Problem Statement
Number of Ways to Stay in the Same Place After Some Steps
You have a pointer at index 0 in an array of size arrLen. At each step, you can move 1 position to the left, 1 position to the right, or stay in the same place (The pointer should not be placed outside the array at any time).
Given two integers steps and arrLen, return the number of ways such that your pointer is at index 0 after exactly steps steps. Since the answer may be too large, return it modulo 10^9 + 7.
Examples
Example 1:
Input: steps = 3, arrLen = 2
Output: 4
Explanation: There are 4 different ways to stay at index 0 after 3 steps:
Right, Left, Stay
Stay, Right, Left
Right, Stay, Left
Stay, Stay, Stay
Example 2:
Input: steps = 2, arrLen = 4
Output: 2
Explanation: There are 2 different ways to stay at index 0 after 2 steps:
Right, Left
Stay, Stay
Example 3:
Input: steps = 4, arrLen = 2
Output: 8
Explanation: There are 8 different ways to stay at index 0 after 4 steps.
Constraints
- 1 <= steps <= 500
- 1 <= arrLen <= 10^6
Problem Breakdown
To solve this problem, we need to:
- This problem can be solved using dynamic programming.
- The key insight is to define dp[i][j] as the number of ways to be at position j after i steps.
- For each step i and position j, we can consider three possible moves: stay, move left, or move right.
- The recurrence relation is: dp[i][j] = dp[i-1][j] + dp[i-1][j-1] + dp[i-1][j+1].
- We can optimize the space complexity by observing that we only need to consider positions up to min(arrLen-1, steps), as we can't go further than steps positions away from the starting point.
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Number of Ways to Stay in the Same Place After Some Steps
You have a pointer at index 0 in an array of size arrLen. At each step, you can move 1 position to the left, 1 position to the right, or stay in the same place (The pointer should not be placed outside the array at any time).
Given two integers steps and arrLen, return the number of ways such that your pointer is at index 0 after exactly steps steps. Since the answer may be too large, return it modulo 10^9 + 7.
Examples
Example 1:
There are 4 different ways to stay at index 0 after 3 steps: Right, Left, Stay Stay, Right, Left Right, Stay, Left Stay, Stay, Stay
Example 2:
There are 2 different ways to stay at index 0 after 2 steps: Right, Left Stay, Stay
Example 3:
There are 8 different ways to stay at index 0 after 4 steps.
Problem Breakdown
Constraints:
- 1 <= steps <= 500
- 1 <= arrLen <= 10^6
Key Insights:
This problem can be solved using dynamic programming.
The key insight is to define dp[i][j] as the number of ways to be at position j after i steps.
For each step i and position j, we can consider three possible moves: stay, move left, or move right.
The recurrence relation is: dp[i][j] = dp[i-1][j] + dp[i-1][j-1] + dp[i-1][j+1].
We can optimize the space complexity by observing that we only need to consider positions up to min(arrLen-1, steps), as we can't go further than steps positions away from the starting point.
Real-World Application
This problem has several practical applications:
Random Walk Problems
Modeling random walks in one dimension, which are used in physics, economics, and other fields to describe paths that consist of a succession of random steps.
Probability Calculations
Calculating the probability of returning to a starting point after a certain number of steps in a stochastic process.
Movement Planning
Planning movement patterns in constrained environments, such as robots navigating in a linear space with boundaries.