Count Palindromic Subsequences

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class Solution:
    def countPalindromes(self, s: str) -> int:
        mod, n, ans = 10 ** 9 + 7, len(s), 0
        pre, cnts = [[[0] * 10 for _ in range(10)] for _ in range(n)], [0] * 10
        for i in range(n):
            c = ord(s[i]) - ord('0')
            if i:
                for j in range(10):
                    for k in range(10):
                        pre[i][j][k] = pre[i - 1][j][k] 
                        if k == c: pre[i][j][k] += cnts[j]
            cnts[c] += 1
        suf, cnts = [[[0] * 10 for _ in range(10)] for _ in range(n)], [0] * 10
        for i in range(n - 1, -1, -1):
            c = ord(s[i]) - ord('0')
            if i < n - 1:
                for j in range(10):
                    for k in range(10):
                        suf[i][j][k] = suf[i + 1][j][k]
                        if k == c: suf[i][j][k] += cnts[j]
            cnts[c] += 1
        for i in range(2, n - 2):
            for j in range(10):
                for k in range(10):
                    ans += pre[i - 1][j][k] * suf[i + 1][j][k]
        return ans % mod

Identifying Problem Isomorphism

“Count Palindromic Subsequences” can be mapped to “Longest Palindromic Subsequence”.

Reasoning:

Both problems involve the concept of subsequences and palindromes. While “Count Palindromic Subsequences” requires you to find the total number of distinct palindromic subsequences, “Longest Palindromic Subsequence” asks you to find the length of the longest possible palindromic subsequence.

The core task in both problems is to identify and manipulate palindromic subsequences within a given string. Dynamic programming is a common solution strategy for both problems, where we build up solutions for smaller subsequences to ultimately solve for the whole string.

“Longest Palindromic Subsequence” is simpler, as it only requires the length of the longest palindromic subsequence, whereas “Count Palindromic Subsequences” needs to handle duplicate subsequences and count the total distinct ones.

10 Prerequisite LeetCode Problems

Here are 10 problems related to subsequences, string manipulation, and dynamic programming that you should solve before “Count Palindromic Subsequences”:

516. Longest Palindromic Subsequence: This problem helps understand the concept of a palindromic subsequence which is fundamental for the given problem.
647. Palindromic Substrings: This problem is a precursor to the given problem and helps you understand how to count palindromic substrings.
5. Longest Palindromic Substring: This problem is a simpler version of the problem at hand, as it asks for the longest palindromic substring rather than all palindromic subsequences.
131. Palindrome Partitioning: This problem can help understand how to generate all possible palindromic partitions of a string.
730. Count Different Palindromic Subsequences: This problem is very similar to the given problem but without the modulo operation.
300. Longest Increasing Subsequence: Understanding this problem can help understand the concept of subsequences.
1143. Longest Common Subsequence: This problem can help understand dynamic programming over subsequences, a key concept for the given problem.
718. Maximum Length of Repeated Subarray: This problem helps understand the concept of subarrays and how they can be used in dynamic programming.
673. Number of Longest Increasing Subsequence: This problem introduces counting with dynamic programming which is important for the given problem.
583. Delete Operation for Two Strings: This problem introduces the concept of string manipulation which can be applied to the given problem.

Clarification Questions

What are the clarification questions we can ask about this problem?

Problem Classification

Problem Statement:Given a string of digits s, return the number of palindromic subsequences of s having length 5. Since the answer may be very large, return it modulo 109 + 7.

Note:

A string is palindromic if it reads the same forward and backward. A subsequence is a string that can be derived from another string by deleting some or no characters without changing the order of the remaining characters.

Example 1:

Input: s = “103301” Output: 2 Explanation: There are 6 possible subsequences of length 5: “10330”,“10331”,“10301”,“10301”,“13301”,“03301”. Two of them (both equal to “10301”) are palindromic. Example 2:

Input: s = “0000000” Output: 21 Explanation: All 21 subsequences are “00000”, which is palindromic. Example 3:

Input: s = “9999900000” Output: 2 Explanation: The only two palindromic subsequences are “99999” and “00000”.

Constraints:

1 <= s.length <= 104 s consists of digits.

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Distilling the Problem to Its Core Elements

In order to have me distill a problem to its core, you could ask questions that prompt for a deeper analysis of the problem, understanding of the underlying concepts, and simplification of the problem’s essence. Here are some examples of such prompts:

Can you identify the fundamental concept or principle this problem is based upon? Please explain.
What is the simplest way you would describe this problem to someone unfamiliar with the subject?
What is the core problem we are trying to solve? Can we simplify the problem statement?
Can you break down the problem into its key components?
What is the minimal set of operations we need to perform to solve this problem?

These prompts guide the discussion towards simplifying the problem, stripping it down to its essential elements, and understanding the core problem to be solved.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Provide names by categorizing these cases

What are the key insights from analyzing the different cases?

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

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

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Code Explanation and Design Decisions

Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?
If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.
If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?

Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.