Number of Ways to Stay in the Same Place After Some Steps

Number of Ways to Stay in the Same Place After Some Steps

What is this problem about?

The Number of Ways to Stay in the Same Place After Some Steps problem asks you to count the number of ways to take exactly steps steps on an array of length arrLen, starting and ending at index 0. Each step you can move left, right, or stay in place (within bounds). Return the count modulo 10^9+7. This hard coding problem uses 2D DP with careful space optimization.

Why is this asked in interviews?

Google asks this because it requires reasoning about reachable positions — after k steps, the maximum reachable index is min(steps/2, arrLen-1), which bounds the state space. The dynamic programming interview pattern with careful state space pruning is the core skill tested.

Algorithmic pattern used

2D DP with state compression. dp[step][pos] = ways to be at position pos after step steps. Transition: dp[s][p] = dp[s-1][p] + dp[s-1][p-1] + dp[s-1][p+1] (stay, came from right, came from left). Key optimization: max reachable position = min(steps//2, arrLen-1). Use rolling array to reduce space. Answer: dp[steps][0].

Example explanation

steps=3, arrLen=2. Positions: 0 and 1. dp[0][0]=1. After step 1: dp[1][0]=1(stay)+1(from 1 but arrLen constraint)=1, dp[1][1]=1(from 0). After step 2: dp[2][0]=dp[1][0]+dp[1][1]=2, dp[2][1]=dp[1][0]+dp[1][1]=2. After step 3: dp[3][0]=dp[2][0]+dp[2][1]=4, dp[3][1]=dp[2][0]+dp[2][1]=4. Answer = dp[3][0] = 4.

Common mistakes candidates make

• Not bounding the maximum reachable index (leads to TLE or MLE).
• Forgetting bounds checks when accessing p-1 and p+1.
• Not applying modular arithmetic at each addition.
• Using O(steps * arrLen) space without the rolling array optimization.

Interview preparation tip

The key insight is that after k steps from position 0, you can never be beyond position k (or arrLen-1). This bounds the effective state space dramatically. Practice similar 1D random walk DP problems where position-bounding prunes the state space. The rolling array space optimization is standard for DP where only the previous row is needed.