#115 Medium Time O(n·sum) Space O(n·sum) Animated

Target Sum

Count ways to assign +/- to reach target sum.

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1 Predict
2 Trace
3 Explain
4 Implement

Pattern Two-dimensional dynamic programming

A table exposes how two changing prefixes or positions depend on smaller cases.

Approach: Dynamic Programming (Memoization)

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Core Insight

A table exposes how two changing prefixes or positions depend on smaller cases.

Common Pitfall

Initialize the first row and column before using neighboring cells.

Description

Count ways to assign +/- to reach target sum.

Constraints

Time Complexity: O(n·sum)
Space Complexity: O(n·sum)

Solution — Dynamic Programming (Memoization)

TimeO(n·sum) SpaceO(n·sum)

Implementation Plan

Translate the invariant into these small, testable moves.

Define the row and column meaning.
Initialize base cases.
Fill cells in dependency order.
Read the answer from the target cell.

def findTargetSumWays(nums, target):
    dp = {} # (index, total) -> # of ways
    
    def backtrack(i, total):
        if i == len(nums):
            return 1 if total == target else 0
        if (i, total) in dp:
            return dp[(i, total)]
            
        dp[(i, total)] = (backtrack(i + 1, total + nums[i]) +
                          backtrack(i + 1, total - nums[i]))
        return dp[(i, total)]
        
    return backtrack(0, 0)

Coin Change II

Interleaving String

Target Sum

Steps

Description

Constraints

Tags

Solution — Dynamic Programming (Memoization)