Number Factors

Explore how to count all possible sequences summing to a target number using a fixed list of numbers. Understand naive recursive approaches and optimize them with top-down memoization and bottom-up tabulation methods in dynamic programming to improve time and space complexity.

We'll cover the following...

Statement
Examples
Try it yourself
Solution
- Naive approach
  - Time complexity
  - Space complexity
- Optimized solution using dynamic programming
  - Top-down solution
    - Time complexity
    - Space complexity
  - Bottom-up solution

Statement

Given a fixed list of numbers, $[1, 3, 4]$ , and a target number, $n$ , count all the possible ways in which $n$ can be expressed as the sum of the given numbers. If there is no possible way to represent $n$ using the provided numbers, return $0$ .

Note: You may assume that you can use a specific number as many times as you want. Additionally, the order in which we select numbers from the list is significant.

Let's say $n$ is 4, then using the given numbers, $1$ , $3$ , and $4$ , the target number $n$ can be expressed in the following four ways:

Using $1$ four times: $1+1+1+1 = 4$ .
Using a $1$ and then a $3$ : $1 + 3 = 4$ .
Using a $3$ and then a $1$ : ...

1.Getting Started

2.0/1 Knapsack

3.Unbounded Knapsack

4.Recursive Numbers

5.Longest Common Substring

6.Palindromic Subsequence

7.Conclusion

Number Factors

Statement