# Algorithm to find the smallest substring containing a set of....

• SlurrerOfSpeech
In summary, the conversation discusses the problem of finding the smallest substring containing a specified amount of characters. The individual is having trouble finding an efficient and elegant solution and is considering restructuring their data to improve computational complexity. They suggest using a binary tree class to classify and order the characters, and also mention using mathematical representations of character frequency and distance to solve an optimization problem. They then mention finding a C++ code that solves the problem and translating it to C#.
SlurrerOfSpeech
characters in a specified amount. I've been working on this for 10+ hours and can't seem to get it right. It passes all the unit tests I've written, but when I use it in the context of the program I'm writing it fails.

The idea is like

ACTAGACC
{ A : 2, C : 1}
------> 4, because smallest substring containing 2 As and 1 C is ACTA.

For some reason, I'm finding this very tricky to write an efficient an elegant solution. The brute force approach (Find all substrings and see which is the smallest that contains the characters) doesn't meet the performance benchmarks that I have.

My code:

Code:
using System;
using System.Collections.Generic;
using System.Linq;

public class MethodTest
{
public Dictionary<char,int> Counter { get; set; }
public string Source { get; set; }
public int ExpectedResult { get; set; }
}

public class Program
{
public static void UnitTests()
{
var tests = new List<MethodTest>() {
new MethodTest() { Source = "ABB", Counter = new Dictionary<char,int>() { { 'B', 1 } }, ExpectedResult = 1 },
new MethodTest() { Source = "ABB", Counter = new Dictionary<char,int>() { { 'A', 1 } }, ExpectedResult = 1 },
new MethodTest() { Source = "A", Counter = new Dictionary<char,int>() { { 'R', 1 } }, ExpectedResult = -1 },
new MethodTest() { Source = "ATTC", Counter = new Dictionary<char,int>() { { 'T', 1 } }, ExpectedResult = 1 },
new MethodTest() { Source = "AAAA", Counter = new Dictionary<char,int>() { { 'A', 3 } }, ExpectedResult = 3 },
new MethodTest() { Source = "AAATTTGC", Counter = new Dictionary<char,int>() { { 'A', 1 }, { 'T', 1 } }, ExpectedResult = 2 },
new MethodTest() { Source =  "GAAATAAA", Counter = new Dictionary<char,int>() { {'A', 4 } }, ExpectedResult = 5 }
};

foreach(var test in tests)
{
int result = SmallestSubstringContainingChars(test.Source,test.Counter);
if(result != test.ExpectedResult)
{
Console.WriteLine("Failed test: Source = {0}, result = {1}, expected = {2}", test.Source, result.ToString(), test.ExpectedResult.ToString());
return;
}
}
Console.WriteLine("All succeeded!");
}

static int SmallestSubstringContainingChars(string source, Dictionary<char, int> counter)
{
int start = 0;
while(start < source.Length && !counter.ContainsKey(source[start]))
++start;
if(start == source.Length)
return -1;

// If here, start is the index of the first character of source that is in the counter.
// Create a pointer end that will increment from start. Keep track of the minimum difference
// between the pointers whenever the pointers have spanned a substring that contains all the
// characters in the counter in their specified coun.

int? minspan = null;
for(int end = start; end < source.Length; ++end)
{
char c = source[end]; // character at end

if(counter.ContainsKey(c)) // ch is in counter
{
// Decrement count of character
counter[c] -= 1;

// If the count is less than 0, meaning we found a surplus of those
// characters in the current range, and if that character was the
// one at the start, we can move the start forward to the next character
// in the string and in the counter.
if(counter[c] < 0 && source[start] == c)
{
while(!counter.ContainsKey(source[++start]));
counter[c] += 1;
}

}

// If we've found all the characters in counter, then from start to end
// we have a substring that fits our criteria. Updated minspan.
if(counter.Values.All(count => count <= 0))
{
int span = end - start;
minspan = minspan == null ? span : Math.Min((int)minspan, span);
}
}

// If didn't find any range that spans all the characters in their specified count, return -1;
// else, return the length of the spanned substring.
return minspan == null ?  -1 : ((int)minspan + 1);
}

public static void Main()
{
UnitTests();
}
}
[CODE]

Hey SlurrerOfSpeech.

If you want to make this more efficient then you will have to re-structure your data to "facilitate" the better computational complexity.

To give you an idea of what I mean I will use the examples of binary trees for sorting and searching.

A binary tree classifies everything to the left and right of something and by continually doing that, it allows you to order things without having to do the brute force search.

You will need to do the same thing for your data structure (that is - a "string") and for the functions you use.

You are looking for the smallest sub-string that contains two particular characters which hints that data structures are to be created to link to the characters themselves and relations to other neighboring characters.

This hints at specifying something where you can project out these characteristics - like a vector where a projection returns the information on character frequency with another projection getting the relation of distance between characters.

I don't know if you know mathematics, but what I am suggesting is that you find a way to represent frequency and distance of characters (i.e. distance between different realizations of "letters") and use that to solve an optimization problem where you are finding whether a string meets the frequency requirements and then if so how you can find out the sub-string that minimizes all possible distances.

Ok, I decided to cheat. I found some C++ code on http://articles.leetcode.com/finding-minimum-window-in-s-which/ that achieves what I want and I translated it to C#. It was basically the same algorithm I was using anyways.

## 1. What is an algorithm to find the smallest substring containing a set of characters?

An algorithm to find the smallest substring containing a set of characters is a step-by-step process that can be used to search for the shortest string that includes all the specified characters. It involves breaking down the input string into smaller substrings and comparing them to the set of characters until a match is found.

## 2. How does the algorithm work?

The algorithm starts by selecting the first character of the input string and checking if it is part of the specified set of characters. If it is, the algorithm then checks the next character in the string, and so on until all the characters in the set have been found. If a character is missing, the algorithm moves on to the next substring until it finds a match. Once a match is found, the algorithm continues to search for a smaller substring that also contains all the characters in the set.

## 3. What is the time complexity of this algorithm?

The time complexity of this algorithm depends on the length of the input string and the number of characters in the specified set. In the worst case scenario, the algorithm would have to check every possible substring in the input string, resulting in a time complexity of O(n^2), where n is the length of the input string.

## 4. Can this algorithm handle large input strings?

Yes, this algorithm can handle large input strings. However, as mentioned earlier, the time complexity may increase significantly with larger input strings, making it less efficient.

## 5. Are there any limitations to this algorithm?

One limitation of this algorithm is that it only finds the smallest substring containing all the specified characters. If there are multiple substrings that meet this criteria, the algorithm will only return the first one it finds. Additionally, the algorithm may not work if the input string contains duplicate characters that are part of the specified set.

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