Question

The American Museum of Natural History in New York City contains more than 32 million specimens and artifacts in its warious collections, including the world's langest collection of dinosaur fossils. Many of these are in storage away from public view, but all must be carefully inventoried. a. Suppose the inventory is unordered (I) and a sequential search is done to locate a specific artifact. Given that the search is executed on a computer that can clo 12,000 comparisons per second, about how much time on the aNerage would the search require? b. Assuming the inventory is sorted, about how much time would a binary search require?

Short answer

Sequential search takes about 1,333.33 seconds; binary search takes about 0.00208 seconds.

Step-by-step solution

Understanding Sequential Search

A sequential search involves going through each item one by one, which makes it O(n) in complexity. This means that, on average, you will have to check half of the items (since the desired item could be anywhere) to find the specific artifact.

Calculating Sequential Search Comparisons

With 32 million items, the average number of comparisons needed is half of them. Therefore, the comparisons will be: \( \frac{32,000,000}{2} = 16,000,000 \) comparisons.

Calculating Time for Sequential Search

The computer can do 12,000 comparisons per second. Thus, the time required for the average case is: \( \frac{16,000,000}{12,000} \approx 1,333.33 \) seconds.

Understanding Binary Search

In a binary search, the list is repeatedly divided in half, which gives it a time complexity of O(log n). This is much more efficient as fewer comparisons are needed.

Calculating Binary Search Comparisons

In a list of 32 million items, the number of comparisons needed is \( \log_2(32,000,000) \). Using the logarithm, we find \( \log_2(32,000,000) \approx 25 \) comparisons.

Calculating Time for Binary Search

Since each comparison takes the same amount of time as in the sequential search, the total time for 25 comparisons is: \( \frac{25}{12,000} \approx 0.00208 \) seconds.

Key concepts

Sequential Search

Sequential search, also known as linear search, is a simple searching algorithm that checks each element one by one until the desired element is found. It doesn't require the list to be sorted. However, its simplicity comes with a downside. The time complexity is directly proportional to the number of elements, denoted as O(n), where "n" is the total number of elements in the list. This means that as the list grows, the time it takes to search grows linearly.

In practical terms, if you have an unordered inventory with 32 million items, on average, 16 million comparisons are needed to find one specific item. Assuming a computer that performs 12,000 comparisons per second, it would take about 1,333.33 seconds to complete the search. This is because on average, you'll be looking through half of the items before you find your target.

Although it's not the most time-efficient, sequential search is useful when dealing with small datasets or unsorted data, as it requires no setup or preprocessing like data sorting.

Binary Search

Binary search significantly increases efficiency when working with sorted lists or datasets. Unlike sequential search, binary search reduces the number of comparisons needed by repeatedly dividing the searchable range in half. This creates a more efficient process characterized by a time complexity of O(log n).

To visualize binary search, imagine starting in the middle of a sorted list. If the middle item isn't what you're looking for, you can determine which half of the list could still contain the target based on whether the target is greater or lesser than the middle item. This action effectively cuts your problem size in half with each step.

For a list of 32 million items, the binary search only needs about 25 comparisons to find an item, which is just a fraction of what sequential search required. Assuming you have the same processing power as before—12,000 comparisons per second—it would take approximately 0.00208 seconds. This is due to its logarithmic nature, which makes binary search immensely powerful for large datasets.

Search Algorithm Efficiency

The efficiency of search algorithms is critical when dealing with large datasets. It directly affects how quickly and effectively a program can access data. Two important factors to consider when selecting a search algorithm are the dataset's size and whether it is sorted.

Sequential search, with its linear time complexity, is simple but not efficient for large datasets. It does not require the data to be sorted and can be useful for smaller lists.

• Advantages: No need for pre-sorting, simpler to implement.
• Disadvantages: Slow for large datasets, inefficient as the list size increases.

Binary search, on the other hand, offers a logarithmic time complexity, making it much more efficient for large, sorted datasets. As the number of elements increases, the difference in time efficiency between sequential and binary search becomes more pronounced.

• Advantages: Fast, especially for large datasets, efficient use of resources.
• Disadvantages: Requires data to be sorted, a bit more complex to implement.

Choosing the right search method involves weighing these factors and considering the specific requirements of your application or problem. Efficient search algorithms save time, computational power, and ultimately improve the performance of software applications.