Problem Description
You are given a 0-indexed array of integers nums, and an integer target. Return the length of the longest subsequence of nums that sums up to target. If no such subsequence exists, return -1. A subsequence is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.
Key Insights
- The problem requires finding subsequences, which can be efficiently approached using dynamic programming.
- We need to consider all possible combinations of elements to determine which combinations can sum to the target.
- The order of elements in the original array must be preserved in the subsequences.
- If multiple subsequences can sum to the target, we need to track the maximum length among them.
Space and Time Complexity
Time Complexity: O(n * target) - where n is the number of elements in nums and target is the target sum.
Space Complexity: O(target) - we use a 1D array to store the maximum lengths for each possible sum up to the target.
Solution
To solve this problem, we can use dynamic programming. We maintain a DP array where dp[i] represents the maximum length of a subsequence that sums up to i. We iterate through each number in the nums array and update the DP array in reverse order, ensuring that each number is only used once per position in the DP array. At the end, we will check the value at dp[target] to determine the length of the longest subsequence that sums to target.