Roman to Integer Conversion in DSA Python

Mental Model

Roman numerals are letters that represent numbers. We add or subtract values based on their order to get the final number.

Analogy: Think of reading a price tag where some discounts apply if a smaller number comes before a bigger number, so you subtract instead of add.

R O M A N  N U M E R A L S
I  V  X  L  C  D  M
1  5 10 50 100 500 1000

Dry Run Walkthrough

Input: Roman numeral: 'MCMIV'

Goal: Convert 'MCMIV' to its integer value

Step 1: Read 'M' (1000), add 1000 to total

Total = 1000

Why: Start with the first symbol's value

Step 2: Read 'C' (100), next is 'M' (1000), since 100 < 1000, subtract 100

Total = 1000 - 100 = 900

Why: Smaller before bigger means subtract

Step 3: Read 'M' (1000), next is 'I' (1), since 1000 > 1, add 1000

Total = 900 + 1000 = 1900

Why: Bigger or equal means add

Step 4: Read 'I' (1), next is 'V' (5), since 1 < 5, subtract 1

Total = 1900 - 1 = 1899

Why: Smaller before bigger means subtract

Step 5: Read 'V' (5), no next symbol, add 5

Total = 1899 + 5 = 1904

Why: Last symbol always adds

Result:

Total = 1904

Annotated Code

DSA Python

class RomanToInteger:
    def __init__(self):
        self.values = {'I': 1, 'V': 5, 'X': 10, 'L': 50, 'C': 100, 'D': 500, 'M': 1000}

    def roman_to_int(self, s: str) -> int:
        total = 0
        for i in range(len(s)):
            value = self.values[s[i]]
            if i + 1 < len(s) and self.values[s[i]] < self.values[s[i + 1]]:
                total -= value  # subtract if smaller before bigger
            else:
                total += value  # add otherwise
        return total


# Driver code
converter = RomanToInteger()
result = converter.roman_to_int('MCMIV')
print(result)

for i in range(len(s)):

iterate over each Roman numeral character

value = self.values[s[i]]

get integer value of current Roman numeral

if i + 1 < len(s) and self.values[s[i]] < self.values[s[i + 1]]:

check if current value is less than next value to decide subtraction

total -= value # subtract if smaller before bigger

subtract current value from total

else:

otherwise add current value

total += value # add otherwise

add current value to total

Output

1904

Complexity Analysis

Time: O(n) because we scan each character once

Space: O(1) because the map of Roman numerals is fixed size

vs Alternative: This is more efficient than parsing substrings or using recursion which may add overhead

Edge Cases

Empty string

Returns 0 as no numerals to convert

DSA Python

for i in range(len(s)):

Single character like 'V'

Returns the value of that single numeral

DSA Python

if i + 1 < len(s) and self.values[s[i]] < self.values[s[i + 1]]:

All numerals in descending order like 'MDCLXVI'

All values are added correctly

DSA Python

else:

When to Use This Pattern

When you see a problem asking to convert Roman numerals to numbers, use the subtract-if-smaller-before-bigger pattern to handle special cases.

Common Mistakes

Mistake: Always adding values without checking the next numeral causes wrong totals for subtractive cases.
Fix: Add a check to subtract when current numeral is smaller than the next one.

Summary

Converts Roman numeral strings to integer numbers by adding or subtracting values based on order.

Use when you need to interpret Roman numerals as numbers in programs.

Remember to subtract when a smaller numeral comes before a larger one, otherwise add.

Practice

1. What is the integer value of the Roman numeral IX?

easy

A. 11

B. 19

C. 6

D. 9

Solution

1. Step 1: Understand Roman numeral rules for subtraction

   In Roman numerals, when a smaller numeral appears before a larger one, it is subtracted. Here, I (1) is before X (10), so 1 is subtracted from 10.

2. Step 2: Calculate the value of IX

   10 - 1 = 9.

3. Final Answer:

   9 -> Option D

4. Quick Check:

   IX = 9 [OK]

Hint: Subtract when smaller numeral is before larger one [OK]

Common Mistakes:

• Adding values instead of subtracting
• Confusing IX with XI
• Ignoring subtraction rule

2. Which of the following Python code snippets correctly initializes a dictionary mapping Roman numerals to integers?

easy

A. roman_map = {'I': 1, 'V': 5, 'X': 10, 'L': 50}

B. roman_map = ['I': 1, 'V': 5, 'X': 10, 'L': 50]

C. roman_map = ('I' = 1, 'V' = 5, 'X' = 10, 'L' = 50)

D. roman_map = {I: 1, V: 5, X: 10, L: 50}

Solution

1. Step 1: Identify correct dictionary syntax in Python

   Python dictionaries use curly braces with key:value pairs, keys as strings must be quoted.

2. Step 2: Check each option

   roman_map = {'I': 1, 'V': 5, 'X': 10, 'L': 50} uses correct syntax with quotes and curly braces. Options A, B, and D use invalid syntax. roman_map = {I: 1, V: 5, X: 10, L: 50} uses unquoted keys which causes NameError.

3. Final Answer:

   roman_map = {'I': 1, 'V': 5, 'X': 10, 'L': 50} -> Option A

4. Quick Check:

   Correct dictionary syntax = roman_map = {'I': 1, 'V': 5, 'X': 10, 'L': 50} [OK]

Hint: Use curly braces and quote keys for dictionaries [OK]

Common Mistakes:

• Using square brackets instead of curly braces
• Not quoting string keys
• Using assignment '=' inside dictionary

3. What is the output of the following Python code?

def roman_to_int(s):
    roman_map = {'I': 1, 'V': 5, 'X': 10}
    total = 0
    for i in range(len(s)):
        if i + 1 < len(s) and roman_map[s[i]] < roman_map[s[i + 1]]:
            total -= roman_map[s[i]]
        else:
            total += roman_map[s[i]]
    return total

print(roman_to_int('XIV'))

medium

A. 14

B. 16

C. 9

D. 11

Solution

1. Step 1: Trace the loop for input 'XIV'

   Characters: X(10), I(1), V(5). Loop index i from 0 to 2.
   i=0: X(10) < I(1)? No, add 10.
   i=1: I(1) < V(5)? Yes, subtract 1.
   i=2: V(5), no next char, add 5.

2. Step 2: Calculate total

   Total = 10 - 1 + 5 = 14.

3. Final Answer:

   14 -> Option A

4. Quick Check:

   X(10) + IV(4) = 14 [OK]

Hint: Subtract if current < next numeral, else add [OK]

Common Mistakes:

• Adding all values without subtraction
• Off-by-one errors in loop
• Ignoring last character addition

4. The following code is intended to convert a Roman numeral string to an integer. What is the error?

def roman_to_int(s):
    roman_map = {'I': 1, 'V': 5, 'X': 10}
    total = 0
    for i in range(len(s)):
        if roman_map[s[i]] < roman_map[s[i + 1]]:
            total -= roman_map[s[i]]
        else:
            total += roman_map[s[i]]
    return total

print(roman_to_int('IX'))

medium

A. SyntaxError due to missing colon

B. KeyError because 'IX' is not in roman_map

C. IndexError due to accessing s[i + 1] without boundary check

D. Returns incorrect value 11 instead of 9

Solution

1. Step 1: Identify loop and index usage

   The loop runs from i=0 to len(s)-1. At last index i, accessing s[i+1] causes IndexError.

2. Step 2: Explain why error occurs

   For input 'IX' (length 2), when i=1, s[i+1] is s[2], which is out of range.

3. Final Answer:

   IndexError due to accessing s[i + 1] without boundary check -> Option C

4. Quick Check:

   Check index bounds before s[i+1] access [OK]

Hint: Always check index before accessing s[i+1] [OK]

Common Mistakes:

• Not checking index bounds in loops
• Assuming s[i+1] always exists
• Ignoring runtime errors

5. Given the Roman numeral string 'MCMXCIV', what integer does the following code return?

def roman_to_int(s):
    roman_map = {'I': 1, 'V': 5, 'X': 10, 'L': 50, 'C': 100, 'D': 500, 'M': 1000}
    total = 0
    prev_value = 0
    for char in reversed(s):
        value = roman_map[char]
        if value < prev_value:
            total -= value
        else:
            total += value
        prev_value = value
    return total

print(roman_to_int('MCMXCIV'))

hard

A. 1494

B. 1994

C. 1949

D. 2094

Solution

1. Step 1: Understand reversed traversal and subtraction logic

   The code processes characters from right to left, subtracting if current value is less than previous.

2. Step 2: Trace values for 'MCMXCIV'

   Characters reversed: V(5), I(1), C(100), X(10), M(1000), C(100), M(1000)
   Calculate total:
   V=5 (add 5)
   I=1 < 5 subtract 1 (total=4)
   C=100 > 1 add 100 (total=104)
   X=10 < 100 subtract 10 (total=94)
   M=1000 > 10 add 1000 (total=1094)
   C=100 < 1000 subtract 100 (total=994)
   M=1000 > 100 add 1000 (total=1994)

3. Final Answer:

   1994 -> Option B

4. Quick Check:

   MCMXCIV = 1994 [OK]

Hint: Traverse right to left, subtract if smaller than previous [OK]

Common Mistakes:

• Processing left to right without subtraction logic
• Confusing numeral values
• Missing reversed traversal step