#1269

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

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

Compute the exact number of ways to remain at index 0 after given steps using state transition dynamic programming.

Problem Statement

You start with a pointer at index 0 in an array of length arrLen. At each step, you can move left by one, move right by one, or stay in the same position. The pointer cannot move outside the array boundaries. Given the number of steps and the array length, compute how many distinct sequences of moves will keep the pointer at index 0 after exactly the specified number of steps.

Return the total number of valid sequences modulo 10^9 + 7. For example, with steps = 3 and arrLen = 2, there are 4 sequences that return the pointer to index 0: Right-Left-Stay, Stay-Right-Left, Right-Stay-Left, and Stay-Stay-Stay. Your solution must handle up to 500 steps and arrays up to length 10^6 efficiently.

Examples

Example 1

Input: steps = 3, arrLen = 2

Output: 4

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

Example 2

Input: steps = 2, arrLen = 4

Output: 2

There are 2 differents ways to stay at index 0 after 2 steps Right, Left Stay, Stay

Example 3

Input: steps = 4, arrLen = 2

Output: 8

Example details omitted.

Constraints

1 <= steps <= 500
1 <= arrLen <= 106

Solution Approach

Dynamic Programming Table Setup

Define a DP table dp[i][j] representing the number of ways to reach position j after i steps. Initialize dp[0][0] = 1 and dp[0][j] = 0 for j > 0. Iterate through steps, updating positions based on previous step transitions: staying, moving left, or moving right within array bounds.

Space Optimization

Since only the previous step's state is needed to compute the current step, use a single array of length min(steps + 1, arrLen) to store current counts. Update values in a temporary array for each step and replace the previous array after each iteration, reducing space complexity to O(min(steps, arrLen)).

Modulo Handling and Boundary Conditions

Ensure every DP update is performed modulo 10^9 + 7 to prevent integer overflow. Limit the pointer's movement to positions less than min(steps, arrLen) since moving beyond that is impossible within the remaining steps. Return dp[steps][0] as the final count of sequences.

Complexity Analysis

Metric	Value
Time	O(n \cdot \min{(n, m)})
Space	O(\min{(n, m)})

Time complexity is O(steps * min(steps, arrLen)) because each step only updates positions reachable within steps. Space complexity is O(min(steps, arrLen)) using optimized DP storage for current and previous step counts.

What Interviewers Usually Probe

Asks about handling large array lengths and step counts.
Checks understanding of dynamic programming state transitions.
Looks for optimization from O(steps arrLen) to O(steps min(steps,arrLen)) space.

Common Pitfalls or Variants

Common pitfalls

Failing to limit positions beyond steps leads to unnecessary computation.
Forgetting modulo operations causes incorrect large-number results.
Using full 2D DP without space optimization can exceed memory limits.

Follow-up variants

Return the number of ways to reach any target index instead of index 0.
Allow only left or right moves, excluding staying in place.
Compute the number of ways with obstacles at certain indices, preventing moves onto them.

FAQ

What pattern does 'Number of Ways to Stay in the Same Place After Some Steps' use?

It uses state transition dynamic programming, tracking the number of ways to reach each index per step.

How do I optimize memory for large arrays in this problem?

Use a single array of length min(steps+1, arrLen) and update iteratively per step instead of a full 2D DP table.

Why do we need modulo 10^9 + 7 in this problem?

The number of sequences can grow extremely large, and modulo arithmetic prevents integer overflow.

Can the pointer move outside the array boundaries?

No, all moves must stay within index 0 to arrLen-1, and any out-of-bounds move is invalid.

How does GhostInterview approach the step-by-step DP solution?

It automatically generates the DP transitions for each step and enforces staying within bounds to compute the correct count.

terminal

Solution

Solution 1: Memoization Search

We observe the data range of the problem and find that $steps$ does not exceed $500$, which means that we can only go to the right for up to $500$ steps.

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class Solution:
    def numWays(self, steps: int, arrLen: int) -> int:
        @cache
        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)

Continue Practicing

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