#12

Medium

LeetCode Problem Workspace

Integer to Roman

Convert a given integer to its Roman numeral representation using hash table mapping and decimal place math operations efficiently.

hash table math string

Problem Statement

Roman numerals consist of seven symbols: I, V, X, L, C, D, M with values 1, 5, 10, 50, 100, 500, and 1000 respectively. Numbers are represented by combining these symbols, adding values, and using subtractive notation for specific cases like IV, IX, XL, XC, CD, and CM. Your task is to convert an integer in the range 1 to 3999 into a valid Roman numeral string while following these rules precisely.

The conversion requires processing each decimal place separately, applying the correct Roman symbols for units, tens, hundreds, and thousands. For example, 3000 becomes MMM, 700 becomes DCC, 40 becomes XL, and 9 becomes IX. The algorithm must ensure the resulting string follows Roman numeral standards, correctly handling subtractive cases and preserving order from largest to smallest symbols, producing the shortest valid representation.

Examples

Example 1

Input: See original problem statement.

Output: See original problem statement.

3000 = MMM as 1000 (M) + 1000 (M) + 1000 (M) 700 = DCC as 500 (D) + 100 (C) + 100 (C) 40 = XL as 10 (X) less of 50 (L) 9 = IX as 1 (I) less of 10 (X) Note: 49 is not 1 (I) less of 50 (L) because the conversion is based on decimal places

Example 2

Input: See original problem statement.

Output: See original problem statement.

50 = L 8 = VIII

Example 3

Input: See original problem statement.

Output: See original problem statement.

1000 = M 900 = CM 90 = XC 4 = IV

Constraints

1 <= num <= 3999

Solution Approach

Hash Table Mapping with Subtractive Combinations

Create a hash table that maps all critical decimal values to their Roman numeral equivalents, including standard values and subtractive cases like 4, 9, 40, 90, 400, and 900. Iterating from largest to smallest key, append the symbol while reducing the integer by the value. This guarantees correct ordering and handles special subtractive scenarios without multiple conditional checks, tightly integrating hash table access and arithmetic to minimize complexity and avoid missing edge cases.

Iterative Reduction by Place Values

Process the integer by iterating through the sorted hash table values in descending order. At each step, divide the integer by the current key to determine how many times the symbol should appear, append that many symbols to the result, and subtract the total value. This approach leverages basic math operations to handle repeated symbols and ensures no invalid Roman numeral sequences are produced, aligning the algorithm closely with the intended Hash Table plus Math pattern.

Concatenating Symbols Efficiently

As each decimal value is processed, build the Roman numeral string by concatenating the corresponding symbols directly. Using a mutable string or list ensures efficient appends, avoiding repeated string reconstruction. This step ensures the final string preserves the correct order of symbols, accounts for subtractive notation, and maintains linear proportionality to the number of symbols, keeping the solution optimized both for performance and memory within the context of integer-to-Roman conversion.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity is O(1) in practice since the input integer is bounded by 3999, iterating through at most 13 keys in the hash table. Space complexity is also O(1) aside from the output string, because the hash table and temporary variables require constant space. These constraints confirm the provided complexities are efficient for this problem's limits.

What Interviewers Usually Probe

Do you handle subtractive notation like IV and IX correctly?
Can you explain how your hash table mapping covers all decimal place values?
Will you avoid building invalid Roman sequences when concatenating symbols?

Common Pitfalls or Variants

Common pitfalls

Ignoring subtractive combinations, leading to outputs like IIII instead of IV.
Appending symbols in the wrong order, producing non-standard Roman numerals.
Failing to reduce the integer properly after each symbol append, causing duplicate or missing symbols.

Follow-up variants

Roman to Integer: Convert a Roman numeral string back to an integer, ensuring proper subtractive handling.
Integer to Roman Extended: Support integers larger than 3999 using overline notation or repeated symbols.
Minimal Roman Representation: Convert integers ensuring the output string uses the fewest symbols possible, optimizing for length.

FAQ

What is the simplest approach for converting an integer to a Roman numeral?

Using a hash table with decimal values and subtractive combinations allows direct mapping of each place value to symbols, ensuring correct order and handling edge cases efficiently.

How do I handle numbers like 4, 9, 40, and 90 in this problem?

Include subtractive cases in your hash table mapping so that 4 maps to IV, 9 to IX, 40 to XL, and 90 to XC, avoiding multiple conditionals or errors in concatenation.

Can this method handle the maximum input 3999 correctly?

Yes, the hash table covers 1000 (M), 900 (CM), 500 (D), and other required values, allowing consistent conversion up to 3999 without violating Roman numeral rules.

What common mistakes should I avoid for this integer-to-Roman conversion?

Avoid missing subtractive mappings, appending symbols in reverse order, or failing to decrement the integer properly after each symbol append, which could produce invalid numerals.

Does the solution pattern rely more on hash table access or arithmetic operations?

It relies on both: the hash table provides symbol mapping, while arithmetic operations determine how many times each symbol is appended, tightly combining Hash Table plus Math logic.

terminal

Solution

Solution 1: Greedy

We can first list all possible symbols $cs$ and their corresponding values $vs$, then enumerate each value $vs[i]$ from large to small. Each time, we use as many symbols $cs[i]$ corresponding to this value as possible, until the number $num$ becomes $0$.

class Solution:
    def intToRoman(self, num: int) -> str:
        cs = ('M', 'CM', 'D', 'CD', 'C', 'XC', 'L', 'XL', 'X', 'IX', 'V', 'IV', 'I')
        vs = (1000, 900, 500, 400, 100, 90, 50, 40, 10, 9, 5, 4, 1)
        ans = []
        for c, v in zip(cs, vs):
            while num >= v:
                num -= v
                ans.append(c)
        return ''.join(ans)

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