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16.11. Diving Board

Description

You are building a diving board by placing a bunch of planks of wood end-to-end. There are two types of planks, one of length shorter and one of length longer. You must use exactly K planks of wood. Write a method to generate all possible lengths for the diving board.

return all lengths in non-decreasing order.

Example: 


Input: 

shorter = 1

longer = 2

k = 3

Output:  {3,4,5,6}

Note:

  • 0 < shorter <= longer
  • 0 <= k <= 100000

Solutions

Solution 1: Case Analysis

If $k=0$, there is no solution, and we can directly return an empty list.

If $shorter=longer$, we can only use a board with length $longer \times k$, so we directly return a list with length $longer \times k$.

Otherwise, we can use a board with length $shorter \times (k-i) + longer \times i$, where $0 \leq i \leq k$. We enumerate $i$ in the range $[0, k]$, and calculate the corresponding length. For different values of $i$, we will not get the same length, because if $0 \leq i \lt j \leq k$, then the difference in length is $(i - j) \times (longer - shorter) \lt 0$. Therefore, for different values of $i$, we will get different lengths.

The time complexity is $O(k)$, where $k$ is the number of boards. Ignoring the space consumption of the answer, the space complexity is $O(1)$.

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class Solution:
    def divingBoard(self, shorter: int, longer: int, k: int) -> List[int]:
        if k == 0:
            return []
        if longer == shorter:
            return [longer * k]
        ans = []
        for i in range(k + 1):
            ans.append(longer