Partition Equal Subset Sum

The problem of partitioning an array into two subsets such that the sum of the elements in both subsets is equal can be solved using dynamic programming. The idea is to use a 2D boolean array to keep track of possible subset sums. Here’s a way to implement this:

1. Check if the total sum of the array is odd. If it’s odd, then it’s not possible to divide it into two equal parts, so return false.
2. Otherwise, we want to find if there’s a subset whose sum is equal to half of the total sum.
3. Create a 2D boolean array dp, where dp[i][j] will be true if there is a subset of the first i elements that has a sum equal to j.
4. Initialize dp[0][0] as true since there is a subset with 0 elements that make a sum of 0.
5. Iterate through the array and update the dp array based on the current element.
6. Return the value of dp[n][sum//2], where n is the length of the array and sum is the total sum.

Here’s the code:

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class Solution:
    def canPartition(self, nums: List[int]) -> bool:
        total_sum = sum(nums)
        # Check if total sum is odd
        if total_sum % 2 != 0:
            return False

        n = len(nums)
        half_sum = total_sum // 2

        # Initialize dp array
        dp = [[False] * (half_sum + 1) for _ in range(n + 1)]
        dp[0][0] = True

        # Iterate through nums and update dp
        for i in range(1, n + 1):
            for j in range(half_sum + 1):
                if j >= nums[i - 1]:
                    dp[i][j] = dp[i - 1][j] or dp[i - 1][j - nums[i - 1]]
                else:
                    dp[i][j] = dp[i - 1][j]

        return dp[n][half_sum]

This code first checks the condition for the total sum and then uses dynamic programming to solve the problem efficiently. It returns true if it’s possible to partition the array into two equal sum subsets and false otherwise.

title: Partition Equal Subset Sum tags: decision-tree include-exclude

Given a non-empty array nums containing only positive integers, find if the array can be partitioned into two subsets such that the sum of elements in both subsets is equal.

Brute Force

We have to find a subset in an array where the sum must be equal to sub set sum. The brute force approach would be to generate all the possible subsets of an array and return true if we find a subset with the required sum.

Mistakes

• Did not recognize that this can be transformed to a similar problem.
• Uses include/exclude form of the decision tree.
• Preprocessing step: Sum of all elements.

Thought Process

Work through the problem by hand. Early termination returns false

1. Define the Interface Input: nums array (positive integers) Output: boolean (true if it can be partitioned into two subsets that are equal in sum)
2. Constraint 1 <= nums.length <= 200
3. The subset does not have to be contiguous (not a subarray)
4. The subset size can be different
5. Invariant Two subsets positive integer 22/2 = 11 The sum must be even, otherwise return false
6. How do we know which elements to pick and sum and compare with some other subset? Should we create all possible subsets and check?
7. Classify the Problem
   • Have you seen a problem like this before?
   • What type of problem ? This is Exhaustive Enumeration. We use Include/Exclude to choose items from the input. Once you pick an element, you know the target that the rest of the subset must be equal to. Prune the tree. What is the bounding function to prune the recursion tree?

0 1 2 3 [1,5,11,5]

0 - rest 0,1 - rest 0,1,2 - rest 0,2 - 1,3 0,3 - 1,2

Subsets of size from 1 to n-1 - subset 1. Remaining elements in subset 2.

1,5,11,5 => 22

We need to set the target as 22/2 = 11

[1,5,11,5]

capacity = 10
sum = 22

Recursion Tree

                             []
    
              {1}                                    {}
       {1,5}               {1}               {5}          {}
    
{11, 1, 5}  {1,5}   {1, 11}  {1}    {5,11}  {5}    {11} {}

22 - 17 = 5

Base Case

When do we stop the recursion? When the sum is equal to the subset target == 0 and there are no elements in nums to choose (nums.size == 0), return true.

Pruning the tree

If the target becomes negative, return. If sum of all elements in nums is not even, return false.

Recursive Cases

We can keep track of the target sum we are looking in the parameter. How do we calculate the target sum?

sum - element picked = target for the child

We need to have an index as a parameter to tell the child which element it needs to include/exclude.

Key Takeaways

• Identifying the target as the capacity that changes for every child based on what element we pick.
• Classifying the problem as Knapsack - Similar but not exactly. Because this is not an optimization problem.
• Identifying the keys for the memoization table.
• We need to capture the output of the subproblem and logical or because. The subset sum can be found in the left or right subtree.
• We need to return true or false for the unit of work after the recursive to decide the result.

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def wrapper(nums, index, target, memo)
  if target == 0 
    return true
  end

  if !memo[index][target].nil?
    return memo[index][target]
  end
  
  # pruning the tree
  if target < 0 || index == nums.length
    return false
  end
    
  # Include
  left = wrapper(nums, index+1, target - nums[index], memo)
    
  # Exclude
  right = wrapper(nums, index+1, target, memo)
    
  result = left || right
    
  memo[index][target] = result
    
  return result
end

# @param {Integer[]} nums
# @return {Boolean}
def can_partition(nums)
  target = nums.sum

  if target.odd?
    return false
  end
  
  # dimensions:  size is m+1 and subs, (target/2) + 1
  memo = Array.new(nums.size+1, nil) { Array.new(target/2, nil) }
    
  wrapper(nums, 0, target/2, memo)
end

The basic idea is that to partition an array into two subsets of equal sum, the total sum of given array must be twice the subset sum. This could also be written as:

1

sub_set_sum = total_sum / 2.

The total sum of an array must be even, only then we can divide it into 2 equal subsets. Finding a subset with a sum equal to a given target is different than subarray sum equals k. Subarray is a contiguous sequence of array elements, whereas the subset could consist of array elements that are not contigous. But each array element must belong to exactly one subset.

Complexity Analysis

Time Complexity: O(m⋅n), where m is the subset sum, and n is the number of array elements. We fill the array of size m⋅n.

Space Complexity : O(m⋅n) , where n is the number of array elements and m is the subset sum.

Mistakes

• Did not recognize that this can be transformed to a similar problem
• Uses include/exclude form of the decision tree
• Preprocessing step: Sum of all elements

By hand work through the problem Early termination, returning false

1. Define the Interface Input: nums array (positive integers) Output: boolean (true if it can be partitioned into two subsets that are equal in sum)

2. Constraints 1 <= nums.length <= 200

3. The subset does not have to be contigous (not a subarray)

4. The subset size can be different

5. Invariant Two subsets positive integer 22/2 = 11 The sum must be even, otherwise return false

6. How do we know which elements to pick and sum and compare with some other subset? Should we create all possible subsets and check?

7. Classify the Problem Have you seen a problem like this before? What type of problem ? Exhaustive Enumeration Include/Exclude Once you pick an element, you know the target that the rest of the subset must be equal to. Prune the tree. What is the bounding function to prune the tree.

   0-1 Bounded Knapsack Problem

   0 1 2 3 [1,5,11,5]

   0 - rest 0,1 - rest 0,1,2 - rest 0,2 - 1,3 0,3 - 1,2

   subsets of size from 1 to n-1 - subset 1 remaining elements in subset 2

1,5,11,5 => 22

We need to set the target as 22/2 = 11

[1,5,11,5]

capacity = 10

sum = 22

                         []

          {1}                                    {}
   {1,5}               {1}               {5}          {}

{11, 1, 5} {1,5} {1, 11} {1} {5,11} {5} {11} {}

22-17 = 5

Base Case

- When do we stop the recursion
    When the sum is equal to the subset
    target == 0 and there are no elements in nums to choose (nums.size == 0), return true
    Pruning the tree
    If the target becomes negative, return
    if sum of all elements in nums is not even, return false
    
Recursion
We can keep track of the target sum we are looking in the parameter

How do we calculate the target sum.
 sum - element picked = target for the child
We need have an index as a parameter to tell the child which
element it needs to include/exclude

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def wrapper(nums, index, target, memo)
   if target == 0 
       return true
   end
   if (!memo[index][target].nil?)
       return memo[index][target]
   end
#     pruning the tree
   if target < 0 || index == nums.length
       return false
   end
    
#    Include
    left = wrapper(nums, index+1, target - nums[index], memo)
    
#   Exclude
    right = wrapper(nums, index+1, target, memo)

    result = left || right

    memo[index][target] = result

    return result
end

# @param {Integer[]} nums
# @return {Boolean}
def can_partition(nums)
    target = nums.sum
    if target.odd?
        return false
    end
#   dimensions:  size is m+1 and subs, (target/2) + 1
    memo = Array.new(nums.size+1, nil) { Array.new(target/2, nil) }
    
    wrapper(nums, 0, target/2, memo)
end

Identifying Problem Isomorphism

“Partition Equal Subset Sum” is approximately isomorphic to “Target Sum”.

In “Partition Equal Subset Sum”, you are asked to determine whether you can partition the array into two subsets such that the sum of numbers in both subsets is equal.

In the “Target Sum” problem, you are given a list of non-negative integers, a target integer S, and two operations you can perform on these integers: + and -. You have to find out how many ways you can assign symbols to make the sum of integers equal to target S.

The connection here is the idea of partitioning a list of numbers into two groups, based on some criteria, and checking something about the sums of the two groups. Although the exact conditions and the operations are different, the strategy of partitioning the array or set and exploring the possible solutions remains similar.

This is an approximate mapping and the strategies or solutions for these problems may not be directly transferable, but the core concepts and approach are somewhat similar.

10 Prerequisite LeetCode Problems

Identify 10 LeetCode problems of lesser complexity, excluding the problem itself that I should solve as preparation for tackling 416. Partition Equal Subset Sum . Include the name of the given problem in the response before the list. Do not add double quotes for the items in the list. Include the reason why that problem is relevant. The format of the response must be:

For the , the following problems is a good preparation: Given an integer array nums, return true if you can partition the array into two subsets such that the sum of the elements in both subsets is equal or false otherwise.

Example 1:

Input: nums = [1,5,11,5] Output: true Explanation: The array can be partitioned as [1, 5, 5] and [11].

Example 2:

Input: nums = [1,2,3,5] Output: false Explanation: The array cannot be partitioned into equal sum subsets.

Constraints:

1 <= nums.length <= 200 1 <= nums[i] <= 100

Problem Classification

Problem Statement:Given an integer array nums, return true if you can partition the array into two subsets such that the sum of the elements in both subsets is equal or false otherwise.

Example 1:

Input: nums = [1,5,11,5] Output: true Explanation: The array can be partitioned as [1, 5, 5] and [11]. Example 2:

Input: nums = [1,2,3,5] Output: false Explanation: The array cannot be partitioned into equal sum subsets.

Constraints:

1 <= nums.length <= 200 1 <= nums[i] <= 100

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.

Clarification Questions

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

Problem Analysis and Key Insights

What are the key insights from analyzing the problem statement?

Problem Boundary

What is the scope of this problem?

How to establish the boundary of this problem?

Distilling the Problem to Its Core Elements

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?

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 edge cases?

How to visualize 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?

Simple Explanation

Can you explain this problem in simple terms or like you would explain to a non-technical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?

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

Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method. How can I recognize these properties by drawing tables or diagrams?

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?

Stepwise Refinement

1. Could you please provide a stepwise refinement of our approach to solving this problem?

2. How can we take the high-level solution approach and distill it into more granular, actionable steps?

3. Could you identify any parts of the problem that can be solved independently?

4. Are there any repeatable patterns within our solution?

Solution Approach and Analysis

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.

Identify Invariant

What is the invariant in this problem?

Identify Loop Invariant

What is the loop invariant in this problem?

Is invariant and loop invariant the same for this problem?

Identify Recursion Invariant

Is there an invariant during recursion in this problem?

Is invariant and invariant during recursion the same for this problem?

Thought Process

Can you explain the basic thought process and steps involved in solving this type of problem?

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.

Establishing Preconditions and Postconditions

1. Parameters:

   • What are the inputs to the method?
   • What types are these parameters?
   • What do these parameters represent in the context of the problem?

2. Preconditions:

   • Before this method is called, what must be true about the state of the program or the values of the parameters?
   • Are there any constraints on the input parameters?
   • Is there a specific state that the program or some part of it must be in?

3. Method Functionality:

   • What is this method expected to do?
   • How does it interact with the inputs and the current state of the program?

4. Postconditions:

   • After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
   • What does the return value represent or indicate?
   • What side effects, if any, does the method have?

5. Error Handling:

   • How does the method respond if the preconditions are not met?
   • Does it throw an exception, return a special value, or do something else?

Problem Decomposition

1. Problem Understanding:

   • Can you explain the problem in your own words? What are the key components and requirements?

2. Initial Breakdown:

   • Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?

3. Subproblem Refinement:

   • For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?

4. Task Identification:

   • Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?

5. Task Abstraction:

   • For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?

6. Method Naming:

   • Can you give each task a simple, descriptive name that makes its purpose clear?

7. Subproblem Interactions:

   • How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?

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

1. Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.

2. 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?

3. 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.

4. 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?

5. Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.

6. 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 code for the solution of this problem.

1. What are the high-level problem-solving strategies or techniques being used by this code?

2. If you had to explain the purpose of this code to a non-programmer, what would you say?

3. Can you identify the logical elements or constructs used in this code, independent of any programming language?

4. Could you describe the algorithmic approach used by this code in plain English?

5. What are the key steps or operations this code is performing on the input data, and why?

6. 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.

1. 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.

2. 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.

3. 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.

1. 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.

2. 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.

3. 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.

Q&A

Similar Problems

Can you suggest 10 problems from LeetCode that require similar problem-solving strategies or use similar underlying concepts as the problem we’ve just solved? These problems can be from any domain or topic, but they should involve similar steps or techniques in the solution process. Also, please briefly explain why you consider each of these problems to be related to our original problem. The response text is of the following format. First provide this as the first sentence: Here are 10 problems that use similar underlying concepts: