K-th Smallest in Lexicographical Order

To solve this problem, you can implement a function to find the kth lexicographically smallest integer in the range [1, n]. Here’s how you can do it:

  1. Start with the current prefix, which is initially 1.
  2. Determine the count of numbers under the current prefix.
  3. If the count is less than k, then move to the next prefix by incrementing it by 1.
  4. If the count is greater or equal to k, move to the next level of the tree by multiplying the prefix by 10, and decrement k by 1.
  5. Repeat steps 2-4 until you reach the required level.

Here’s the code:

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class Solution:
    def findKthNumber(self, n: int, k: int) -> int:
        prefix = 1
        while k > 1:
            count = 0
            next_prefix = prefix + 1
            curr = prefix
            while curr <= n:
                count += min(n + 1, next_prefix) - curr
                curr *= 10
                next_prefix *= 10

            if k > count:
                prefix += 1
                k -= count
            else:
                prefix *= 10
                k -= 1

        return prefix

Explanation:

  • The variable prefix keeps track of the current prefix of the number.
  • The variable count counts the numbers under the current prefix.
  • If the count of numbers is less than k, it means we need to move to the next prefix.
  • If the count of numbers is greater or equal to k, it means the kth smallest number is under the current prefix, so we go one level deeper in the tree and decrement k by 1.
  • When k reaches 1, we’ve found the kth smallest number.

Identifying Problem Isomorphism

“Kth Smallest Element in a Sorted Matrix” is approximately isomorphic to “K-th Smallest in Lexicographical Order”.

Identifying Problem Isomorphism

In “K-th Smallest in Lexicographical Order”, the objective is to find the kth smallest number in the lexicographical ordering of all numbers from 1 to n.

In “Kth Smallest Element in a Sorted Matrix”, the task is to find the kth smallest element in a sorted matrix, where each row and column are sorted in ascending order.

The common aspect is the concept of finding the kth smallest element in a sorted sequence. In both problems, you have to navigate through sorted data structures to find the kth smallest element.

However, the key difference is in the data structure being used. The lexicographical problem implies a sequence based on string comparison, whereas the sorted matrix problem involves a 2-dimensional data structure.

“K-th Smallest in Lexicographical Order” is more complex due to the need to understand lexicographical order and how to navigate through it efficiently. “Kth Smallest Element in a Sorted Matrix” is easier as it involves navigating a 2D matrix with sorted rows and columns, a simpler data structure.

10 Prerequisite LeetCode Problems

«««< Updated upstream “K-th Smallest in Lexicographical Order” involves recursion, binary search, and number theory. Here are 10 problems for tackling this problem:

“K-th Smallest in Lexicographical Order” involves recursion, binary search, and number theory. Here are 10 problems to prepar for tackling this problem:

Stashed changes

  1. “70. Climbing Stairs”

    • Introduces the concept of recursion, which is crucial to understand the main problem.

  2. “278. First Bad Version”

    • This problem reinforces the concept of binary search.

  3. “33. Search in Rotated Sorted Array”

    • Another problem that uses binary search, but in a slightly more complex context.

  4. “367. Valid Perfect Square”

    • This problem reinforces the concept of binary search in a number theory context.

  5. “441. Arranging Coins”

    • This problem gives you a sense of how to approach number theory problems.

  6. “69. Sqrt(x)”

    • This problem applies binary search in a different way, looking for a square root.

  7. “744. Find Smallest Letter Greater Than Target”

    • This is another problem that helps understand binary search.

  8. “374. Guess Number Higher or Lower”

    • This problem is another good binary search practice.

  9. “153. Find Minimum in Rotated Sorted Array”

    • This problem involves binary search in a complex setting which can help develop problem-solving skills needed for the main problem.

  10. “34. Find First and Last Position of Element in Sorted Array”

    • This problem also uses the concept of binary search but adds an additional level of complexity in having to find both the start and end position.

These cover binary search, recursion, and number theory, which are crucial for solving the “K-th Smallest in Lexicographical Order” problem.

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class Node:
    def __init__(self, val):
        self.val = val
        self.nextNodes = []

    def __eq__(self, other):
        return self.val == other

class Solution:
    def generateTrieTree(self, data: list):
        root = Node(None)
        for intChar in data:
            pointer = root
            for char in intChar:
                try:
                    existedNodeIdx = pointer.nextNodes.index(char)
                    pointer = pointer.nextNodes[existedNodeIdx]
                except:
                    newNode = Node(char)
                    pointer.nextNodes.append(newNode)
                    pointer = newNode
        return root

    def findKthNumber(self, n: int, k: int) -> int:
        result = 1
        k -= 1
        while k > 0:
            count = 0
            interval = [result, result + 1]
            while interval[0] <= n:
                count += (min(n + 1, interval[1]) - interval[0])
                interval = [10 * interval[0], 10 * interval[1]]

            if k >= count:
                result += 1
                k -= count
            else:
                result *= 10
                k -= 1
        return result
        
        
        root = self.generateTrieTree([str(i) for i in range(1, n + 1)])
        output = []

        def traverse(pointer, previousVal):
            nextNodes = pointer.nextNodes
            if not nextNodes:
                return

            for node in nextNodes:
                if len(output) == k:
                    return

                newVal = previousVal + node.val
                output.append(newVal)
                traverse(node, newVal)

        traverse(root, "")
        return int(output[-1])

Language Agnostic Coding Drills

  1. Data Types - Understanding primitive data types in Python such as int, str, list, etc.

  2. Custom Data Structures - Ability to define your own data structures, like the Node class in this case. This class represents a node in a trie, with a value and a list of next nodes.

  3. Classes and Objects - Understanding the concept of classes, objects, and methods. This is evident in the definition and use of the Node and Solution classes.

  4. Iteration - Looping over data using ‘for’ loops to perform actions on each item. This is done in several places such as generating the trie tree and searching for the kth number.

  5. Exception Handling - Catching and handling exceptions using try/except blocks. In this case, it’s used when trying to find a node in the list of next nodes.

  6. Recursion - Function calling itself. In this script, the ’traverse’ function is a recursive function.

  7. Sorting and Searching - Understanding how to sort a list or how to search for an item in a list.

  8. Mathematical Operations - Performing basic mathematical operations such as multiplication and addition.

Now let’s describe the problem-solving approach step by step:

  1. The problem starts by creating a trie tree of all the numbers up to ’n’. It does so by converting each number into a string, then iterating over each character in the string, and adding nodes to the trie tree accordingly.

  2. To find the kth smallest number, it performs a depth-first search (DFS) on the trie. It starts from the root, then explores as far as possible along each branch before backtracking.

  3. It maintains a count of how many numbers it has encountered and stops once it has encountered ‘k’ numbers.

  4. The value of the last encountered number is the kth smallest number.

  5. This method of generating all numbers and sorting them would take considerable time and memory, especially for large ’n’ and ‘k’. So, it also uses a binary search mechanism to reduce the search space, thereby making the solution more efficient.

  6. This problem combines several concepts including custom data structures, recursion, searching, and sorting algorithms, making it a challenging one that tests multiple areas of knowledge and skills.

Targeted Drills in Python

  1. Data Types: Understand basic data types by creating variables of different types.

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    int_val = 5
str_val = 'hello'
list_val = [1, 2, 3, 4]
print(type(int_val), type(str_val), type(list_val))

  2. Custom Data Structures: Create a basic Node class for a linked list.

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    class Node:
    def __init__(self, value):
        self.value = value
        self.next = None
node = Node(5)
print(node.value)

  3. Classes and Objects: Create a class with methods.

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    class MyClass:
    def __init__(self, value):
        self.value = value

    def print_value(self):
        print(self.value)

obj = MyClass(5)
obj.print_value()

  4. Iteration: Use a for loop to iterate over a list.

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    for i in range(5):
    print(i)

  5. Exception Handling: Write a function that handles an exception.

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    def safe_division(x, y):
    try:
        return x / y
    except ZeroDivisionError:
        return None
print(safe_division(1, 0))

  6. Recursion: Write a recursive function to calculate factorial.

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    def factorial(n):
    if n == 0:
        return 1
    else:
        return n * factorial(n-1)
print(factorial(5))

  7. Sorting and Searching: Sort a list and search for an item in the list.

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    numbers = [6, 4, 9, 2]
sorted_numbers = sorted(numbers)
print(sorted_numbers)
print(4 in sorted_numbers)

  8. Mathematical Operations: Perform mathematical operations.

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    add = 5 + 3
subtract = 5 - 3
multiply = 5 * 3
divide = 5 / 3
print(add, subtract, multiply, divide)

After you’re comfortable with these drills, you should be able to approach the given problem.