Problem Decomposition

The problem might be decomposable into smaller problems that are relatively easy to solve, but which requires reassembling all the solutions to these smaller problems into an overall solution. Analyzing complex problems consists of breaking the ideas into smaller steps. Problem decomposition is about breaking hard problems into smaller pieces. Big problems are hard to solve. We deconstruct the problem by isolating the important parts of the problem.

Start from where you can make some sense of the problem and proceed from there. Break a complex problem into subproblems. Deal first with one step and then another. It is often easier to solve the smaller problems and then build an answer to the full problem from the answers to the smaller problems.

The strategy of “problem decomposition” is an essential part of problem-solving, particularly in computer science and software development. Let’s break down the main points:

  1. Problem Decomposition: Also known as “problem breaking” or “sub-problem mapping,” this is a strategy where a complex problem is divided into smaller, more manageable sub-problems. Each of these sub-problems can be addressed independently, and the solutions can then be combined to solve the overall problem. This technique is a cornerstone of many programming paradigms, including procedural, object-oriented, and functional programming.

  2. Solving Smaller Problems: Often, the smaller sub-problems are easier to solve than the original, larger problem. Each smaller problem is less complex, so you can focus on a single issue without being overwhelmed by the rest of the problem. Once you’ve solved all the smaller problems, you can assemble their solutions to solve the overall problem.

  3. Reassembling Solutions: After solving each of the smaller problems, the next step is to combine these solutions to form the solution for the original problem. This can often involve coordinating the outputs and side-effects of each sub-problem solution, or using the solutions of the sub-problems as inputs or building blocks for other parts of the overall problem’s solution.

  4. Starting Where It Makes Sense: Problem decomposition also allows you to start solving the problem in an area where you feel most comfortable or where the solution seems most apparent. This can help you make progress and build momentum towards solving the other parts of the problem.

Overall, problem decomposition is about managing complexity. By breaking a large problem down into smaller parts, you can reduce the cognitive load required to understand and solve the problem. This technique is foundational to many areas of computer science and software development, from designing algorithms to structuring large software systems.

The solution or at least a critical component is frequently seen when the problem condition is transformed toward an extreme version. This is especially true if some aspect of the problem approaches zero. The important role of zero can be seen in many problems.

For instance the number of unique ways to traverse a two dimensional grid can be broken into two simpler versions where we have one row with many columns and one column with many rows. Thus we reduce the two dimensional problem into a one dimensional problem. These two smaller instances of the problem become the base cases. Applying this principle makes the problem decomposition step a bit easier.

This is an approach to problem-solving often used in mathematics and computer science, which involves examining extreme cases, often involving zero or one, to simplify the problem.

In the example given, we’re considering the problem of determining the number of unique ways to traverse a two-dimensional grid. Without any simplification, this could potentially be quite a complex problem. But we can simplify it by transforming it into two one-dimensional problems: traversing a grid with one row and multiple columns, and traversing a grid with one column and multiple rows.

Why does this simplification help? Because the number of unique ways to traverse a one-dimensional grid is much easier to compute. In fact, there’s only one way: you just move straight from one end to the other.

We can then use these one-dimensional solutions as “base cases” in a recursive or dynamic programming approach to solve the original two-dimensional problem. This is because the number of ways to traverse the two-dimensional grid can be calculated by adding up the number of ways to traverse smaller parts of the grid, each of which is effectively a one-dimensional problem.

So, by transforming the problem to an “extreme” version where some aspect of the problem approaches zero (in this case, reducing the number of dimensions from two to one), we can make the problem easier to solve. This concept can be applied to many different problems in computer science, and is a powerful tool for problem decomposition.

The process being described here is known as “Problem Reduction” or “Problem Decomposition.” It is a common strategy in computer science and algorithm design where a complex problem is broken down into simpler, more manageable sub-problems. This often involves simplifying the problem to an extreme case (such as reducing the dimensions or reducing the problem size to zero or one) to understand the base cases, which can then be built upon to solve the original, more complex problem. When this approach is used in a recursive manner, it forms the basis of techniques such as “Divide and Conquer” and “Dynamic Programming”.

Here is the content on Problem Decomposition following the requested structure:

Problem Decomposition

Description

Problem decomposition is the process of breaking down a complex problem into smaller, manageable sub-problems. This divides the problem space and allows focusing on specific sub-problems independently.

The solutions to the sub-problems are then combined to form the solution for the overall original problem. This makes problem solving more efficient and manageable.

For example, sorting an array can be decomposed into sub-problems of sorting smaller partitions which are then merged.

Decomposition and abstraction are key concepts in software engineering and algorithm design. Decomposition allows tackling complexity.

Solution

Here is code to implement a decimal to binary conversion using decomposition:

Java

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13

String decToBinary(int n) {
  if (n == 0) 
    return "0";

  String bin = "";

  while (n > 0) {
    bin = (n % 2) + bin;
    n = n / 2;
  }

  return bin;
}

C++

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12

string decToBinary(int n) {
  if (n == 0)
    return "0"; 
  
  string bin = "";
  while (n > 0) {
    bin = to_string(n % 2) + bin;
    n = n / 2;
  }

  return bin;
}

Python

 1
 2
 3
 4
 5
 6
 7
 8
 9
10

def dec_to_binary(n):
  if n == 0:
    return "0"

  bin_str = ""
  while n > 0:
    bin_str = str(n % 2) + bin_str
    n = n // 2
  
  return bin_str

The complex decimal to binary conversion is decomposed into sub-problems of extracting remainder repeatedly and preprocessing remainder.

Decomposition breaks down complexity into manageable steps.

Description: Problem Decomposition

Problem decomposition is the act of breaking down a complex problem into smaller, more manageable sub-problems. The main goal is to reduce complexity and make the problem easier to understand and solve. Decomposing a problem can involve multiple steps, like dividing it into smaller parts, solving each part separately, and then combining the solutions to get the final answer. This technique is commonly used in divide-and-conquer algorithms, as well as in object-oriented programming.

Solution:

We’ll implement the Merge Sort algorithm to demonstrate problem decomposition. Merge Sort is an example of a divide-and-conquer algorithm where the main array is divided into two halves, sorted individually, and then merged.

Java

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19

public class ProblemDecomposition {
    public static void merge(int[] arr, int l, int m, int r) {
        // Merge two sorted sub-arrays
    }

    public static void mergeSort(int[] arr, int l, int r) {
        if (l < r) {
            int m = (l + r) / 2;
            mergeSort(arr, l, m);
            mergeSort(arr, m + 1, r);
            merge(arr, l, m, r);
        }
    }

    public static void main(String[] args) {
        int[] arr = {12, 11, 13, 5, 6, 7};
        mergeSort(arr, 0, arr.length - 1);
    }
}

C++

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21

#include <iostream>
#include <vector>
using namespace std;

void merge(vector<int>& arr, int l, int m, int r) {
    // Merge two sorted sub-arrays
}

void mergeSort(vector<int>& arr, int l, int r) {
    if (l < r) {
        int m = (l + r) / 2;
        mergeSort(arr, l, m);
        mergeSort(arr, m + 1, r);
        merge(arr, l, m, r);
    }
}

int main() {
    vector<int> arr = {12, 11, 13, 5, 6, 7};
    mergeSort(arr, 0, arr.size() - 1);
}

Python

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12

def merge(arr, l, m, r):
    # Merge two sorted sub-arrays
    
def mergeSort(arr, l, r):
    if l < r:
        m = (l + r) // 2
        mergeSort(arr, l, m)
        mergeSort(arr, m + 1, r)
        merge(arr, l, m, r)

arr = [12, 11, 13, 5, 6, 7]
mergeSort(arr, 0, len(arr) - 1)

Key Takeaways:

  • Problem decomposition simplifies complex problems by breaking them down into smaller parts.
  • Merge Sort is an example that uses problem decomposition via the divide-and-conquer approach.
  • The core algorithmic logic is consistent across Java, C++, and Python; only the syntax varies.
  • Each part of the problem is tackled individually, and the solutions are combined for the final answer.