I started with recursion, which felt natural, but then they asked me to compare it against memoization and a full iterative DP solution.
Start by clarifying the problem: we need to count distinct sign assignments, not list them. Then propose a dynamic programming solution that tracks the number of ways to reach each possible sum, using the fact that the total sum is bounded. Finally, discuss trade-offs between time and space complexity and possible optimizations.
Pro tip: Mention that this is a variation of the Partition Equal Subset Sum problem and that you can reduce it to counting subsets with a specific sum, which shows deeper insight. Also, proactively discuss edge cases like zeros and large targets to demonstrate thoroughness.
Confirm that we need to count distinct sign assignments (order matters? no, assignments are per element) and that the array contains non-negative integers. Ask about constraints on array size and target value.
Recognize that this is a subset sum counting problem: assign + to one subset and - to the complement, so the expression equals (sum of + subset) - (sum of - subset) = T. This implies sum(+) = (total_sum + T)/2.
Use a 1D DP array where dp[s] = number of ways to achieve sum s using a subset of the numbers. Iterate through each number and update dp in reverse to avoid reusing the same number.
Check if (total_sum + T) is odd or negative, then return 0. Also handle zeros correctly (they double the number of ways). Discuss space optimization and potential overflow.
Time complexity O(n * sum) and space O(sum). Mention that if sum is large, this may be inefficient, and discuss alternative approaches like meet-in-the-middle for smaller n.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Zeros tripped me up for a second because they don't change the expression value but they do double the number of valid assignments.
First, clarify the specific algorithm or problem context (e.g., prefix sums, sliding window, or division) to tailor your answer. Then, explain how zeros affect the algorithm's behavior and how large sums impact data types and performance. Finally, discuss trade-offs and mitigation strategies, such as using appropriate data types or handling edge cases explicitly.
Pro tip: Demonstrate awareness of Amazon's Leadership Principles by emphasizing customer impact: e.g., ensuring correctness for zero values prevents bugs that affect users, and handling large sums avoids overflow that could lead to incorrect results or system failures.
Ask or state the specific algorithm or problem being solved (e.g., prefix sums, sliding window, division) to ground your answer. This shows you don't make assumptions and tailor solutions to the actual use case.
Describe how zeros affect the algorithm: e.g., in prefix sums, zeros don't change cumulative sums; in division, zeros cause division by zero; in sliding window, zeros may affect window validity. Mention any special handling needed.
Discuss potential issues with large sums: integer overflow, precision loss with floating-point, or performance degradation. Explain how to choose appropriate data types (e.g., 64-bit integers, BigInteger) or algorithms that avoid large sums.
Compare solutions: e.g., using modulo arithmetic to avoid overflow, or two-pointer techniques instead of prefix sums. Highlight trade-offs between time, space, and correctness.
Conclude by mentioning edge cases (all zeros, very large values) and how you would test them. This shows thoroughness and a quality-focused mindset.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Backtracking through the DP table to reconstruct a path.
Explain that you will augment the DP table to store not just counts but also the previous state and chosen sign, then backtrack from the target to reconstruct one valid assignment. Emphasize that this adds O(n * sum) space but keeps the same time complexity, and that you can also use a parent pointer array to save memory.
Pro tip: Mention that you can avoid storing the entire DP table by using a hash map of reachable sums with parent pointers, which is more memory-efficient for large sums. Also, clarify that if multiple valid assignments exist, returning any one is acceptable, so you can stop early during backtracking.
Confirm that the input is an array of numbers and a target sum, and that each number can be assigned a '+' or '-' sign. Ask about constraints (e.g., array size, sum range) to decide on the DP approach and memory usage.
Define dp[i][s] as the number of ways to assign signs to the first i numbers to reach sum s. Additionally, store a parent pointer or a choice array indicating which sign was used to reach s from the previous state.
Iterate through the numbers and sums, updating dp[i][s] based on dp[i-1][s - num] (for '+') and dp[i-1][s + num] (for '-'). When a transition is valid, record the chosen sign and the previous sum in auxiliary arrays.
Starting from dp[n][target], follow the recorded choices backwards to determine the sign for each number. If the target is unreachable, return an empty list or indicate no solution.
State that time complexity is O(n * sum) and space is O(n * sum) for the full table, but can be reduced to O(sum) using a 1D DP with parent pointers or a hash map. Mention that early termination during backtracking can save time.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.