This document presents a C++ implementation of a substring searching function that finds occurrences of a given substring within a target string using an efficient O(n+m) time complexity algorithm. The function utilizes cumulative sums and performs character-by-character verification in case of matching sums to ensure accuracy, especially in the presence of anagrams. The code includes examples that demonstrate the functionality and bounds checks for various cases.

1. ...string_searchingstring_searchingstring_searching.cpp 1
#include <vector>
#include <string>
#include <algorithm>
#include <iostream>
/**
*
* @copyright Copyright Dr Russell John Childs, Phd, 2020-01-14
*
* Find occurrences of a substr S of length m, in a target string T of length n.
* Order-of-complexity: O(n+m) in time, O(1) in space.
*
* Operation:
* (1) Convert char to integer, I.
* (2) Use std::pair sliding window, w, with cumulative sums of I at positions
* p and p-m of target, where p is the range (substr length, target length).
* (3) Compare cumulative sum of I for substr against w.second - w.first
* (4) If (3) succeeds, perform O(m) char-by-char check that S=T[p-m]...T[m].
* (5) if (4) succeeds, append p-m to return vector of "found positions".
*
* Notes: Code is simpler than trie or KMP. Compiled under Visual Studio.
* The O(m) verification step (4) is only necessary because "+" commutes, which
* poses a problem for anagrams, a+b = b+a. There is a Group Theoretic way of
* creating an O(1)-updatable hash function that eliminates the need for (4).
* A char encoding, such as c -> (c-' ')*('~' - ' '), may differentiate between
* some same-length strings with different chars and product(c->prime) may
* differentiate between all same-length strings with differents chars, but not
* anagrams.
*
* @param substr : const std::string& - substring sought in target
* @param target : const std::string& - target string being searched
* @return std::vector<unsigned> - positions in target of matches, none -> empty
*
*/
std::vector<unsigned>
find(const std::string& substr, const std::string& target)
{
std::vector<unsigned> ret_val;
std::pair<unsigned long long, unsigned long long> window(0, 0);
unsigned signature = 0;
for (unsigned i = 0; (target.length() >= substr.length())
&& (i < target.length()); ++i)
{
//Get cumulative sum at pos i of target string being searched
window.second += target[i];
//Get cumulative sum for substring being sought
if (i < substr.length())
{
signature += substr[i];
}
if (i >= substr.length()-1)
{
//Check if cumul sum of substr == cumul sum of window
if ((window.second - window.first) == signature)
{

2. ...string_searchingstring_searchingstring_searching.cpp 2
//Perform O(m) check of letter-by-letter match
bool match = true;
for (unsigned j = 0; j < substr.length(); ++j) match &=
(substr[j] == target[i - substr.length() + j + 1]);
//If matched, add index to vector of found positions
if (match) ret_val.push_back(i - substr.length() + 1);
}
//Get cumul sum at pos i - m of target string being searched
window.first += target[i - substr.length() + 1];
}
}
return ret_val;
}
int main(void)
{
//Bounds check on target length < substr length, expect empty list
std::string substr = "Amazon";
std::string target = "Amazo";
std::cout << substr << " found in " << target << " at ";
std::cout << std::endl;
//Check for substr at beginning and end, expect 0 and 19
target = "Amazon----Amazo----Amazon";
std::cout << substr << " found in " << target << " at ";
for (auto& elem : find(substr, target)) std::cout << elem << " ";
std::cout << std::endl;
//Check for substr in middle, expect 9
target = "Amazo----Amazon----mazon";
std::cout << substr << " found in " << target << " at ";
for (auto& elem : find(substr, target)) std::cout << elem << " ";
std::cout << std::endl;
//Check for repeated substr, expect 0,6,12,18,24,30
target = "AmazonAmazonAmazonAmazonAmazonAmazon";
std::cout << substr << " found in " << target << " at ";
for (auto& elem : find(substr, target)) std::cout << elem << " ";
std::cout << std::endl;
//Check for substr not matched, expect empty list
target = "Ama----z----on";
std::cout << substr << " found in " << target << " at ";
for (auto& elem : find(substr, target)) std::cout << elem << " ";
std::cout << std::endl;
}