Rings and Rods

The goal is to find the number of rods that have all three colors of rings on them. We need to count the number of unique colors for each rod. To accomplish this, we’ll use a dictionary where the keys are the rods and the values are sets of colors.

This is the algorithm:

  1. Initialize an empty dictionary rods.
  2. Iterate over the rings string by 2 characters at a time.
    1. Parse the color and the rod from the current pair of characters.
    2. If the rod is not yet in the dictionary, add it with a new set that contains the current color.
    3. If the rod is already in the dictionary, add the current color to its set.
  3. After processing all the rings, count the rods that have all three colors on them and return this count.

Here is the Python code that implements this algorithm:

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class Solution:
    def countPoints(self, rings: str) -> int:
        rods = {}

        # Iterating through rings
        for i in range(0, len(rings), 2):
            # Extracting color and rod from each pair
            color = rings[i]
            rod = rings[i+1]

            # If rod is not in the dictionary, add it
            if rod not in rods:
                rods[rod] = {color}
            # If rod is in the dictionary, add color to the set
            else:
                rods[rod].add(color)

        # Counting the rods that have all 3 colors
        count = 0
        for colors in rods.values():
            if len(colors) == 3:
                count += 1

        return count

In this code, we start by initializing an empty dictionary rods. Then, we start a loop that iterates over each pair of characters in rings. For each pair, we parse the color and the rod and add the color to the set of colors for the current rod in the rods dictionary. After processing all the rings, we count the rods that have all three colors on them and return this count.

Identifying Problem Isomorphism

“Counting Rods with All Colors” can be approximately mapped to “Jewels and Stones”.

In “Jewels and Stones”, you’re given a string representing types of jewels, and another string representing stones you have, and you need to find out how many of your stones are also jewels. This is similar to our problem where you are given a string representing rings of different colors (analogous to types of jewels) placed on rods (analogous to stones), and you have to find out how many rods have all colors of rings (analogous to counting stones that are jewels).

The reason behind this selection is the structure of the problem - a set of elements (rings or jewels), and another set (rods or stones), where the task is to count elements of the second set (rods or stones) that have certain properties regarding the first set (rings or jewels).

The mapping is approximate because in “Jewels and Stones”, we just need to find out if a stone is a jewel or not, whereas in “Counting Rods with All Colors”, we need to find rods that have all colors of rings.

“Jewels and Stones” is simpler because it does not involve the additional complexity of needing all colors on a rod. As we move to more complex problems, we could consider other isomorphic problems where we need to check for the presence of all elements of a certain set, which increases the complexity.

The abstract representation of these problems is not the same.

The abstract representation of the problem “Counting Rods with All Colors” is:

Given a string representing a sequence of labeled items (rings of different colors) each associated with a category (rod number), count the number of categories that contain all types of labeled items.

The abstract representation of the problem “Jewels and Stones” is:

Given a string representing types of valuable items (jewels) and another string representing a collection of items (stones), count the number of items in the collection that are also valuable.

While both problems involve strings and counting, the conditions for counting differ. In “Counting Rods with All Colors”, we count categories (rods) that contain all types of items (rings of all colors). In “Jewels and Stones”, we count items (stones) that are also valuable (jewels). So, although there are similarities, the abstract representations are not identical.

Abstract Representation of the Problem

We are given an input string of length 2n that is a sequence of n pairs. Each pair consists of a character and a digit. The character represents a category and the digit represents a group. Each group can have one or more categories associated with it. We need to count how many groups have all distinct categories.

In terms of structure, we have:

  1. A sequence (string) of length 2n.
  2. Each adjacent pair of elements in the sequence represents an association of a category (character) with a group (digit).
  3. There are 3 distinct categories (represented by characters ‘R’, ‘G’, ‘B’) and 10 distinct groups (represented by digits ‘0’ to ‘9’).
  4. A group can have multiple categories associated with it.
  5. The task is to count the number of groups associated with all distinct categories.

Key elements involved in the problem:

  1. A string of length 2n, where each adjacent pair represents an association.
  2. Categories, represented by distinct characters.
  3. Groups, represented by digits.
  4. The process of parsing the sequence, creating associations between categories and groups.
  5. The process of counting groups that are associated with all categories.