#2400

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Number of Ways to Reach a Position After Exactly k Steps

Find the number of ways to reach a position after exactly k steps on an infinite number line using dynamic programming.

Problem Statement

You are given two integers, startPos and endPos, representing your starting and target positions on an infinite number line. You can move one position to the left or one position to the right with each step. Given a positive integer k, the task is to find how many different ways you can reach the position endPos starting from startPos in exactly k steps.

Since the number of ways can be very large, return the result modulo 10^9 + 7. Two ways are considered different if the order of the steps is not identical, meaning the sequence of left and right moves matters.

Examples

Example 1

Input: startPos = 1, endPos = 2, k = 3

Output: 3

We can reach position 2 from 1 in exactly 3 steps in three ways:

1 -> 2 -> 3 -> 2.
1 -> 2 -> 1 -> 2.
1 -> 0 -> 1 -> 2. It can be proven that no other way is possible, so we return 3.

Example 2

Input: startPos = 2, endPos = 5, k = 10

Output: 0

It is impossible to reach position 5 from position 2 in exactly 10 steps.

Constraints

1 <= startPos, endPos, k <= 1000

Solution Approach

State Transition Dynamic Programming

Start by defining a DP table where dp[i][j] represents the number of ways to reach position j in exactly i steps. The transitions are based on the fact that each step either moves left or right, so the state for a given step can be calculated from the previous step's states.

Balancing Left and Right Moves

The number of right moves can be denoted as x, and the number of left moves as y. To reach the target, the difference between right and left moves must be equal to the target offset. Ensure the total number of steps k is distributed between these two types of moves such that the difference is satisfied.

Modulo 10^9 + 7

Since the number of ways can be large, every intermediate result must be taken modulo 10^9 + 7 to avoid overflow and adhere to 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 chosen to fill the DP table, typically O(k * n), where n is the number of positions that can be reached. The space complexity will depend on whether the table is optimized to reduce memory usage, but it is typically O(k * n). Both time and space can be reduced with optimized dynamic programming approaches.

What Interviewers Usually Probe

The candidate can handle dynamic programming approaches and optimizes for space and time efficiency.
The candidate shows an understanding of state transitions and can apply them effectively to solve combinatorics problems.
The candidate demonstrates the ability to handle modular arithmetic for large numbers, which is a key aspect of the problem.

Common Pitfalls or Variants

Common pitfalls

Not handling the case where the number of left and right moves cannot sum to the required target difference.
Overcomplicating the problem by not utilizing dynamic programming efficiently, resulting in unnecessary computations.
Failing to apply the modulo operation at each step, leading to potential overflow or incorrect results.

Follow-up variants

Change the number of steps k to a larger value and analyze performance.
Introduce additional constraints, such as limiting the number of right or left moves.
Extend the problem to multidimensional grids, where you can move in more directions than just left or right.

FAQ

What is the primary approach to solving the 'Number of Ways to Reach a Position After Exactly k Steps' problem?

The problem is best solved using dynamic programming, specifically state transition DP, where you calculate the number of ways to reach a position based on the previous step's results.

How do I handle large numbers in this problem?

The solution requires applying modulo 10^9 + 7 to each result to prevent overflow and to meet the problem's constraints.

What is the time complexity for the 'Number of Ways to Reach a Position After Exactly k Steps' problem?

The time complexity typically depends on the size of the DP table, generally O(k * n), where n is the number of reachable positions.

Can I use a brute force approach for this problem?

Brute force is not efficient for this problem, as it would result in excessive computations. Dynamic programming significantly reduces time complexity.

How do I optimize space complexity for this problem?

You can optimize space complexity by using a rolling array or a 1D array instead of a full 2D DP table, as the current state only depends on the previous step.

terminal

Solution

Solution 1: Memorization Search

We design a function $dfs(i, j)$, which represents the number of ways to reach the target position when the current position is $i$ distance from the target position and there are $j$ steps left. The answer is $dfs(abs(startPos - endPos), k)$.

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class Solution:
    def numberOfWays(self, startPos: int, endPos: int, k: int) -> int:
        @cache
        def dfs(i: int, j: int) -> int:
            if i > j or j < 0:
                return 0
            if j == 0:
                return 1 if i == 0 else 0
            return (dfs(i + 1, j - 1) + dfs(abs(i - 1), j - 1)) % mod

        mod = 10**9 + 7
        return dfs(abs(startPos - endPos), k)

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