What is the time complexity of Number of Ways to Rearrange Sticks With K Sticks Visible?

The optimal solution for Number of Ways to Rearrange Sticks With K Sticks Visible runs in O(n * k) time and uses O(n * k) space. The complexity is O(n * k) due to the nested loops filling the DP table, which is efficient compared to the brute force approach.

What is the brute force solution for Number of Ways to Rearrange Sticks With K Sticks Visible?

The brute force approach for Number of Ways to Rearrange Sticks With K Sticks Visible is Brute Force, with O(n!) time complexity and O(n) space complexity. The brute force approach tries all possible arrangements of the sticks and counts how many of those arrangements have exactly k sticks visible from the left. This is straightforward but inefficient due to the factorial growth of permutations. The optimal Optimal Solution approach improves this to O(n * k).

What algorithm or data structure is used to solve Number of Ways to Rearrange Sticks With K Sticks Visible?

Number of Ways to Rearrange Sticks With K Sticks Visible uses the following concepts: Math, Dynamic Programming, Combinatorics. The recommended patterns to study are: Dynamic Programming, Combinatorics.

What are the interview tips for Number of Ways to Rearrange Sticks With K Sticks Visible?

Always start with a simple approach to understand the problem better. Be prepared to explain your thought process and how you arrived at the optimal solution. Practice writing out your dynamic programming transitions clearly.

What are common mistakes when solving Number of Ways to Rearrange Sticks With K Sticks Visible?

Not considering the base cases in dynamic programming. Overcomplicating the brute force solution instead of focusing on counting visible sticks correctly.

#1866

Number of Ways to Rearrange Sticks With K Sticks Visible

Hard

MathDynamic ProgrammingCombinatoricsDynamic ProgrammingCombinatorics

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n!)	O(n * k)
Space	O(n)	O(n * k)

💡

Intuition

Time O(n * k)Space O(n * k)

The optimal solution uses dynamic programming to build the number of arrangements based on previously computed values. It leverages combinatorial principles to efficiently calculate the number of ways to arrange sticks with exactly k visible.

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Algorithm

5 steps

1Step 1: Create a DP table dp[i][j] where dp[i][j] represents the number of ways to arrange i sticks with exactly j visible.
2Step 2: Initialize base cases: dp[0][0] = 1 (0 sticks, 0 visible).
3Step 3: Use the relation: dp[i][j] = dp[i-1][j-1] * (i - 1) + dp[i-1][j] * (i - 1), where the first term accounts for placing the longest stick visible and the second for not placing it.
4Step 4: Iterate through the DP table to fill it up to dp[n][k].
5Step 5: Return dp[n][k] modulo 10^9 + 7.

solution.py8 lines

1def count_visible_sticks(n, k):
2    MOD = 10**9 + 7
3    dp = [[0] * (k + 1) for _ in range(n + 1)]
4    dp[0][0] = 1
5    for i in range(1, n + 1):
6        for j in range(1, min(k, i) + 1):
7            dp[i][j] = (dp[i - 1][j - 1] * (i - 1) + dp[i - 1][j] * (i - 1)) % MOD
8    return dp[n][k]

ℹ

Complexity note: The complexity is O(n * k) due to the nested loops filling the DP table, which is efficient compared to the brute force approach.

1The number of visible sticks is determined by the placement of the longest stick.
2Dynamic programming allows us to build solutions incrementally based on smaller subproblems.

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