Question

Show that the worst-case time complexity for Binary Search (Algorithm 2.1) is given by \(W(n)=[\lg n]+1\) When \(n\) is not restricted to being a power of \(2 .\) Hint. First show that the recurrence equation for \(W(n)\) is given by \(W(n)=1+W\left(\left\lfloor\frac{n}{2}\right\rfloor\right) \quad\) for \(n>1\) \(W(1)=1\) To do this, consider even and odd values of \(n\) separately. Then use induction to solve the recurrence equation.

Short answer

The worst-case time complexity for Binary Search is given by \(W(n)=[\lg n]+1\) when \(n\) is not restricted to being a power of 2. This has been proven through using a recurrence relation and proof by induction.

Step-by-step solution

Understand the Binary Search Algorithm

Binary search algorithm is an efficient search algorithm that finds the position of a target value within a sorted array. It compares the target value to the middle element of the array; if they are unequal, the half in which the target cannot be located is eliminated and the search continues on the remaining half, again taking the middle element to compare to the target value, and repeating this until the target value is found. If the search ends with the remaining half being empty, the target is not in the array.

Calculate the Worst-case Time Complexity

Worst-case time complexity is defined as the maximum amount of time taken by an algorithm to execute, and in case of Binary search, it occurs when the target value is the largest element in the array or doesn't exist in the array. Every comparison eliminates half the remaining possibilities. This makes the worst-case search time for Binary search \(\log n\) if n is a power of 2, and one additional step is required if \(n\) is not a power of 2.

Formulate the Recurrence Equation

The worst-case scenario for Binary Search can be expressed as a recurrence relation. For \(n>1\), we have the equation \(W(n)=1+W\left(\left\lfloor\frac{n}{2}\right\rfloor\right)\). This equation accounts for every time we divide the array in two (rounding down if \(n\) is not a power of two), and the '+1' accounts for the division step itself. For \(n=1\), we have the base case \(W(1)=1\). This accounts for the case when we have only one element left in the array.

Use Induction to Prove the Recurrence

To show that the given relation \(W(n)=[\lg n]+1\) solves the recurrence, we can use proof by induction. According to the induction hypothesis, assume the result holds true for all values less than \(n\).\n\nBase case: For \(n=1\), \(W(n)=W(1)=1\) which equals \([\lg 1] + 1 = 1\), so the base case holds true.\n\nInductive Step: Assuming the result holds for all values less than \(n\), we need to prove it for \(n\). We consider two cases- \(n\) is odd and \(n\) is even. \n\nWhen \(n\) is even, \(n=2k\), so \(\left\lfloor\frac{n}{2}\right\rfloor=k\). Thus, \(W(n)=1+W(k)=1+[\lg k] + 1 = [\lg 2k] + 1 = [\lg n] + 1\). \n\nWhen \(n\) is odd, \(n=2k+1\), so \(\left\lfloor\frac{n}{2}\right\rfloor=k\). Thus, \(W(n)=1+W(k)=1+[\lg k] + 1 = [\lg (2k)] + 1 < [\lg (2k+1)] + 1 = [\lg n] + 1\). \n\nIn both cases, the hypothesis holds, so the proof by induction is complete.

Key concepts

Algorithm Analysis

In computer science, Algorithm Analysis is a vital process for understanding how an algorithm behaves in terms of its time and space requirements. By analyzing algorithms, we assess their efficiency and compare them to alternative solutions. Binary Search, for instance, is often analyzed to determine its worst-case scenario behavior. This is fundamental for students and developers alike to ensure that the algorithm will meet performance expectations, especially with large datasets. By calculating its worst-case time complexity, we are essentially determining the maximum number of steps required to find an element in a sorted list, which we denote as O(log n).

Understanding this concept not only helps you grasp particular algorithm performance but also empowers you to make better decisions regarding which algorithms to employ based on the problem's constraints and the resources available.

Recurrence Relations

Recurrence Relations are equations used to define sequences recursively, where the next term is a function of the preceding terms. They play a significant role in algorithm analysis, especially for divide-and-conquer algorithms like Binary Search. The recurrence for Binary Search expresses how the problem size reduces at each step of the algorithm. In Binary Search, each step cuts the problem size in half until the size becomes 1. It’s crucial for proper understanding because it sets the stage for determining the time complexity of the algorithm with the aid of mathematical tools, illustrating how an algorithm scales with the input size. Students usually find the concept of recurrences challenging, but once mastered, it's a powerful tool in algorithm design and analysis.

Proof by Induction

Proof by Induction is a mathematical technique used to prove that a given property holds true for all natural numbers. Induction consists of two parts: the base case and the inductive step. You verify the property for the initial number (usually 1), known as the base case. Then, you assume the property holds for some natural number, and demonstrate that if it's true for that number, it's also true for the next one – that's the inductive step. This approach can confirm, for instance, that our formula for the Binary Search time complexity holds for all possible input sizes, not just those easily computed or observed. It's a crucial step in closing the argument and ensuring the recurrence relation we formulated truly represents the Binary Search's worst-case scenario.

Logarithmic Time Complexity

The term Logarithmic Time Complexity, denoted as O(log n), refers to an algorithm where the time to complete the task is proportional to the logarithm of the input size. Logarithmic complexity is highly efficient and is characteristic of algorithms that divide the problem space in half each step, like Binary Search. To put it in practical terms, even if the size of the sorted list increases massively, the number of steps needed to find an element or conclude it’s not present won’t increase all that much. For students, grasping the concept of logarithmic time complexity can sometimes be the difference between writing an okay program and one that's extremely efficient and scalable. It's a concept you will encounter repeatedly in more advanced data structure and algorithm courses as well as in real-world programming scenarios.