of an algorithm, understanding its worst-case time complexity is crucial. For MergeSort, which is a popular sorting algorithm, the worst-case time complexity defines the maximum amount of time taken to sort a list in the least favorable scenario.
MergeSort's worst-case time complexity can be expressed through the recurrence relation:
\[W(n) = W\left(\left\lfloor\frac{n}{2}\right\rfloor\right) + W\left(\left\lceil\frac{n}{2}\right\rceil\right) + n - 1\]
This equation explains that the time it takes to solve a problem of size \(n\) involves:

• solving two subproblems of sizes roughly \(\lfloor n/2 \rfloor\) and \(\lceil n/2 \rceil\)
• and then an additional time of \(n - 1\) operations for merging.

In simpler terms, it captures splitting the list, solving the resulting half-sized problems, and combining the results. The term \(n - 1\) reflects the maximum number of comparisons needed during the merging phase at each recursion level.

Recurrence Relation

In understanding the behavior of recursive algorithms like MergeSort, recurrence relations are pivotal. They provide a way to formally express the time complexity of each recursive step.
For MergeSort, the recurrence is given by:
\[W(n) = W\left(\left\lfloor\frac{n}{2}\right\rfloor\right) + W\left(\left\lceil\frac{n}{2}\right\rceil\right) + n - 1\]
This equation essentially breaks down the computational process:

• \(W\left(\left\lfloor\frac{n}{2}\right\rfloor\right)\) and \(W\left(\left\lceil\frac{n}{2}\right\rceil\right)\) denote the time complexity of recursively sorting the two halves.
• The time \(n - 1\) accounts for comparing and merging the elements of the two halves.

This translates to a smaller, yet similar, version of the original problem being solved at each stage, which is a hallmark of recursive algorithms. When the subproblems reach their base cases, typically when the size becomes one, the solution compounds upward.

Divide and Conquer Algorithm

MergeSort exemplifies the divide and conquer strategy, a fundamental concept in computer science. This method works by breaking down a problem into smaller subproblems, solving the subproblems independently, and then combining their solutions to tackle the original problem.
In the context of MergeSort:

• **Divide:** The list is split into two roughly equal halves.
• **Conquer:** Each half is sorted recursively.
• **Combine:** The sorted halves are merged into a single sorted list.

This approach is powerful because it exploits the simplicity of sorting smaller lists. The merging step is efficient, requiring at most \(n - 1\) comparisons to unite the halves if its size was \(n\). Divide and conquer not only helps manage complex issues by simplifying them but also forms the backbone of efficient algorithms, like MergeSort, by transforming time-consuming operations into manageable computations.