Problem Description

In this problem, we are given two strings, jewels and stones. The jewels string represents the types of stones that are considered to be jewels. Each character in the jewels string is a unique jewel type. The stones string contains the stones that we have, and each character represents a type of stone. Our task is to count how many stones we have that are also jewels. It is important to note that the identification of jewels is case sensitive, meaning a and A would be treated as different types of stones.

Intuition

The solution approach involves using a set data structure in Python, which contains only unique items and allows for fast membership testing. By converting the jewels string into a set, we can quickly check if a stone (character in stones string) is a jewel by seeing if it exists in the set.

Here is the intuition behind the steps:

1. Convert the jewels string into a set. This is important to remove any duplicate characters, which can occur in the input, and to allow for O(1) (constant time) look-ups.
2. Iterate through each character in the stones string. For every stone, we will check if it's also a jewel.
3. We check if a stone is a jewel by determining if that stone (character) is present in the jewels set.
4. Count the number of stones that are jewels. This is done by summing the number of True values yielded by the expression c in s for each stone c in the stones string, where s is the set of jewels.
5. Return the count as the final answer.

The reasoning behind this approach is centered on set theory—by comparing elements of one set (our stones) to another (our jewels), we're looking for the intersection—the items that are common to both. The use of a set data structure simplifies this search, making the algorithm efficient and concise.

Solution Approach

The implementation of the solution relies on several concepts and data structures, which optimize the process of identifying jewels within the stones. Let's dive into the approach based on the reference solution:

1. Set Data Structure: The solution begins with creating a set s from the jewels string. Sets are an appropriate choice for two main reasons: they inherently prevent duplicate elements and provide constant time complexity (O(1)) for membership checking. This is crucial because it prevents the algorithm's performance from degrading, even with a large number of jewel types.

   s = set(jewels)

2. For Loop and Membership Testing: We iterate through each character c in the string stones, which allows us to examine each stone we possess.

   for c in stones:

3. Boolean Expression in Sum: For every stone represented by character c, we test if it is in the set s (the jewels). The expression c in s evaluates to a boolean (True if c is a jewel, False if not).

   c in s

4. Sum to Count Trues: The sum function in Python conveniently treats True as 1 and False as 0. Thus, when we sum up the boolean expression results, we effectively count how many times c in s evaluates to True, which corresponds to the number of stones that are jewels.

   sum(c in s for c in stones)

5. Return the Result: The result of the sum operation is the final answer, which is the number of stones that are also jewels.

Combining these steps in a single expression leads to a compact and efficient solution. The solution's elegance lies in its use of Python's set and iteration mechanisms to directly tie together the question (is this a jewel?) with the answer (how many jewels?).

class Solution:
    def numJewelsInStones(self, jewels: str, stones: str) -> int:
        s = set(jewels)
        return sum(c in s for c in stones)

This Python code snippet achieves our goal using advanced concepts in an easy-to-understand and concise way.

Example Walkthrough

Let's take a simple example to illustrate the solution approach. Suppose we have the following inputs:

• jewels: "aA"
• stones: "aAAbbbb"

As per the problem statement, each character in the jewels string is a unique jewel type, and they are case-sensitive. So in this case, we have two types of jewels: 'a' and 'A'. We have to find out how many of the stones are actual jewels.

Following the steps from the solution approach:

1. Convert the jewels string into a set. Thus, we create a set of jewels s:

   s = set('aA')  # The set will contain {'a', 'A'}

2. Iterate through each character in the stones string using a for loop:

   for c in "aAAbbbb":

3. For each character c in stones, check if it is present in the set of jewels s. We'll do this using the expression c in s, which returns True if c is a jewel, otherwise False:

   'a' in s  # True
   'A' in s  # True
   'A' in s  # True
   'b' in s  # False
   'b' in s  # False
   'b' in s  # False
   'b' in s  # False

4. We evaluate the expression c in s inside a sum() function to add up the True values (which are counted as 1):

   sum(c in s for c in "aAAbbbb")  # This will equal 3

5. The result of the sum expression (3 in this case) is the number of stones that are also jewels, which is what we return as the answer.

Following this process, we were able to determine that out of the stones we possess ("aAAbbbb"), three of them are jewels ("aAA").

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the code is O(n + m) where n is the length of the jewels string and m is the length of the stones string. The creation of the set s from the jewels string takes O(n) time. Then, we iterate over each character c in stones, which takes O(m) time, to check whether it is in the set s. Since set lookup is O(1) on average, the time for the sum operation is also O(m).

Space Complexity

The space complexity is O(n), where n is the length of the jewels string. This is because we create a set s to store the distinct characters in jewels. There is no additional significant space used as we only store a single set.

Learn more about how to find time and space complexity quickly using problem constraints.