Prefixes and Suffixes

December 11, 2024 | Algorithms

When it comes to solving algorithmic problems, prefixes and suffixes often play a crucial role in simplifying complex computations and optimizing performance. These concepts aren’t just theoretical—they’re practical tools used in a wide range of programming tasks, from string manipulation to dynamic programming and more.

What Are Prefixes and Suffixes?

Prefix

A prefix is a substring that starts at the beginning of a string and ends at any position within the string. For example:

String: "programming"
Prefixes: "p", "pr", "pro", "prog", ..., "programming"

Suffix

A suffix is a substring that ends at the end of a string and starts at any position within the string. For example:

String: "programming"
Suffixes: "g", "ng", "ing", "ming", ..., "programming"

Why Are Prefixes and Suffixes Useful?

1. Search Optimization: Prefixes are essential for building prefix trees (tries)., which are data structures that enable efficient searching, especially in dictionary-like scenarios or autocomplete systems.

2. Pattern Matching: Both prefixes and suffixes are integral to string matching algorithms like the Knuth-Morris-Pratt (KMP). algorithm, which uses a prefix-suffix table to skip unnecessary comparisons.

3. Dynamic Programming: Prefix sums and suffix sums help break down problems into smaller, manageable parts. For example:

   • Prefix sum: Helps compute cumulative sums quickly for range queries.
   • Suffix sum: Similarly helps in problems where you need cumulative sums from the end of a list.

Common Applications

1. String Matching

Finding occurrences of one string (pattern) within another can be optimized using the concept of prefix-suffix matches. The KMP algorithm builds a prefix table to find overlaps between a string and its prefixes and suffixes.

2. Subarray Problems

Prefix sums make range queries efficient. For instance, given an array, you can compute the sum of any subarray using:

prefix_sum[j] - prefix_sum[i - 1]

where i and j are the bounds of the subarray.

3. Palindrome Problems

To find the longest palindromic prefix or suffix in a string, algorithms like Manacher’s algorithm or dynamic programming leverage the properties of prefixes and suffixes.

4. Text Completion and Suggestion

Autocomplete systems use tries to store prefixes of words, enabling fast lookups for partial strings.

Code Examples

Prefix Sum

def calculate_prefix_sum(arr):
    prefix_sum = [0] * len(arr)
    prefix_sum[0] = arr[0]
    for i in range(1, len(arr)):
        prefix_sum[i] = prefix_sum[i - 1] + arr[i]
    return prefix_sum

# Example usage:
arr = [1, 2, 3, 4, 5]
print(calculate_prefix_sum(arr))  # Output: [1, 3, 6, 10, 15]

Suffix Sum

def calculate_suffix_sum(arr):
    suffix_sum = [0] * len(arr)
    suffix_sum[-1] = arr[-1]
    for i in range(len(arr) - 2, -1, -1):
        suffix_sum[i] = suffix_sum[i + 1] + arr[i]
    return suffix_sum

# Example usage:
arr = [1, 2, 3, 4, 5]
print(calculate_suffix_sum(arr))  # Output: [15, 14, 12, 9, 5]