970. Powerful Integers
Description
Given three integers x
, y
, and bound
, return a list of all the powerful integers that have a value less than or equal to bound
.
An integer is powerful if it can be represented as xi + yj
for some integers i >= 0
and j >= 0
.
You may return the answer in any order. In your answer, each value should occur at most once.
Example 1:
Input: x = 2, y = 3, bound = 10 Output: [2,3,4,5,7,9,10] Explanation: 2 = 20 + 30 3 = 21 + 30 4 = 20 + 31 5 = 21 + 31 7 = 22 + 31 9 = 23 + 30 10 = 20 + 32
Example 2:
Input: x = 3, y = 5, bound = 15 Output: [2,4,6,8,10,14]
Constraints:
1 <= x, y <= 100
0 <= bound <= 106
Solutions
Solution 1: Hash Table + Enumeration
According to the description of the problem, a powerful integer can be represented as $x^i + y^j$, where $i \geq 0$, $j \geq 0$.
The problem requires us to find all powerful integers that do not exceed $bound$. We notice that the value range of $bound$ does not exceed $10^6$, and $2^{20} = 1048576 \gt 10^6$. Therefore, if $x \geq 2$, then $i$ is at most $20$ to make $x^i + y^j \leq bound$ hold. Similarly, if $y \geq 2$, then $j$ is at most $20$.
Therefore, we can use double loop to enumerate all possible $x^i$ and $y^j$, denoted as $a$ and $b$ respectively, and ensure that $a + b \leq bound$, then $a + b$ is a powerful integer. We use a hash table to store all powerful integers that meet the conditions, and finally convert all elements in the hash table into the answer list and return it.
Note that if $x=1$ or $y=1$, then the value of $a$ or $b$ is always equal to $1$, and the corresponding loop only needs to be executed once to exit.
The time complexity is $O(\log^2 bound)$, and the space complexity is $O(\log^2 bound)$.
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