Combination Sum II

tags: sort backtracking combination pruning-recursion-tree narrowing-the-domains-of-the-variables

Given a collection of candidate numbers (candidates) and a target number (target), find all unique combinations in candidates where the candidate numbers sum to target. Each number in candidates may only be used once in the combination.

Note: The solution set must not contain duplicate combinations.

Example 1:
Input: candidates = [10,1,2,7,6,1,5], target = 8
Output: 
[
[1,1,6],
[1,2,5],
[1,7],
[2,6]
]

Example 2:
Input: candidates = [2,5,2,1,2], target = 5
Output: 
[
[1,2,2],
[5]
]

Constraints

  • 1 <= candidates.length <= 100
  • 1 <= candidates[i] <= 50
  • 1 <= target <= 30

Thinking Process

  • Sort the candidates outside the loop (n log n).
  • We can use another data structure to keep track of what element has been visited.
  • To maintain the invariant, we never look back (the index will keep going from left to right).
  • Depth first traversal is used to solve this problem.
  • We will start at index+1 and go till the end.
  • We do it on the fly, no auxiliary data structure is needed.

Classify the Problem

  • Fractional or 0-1? It’s 0-1
  • Unbounded or bounded? Bounded
  • 0-1 Bounded Knapsack

We have to find all solutions for this problem.

Constraints

  • Unique combinations: (1,1,2). Not to be included (1,2,1) (2,1,1).
  • Membership matters.
  • Order of the elements does not matter.

Invariant: The sum of all values in the combination has to be equal to the target

Define the interface

Input: candidates (positive numbers), target (integer)

Brute Force Approach

  1. For each element, we can either include it or exclude it.
  2. Therefore there are 2^n possible combinations that need to be checked if the sum meets the target.
  3. Once we have exceeded the capacity, we can prune the recursion tree to discard infeasible solutions.

How are we going to build the combination? We are making the choice at an index - do we include or exclude the element? If we pick 10, it exceeds the target, we can prune the tree by using a bounding function. The height of the tree is equal to the number of levels (the index value). We can express the combination as a series of include/exclude decisions.

We have an empty collection, for each branch (representing one of the elements). If we choose 1, we have capacity of 7 left. At the next level, we have another series of decisions to make. We are looping through all of our indices starting from 0 to size-1 (first level). We can have duplicate elements in the candidates. We cannot have duplicate combinations

Can we draw a decision tree to analyze the problem? Do we vary the value of the index that is used to pick an element from the candidate?

Base Case: When we meet the target, we can add the combination to the output

How do we prune the tree? Bounding function: When the remaining sum becomes negative, return

Recursive Cases: Parameters to add to the dfs method remaining, start index

1

if i > start and candidates[i] == candidates[i-1]

Time and Space Complexity

Time: O( ) Space: O( )

Implementation

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def backtrack(candidates, target, start, combination, output)
  if target == 0
    output << combination.dup
    return
  end
    
  if target < 0
    return
  end
    
  for i in (start..candidates.size-1)
    if (i > start && candidates[i] == candidates[i-1])
      next
    end
    combination << candidates[i]
    backtrack(candidates, target - candidates[i], i+1, combination, output)
    combination.pop
  end
end

# @param {Integer[]} candidates
# @param {Integer} target
# @return {Integer[][]}
def combination_sum2(candidates, target)
  output = []
  combination = []
  candidates.sort!
  backtrack(candidates, target, 0, combination, output)
  output
end

For:

[1, 1, 1, 2, 2]
4

The output is [[1,1,2],[1,1,2],[1,1,2],[1,1,2],[1,1,2],[1,1,2],[2,2]] without the skipping condition for duplicate combination.

With condition:

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

This is the correct answer.

When the dfs traversal first handles the first 1 and adds (1,1,2) as a combination. In any other paths if the number begins with 1 again, it should be skipped.

Add the check:

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if candidates[i] > target
  break
end

Inside the loop to get a big performance boost.

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def backtrack(candidates, target, start, combination, output)
  if target == 0
    output << combination.dup
    return
  end
        
  for i in (start..candidates.size-1)
    # prevent duplicate combination from getting generated
    if (i > start && candidates[i] == candidates[i-1]) 
      next
    end
    # prune the tree (bounding function)
    if candidates[i] > target
      return
    end

    combination << candidates[i]
    backtrack(candidates, target - candidates[i], i+1, combination, output)
    combination.pop
  end
end

# @param {Integer[]} candidates
# @param {Integer} target
# @return {Integer[][]}
def combination_sum2(candidates, target)
  output = []
  combination = []
  candidates.sort!
  backtrack(candidates, target, 0, combination, output)
  output
end

This problem is similar to the previous one but with the added constraint that each number in candidates may only be used once in the combination, and we must eliminate duplicate combinations.

Here’s a step-by-step guide to tackle this problem:

  1. Sort the Candidates: This will help us to identify and avoid duplicates.

  2. Initialize a Result List: This will hold all the unique combinations.

  3. Define a Recursive Function: This function will take the current combination, the remaining target, and the starting index of the candidates as parameters.

  4. Base Case: If the remaining target is zero, add the current combination to the result list.

  5. Recursive Case: Iterate through the candidates starting from the given index. For each candidate:

    • If the current candidate is the same as the previous one, skip to avoid duplicates.
    • Add it to the current combination.
    • Make a recursive call with the updated combination, reduced target, and the next index.
    • Remove the last added candidate from the current combination to backtrack.

  6. Return the Result List: Call the recursive function initially with an empty combination, the given target, and starting index 0. Return the result list.

Here’s the code:

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class Solution:
    def combinationSum2(self, candidates: List[int], target: int) -> List[List[int]]:
        candidates.sort() # Sort the candidates to identify duplicates
        def backtrack(combination, remaining_target, start):
            if remaining_target == 0:
                result.append(list(combination))
                return
            for i in range(start, len(candidates)):
                if i > start and candidates[i] == candidates[i - 1]: # Skip duplicates
                    continue
                if remaining_target - candidates[i] >= 0:
                    combination.append(candidates[i])
                    # Recursive call with updated combination, reduced target, and next index
                    backtrack(combination, remaining_target - candidates[i], i + 1)
                    # Remove last added candidate to backtrack
                    combination.pop()

        result = []
        # Initial call to the recursive function
        backtrack([], target, 0)
        return result

This code returns all the unique combinations of candidates where the chosen numbers sum to the target, avoiding duplicates. The time complexity is (O(2^N \cdot k)), and the space complexity is (O(k)), where (N) is the number of candidates and (k) is the size of the combination.

Identifying Problem Isomorphism

“Combination Sum II” has an approximate isomorphism “Combination Sum III”.

“Combination Sum II” asks for all unique combinations of a list of candidate numbers where each combination sums up to a target number, and each number in the list can be used only once.

“Combination Sum III” is a slight variant where you are asked to find all possible combinations of k numbers that add up to a number n. The numbers are from 1 to 9, and each number is used only once.

Both problems share the same structure of exploring combinations of numbers that add up to a specific target. The difference is in the constraints and specific details - while “Combination Sum II” has a list of candidate numbers, “Combination Sum III” restricts the numbers from 1 to 9 and requires that each combination must contain exactly k numbers.

Understanding the solution for “Combination Sum II” can therefore provide insights into solving “Combination Sum III”, and vice versa, due to their similarities in problem-solving structure.

Sort the candidates outside the loop (n log n) We can use another data structure to keep track of what element has been visited To maintain the invariant, we never look back (the index will keep going from left to right) Depth first traversal is used to solve this problem We will start at index+1 and go till the end We do it on the fly, no auxiliary data structure is needed

  1. Classify the Problem

    • Fractional or 0-1? It’s 0-1
    • Unbounded or bounded? Bounded

    0-1 Bounded Knapsack

  2. Find all solutions

  3. Contraints Unique combinations: (1,1,2) Not to be included (1,2,1) (2,1,1) Membership Order of the elements does not matter

  4. Invariant The sum of all values in the combination has to be equal to the target

  5. Define the interface Input: candidates (positive numbers), target (integer)

  6. Brute Force Approach For each element, we can either include it or exclude it Therefore there are 2^n possible combinations that needs to be checked if the sum meets the target. Once we have exceeded the capacity, we can prune the recursion tree to discard infeasible solutions.

  7. How are we going to build the combination? We are making the choice at a index - do we include or exclucde the element? If we pick 10, it exceeds the target, we can prune the tree by using a bounding function. The height of the tree is equal to the number of levels (the index value) We can express the combination as a series of include/exclude decisions

    We have an empty collection, for each branch (representing one of the elements) If we choose 1, we have capacity of 7 left At the next level, we have another series of decisions to make We are looping through all of our indices starting from 0 to size-1 (first level)

    We can have duplicate elements in the candidates We cannot have duplicate combinations

  8. Can we draw a decision tree to analyze the problem?

    • Do we vary the value of the index that is used to pick an element from the candidate?

  9. Base Case

    • When we meet the target, we can add the combination to the output

  10. How do we prune the tree?

    • Bounding function When the remaining sum becomes negative, return

  11. Recursive Cases

    • Parameters to add to the dfs method remaining, start index if i > start and candidates[i] == condidates[i-1]
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def backtrack(candidates, target, start, combination, output)
  if target == 0
      output << combination.dup
      return
  end

    if target < 0
        return
    end

    for i in (start..candidates.size-1)
       if (i > start && candidates[i] == candidates[i-1]) 
         next
       end
        combination << candidates[i]
        backtrack(candidates, target - candidates[i], i+1, combination, output)
        combination.pop
    end
end

# @param {Integer[]} candidates
# @param {Integer} target
# @return {Integer[][]}
def combination_sum2(candidates, target)
    output = []
    combination = []
    candidates.sort!
    backtrack(candidates, target, 0, combination, output)
    output
end

Sort the candidates outside the loop (n log n) We can use another data structure to keep track of what element has been visited To maintain the invariant, we never look back (the index will keep going from left to right) Depth first traversal is used to solve this problem We will start at index+1 and go till the end We do it on the fly, no auxiliary data structure is needed

  1. Classify the Problem

    • Fractional or 0-1? It’s 0-1
    • Unbounded or bounded? Bounded

    0-1 Bounded Knapsack

  2. Find all solutions

  3. Contraints Unique combinations: (1,1,2) Not to be included (1,2,1) (2,1,1) Membership Order of the elements does not matter

  4. Invariant The sum of all values in the combination has to be equal to the target

  5. Define the interface Input: candidates (positive numbers), target (integer)

  6. Brute Force Approach For each element, we can either include it or exclude it Therefore there are 2^n possible combinations that needs to be checked if the sum meets the target. Once we have exceeded the capacity, we can prune the recursion tree to discard infeasible solutions.

  7. How are we going to build the combination? We are making the choice at a index - do we include or exclucde the element? If we pick 10, it exceeds the target, we can prune the tree by using a bounding function. The height of the tree is equal to the number of levels (the index value) We can express the combination as a series of include/exclude decisions

    We have an empty collection, for each branch (representing one of the elements) If we choose 1, we have capacity of 7 left At the next level, we have another series of decisions to make We are looping through all of our indices starting from 0 to size-1 (first level)

    We can have duplicate elements in the candidates We cannot have duplicate combinations

  8. Can we draw a decision tree to analyze the problem?

    • Do we vary the value of the index that is used to pick an element from the candidate?

  9. Base Case

    • When we meet the target, we can add the combination to the output

  10. How do we prune the tree?

    • Bounding function When the remaining sum becomes negative, return

  11. Recursive Cases

    • Parameters to add to the dfs method remaining, start index if i > start and candidates[i] == condidates[i-1]
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def backtrack(candidates, target, start, combination, output)
  if target == 0
      output << combination.dup
      return
  end

    for i in (start..candidates.size-1)
       if (i > start && candidates[i] == candidates[i-1]) 
         next
       end
       if candidates[i] > target
         return
       end

        combination << candidates[i]
        backtrack(candidates, target - candidates[i], i+1, combination, output)
        combination.pop
    end
end

# @param {Integer[]} candidates
# @param {Integer} target
# @return {Integer[][]}
def combination_sum2(candidates, target)
    output = []
    combination = []
    candidates.sort!
    backtrack(candidates, target, 0, combination, output)
    output
end

10 Prerequisite LeetCode Problems

Identify 10 LeetCode problems of lesser complexity, excluding the problem itself that I should solve as preparation for tackling 40. Combination Sum II . 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 is a good preparation:

Problem Classification

Problem Statement: 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?

Identifying Problem Isomorphism

Can you help me with finding the isomorphism for this problem?

Which problem does it map to on Leetcode for 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. Do not include the 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: