Combination Sum IV: A Dynamic Programming Solution

The "Combination Sum IV" problem is a dynamic programming challenge that focuses on finding the total number of possible combinations that add up to a given target number, using elements from an array.

Problem Statement

Given an array of distinct integers nums and a target integer target, return the number of possible combinations that add up to target.

Example

• Input: nums = [1, 2, 3], target = 4 Output: 7 Explanation: The possible combination ways are:
  • (1, 1, 1, 1)
  • (1, 1, 2)
  • (1, 2, 1)
  • (1, 3)
  • (2, 1, 1)
  • (2, 2)
  • (3, 1)

Dynamic Programming Solution

Breaking Down the Solution

• Initialize dp Array: Create a dp array of length target + 1 and initialize it with zeros. Set dp[0] to 1, representing the base case.

• Dynamic Programming Iteration: Iterate through each possible sum from 1 to target. For each sum, iterate through the numbers in nums and add to dp[i] the number of ways to reach i - num.

• Calculate Total Combinations: By the end of the iterations, dp[target] contains the total number of ways to reach the target sum using numbers from nums.

Conclusion

The Combination Sum IV problem is a valuable exercise in dynamic programming, demonstrating how to efficiently solve problems related to counting and combinations. It illustrates the importance of building up solutions for smaller subproblems and combining them to form the solution to the overall problem.

Rust Solution

Rust's as usize / as i32 casts carry the whole solution: Vec indices must be usize while the values are i32, so every crossing between the two is spelled out — (target + 1) as usize for the length, i as i32 >= num for the guard, and num as usize inside dp[i - num as usize] — unlike JS where a single number type covers both roles. The vec![0i32; (target + 1) as usize] macro allocates and zero-fills in one step, and for &num in &nums borrows the vector and copy-destructures each element so nums is never consumed. The final dp[target as usize] carries no semicolon, making it the tail expression the function returns.

Go Solution

Go keeps every quantity as int, so no casts appear anywhere — i >= num and dp[i-num] compare and index freely because slice indices and values share the one type, sidestepping the signed/unsigned juggling Rust needs. make([]int, target+1) returns the slice already zero-filled (0 is the zero value for int), so dp[0] = 1 just overwrites the base case. The for _, num := range nums loop discards the index via the blank identifier _, since only each value matters here.