To solve this problem, simulate the process of updating an array based on prefix sum updates. After each second, each element becomes the sum of itself and its preceding elements. The challenge lies in applying this transformation efficiently for K seconds and returning the N-th value of the array.
Problem Statement
You are given two integers, n and k. You start with an array a of n integers, where each element is initially 1. After every second, each element of the array is updated to be the sum of all its preceding elements plus itself. For example, after one second, the array updates such that a[0] stays the same, a[1] becomes a[0] + a[1], a[2] becomes a[0] + a[1] + a[2], and so on.
Your task is to return the value of a[n - 1] after performing k seconds of these updates.
Examples
Example 1
Input: n = 4, k = 5
Output: 56
Example 2
Input: n = 5, k = 3
Output: 35
Constraints
- 1 <= n, k <= 1000
Solution Approach
Simulate the Array Updates
For each second, update the array by iterating through all elements and adjusting each element to the sum of itself and its preceding elements. This direct simulation helps to understand the mechanics of the problem, though it can be inefficient for large values of n and k.
Use Prefix Sum Optimization
Instead of performing full updates for each element, maintain a running sum of the elements to optimize the computation of each element's new value. This prefix sum approach significantly reduces the time complexity and helps avoid redundant calculations.
Efficient Calculation Through Iteration
To calculate the final value of a[n - 1] efficiently, iterate the array up to k times and update each element using the prefix sum technique. This iterative process ensures that you can handle larger values of n and k within the problem’s constraints.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time complexity depends on the approach used. A naive simulation approach results in O(n * k), while using prefix sums can reduce the complexity to O(n * k) with optimizations. The space complexity is O(n), as the array requires storage for n elements.
What Interviewers Usually Probe
- Can the candidate optimize a brute-force simulation approach using mathematical tricks like prefix sums?
- Does the candidate understand the pattern of updating elements based on their predecessors?
- Is the candidate comfortable with trade-offs between time and space complexity?
Common Pitfalls or Variants
Common pitfalls
- Overlooking the need for prefix sum optimization and resorting to a brute-force simulation, leading to inefficiency.
- Not correctly handling the simulation for large values of k, resulting in time limit exceeded errors.
- Misunderstanding the problem's structure and failing to implement the running sum technique effectively.
Follow-up variants
- What if the array starts with different initial values instead of 1?
- What if the problem only asks for the sum of the array instead of the last element after k seconds?
- Can this approach be adapted for multidimensional arrays or more complex data structures?
How GhostInterview Helps
- GhostInterview provides insight into how to optimize simulations using mathematical techniques like prefix sums.
- The solver focuses on providing both brute-force and optimized solutions, aiding the understanding of trade-offs.
- GhostInterview guides you through efficiently calculating updates without redundant computations, helping improve performance.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the best way to approach 'Find the N-th Value After K Seconds'?
Start by simulating the array updates, then optimize using prefix sums to avoid redundant recalculations and improve performance.
How can I optimize my solution for large n and k values?
Use the prefix sum technique to reduce unnecessary calculations, which helps in achieving better time complexity.
What is the time complexity of this problem?
The time complexity depends on the solution. A direct simulation is O(n * k), while prefix sum optimization can reduce it to O(n * k) with better performance.
How does prefix sum help in this problem?
Prefix sum allows for faster computation of each element's updated value by maintaining a cumulative sum, thus reducing the need for multiple iterations.
Is it possible to solve this problem without a prefix sum approach?
Yes, but it would be inefficient, especially for large values of n and k, and would lead to time limit exceeded errors.
Need direct help with Find the N-th Value After K Seconds instead of spending more time grinding it?
Download GhostInterview when you want a LeetCode solver, not another long practice loop. Capture Find the N-th Value After K Seconds from a screenshot, get the answer path and complexity, and use supported stealth workflows that stay outside captured layers.
Capture the prompt fast instead of rewriting the problem by hand.
Get the solution path, trade-offs, and complexity summary in one pass.
Stay outside captured layers on supported screen-share workflows.
Stay in the same pattern family
Compute the number of monotonic pairs in an integer array using state transition dynamic programming efficiently.
Open problem page#3251 Find the Count of Monotonic Pairs IIThis problem involves finding the count of monotonic pairs in an array using dynamic programming and combinatorics techniques.
Open problem page#3312 Sorted GCD Pair QueriesSolve the Sorted GCD Pair Queries problem by efficiently counting and locating GCDs of all array pairs using hash mapping.
Open problem page