Leetcode #1269: Number of Ways to Stay in the Same Place After Some Steps
In this guide, we solve Leetcode #1269 Number of Ways to Stay in the Same Place After Some Steps in Python and focus on the core idea that makes the solution efficient.
You will see the intuition, the step-by-step method, and a clean Python implementation you can use in interviews.
Problem Statement
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 in the array, or stay in the same place (The pointer should not be placed outside the array at any time).
Quick Facts
- Difficulty: Hard
- Premium: No
- Tags: Dynamic Programming
Intuition
The problem breaks into overlapping subproblems, so caching results prevents exponential repetition.
A carefully chosen DP state captures exactly what we need to build the final answer.
Approach
Define the DP state and recurrence, then compute states in the correct order.
Optionally compress space once the recurrence is clear.
Steps:
- Choose a DP state definition.
- Write the recurrence and base cases.
- Compute states in the correct order.
Example
Input: steps = 3, arrLen = 2
Output: 4
Explanation: There are 4 differents ways to stay at index 0 after 3 steps.
Right, Left, Stay
Stay, Right, Left
Right, Stay, Left
Stay, Stay, Stay
Python Solution
class Solution:
def numWays(self, steps: int, arrLen: int) -> int:
def dfs(i, j):
if i > j or i >= arrLen or i < 0 or j < 0:
return 0
if i == 0 and j == 0:
return 1
ans = 0
for k in range(-1, 2):
ans += dfs(i + k, j - 1)
ans %= mod
return ans
mod = 10**9 + 7
return dfs(0, steps)
Complexity
The time complexity is , and the space complexity is . The space complexity is .
Edge Cases and Pitfalls
Watch for boundary values, empty inputs, and duplicate values where applicable. If the problem involves ordering or constraints, confirm the invariant is preserved at every step.
Summary
This Python solution focuses on the essential structure of the problem and keeps the implementation interview-friendly while meeting the constraints.