#3179

Medium

LeetCode Problem Workspace

Find the N-th Value After K Seconds

Solve for the N-th value after K seconds by simulating array updates and using prefix sum techniques.

array math simulation combinatorics prefix sum

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?

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.

terminal

Solution

Solution 1: Simulation

We notice that the range of the integer $n$ is $1 \leq n \leq 1000$, so we can directly simulate this process.

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class Solution:
    def valueAfterKSeconds(self, n: int, k: int) -> int:
        a = [1] * n
        mod = 10**9 + 7
        for _ in range(k):
            for i in range(1, n):
                a[i] = (a[i] + a[i - 1]) % mod
        return a[n - 1]

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