Combination Sum III

tags: combination backtracking narrowing-the-domains-of-the-variables

Find all valid combinations of k numbers that sum up to n such that the following conditions are true:

  • Only numbers 1 through 9 are used.
  • Each number is used at most once.

Return a list of all possible valid combinations. The list must not contain the same combination twice, and the combinations may be returned in any order.

Example 1:
Input: k = 3, n = 7
Output: [[1,2,4]]
Explanation:
1 + 2 + 4 = 7
There are no other valid combinations.

Example 2:
Input: k = 3, n = 9
Output: [[1,2,6],[1,3,5],[2,3,4]]
Explanation:
1 + 2 + 6 = 9
1 + 3 + 5 = 9
2 + 3 + 4 = 9
There are no other valid combinations.

Example 3:
Input: k = 4, n = 1
Output: []
Explanation: There are no valid combinations. [1,2,1] is not valid because 1 is used twice.

Example 4:
Input: k = 3, n = 2
Output: []
Explanation: There are no valid combinations.

Example 5:
Input: k = 9, n = 45
Output: [[1,2,3,4,5,6,7,8,9]]
Explanation:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

​​​​​​​There are no other valid combinations.

Constraints

  • 2 <= k <= 9
  • 1 <= n <= 60

Thinking Process

Define the Interface

Input: n, k (both are integers)
Output: array of combinations (array of arrays)

Implicit input in the problem statement, we can use only 1,2,3,4,5,6,7,8 and 9 to generate the combinations.

Analyze the Inputs and Outputs

What is the relationship between the input and output?

  • The size of the combination in the output cannot exceed k.
  • The sum of the numbers must be equal to n.
  • The combinations in the output must be unique.

This means: (1,2,3) is the same as (2,1,3) or (3,2,1) or these numbers can appear in any order.

We want to choose k objects from a set of n objects. We are not interested in ordering them. The number of different ways that k objects can be chosen from a set of n objects (when order doesn’t matter) is called n choose k. The formula that relates n and k

There are 35 combinations. We can use the n choose k formula to find out how many different ways you can choose k items from n items set without repetition and without order. This number is also called combination number or n choose k or binomial coefficient or simply combinations.

Classify the Problem

We use each number at most once. So this is bound by one. We can either pick an element or not pick an element, so it is 0-1. This problem is 0-1 Bounded Knapsack problem.

Identify the Invariant

  • Sum of all the numbers in the combination must be equal to n (capacity)
  • Length must be equal to k
  • Combination must be unique

Solve the Problem By Hand

How to generate the combinations by hand? Let’s work through example 1. What is the starting point?

k = 3, n = 9

0 1 2 3 4 5 6 7 8 [1, 2, 3, 4, 5, 6, 7, 8, 9]

The numbers are in ascending order.

  • Pick the first number (1) - at most once.
  • From numbers 2 to 9, we can pick 2, so we now have (1, 2).
  • 2 is now taken, now we have (3 to 9) available for the next choice.
  • Select 3, 3 does not sum up to 7 (1+2+3 != 7).
  • We try 4 in place of 3, we now have a sum of 7 (1+2+4 == 7)

If the sum exceeds n, we don’t have a valid combination. We can define a bounding function to avoid unnecessary work.

Choose an Approach

  • Recursion
  • Backtracking

Identify the Base Case

When the current list size == k and remaining sum == 0, we have found a valid combination. Add it to the output array.

Pruning the Tree

(don’t explore paths leading to infeasible solutions). If the remaining sum is < 0 or if the combination exceeds, return.

Wrapper Method Parameters

How do we keep track of the remaining sum? We can need the following parameters in the backtrack method:

  • remaining
  • start
  • k

Narrowing the Domain of the Variable

  • How to control the children’s index so that we don’t look back?
  • Should we increment the index?

The reason is that we want to know whether I need to include myself or not. This is an example for narrowing the domain of the variable. We have the numbers that we choose in order: [1,2,3,4,5,6,7,8,9]. At any given step, once we pick a digit, e.g 4, we will not consider any digits before 4 for the subsequent steps. The candidates are reduced down to [5,6,7,8,9]. This illustrates narrowing the domain of the variable.

Implementation

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def backtrack(n, k, combination, start, output)    
  if combination.sum == n && combination.size == k
    # We need to copy the combination to output, otherwise it will be overwritten by
    # other recursive calls
    output << combination.dup
    return
  end
  # Pruning function: Avoid exploring paths that lead to infeasible solutions
   if (combination.sum != n && combination.size == k)
    return
  end
  # The start index keeps increasing in every recursive call. 
  # This is narrowing the domain of the variable. 
  for i in (start..9) 
    # choose
    combination << i
    # explore
    backtrack(n, k, combination, i+1, output)
   # unchoose
    combination.pop
  end
end

# @param {Integer} k
# @param {Integer} n
# @return {Integer[][]}
def combination_sum3(k, n)
  output = []
  combination = []
  start = 1
  backtrack(n, k, combination, start, output)
  output
end

Modify the code as follows. Does it run faster? Why?

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def backtrack(n, k, combination, start, output)
  if n == 0 && combination.size == k
    output << combination.dup
    return
  elsif (n < 0 || combination.size == k)
    return
  end
    
  for i in (start..9)
    combination << i+1
    backtrack(n-i-1, k, combination, i+1, output)
    combination.pop
  end
end

# @param {Integer} k
# @param {Integer} n
# @return {Integer[][]}
def combination_sum3(k, n)
  output = []
  combination = []
  backtrack(n, k, combination, 0, output)
  output
end

Alternative Implementation

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def backtrack(n, k, combination, start, output)
  # Instead of computing the sum over and over again, we can check the value of n
  if n == 0 && combination.size == k
    output << combination.dup
    return
  elsif (n < 0 || combination.size > k)
    return
  end
    
  for i in (start..9)
    combination << i
    # Reduce the value of n by the chosen value i
    backtrack(n-i, k, combination, i+1, output)
    combination.pop
  end
end

# @param {Integer} k
# @param {Integer} n
# @return {Integer[][]}
def combination_sum3(k, n)
  output = []
  combination = []
  start = 1
  backtrack(n, k, combination,start, output)
  output
end

Complexity Analysis

Time: O( ) Space: O(K)

Here’s a solution to the problem of finding all valid combinations of k numbers that sum up to n, using numbers 1 through 9 at most once:

  1. Input Constraints: k represents the number of numbers in each combination, and n is the target sum.
  2. Output: A list of all possible valid combinations.

We can achieve this by recursion and backtracking:

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class Solution:
    def combinationSum3(self, k: int, n: int) -> List[List[int]]:
        def backtrack(start, k, n, path):
            # If there are k numbers and the sum is n, add the path to the result
            if k == 0 and n == 0:
                result.append(path[:])
                return
            # If k reaches 0 or n reaches 0, the path is not valid
            if k == 0 or n == 0:
                return
            # Iterate from the current number to 9
            for i in range(start, 10):
                # Add the current number to the path
                path.append(i)
                # Continue the search with the updated k and n
                backtrack(i + 1, k - 1, n - i, path)
                # Remove the current number from the path to backtrack
                path.pop()

        result = []
        backtrack(1, k, n, [])
        return result

This code sets up a recursive function backtrack to try all possible combinations and add the valid ones to the result list. It starts from 1, and for each number i, it adds it to the current path and continues the search with k - 1 numbers remaining and n - i as the new target sum. The path is then backtracked by removing the current number, and the process continues with the next number.

The final result is a list of all valid combinations satisfying the given conditions.

  1. Classify the Problem 0-1 Bounded Knapsack Problem

  2. Identify the Invariant

    • Sum of all the numbers in the combination must be equal to n (capacity)
    • Length must be equal to k
    • Combination must be unique

  3. How to Generate the Combinations by hand?

    • What is the starting point? Input: n, k Output: array of combinations

      0 1 2 3 4 5 6 7 8 [1, 2, 3, 4, 5, 6, 7, 8, 9]

      Pick the first number (1) - at most once From 2 to 9 (1, 2) Second is also taken now we have (3 to 9) If 3 is not working (1+2+3 != 7) Try 4, (1+2+4 == 7)

      If the sum exceeds n, we don’t have a valid combination We can define a bounding function to avoid unnecessary work The numbers are in ascending order

      k = 3, n = 9

      0 1 2 3 4 5 6 7 8 [1, 2, 3, 4, 5, 6, 7, 8, 9]

  4. Recursion Backtracking

  5. Base Case

    • The current list size == k and remaining sum == 0 We found a valid combination and add it to the output array

    Pruning the Tree (don’t explore paths leading infeasible solution)

    • If the remaining sum is < 0 or if the combination exceeds Return

  6. How do we keep track of the remaining sum

    • remaining
    • start
    • k

  7. How to control the children’s index so that we don’t look back

    • Should I increment the index? The reason is that we want to know whether I need to include myself or not.

n choose k formula

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def backtrack(n, k, combination, start, output)
   if combination.sum == n && combination.size == k
       output << combination.dup
       return
   elsif (combination.sum != n && combination.size == k)
       return
   end
    
    for i in (start..9)
       combination << i
        backtrack(n, k, combination, i+1, output)
        combination.pop
    end
end

# @param {Integer} k
# @param {Integer} n
# @return {Integer[][]}
def combination_sum3(k, n)
    output = []
    combination = []
    backtrack(n, k, combination, 1, output)
    output
end

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def backtrack(n, k, combination, start, output)
    if n == 0 && combination.size == k
        output << combination.dup
        return
    elsif (n < 0 || combination.size == k)
        return
    end
    
    for i in (start..9)
       combination << i
       backtrack(n-i, k, combination, i+1, output)
       combination.pop
    end
end

# @param {Integer} k
# @param {Integer} n
# @return {Integer[][]}
def combination_sum3(k, n)
  output = []
  combination = []
  backtrack(n, k, combination, 1, output)
  output
end

Identifying Problem Isomorphism

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

In “Combination Sum III”, you have 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 can only be used once.

“Combination Sum IV” asks for the number of possible combinations that add up to a target number, given a list of candidate numbers. The difference here is that the numbers can be used unlimited times.

While “Combination Sum III” and “Combination Sum IV” differ in their constraints and details - one has limited numbers (1-9) and each can be used only once, the other has a list of candidate numbers and each number can be used unlimited times - they both involve finding combinations of numbers that sum to a target.

Thus, understanding the solution for “Combination Sum III” can provide valuable insights into solving “Combination Sum IV”, and vice versa, due to their shared theme of combination and summation.

10 Prerequisite LeetCode Problems

Identify 10 LeetCode problems of lesser complexity, excluding the problem itself that I should solve as preparation for tackling 216. Combination Sum III . 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?

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: