Add Bold Tag in String

Identifying Problem Isomorphism

“Add Bold Tag in String” requires string manipulation and recognition of certain substrings within a larger string. The idea is to find certain words within a given string and then add HTML bold tags around the sections of the string that should be bolded.

A problem that is similar in its approach is “Merge Intervals”. Now, it might not seem directly related as it’s not a string problem, but the concept of merging overlapping intervals is somewhat akin to the process of determining which sections of the string should be bolded. In “Add Bold Tag in String”, you determine overlapping ‘bold’ areas and merge them, while in “Merge Intervals”, you’re merging overlapping intervals.

On the more complex side, we have “Insert Interval”. This problem is a bit more complicated because you need to find the correct spot to insert a new interval, and then also merge any intervals that overlap as a result.

The reasoning behind this selection:

“Merge Intervals” and “Add Bold Tag in String” both require the ability to identify and handle overlapping segments, whether it’s time intervals or substrings within a string.
“Add Bold Tag in String” adds a layer of complexity with the requirement of recognizing words within a string.
“Insert Interval” adds further complexity by requiring the correct placement of a new interval and handling subsequent merges.

So, in increasing order of complexity:

“Merge Intervals”
“Add Bold Tag in String”
“Insert Interval”

This mapping is approximate. The specific coding techniques required to solve each problem may differ, but there are similarities in the underlying concepts of overlap and merging.

10 Prerequisite LeetCode Problems

This involves both string manipulation and string searching techniques. Here are 10 problems to prepare:

28. Implement strStr(): This problem asks you to implement the basic string searching algorithm, which is helpful for understanding how to find substrings in a string.
459. Repeated Substring Pattern: This problem involves identifying repeated substrings within a string, which is related to the task of identifying words within the input string in the target problem.
344. Reverse String: This problem involves basic string manipulation, which could be helpful for understanding how to manipulate strings in general.
14. Longest Common Prefix: Finding the longest common prefix between several strings could help with the understanding of how to compare strings and find commonalities.
387. First Unique Character in a String: This problem is about identifying unique characters in a string, which is a more simple version of finding unique substrings or words in a string.
20. Valid Parentheses: While not directly related, the concept of using a stack to track open and closed tags is somewhat analogous to tracking open and closed parentheses in this problem.
76. Minimum Window Substring: Understanding the sliding window technique in this problem could be useful for identifying words within the input string in the target problem.
125. Valid Palindrome: This problem could help in understanding how to traverse strings in different ways, which might be useful in the target problem.
680. Valid Palindrome II: This problem extends on the previous one and includes more complex string manipulation.
58. Length of Last Word: This is a simpler problem about finding words within a string.

These cover string manipulation and string searching, which will be useful for the “Add Bold Tag in String” problem.

Clarification Questions

Here are some questions we could ask for clarification:

What should be the output if the string s or the list words is empty?
How should we handle case sensitivity? For example, if a word in words is “apple” and in s we have “Apple”, should this be considered a match?
Are there any restrictions on the characters that can appear in s and words? The problem mentions English letters and digits, but can we expect punctuation, whitespace, or special characters?
In the case where multiple words in words can match the same substring in s, how should this situation be handled? For example, if words contains “ab” and “abc”, and s is “abc”, should we bold the entire “abc” or just “ab”?
What should be the output if the same word appears in words multiple times?
How should we handle situations where one word in words is a substring of another word in words? For example, if words contains both “abc” and “bc”, should both be highlighted in the string if they appear?

Problem Analysis and Key Insights

The key insights from analyzing the problem statement include:

Substrings in s that exist in words need to be wrapped in bold tags: The problem is essentially about identifying those substrings in s that are present in words and wrapping them in bold tags  and .
Overlapping substrings should be wrapped together: If two or more substrings that should be wrapped are overlapping, they need to be considered as one single unit and wrapped together. This suggests that we need to keep track of the positions in s that need to be wrapped, and combine consecutive or overlapping positions.
The order of wrapping is important: If two substrings that need to be wrapped are consecutive, we need to wrap them together as well. This indicates that the process of wrapping the substrings is not independent, but is influenced by the positions of the other substrings to be wrapped.
The final output is a string: The output of the problem is a modified version of the input string s. We need to construct this string based on the positions identified for wrapping.
All the values of words are unique: This suggests that we don’t need to handle duplicates in words.

These insights guide us towards a solution where we need to scan the string s, identify the positions that need to be wrapped based on words, combine overlapping or consecutive positions, and finally construct the output string by inserting the bold tags at the appropriate positions.

Problem Boundary

The scope of this problem is to create an algorithm that can:

Take a string s and an array of strings words as inputs.
Identify all substrings in s that exist in words.
Add a pair of bold tags ( and ) to wrap those identified substrings in s.
If two such substrings overlap, wrap them together with only one pair of closed bold-tag.
If two substrings wrapped by bold tags are consecutive, combine them.
Return the modified string after adding the bold tags.

The algorithm needs to handle all cases within the given constraints, including edge cases. The constraints include s and words[i] consisting of English letters and digits, words containing unique values, and the lengths of s and words[i] being within the defined limits.

The algorithm doesn’t need to handle cases outside of these constraints. For example, it doesn’t need to handle cases where s or words[i] contain special characters, or words contains duplicate values.

The boundaries of this problem are established by the constraints given in the problem statement. Specifically:

The string s and each string in the words array consist only of English letters and digits. This means we don’t have to consider special characters, spaces, or non-English alphabets.
The length of s is between 1 and 1000, inclusive. This means we don’t have to handle empty strings or strings longer than 1000 characters.
The length of the words array is between 0 and 100, inclusive. This means we don’t have to handle cases where the words array is larger than 100.
Each string in the words array has a length between 1 and 1000, inclusive. This means we don’t have to handle empty strings or strings longer than 1000 characters in words.
All the values of words are unique. This means we don’t have to consider situations where there are duplicate words in words.
We only need to add bold tags around substrings of s that exist in words. Other formatting requirements are not within the scope of this problem.
If two substrings that need to be wrapped by bold tags overlap or are consecutive, they should be combined into a single pair of bold tags. This implies that we cannot have nested or consecutive bold tags in the output.

By considering these constraints, we can establish the boundaries of the problem. Any inputs that fall outside these constraints are not within the problem’s scope and we do not need to account for them in our solution.

Problem Classification

This problem belongs to the domain of “String Processing”. The main task is to perform certain operations on the input string based on the given words.

‘What’ Components:

A string ’s’.
An array of strings ‘words’.

The goal is to add HTML bold tags (’’ and ‘’) around the substrings in ’s’ that match any string in ‘words’. If two such substrings overlap or are consecutive, they should be wrapped together with a single pair of bold tags.

This problem could be further classified as a “String Matching” problem. It requires us to identify occurrences of certain substrings (the words in ‘words’) within the main string ’s’. The operation of adding HTML tags based on these matches adds an extra layer of complexity to the problem.

Distilling the Problem to Its Core Elements

Fundamental Concept: This problem is essentially a string manipulation task that relies on substring matching. The main principle is to identify all occurrences of a set of specified substrings within a larger string, and then mark these occurrences in some way—in this case, by adding HTML bold tags.
Simplest Description: Imagine you’re reading a book and you want to highlight certain words. You have a list of words that you’re interested in, and you want to go through the book and highlight those words wherever they appear. But, instead of using a highlighter, you’re adding HTML bold tags around the words to make them stand out.
Core Problem: The key problem is to identify all instances of the given words within the string, even if they overlap or are consecutive, and then add HTML bold tags around them in the correct manner, ensuring that overlapping or consecutive occurrences are enclosed within a single pair of tags.
Key Components:
- Identify all instances of each word within the string.
- Handle overlapping and consecutive instances.
- Add HTML bold tags around each instance or group of instances.
Minimal Set of Operations:
- Scan the string for occurrences of each word in the list.
- Mark the start and end of each occurrence.
- Merge overlapping or consecutive marked sections.
- Insert HTML bold tags at the start and end of each final marked section.

Visual Model of the Problem

To visualize this problem, we can imagine the input string as a sequence of characters on a line. Each character has a specific position in this sequence.

Suppose our input string is “abcxyz123” and our words are [“abc”,“123”].

We can represent the string and the words like this:

abcxyz123
|   |   |
abc 123

The vertical bars denote the start and end of the words in the string.

The problem is asking us to put HTML bold tags around the words in the string. This would look like this:

<b>abc</b>xyz<b>123</b>

Now, imagine if the words were overlapping or consecutive. For example, if the string is “aaabbb” and the words are [“aa”,“b”]. We would initially identify the words like this:

aaabbb
||  | |
aa  b

However, since these marked sections are overlapping or consecutive, we need to merge them and wrap them together:

<b>aaabbb</b>

This visual representation can help us understand how to approach the problem. We need to find a way to identify the start and end of each word in the string, merge overlapping or consecutive sections, and then insert the HTML tags at the appropriate positions.

Problem Restatement

We’re given a string and a list of words. We need to go through the string and for each substring that matches one of the words in the list, we need to wrap it with the HTML bold tags  and . If there are any overlapping or adjacent bolded sections, we need to merge them into a single bolded section. The goal is to return the modified string.

A couple of things to note are that all the words are unique, and both the string and the words consist only of English letters and digits. The length of the string and each word can be up to 1000 characters, and there can be up to 100 words.

The primary challenge here is to figure out how to identify the words in the string and handle overlapping or adjacent bolded sections correctly.

Abstract Representation of the Problem

We are given a sequence of characters (MainSequence) and a set of unique sequences (SubSequences). Our task is to mark certain sections of the MainSequence based on whether they match any of the SubSequences. If these marked sections overlap or are adjacent, they should be consolidated into a single marked section.

The marking is done by adding a special symbol at the beginning and end of the matching section in the MainSequence. After traversing the entire MainSequence and marking appropriate sections, we return the modified MainSequence.

In this abstraction, we’ve removed the specific details of strings, words, and HTML tags, and instead used more general terms like MainSequence, SubSequences, and special symbols. This highlights the underlying structure and elements of the problem and helps us focus on the core task of identifying, marking, and consolidating matching sections.

Terminology

There are a few fundamental concepts that are important for understanding the problem and its solution:

Substring: A substring is a contiguous sequence of characters within a string. For example, “abc” is a substring of “abcxyz”.
Overlap: In this context, overlap refers to two or more substrings that share some common characters. For example, the substrings “abc” and “bcd” overlap in the string “abcd” because they both include the characters “bc”.
Adjacent: Two substrings are adjacent if they occur next to each other with no intervening characters. For example, “abc” and “def” are adjacent in the string “abcdef”.
HTML tags: HTML tags are used to mark up text in HTML documents. The  and  tags in particular are used to make text bold.
String Matching: String matching is the process of finding one or more occurrences of a smaller string (the “pattern”) within a larger string. This is a core concept in many string processing algorithms.

These concepts are crucial to understanding the problem, which involves identifying substrings (from the words array) within the main string, dealing with overlaps and adjacencies, and marking them with HTML tags.

Problem Simplification and Explanation

Let’s imagine that the string s is a road and the words in the words array are different types of cars. The problem asks us to find out where each car (word) is parked on the road (the string).

However, there are a few special rules for parking:

If two cars overlap, meaning they occupy some of the same space on the road, they need to be wrapped together in a protective cover (the bold tags). This is like saying if two words share some of the same characters in the string, they should be enclosed within the same pair of bold tags.
If two cars are parked right next to each other, they also need to be covered together in a single protective cover. So, if two words are adjacent to each other in the string, they too should be enclosed within the same pair of bold tags.

At the end of the process, we should be able to see all the cars (words) that are parked on the road (the string), each of them either individually wrapped or grouped together with others in a protective cover (bold tags), according to the rules.

The key concepts involved here are string matching (finding where each car is parked), handling overlaps (grouping overlapping cars together), and handling adjacencies (grouping adjacent cars together). The interaction between these concepts is what makes this problem a bit challenging.

Constraints

Here are some characteristics that we can exploit to devise an efficient solution:

All Values of Words are Unique: This means that we don’t need to check for duplicate words in the words array. This can save us from unnecessary computations.
Words and s Consist of English Letters and Digits: This means that we don’t need to handle special characters or different languages. This simplifies the pattern matching task.
Length Constraints: The lengths of s and words are within a reasonable range (1 to 1000), which means we can use methods that scale with the size of the input, such as iteration or string manipulation.
Bold Tags Need to be Closed: The problem states that we need to close the bold tags once we start them. We can use this information to keep track of whether we’re currently in a bold section or not, which can simplify the logic of when to insert tags.
Merging Overlapping and Consecutive Tags: The problem requires us to merge overlapping and consecutive bold tags. This condition allows us to simplify our final string by reducing the number of opening and closing tags we need to include.

By recognizing and leveraging these constraints, we can create an optimized solution that efficiently finds and marks all instances of the words in s.

The constraints of the problem provide the following insights:

Unique Words: As every word in the words array is unique, we don’t need to account for any duplicates when scanning the string s. This simplifies our problem as we know every word in words will only need to be considered once.
Character Limitations: The characters in s and words are limited to English letters and digits. This means we don’t need to handle any special characters or whitespace, simplifying the pattern matching and string manipulation tasks.
Length Restrictions: The size of the string s and the words array are bounded. This means that any solution that scales with the size of the inputs will be reasonable and should execute within an acceptable time frame.
Bold Tag Rules: The rules for applying the bold tags, such as the need to close them and the requirement to merge overlapping and consecutive tags, help guide the design of our solution. We know that we’ll need to keep track of whether we’re within a bold section and will need to merge sections as necessary.

Overall, these constraints and the insights they provide help to guide our solution approach and ensure that it will be both correct and efficient.

Case Analysis

Let’s consider some additional examples and edge cases:

Minimum Input Case:
- Input: s = "a", words = ["a"]
- Output: "a"
- Here, the string s and the words array both contain the minimum possible values as per the problem’s constraints. The single character in s is present in words, so it should be enclosed within bold tags.
Case with No Matches:
- Input: s = "abc", words = ["xyz"]
- Output: "abc"
- In this case, none of the words in words are present in s, so the output should be the same as the input.
Case with Partial Overlapping Words:
- Input: s = "abcdef", words = ["abc", "cde"]
- Output: "abcdef"
- Here, the words “abc” and “cde” both occur in s and overlap at “c”. Hence, they should be merged into a single bold tag.
Case with Non-Overlapping Words:
- Input: s = "abcdef", words = ["abc", "def"]
- Output: "abcdef"
- The words “abc” and “def” are present in s but don’t overlap, so they should be enclosed in separate bold tags.
Case with Overlapping and Non-Overlapping Words:
- Input: s = "abcdefabc", words = ["abc", "cde", "abc"]
- Output: "abcdefabc"
- Here, the words “abc” and “cde” overlap and are enclosed within a single bold tag. The second occurrence of “abc” is not overlapping with any other word and is enclosed within a separate bold tag.
Case with All Characters in Words:
- Input: s = "abc", words = ["a", "b", "c"]
- Output: "abc"
- All characters of s are present in words, so the entire string s should be bolded.

These cases cover a wide range of scenarios and help ensure that the solution correctly handles edge cases, overlapping and non-overlapping words, and instances where no words are found in s.

Visualizing these cases can help better understand the problem and how the different situations interact. Let’s consider the provided examples and visualize the strings and the words:

Minimum Input Case:
- s = "a", words = ["a"]
- Here, the single character in s is exactly the same as the single word in words. So we can visualize it as follows:
```
a
|
v
[a]
```
Case with No Matches:
- s = "abc", words = ["xyz"]
- In this case, none of the words in words are present in s. The visualization would be:
```
abc
|
X
[xyz]
```
Case with Partial Overlapping Words:
- s = "abcdef", words = ["abc", "cde"]
- Here, the words “abc” and “cde” both occur in s and overlap at “c”. The visualization could look like this:
```
abcdef
|   |
v   v
[abc, cde]
```
Case with Non-Overlapping Words:
- s = "abcdef", words = ["abc", "def"]
- The words “abc” and “def” are present in s but don’t overlap. The visualization would be:
```
abcdef
|   |
v   v
[abc, def]
```
Case with Overlapping and Non-Overlapping Words:
- s = "abcdefabc", words = ["abc", "cde", "abc"]
- Here, the words “abc” and “cde” overlap and the second occurrence of “abc” is not overlapping with any other word. The visualization could be:
```
abcdefabc
|   |   |
v   v   v
[abc, cde, abc]
```
Case with All Characters in Words:
- s = "abc", words = ["a", "b", "c"]
- All characters of s are present in words. The visualization would be:
```
abc
| | |
v v v
[a, b, c]
```

In these visualizations, the arrows indicate the presence of the words in the string s. This helps in understanding how the words are located in s and how they may overlap or exist independently.

Analyzing these cases provides several key insights:

Overlap Handling: When two words overlap in the string (as in the ‘Partial Overlapping Words’ case), they should be wrapped together in a single pair of bold tags. This insight is critical for handling cases where there’s an overlap, and it suggests that we need a way to track and merge overlapping regions.
Non-Overlap Handling: When the words do not overlap (as in the ‘Non-Overlapping Words’ case), they should each be wrapped separately in bold tags. This insight tells us that non-overlapping words must be treated independently.
Full Coverage: When every character of the string is in the list of words (as in the ‘All Characters in Words’ case), the entire string should be wrapped in bold tags. This insight could help us optimize the solution for situations where all characters are in the words list.
Repetition Handling: If a word appears more than once in the string (as in the ‘Overlapping and Non-Overlapping Words’ case), each occurrence should be considered independently. This insight suggests that our solution should handle repetitions correctly, regardless of whether they overlap with other words.
No Match Handling: When none of the words appear in the string (as in the ‘No Matches’ case), the string should remain unchanged. This insight could be useful for quickly handling cases where there’s no match and returning the string as it is.
Minimum Input: In the minimum input case, the single character string matches the single word. This insight emphasizes the need to consider the possibility of having only one character in the string and a single word in the words list, and ensuring that the solution handles this scenario correctly.

These insights play a crucial role in guiding the development of a comprehensive and efficient solution to the problem.

Identification of Applicable Theoretical Concepts

There are a few concepts that can simplify this problem:

String Matching: This problem involves finding occurrences of multiple words within a given string. This is a common problem in computer science and there are multiple algorithms to handle it, such as Rabin-Karp or Knuth-Morris-Pratt (KMP). However, given the constraints of this problem, a simple linear scan can work effectively.
Interval Merging: The problem involves finding regions in the string that match the words and potentially merging these regions if they overlap. This is similar to the classical interval merging problem, where you’re given a collection of intervals and the goal is to merge all overlapping intervals.
Boolean Array or List: We can use a boolean list to mark the positions in the string that match any word. This simplifies the problem by allowing us to identify and handle overlapping and non-overlapping regions more easily.
Two Pointers Technique: We could use this technique to scan the string and apply the bold tags. One pointer can indicate the start of a region to be bolded, and the other can mark the end of this region. This technique helps in efficiently traversing the string and applying the bold tags where necessary.
Prefix and Suffix Checking: When creating the final string, we need to ensure that we only keep the first five and last five characters of a bolded region, if the region’s length exceeds 10. This is essentially a problem of checking prefixes and suffixes, which is a common operation on strings.

By applying these concepts, we can develop a more efficient solution to the problem.

Simple Explanation

Imagine you’re working on a school project where you have to highlight certain words in a book. You have a list of words that need to be highlighted, and you’ve been given a marker to do so.

Now, there are a few rules you need to follow:

If a word from your list appears in the book, you need to highlight it.
If two words from your list overlap or are adjacent to each other in the book, you must highlight them together as if they were one word.
Because your marker is quite wide, if the highlighted area becomes longer than 10 letters, you should only fully highlight the first five letters and the last five letters of that area. The middle part should be left unhighlighted.

Once you’ve finished highlighting, you will represent the highlighted areas in a special way: you’ll write down the highlighted letters and, at the end, note down the total length of the area you’ve highlighted, even if some of the middle part was left unhighlighted due to the marker’s width.

The task here is to read through the book and highlight the words from your list following these rules. In the end, you’ll have a version of the book’s content with the important words highlighted and the highlighting length noted down.

Problem Breakdown and Solution Methodology

Let’s take a step-by-step approach to solve this problem.

Identifying Words to Highlight: The first step in our process is to scan through the input string and identify any substrings that match the words in our list. We can think of this like using a magnifying glass to scrutinize each word in our string and see if it matches any of our target words.
Marking Words for Highlighting: After identifying the words, we need to mark them for highlighting. We can imagine this as using a pencil to lightly underline the words we want to highlight. However, instead of using a pencil, we will use a boolean array of the same length as the input string. Each element of this array will be set to True if the corresponding character in the string is part of a word to be highlighted, and False otherwise.
Applying Highlighting Rules: Now that we have marked the words to be highlighted, we need to apply the rules for highlighting. This is similar to going back with our marker and making the pencil marks permanent. We need to handle overlapping or consecutive words and highlight them together. To do this, we can iterate through the boolean array and whenever we transition from False to True, we add a , and when we transition from True to False, we add a  to our result string.
Generating the Final String: The last step is to generate our final string. We do this by iterating through the input string and the boolean array together. If a character is part of a word to be highlighted (the corresponding element in the boolean array is True), we add it to the result string as is. If it’s not part of a word to be highlighted, we still add it to the result string, but without any highlighting tags.

Let’s consider an example:

s = "abcxyz123"
words = ["abc", "123"]

The step-by-step process would look like this:

We scan through s and identify “abc” and “123” as words to highlight.
We create a boolean array: [True, True, True, False, False, False, True, True, True].
We apply our highlighting rules and get: abcxyz123.

Changes in the problem’s parameters (the string s or the words array) would result in different words being highlighted and different final strings. For instance, if we added “xyz” to our words array in the above example, our final string would become abcxyz123, as “abc”, “xyz”, and “123” are all consecutive words to be highlighted.

Inference of Problem-Solving Approach from the Problem Statement

Let’s identify the key terms that guide our approach to solving this problem:

String: In this problem, we are given a string s and an array of words. Strings are one of the fundamental data types that we need to be comfortable with. Our main task is to manipulate and traverse the string s in order to solve the problem. This informs our approach that we will need to iterate through s and perform some operations based on its characters.
Array/List of Words: The problem also provides us with a list of words. These are the words that we need to identify and highlight in our main string. The fact that these words are provided in an array tells us that we will need to use some sort of searching or matching technique to find these words in our main string.
Substrings: The words that we need to highlight in s are actually substrings of s. This concept of substrings is critical to our approach because it tells us that we need to look at portions of s and not just individual characters.
Bold Tags: The problem asks us to add HTML-like bold tags ( and ) around the words we need to highlight. This indicates that we need to modify the original string s and create a new string that includes these tags.
Overlap and Consecutiveness: The problem mentions that if two substrings to be highlighted are overlapping or consecutive, they should be wrapped together. This informs us that we need to keep track of these conditions as we iterate through the string.
Boolean Array: We use a boolean array to mark the characters that belong to a word in s that needs to be highlighted. This concept of using a separate data structure to keep track of certain properties of the elements in our main string is a common technique in many problems and is a key part of our approach here.

Let’s use diagrams to represent some of the key concepts and properties in this problem:

String s and List of Words: We can represent the string s as a sequence of characters and the list of words as a sequence of strings.

s = "abcxyz123"

words = ["abc", "123"]

Substrings: Each word we need to highlight is a substring of s. We can visualize this by underlining the corresponding section of s for each word.

s = "abcxyz123"
      ___   ___
words = ["abc", "123"]

Bold Tags: When we add bold tags around the words to be highlighted, we’re modifying s. We can visualize this by showing s after the bold tags have been added.

s = "<b>abc</b>xyz<b>123</b>"

Overlap and Consecutiveness: If two words to be highlighted are overlapping or consecutive, they should be wrapped together. We can visualize this with an example. Let’s say s = "aaabbb" and words = ["aa", "b"]. Here, “aa” and “b” overlap and are consecutive.

s = "aaabbb"
     __  _  _  _
words = ["aa", "b"]
After adding bold tags separately:
s = "<b>a<b>a</b>a</b><b>b</b><b>b</b><b>b</b>"
After merging:
s = "<b>aaabbb</b>"

Boolean Array: We can visualize the boolean array as a table with the characters of s on one axis and a boolean value (True/False or 1/0) on the other axis. Each entry in the table indicates whether the corresponding character should be highlighted.

s = "abcxyz123"
bool_array = [1, 1, 1, 0, 0, 0, 1, 1, 1]

This indicates that the characters "abc" and "123" should be highlighted.

These visualizations should help you understand the properties of the problem and how we can use them to guide our approach.

How did you infer from the problem statement that this problem can be solved using ?

Simple Explanation of the Proof

I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?

Let’s break down our approach to this problem:

Recognize Substrings: The first step is to recognize all the substrings in s that match any word in the words array. For every word in words, we can iterate over s to find all occurrences of that word. This can be done using a sliding window approach, where the size of the window is equal to the length of the word. We slide the window over s, and at each step, we compare the window’s content with the word.
Build Boolean Array: Once we’ve identified all occurrences of each word in s, we can build a boolean array of the same length as s. For each character in s, we set the corresponding entry in the boolean array to True if that character should be highlighted, and False otherwise.
Merge Overlapping and Consecutive Highlights: Next, we need to merge any overlapping or consecutive True values in the boolean array. We can do this by iterating over the array and keeping track of the start and end points of each continuous block of True values.
Add Bold Tags: Finally, we can construct the output string by iterating over s and the boolean array together. Whenever we encounter a True value in the boolean array, we add a “” tag to the output. Whenever we encounter a False value immediately after a True value, we add a “” tag to the output.

These steps represent a refined, granular breakdown of our approach to this problem. The task of identifying substrings can be performed independently for each word in words. Within each step, there are repeatable patterns, such as sliding a window over s or iterating over the boolean array.

Solution Approach and Analysis

Let’s break this problem down into manageable steps:

Identify Matches: The first step is to iterate over the string s and for each position, check if a substring of s starting at that position matches any word in words. If there’s a match, we note the start and end positions of the match in s. This can be imagined as highlighting the matched words in a physical copy of s with a highlighter.
Create Boolean Array: Next, we create a boolean array, mark, with the same length as s. For each position in s that’s part of a match, we set the corresponding entry in mark to True. This array can be thought of as a series of flags, indicating whether each character in s is part of a highlighted (matched) section or not.
Build Output String: Now, we build the final output string. We iterate over s and mark together. Whenever we see a False-to-True transition in mark, we add a “” tag to the output. Whenever we see a True-to-False transition, we add a “” tag. In between transitions, we simply copy characters from s to the output. This process is analogous to going through the marked-up physical copy of s and adding opening and closing bold tags at the start and end of each highlighted section.

Let’s illustrate this process with an example:

Suppose s = "aaabbb" and words = ["aa", "b"].

After step 1, we have identified all matches of words in s and noted their start and end positions. For “aa”, the positions are (0, 1) and (1, 2). For “b”, the positions are (3, 3), (4, 4), and (5, 5).
In step 2, we construct the mark array. For the given s and words, mark would be [True, True, True, True, True, True] because every character in s is part of a match.
In step 3, we build the final output string. Since all characters in s are part of a match, the output is “aaabbb”.

If the words array changes, it would affect the matches found in step 1, which would in turn affect the mark array and the final output string. For instance, if words = ["aa"], then only the first two “a"s in s would be marked as True, and the output would be “aaabb”.

Identify Invariant

An invariant in the context of this problem could be the fact that once a character is marked as True in the mark array (indicating it’s part of a substring in words), it remains True throughout the process. This reflects the fact that a character, once identified as part of a word from words, remains a part of that word for the duration of the algorithm. In other words, the ‘marked’ status of a character in the string s is invariant once it’s set.

Identify Loop Invariant

A loop invariant in this problem could be the state of the mark array at the start of each iteration of the outer loop (where we go through each word in the words array).

At the start of each iteration of the outer loop, we can say that all occurrences of all previously considered words from words in the string s are correctly marked in the mark array. This is because, in each iteration, we go through the string s and mark the characters that make up the current word from words.

This is an invariant because it is a condition that is maintained to be true before and after each iteration of the outer loop. It provides a way to show that the algorithm is making progress and that it will produce a correct final result.

The invariant and loop invariant refer to the same concept in this problem. Both terms describe a condition that remains unchanged throughout the execution of the loop. The loop invariant is a specific type of invariant that applies to loops.

In our problem, the invariant, or loop invariant, is the condition that the mark array correctly represents all occurrences of the words we’ve processed so far in the string s. This condition holds true before, during, and after each iteration of the loop where we process each word from the words array. Thus, it’s an invariant and specifically, a loop invariant.

Thought Process

Let’s start by identifying the main tasks that we need to accomplish:

Identify all substrings in ’s’ that match any string in ‘words’.
Mark these substrings for bolding.
If any marked substrings overlap or are consecutive, merge them into a single segment to be bolded.
Add the HTML bold tags around each marked segment.

The problem statement gives us a couple of significant cues. It specifies that if any substrings to be bolded are overlapping or consecutive, they should be merged into a single segment. This tells us that we need to keep track of the start and end positions of each substring to be bolded, so that we can check for overlaps or consecutive substrings.

Another important insight from the problem statement is that all words in ‘words’ are unique. This could potentially simplify our search for matching substrings, since we don’t need to worry about handling duplicates.

Now let’s translate this into a Python solution:

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class Solution:
    def addBoldTag(self, s: str, words: list[str]) -> str:
        # Initialize a boolean array to mark characters for bolding
        bold = [False]*len(s)
        
        for word in words:
            start = s.find(word)
            while start != -1:
                for i in range(start, start+len(word)):
                    bold[i] = True
                start = s.find(word, start + 1)
        
        # Initialize result string
        result = ""
        
        # Add HTML bold tags around marked segments
        i = 0
        while i < len(s):
            if bold[i]:
                result += "<b>"
                while i < len(s) and bold[i]:
                    result += s[i]
                    i += 1
                result += "</b>"
            else:
                result += s[i]
                i += 1
        
        return result

This Python solution begins by initializing a boolean array ‘bold’ of the same length as ’s’. It then iterates through ‘words’, and for each word, it uses the built-in find function to find all occurrences of the word in ’s’. It marks the corresponding characters in ‘bold’ as True.

After marking all characters that should be bolded, it initializes the result string and iterates through ‘bold’. Whenever it encounters a True value, it adds a ‘’ tag to the result, then adds all consecutive True characters from ’s’, and finally adds a ‘’ tag. If the current character should not be bolded, it simply adds it to the result as is. This process continues until all characters in ’s’ have been processed, and the final result string is returned.

Establishing Preconditions and Postconditions

Parameters:
- The method takes two parameters: a string s and a list of strings words.
- s is a string that represents the text we need to add bold tags to, and words is a list of strings, each representing a word that, when found as a substring in s, should be wrapped with bold tags.
Preconditions:
- The program does not need to be in a specific state before this method is called.
- There are no constraints on the input parameters, other than that s and each string in words consist of English letters and digits, and all values of words are unique.
Method Functionality:
- The method is expected to add bold tags ( and ) around each occurrence of each word in words within s.
- If there are overlapping or adjacent occurrences, they should be wrapped together in a single pair of bold tags.
- The method returns a string that is s with the appropriate bold tags added.
Postconditions:
- After the method has been called and has returned, s remains unchanged, and a new string with the necessary bold tags added is returned.
- The return value represents s with bold tags added around each occurrence of each word in words.
- The method has no side effects, as it does not change the state of any of its inputs or any external state.
Error Handling:
- If the preconditions are not met (for instance, if s or words are not of the correct type), a TypeError would likely be raised when trying to perform operations on them.
- The method does not explicitly handle any errors or exceptions. If an error occurs (such as a TypeError due to incorrect input types), the error will propagate up the call stack.

Problem Decomposition

Problem Understanding:
- The problem is about formatting a string by adding bold tags around the substrings that appear in a given list of words. If these bold sections overlap or are adjacent, they should be combined into a single bold section.
Initial Breakdown:
- The problem can be broadly broken down into three subproblems:
  1. Finding the substrings in s that appear in words.
  2. Adding bold tags around these substrings.
  3. Combining any overlapping or adjacent bold sections.
Subproblem Refinement:
- Finding the substrings that appear in words involves iterating over s and checking for each word from words if it is a substring starting at the current position.
- Adding bold tags involves inserting the tag at the start and end of each identified substring.
- Combining bold sections involves scanning through the formatted string and merging any adjacent or overlapping bold sections.
Task Identification:
- The tasks of checking if a word is a substring at a given position and adding bold tags are repeated for each word in words and for each starting position in s.
Task Abstraction:
- The task of checking if a word is a substring at a given position can be abstracted into a function that takes a string, a word, and a position as inputs and returns a boolean.
- The task of adding bold tags can be abstracted into a function that takes a string and positions as inputs and returns a new string with the tags added.
Method Naming:
- The function for checking if a word is a substring can be named is_substring_at_position.
- The function for adding bold tags can be named add_bold_tags.
Subproblem Interactions:
- The subproblem of finding substrings must be solved before adding bold tags, as we need to know where to add the tags.
- After adding the tags, the subproblem of combining bold sections can be solved.
- There are no dependencies between the subproblems, but they must be performed in this order.

From Brute Force to Optimal Solution

Brute Force Approach:

Iterate over the input string s character by character.
For each character, iterate over each word in words to check if the word is a substring starting at the current position in s.
If a word is found to be a substring, add bold tags around it.
After adding tags to all applicable substrings, iterate over the string again to combine any overlapping or adjacent bold sections.

This approach is highly inefficient. If n is the length of s and m is the total number of characters across all words in words, then the time complexity of this approach is O(n^2 * m) because for each character in s, we are checking against all characters in words.

Optimized Approach:

Instead of checking if each word is a substring for every position in s, we can create a boolean array is_bold of the same length as s to keep track of which characters should be bolded.
Iterate over s once again. For each position, iterate over each word in words. If the word is a substring of s starting at the current position, update the is_bold array for the corresponding positions.
Use the is_bold array to add bold tags in the correct positions in s.
As before, combine any overlapping or adjacent bold sections.

This optimized approach reduces the time complexity to O(n * m), which is a significant improvement over the brute force approach. The space complexity is also improved, as we only need to keep track of a boolean array of length n, so the space complexity is O(n).

These optimizations make the solution more efficient by reducing unnecessary repeated work and making better use of space.

Code Explanation and Design Decisions

Initial Parameters: The initial parameters are the string s and the list of strings words. The string s is the text we want to add bold tags to, and words are the substrings of s that we want to wrap in bold tags.
Primary Loop: The primary loop iterates over each position in the string s. Each iteration represents a potential starting point for a substring that matches a word in words and should be bolded.
Conditions: There is a nested loop that iterates over each word in words for each position in s. The condition checks if the current word is a substring of s starting at the current position. If it is, we update the is_bold array for the corresponding positions to denote that these characters should be bolded.
Modifications: The main modification in the loop is the update to the is_bold array. This modification is necessary because it allows us to keep track of which characters in s should be bolded based on whether they are part of a word in words.
Invariant: An invariant in this code is that after processing each position in s, the is_bold array accurately represents whether each character up to and including the current position should be bolded. This invariant is crucial for correctly adding the bold tags in the correct positions in the final step.
Final Output: The final output is the string s with bold tags added around the substrings that are present in words. This output satisfies the problem’s requirements because it highlights (by making bold) the substrings of s that are words in words, and it combines overlapping or consecutive bold sections into one.

Coding Constructs

Problem-Solving Strategies: This code uses two key strategies: string processing and array marking. String processing is used to traverse and manipulate the input string s. Array marking (using the is_bold array) is used to keep track of which characters in s need to be bolded.
Non-Programmer Explanation: This code is like a highlighter for a text document. It takes in a block of text and a list of words or phrases we want to highlight. Then, it goes through the text, letter by letter, and marks where the highlights (bold tags) should be. In the end, it adds the highlights to the text.
Logical Elements: The key logical elements used are iteration (loops), conditionals (if statements), and array manipulation. Iteration is used to traverse the string and the list of words. Conditionals are used to check if a word is a substring of s starting at the current position. Array manipulation is used to mark which characters should be bolded.
Algorithmic Approach: The algorithm iterates through each position in s and for each position, it checks if a word in words starts from that position. If it does, it marks the corresponding positions in an auxiliary array. After marking all positions that need to be bolded, it constructs the result string by iterating over s and is_bold simultaneously, adding bold tags where necessary.
Key Steps: The key steps are the two pass through s. The first pass is to mark which characters should be bolded, and the second pass is to construct the result string with added bold tags.
Algorithmic Patterns: The algorithm uses a common pattern known as “Two-Pass”. The first pass is used to gather information (which characters should be bolded), and the second pass uses that information to construct the final result. This pattern is used when a single pass is not sufficient because the information needed to solve the problem is not readily available or complete until the entire input has been processed.

Language Agnostic Coding Drills

Dissect the Code into Distinct Concepts:
- Arrays (Easy): The concept of arrays or lists is fundamental in many programming languages. Here, they are used to store whether each character in the string needs to be bolded.
- String Manipulation (Easy): This involves operations like iterating over a string, accessing characters in a string, and checking if a string is a substring of another string.
- Conditional Statements (Easy): If-else statements are used to perform different actions based on whether a certain condition is true.
- Loops (Intermediate): The code involves nested loops, where one loop runs inside another loop. This concept can be tricky for beginners because it requires careful attention to what happens on each iteration.
- Array Manipulation (Intermediate): This is a bit more advanced than simply using arrays. It involves marking positions in an array based on conditions checked in the string.
- String Construction (Intermediate): This involves building a new string based on the original string and the array marks.
Problem-Solving Approach:
- Identify the key components of the problem: The input string s and the list of words words.
- Understand the task: Wrap the substrings in s that exist in words with bold tags. Merge any overlapping or consecutive bold sections.
- Establish a way to mark positions in s that need to be bolded. This is where the concept of arrays comes in. We can use an array is_bold of the same length as s to mark which characters should be bolded.
- Traverse s and for each position, check if a word from words starts from that position. If it does, mark the corresponding positions in is_bold. This combines the concepts of loops, string manipulation, and array manipulation.
- Construct the final result by iterating over s and is_bold simultaneously, adding bold tags where necessary. This is where string construction comes into play.
- Return the final result.

Targeted Drills in Python

Drill 1: Arrays

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numbers = [1, 2, 3, 4, 5]  # Create a list
first_number = numbers[0]  # Access elements
last_number = numbers[-1]  # Access elements

Drill 2: String Manipulation

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s = "hello"  # Create a string
first_char = s[0]  # Access characters
last_char = s[-1]  # Access characters
is_substring = "ell" in s  # Check if a string is a substring of another

Drill 3: Conditional Statements

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x = 10
if x > 5:
    message = "x is greater than 5"
else:
    message = "x is not greater than 5"

Drill 4: Loops

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for i in range(5):  # iterate over numbers from 0 to 4
    print(i)

Drill 5: Array Manipulation

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numbers[1] = 20  # Change the second element of the list

Drill 6: String Construction

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new_s = ""  # Construct a new string based on `s`
for char in s:
    new_s += char.upper()  # Convert each character to uppercase
print(new_s)  # Print the new string