Count Number of Special Subsequences

The problem asks for the number of different subsequences in an array that are special. A special sequence consists of positive counts of 0s, followed by 1s, then 2s.

A dynamic programming (DP) approach can be used to solve this problem. We’ll maintain three variables representing the count of special subsequences ending with 0, 1, and 2, respectively.

Here’s a step-by-step guide to the approach:

Initialize Variables: Initialize three variables zero_count, one_count, and two_count to represent the count of special subsequences ending with 0, 1, and 2, respectively.
Iterate Through the Array: Iterate through the given array, and for each number:
- If it’s a 0, the count of special subsequences ending with 0 is updated.
- If it’s a 1, the count of special subsequences ending with 1 is updated.
- If it’s a 2, the count of special subsequences ending with 2 is updated.
Calculate the Result: The final result is the value in two_count.
Return the Result Modulo (10^9 + 7): As the result can be large, we return it modulo (10^9 + 7).

Here’s the code:

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from typing import List

class Solution:
    def countSpecialSubsequences(self, nums: List[int]) -> int:
        MOD = 10**9 + 7
        zero_count, one_count, two_count = 0, 0, 0

        for num in nums:
            if num == 0:
                zero_count = (2 * zero_count + 1) % MOD
            elif num == 1:
                one_count = (2 * one_count + zero_count) % MOD
            elif num == 2:
                two_count = (2 * two_count + one_count) % MOD

        return two_count

Explanation:

For 0: We can either include the new 0 or not, and also create a new sequence with a single 0. Hence, zero_count = 2 * zero_count + 1.
For 1: We can either include the new 1 in existing special sequences ending with 1 or not, or extend the special sequences ending with 0. Hence, one_count = 2 * one_count + zero_count.
For 2: We can either include the new 2 in existing special sequences ending with 2 or not, or extend the special sequences ending with 1. Hence, two_count = 2 * two_count + one_count.

The code provides an efficient solution with a time complexity of (O(n)), where (n) is the length of the given array.

Identifying Problem Isomorphism

“Count Number of Special Subsequences” can be mapped to “Distinct Subsequences II”.

Both require finding the number of distinct subsequences in a given sequence. In “Count Number of Special Subsequences”, the task is to count the number of subsequences that are of the form [0,1,2] with possibly repeating numbers. In “Distinct Subsequences II”, the task is to count the number of distinct, non-empty subsequences of a string.

Despite the specifics of what classifies as a “special” subsequence is different, the core task in both problems involves counting specific subsequences which require similar logical reasoning and understanding of subsequence properties.

“Distinct Subsequences II” is simpler as it involves counting all distinct subsequences, whereas “Count Number of Special Subsequences” involves more specific conditions for the subsequences to be considered “special”, which adds an extra layer of complexity to the problem.

This is an approximate mapping as the specifics and constraints of the problems are different.

To solve, “Count Number of Special Subsequences”, you need a good understanding of dynamic programming, arrays, and subsequence problems. Here are 10 problems to get ready:

LeetCode 300. Longest Increasing Subsequence
- This problem will give you a good understanding of the concept of subsequence and how dynamic programming can be used to solve these kind of problems.
LeetCode 53. Maximum Subarray
- This is a simpler dynamic programming problem that will get you accustomed to working with arrays and subarrays.
LeetCode 198. House Robber
- Another simpler dynamic programming problem. Solving this will help you understand how to consider different states in dynamic programming.
LeetCode 62. Unique Paths
- This problem requires dynamic programming to find all possible paths from the top-left to the bottom-right corner of a grid. This will strengthen your understanding of DP.
LeetCode 416. Partition Equal Subset Sum
- This problem also involves dynamic programming. It will improve your ability to reason about subsets, which is an important skill for problem 1955.
LeetCode 1143. Longest Common Subsequence
- This problem requires you to find the longest common subsequence in two strings. It will further improve your understanding of subsequences and dynamic programming.
LeetCode 322. Coin Change
- This problem introduces you to the concept of solving a problem by trying out all possible choices, which is fundamental to dynamic programming.
LeetCode 518. Coin Change 2
- This is a follow-up to the Coin Change problem. It introduces you to the concept of order of computation in dynamic programming, which is crucial to problem 1955.
LeetCode 121. Best Time to Buy and Sell Stock
- Another dynamic programming problem that focuses on maximizing profit given an array of prices.
LeetCode 139. Word Break
- This is a dynamic programming problem that focuses on partitioning a string into substrings, which can be thought of as a form of subsequence problem.

Understanding the solutions to these problems and the concepts behind them is the key to learning. After these problems, you should be prepared to tackle problem 1955.

The problem “1955. Count Number of Special Subsequences” involves understanding of dynamic programming and sequence problems. Here are 8 problems to prepare:

70. Climbing Stairs
376. Wiggle Subsequence
646. Maximum Length of Pair Chain
673. Number of Longest Increasing Subsequence
718. Maximum Length of Repeated Subarray
746. Min Cost Climbing Stairs
516. Longest Palindromic Subsequence: In this problem, you’re required to find the longest palindromic subsequence in a string. This problem will further strengthen your understanding of subsequences.
376. Wiggle Subsequence: This problem asks you to find a longest ‘wiggle’ subsequence, which requires understanding of sequence handling and dynamic programming.

These problems will make you familiar with dynamic programming concepts and handling sequence-based problems, which are crucial for solving the problem “1955. Count Number of Special Subsequences”.

Problem Classification

Problem Statement:A sequence is special if it consists of a positive number of 0s, followed by a positive number of 1s, then a positive number of 2s.

For example, [0,1,2] and [0,0,1,1,1,2] are special. In contrast, [2,1,0], [1], and [0,1,2,0] are not special. Given an array nums (consisting of only integers 0, 1, and 2), return the number of different subsequences that are special. Since the answer may be very large, return it modulo 109 + 7.

A subsequence of an array is a sequence that can be derived from the array by deleting some or no elements without changing the order of the remaining elements. Two subsequences are different if the set of indices chosen are different.

Example 1:

Input: nums = [0,1,2,2] Output: 3 Explanation: The special subsequences are bolded [0,1,2,2], [0,1,2,2], and [0,1,2,2].

Example 2:

Input: nums = [2,2,0,0] Output: 0 Explanation: There are no special subsequences in [2,2,0,0].

Example 3:

Input: nums = [0,1,2,0,1,2] Output: 7 Explanation: The special subsequences are bolded:

[0,1,2,0,1,2]
[0,1,2,0,1,2]
[0,1,2,0,1,2]
[0,1,2,0,1,2]
[0,1,2,0,1,2]
[0,1,2,0,1,2]
[0,1,2,0,1,2]

Constraints:

1 <= nums.length <= 105 0 <= nums[i] <= 2

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.

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.

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?

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.