Write an algorithm that searches a sorted list of n items by...

Write an algorithm that searches a sorted list of $n$ items by dividing it into three sublists of almost $n / 3$ items. This algorithm finds the sublist that might contain the given item and divides it into three smaller sublists of almost equal size. The algorithm repeats this process until it finds the item or concludes that the item is not in the list. Analyze your algorithm and give the results using order notation.

Key Concepts

Divide and Conquer

Divide and conquer is a strategy that involves breaking a problem into smaller subproblems, solving each subproblem independently, and then combining their solutions to solve the original problem. This approach is effective in reducing complex problems into manageable parts and is commonly used in the design of efficient algorithms.

Ternary Search

Ternary search is a search technique that divides the input into three parts and determines which part may contain the target value. This method is a generalization of binary search and, when applied to a sorted list, can locate a target element by recursively narrowing down the interval, resulting in a logarithmic time complexity.

Recurrence Relation

A recurrence relation expresses the running time of a recursive algorithm in terms of its running time on smaller inputs. For example, when a problem of size n is divided into one subproblem of size n/3 with additional constant work, the recurrence relation helps analyze the overall time complexity of the algorithm.

Big-O Notation

Big-O notation provides an upper bound on an algorithm's running time or space requirements, allowing the comparison of algorithm efficiency by focusing on the most significant factors. In this context, the algorithm's time complexity is described by Big-O notation, indicating its performance relative to the input size.

Transcript

00:01 So for this problem, we're just going to call this procedure ternary.

00:07 And ternary takes in a list of integers, so some n number of integers.

00:19 And there's at least going to need to be minimum of one of them.

00:25 All right.

00:25 And then we also have the search algorithm, the item we're searching for, which is also an okay, so first we want to define our search interval.

00:50 So we'll set i equal to 1 and j equal to n.

00:59 So i is going to be the position of the first element of our search interval and j will be the last element of our search interval.

01:09 So at the moment we've just initialized it to the entire list.

01:16 Okay, and so then we'll consider the case for while i is less than j minus one okay and so this ensures that the algorithm would stop when there's at most two elements in the search interval we also want to define a couple of new variables okay so lots of definitions here so let's let l lowercase l be equal to the floor of i plus j divided by three okay and we're going to do essentially the same thing with you okay so it's going to be the floor except it's going to be two times i plus j divided by three and so these are also setting positions in our list okay so then we can determine in which sub interval x lies right so if we think about all these variables if we have a list of so a1, all the way up to a .l.

03:02 It's going to go all the way up to au, and then up to a subj, right? so we're just sort of creating roughly evenly distributed intervals here...

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