Target Sum

LeetCode 494 | Difficulty: Medium

Medium

Problem Description

You are given an integer array nums and an integer target.

You want to build an expression out of nums by adding one of the symbols '+' and '-' before each integer in nums and then concatenate all the integers.

- For example, if `nums = [2, 1]`, you can add a `'+'` before `2` and a `'-'` before `1` and concatenate them to build the expression `"+2-1"`.

Return the number of different expressions that you can build, which evaluates to target.

Example 1:

Input: nums = [1,1,1,1,1], target = 3
Output: 5
Explanation: There are 5 ways to assign symbols to make the sum of nums be target 3.
-1 + 1 + 1 + 1 + 1 = 3
+1 - 1 + 1 + 1 + 1 = 3
+1 + 1 - 1 + 1 + 1 = 3
+1 + 1 + 1 - 1 + 1 = 3
+1 + 1 + 1 + 1 - 1 = 3

Example 2:

Input: nums = [1], target = 1
Output: 1

Constraints:

- `1 <= nums.length <= 20`

- `0 <= nums[i] <= 1000`

- `0 <= sum(nums[i]) <= 1000`

- `-1000 <= target <= 1000`

Topics: Array, Dynamic Programming, Backtracking

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Approach

Dynamic Programming

Break the problem into overlapping subproblems. Define a state (what information do you need?), a recurrence (how does state[i] depend on smaller states?), and a base case. Consider both top-down (memoization) and bottom-up (tabulation) approaches.

When to use

Optimal substructure + overlapping subproblems (counting ways, min/max cost, feasibility).

Backtracking

Explore all candidates by building solutions incrementally. At each step, choose an option, explore further, then unchoose (backtrack) to try the next option. Prune branches that can't lead to valid solutions.

When to use

Generate all combinations/permutations, or find solutions that satisfy constraints.

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Solutions

Solution 1: C# (Best: 120 ms)

Metric	Value
Runtime	120 ms
Memory	40.7 MB
Date	2020-01-01

Solution

public class Solution {
    public int FindTargetSumWays(int[] nums, int S) {
        int[] dp = new int[2001];
    dp[nums[0] + 1000] = 1;
    dp[-nums[0] + 1000] += 1;
    for (int i = 1; i < nums.Length; i++)
    {
        int[] next = new int[2001];
        for (int sum = -1000; sum <= 1000; sum++)
        {
            if (dp[sum + 1000] > 0)
            {
                next[sum + nums[i] + 1000] += dp[sum + 1000];
                next[sum - nums[i] + 1000] += dp[sum + 1000];
            }
        }
        dp = next;
    }
    return S > 1000 ? 0 : dp[S + 1000];
    }
}

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Complexity Analysis

Approach	Time	Space
Dynamic Programming	$O(n)$	$O(n)$
Backtracking	$O(n! or 2^n)$	$O(n)$

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Interview Tips

Key Points

• Discuss the brute force approach first, then optimize. Explain your thought process.
• Define the DP state clearly. Ask: "What is the minimum information I need to make a decision at each step?"
• Consider if you can reduce space by only keeping the last row/few values.
• Identify pruning conditions early to avoid exploring invalid branches.