In this assignment, we will discuss the recursive approach to solving the "Valid Palindrome String" problem:
// Given a string, determine whether it is a valid palindrome or not.
// A palindrome string is a string that reads the same backwards as forwards.
// You may assume that the input will be in lowercase, containing
// valid English alphabet characters without white spaces.
// Input: "madam"
// Output: true
// Input: "abcbea"
// Output: false
Although this problem can also be solved iteratively, we will focus on the recursive solution.
As we've seen, recursive functions require a base case and a recursive case. The base case represents the simplest scenario where the solution is evident and serves as the termination condition for the recursion. The recursive case is where the function calls itself to solve smaller subproblems.
Let's begin by identifying the base case. When can we determine that a given string is a palindrome? By answering this question, we can define the base case. The base case for the "Valid Palindrome String" problem occurs when we have an empty string (a string with no characters) or a string with only one character. In both cases, we can easily determine that it is a valid palindrome and return true.
Now, let's focus on the recursive case. To define the recursive case, we need to create a recursive definition of the problem. The recursive definition involves describing a problem in the English language declaratively. This means that we explain what should happen rather than how it should happen, which is imperative language. Let's use the following template to help us create the recursive definition:
Let's break down each part enclosed in square brackets:
[data structure]: This refers to the data structure given in the problem. In the case of the "Valid Palindrome String" problem, the data structure is a string. However, it could also be an array or another data structure like a binary tree.
[problem description]: This represents the name of the problem. In the case of "Valid Palindrome String," the problem description is "valid palindrome."
[some condition is true]: This part is often the most challenging to determine, and it varies for each problem. We will explore what this condition is specifically for the valid palindrome problem.
Let's fill in the template using the specific details of the "Valid Palindrome String" problem:
This looks much better. Now, we need to consider the condition. We know that a string is a valid palindrome if it reads the same backward and forwards. This implies that the first and last characters of the string must be the same for it to be a valid palindrome. Thus, our condition is: "The first and the last characters of the string are the same."
Plugging this condition into our template, we get:
Any mention of [problem definition], or "is a valid palindrome" in this case, indicates a recursive call to the function. When we think about what 'the rest of the string' means, we can surmise that since our condition is checking the first and last character of the string, 'the rest' means the entire string except it's first and last character.
Let's see this in code:
function isValidPalindrome(str) {
if (str.length < 2) {
return true;
}
return (
str[0] === str[str.length - 1] &&
isValidPalindrome(str.slice(1, str.length - 1))
);
}
In this solution, we check the base case first, which handles strings with zero or one character. If the length of the string is less than 2, we immediately return true since an empty string or a string with a single character is considered a valid palindrome.
In the recursive case, we compare the first and last characters of the string (str[0] and str[str.length - 1]). If they are equal, we make a recursive call to the function with the substring obtained by removing the first and last characters using str.slice(1, str.length - 1).
It's important to be aware that in the provided solution, each recursive call involves using the slice method to obtain a new substring. It's reasonable to assume that slice has a time complexity of O(N) and may also allocate new memory for the sliced substring.
Due to this slicing operation, the time and space complexity of the solution become O(N^2). This is because, in the worst case, for each recursive call, we create a new substring that is nearly the same length as the original string. As strings are immutable in JavaScript, the engine may perform optimizations internally, but it's not something we can rely on consistently.
To optimize the solution and avoid unnecessary slicing and memory allocation, we can use pointers or indices to track the start and end positions of the substring. By moving these pointers inward while comparing the characters, we can achieve a more efficient solution with a time and space complexity of O(N).
Let's update the code to utilize pointers instead of using slice:
function isValidPalindrome(str) {
return isValidPalindromeHelper(str, 0, str.length - 1);
}
function isValidPalindromeHelper(str, start, end) {
if (end <= start) {
return true;
}
return (
str[start] === str[end] && isValidPalindromeHelper(str, start + 1, end - 1)
);
}
In this updated implementation, the base case remains the same: when the start pointer is greater than or equal to the end pointer. This condition indicates that we have processed all the characters in the substring. The recursive case involves comparing the characters at the start and end positions and recursively calling the helper function while moving the pointers inward.
With this approach, we avoid the need for slicing and instead rely on efficient pointer manipulation. This results in improved time and space complexity, both of which are O(N) for the given problem.
We hope that this walkthrough of the "Valid Palindrome String" problem will serve as a helpful guide when you attempt to solve other recursive problems. Remember that developing a recursive definition requires practice, and with time, you will improve. Additionally, it's important to avoid allocating unnecessary memory or performing unnecessary work in each function call. If you require additional parameters, utilize helper functions to manage them effectively.