Question

You are given a sequence of lists of words, representing the pages of a book. Your task is to build an index (a sorted list of words), each clement of which has a list of sorted numbers representing the pages on which the word appears. Describe an algorithm for building the index and give its big-Oh running time in terms of the total number of words.

Short answer

An algorithm involves collecting, sorting pages per word, sorting words, and combining them. It's O(n log n).

Step-by-step solution

Collect All Words with Page Numbers

Start by scanning each page of the book, represented as a list of words, to collect information. For every word encountered, check if it already exists in a dictionary (or hash map). If it doesn't exist, create a new entry with the word as the key and initialize a set to store pages numbers. If it does exist, simply add the current page number to the set associated with that word. The use of a set ensures that each page number is stored uniquely even if the word appears multiple times on the same page.

Sort the Pages for Each Word

Once you have collected all the word-page associations, iterate over each word in the dictionary. Convert the set of page numbers into a sorted list. This step involves sorting the unique page numbers for each word, which typically takes O(k log k) time for k pages associated with a word.

Sort All Words Alphabetically

Extract all the words from the dictionary keys and sort them alphabetically. This sorting step ensures that the final index is ordered by the words. Sorting n words typically takes O(n log n) time.

Compile the Final Index

Create the final output by combining the sorted words and their associated sorted lists of page numbers. This can be done by iterating through the sorted list of words and appending each word along with its sorted list of page numbers from the dictionary.

Determine the Big-Oh Complexity

The overall complexity involves scanning through all words, inserting page numbers, sorting pages for each word, and sorting the entire list of words. The dominant operation is the sorting of words and pages. The complexity for scanning and inserting is closest to linear time, O(n), because inserting into a set is O(1) on average. Thus, the sorting part dictates the complexity, leading to an overall complexity of O(n log n), where n is the total number of words in the book.

Key concepts

Data Structures

Data structures are essential tools for organizing and storing data, allowing efficient data retrieval and manipulation. In the problem of building an index from a book, we utilize a dictionary, also known as a hash map, to store each word along with the pages it appears on.

• The dictionary holds each word as a key.
• The value for each key is a set of pages where the word is found.

Using a set is clever for two reasons:
1. It automatically handles duplicates, ensuring that if a word appears multiple times on the same page, it is counted only once.
2. It provides efficient time complexity for insertion and checking membership—specifically, average O(1) time complexity due to the underlying hash map structure. This is how data structures like dictionaries and sets simplify the problem-solving process.

Hash Maps

Hash maps, or dictionaries in Python, allow for fast access to data using keys. These are incredibly effective for scenarios where you need to store and quickly look up values associated with unique keys, such as building an index.

• They use a hash function to compute an index into an array of buckets or slots, from which the desired value can be found.
• Inserting and retrieving data often takes constant time, O(1), making them excellent for applications requiring fast lookups, like counting occurrences or grouping data.

In our index-building task, every word serves as a key in our hash map. A set or list of page numbers is the value associated with each key. This allows efficient storage and retrieval of page numbers for any given word.

Big-Oh Notation

Big-Oh notation is a mathematical concept used to describe the upper bound of an algorithm's runtime, giving us an idea of its efficiency as input size grows. In our problem, the Big-Oh notation helps us evaluate the algorithm's performance for creating a book index.

• We consider the time complexity of scanning each word and inserting page numbers. This is approximately O(n), where n is the total number of words in the book.
• The most time-consuming parts are sorting operations. Sorting the list of words and sorting page numbers for each word, both of which generally operate in O(n log n) time.

Thus, the overall complexity is dominated by the sorting operations, leading to an effective time complexity of O(n log n), which is efficient for most practical purposes.

Sorting Algorithms

Sorting is fundamental in computing for organizing data. In the context of our index-building task, we perform sorting in two critical areas:

• Sorting page numbers for each word ensures our lists are well-ordered and easy to navigate.
• Sorting the entire list of words alphabetically is essential for creating a readable and user-friendly index.

Standard sorting algorithms, such as quicksort and mergesort, typically operate in O(n log n) time. This is because they divide the problem into smaller segments (divide and conquer approach) and are very efficient with larger datasets.
Implementing these sorts not only provides order and aesthetics but also enhances readability and searchability of the final index.