1780. Check if Number is a Sum of Powers of Three - Detailed Explanation
Problem Statement
Given an integer n, determine whether it is possible to represent n as a sum of distinct powers of three. In other words, check if there exists a subset of numbers {3⁰, 3¹, 3², …} (each used at most once) such that their sum equals n.
Example 1
- Input:
n = 12 - Output:
true - Explanation:
12 can be written as 9 + 3, which corresponds to 3² + 3¹.
Example 2
- Input:
n = 91 - Output:
true - Explanation:
91 can be written as 81 + 9 + 1, which is 3⁴ + 3² + 3⁰.
Example 3
- Input:
n = 21 - Output:
false - Explanation:
There is no way to represent 21 as a sum of distinct powers of three.
Constraints
- 1 ≤ n ≤ 10⁷ (or similar range, as given by the problem)
Hints and Intuition
-
Hint 1:
Think about the properties of powers of three. The numbers are 1, 3, 9, 27, … and each power is distinct and grows exponentially. -
Hint 2:
One efficient observation is related to the base-3 (ternary) representation of n. If n can be represented as a sum of distinct powers of three, then its ternary representation should consist only of the digits 0 and 1.- If any digit is 2 (or higher), then the number uses a power more than once or cannot be expressed using distinct powers.
-
Hint 3:
Alternatively, you can use a greedy algorithm by subtracting the largest power of three that does not exceed n and repeating the process. If you reach 0 exactly, then n is a valid sum.
Detailed Approaches
Approach A: Ternary Representation Check
-
Convert to Base-3:
Convert the given integer n to its ternary (base-3) string. -
Check Digits:
Scan through the digits of the ternary representation.- If any digit is '2' (or greater), return false.
- Otherwise, return true.
Why This Works:
- When you write a number as a sum of distinct powers of three, each power (3⁰, 3¹, 3², …) can only appear at most once. In base-3, this corresponds to having either a 0 (power not used) or a 1 (power used). A digit '2' would indicate that a particular power is being used twice, which is not allowed.
Approach B: Greedy Subtraction
-
Find the Largest Power:
Identify the largest power of three that is less than or equal to n. -
Subtract and Iterate:
Subtract that power from n and repeat the process with the new value of n. -
Check for Completion:
- If n becomes 0, return true.
- If no power can be subtracted (or if n becomes negative), return false.
Why This Works:
- The greedy approach mimics the selection of digits in the ternary representation. By always subtracting the largest possible power, you ensure that you are using each power at most once. However, this approach may be less direct than the base conversion method.
Code Implementation
Python Implementation (Using Ternary Check)
Java Implementation (Using Ternary Check)
Complexity Analysis
Time Complexity:
- Ternary Conversion:
The process of converting n to base-3 takes O(log₃ n) iterations. - Digit Check:
Scanning the resulting string is also O(log₃ n). - Overall:
The overall time complexity is O(log n).
Space Complexity:
- Ternary String:
The space used for storing the ternary representation is O(log₃ n), which is O(log n). - Overall:
The space complexity is O(log n).
Common Pitfalls and Edge Cases
Common Pitfalls:
- Improper Base Conversion:
Failing to convert the number to a proper ternary string can lead to incorrect results. - Edge Conditions:
Ensure that the loop terminates properly when n becomes 0.
Edge Cases:
- n = 1:
The ternary representation is "1" (which is valid since it contains only a single '1'). - n = 0:
Although the problem usually guarantees n is positive, if n is 0, you can define the answer as true (since 0 is the sum of zero powers) or handle it as a special case. - Large n:
The conversion and check operations should be efficient even for larger values of n due to the logarithmic complexity.
Related Problems for Further Practice
- Sum of Powers of Two:
Similar to this problem but with powers of two, often checking the binary representation. - Check if Number is a Sum of Distinct Primes:
Although more complex, it involves representing a number as a sum of distinct prime numbers. - Subset Sum Problems:
Problems that involve determining if a subset of numbers can sum to a target value.
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