Module 3: Technical Preparation - Advanced Challenges 3

This final technical module builds upon the foundations established in Modules 1 and 2, introducing expert-level algorithm techniques and optimization strategies. These advanced concepts will help you tackle the most complex technical interview questions and excel in your General Code Assessment (GCA).

You'll learn to recognize advanced problem patterns, implement sophisticated algorithms, and optimize solutions for both time and space efficiency. We'll focus on hard-level interview questions that require multiple techniques to be combined for optimal solutions.

By the end of this module, you'll be equipped with the knowledge and skills to approach even the most challenging technical interview questions with confidence and demonstrate expert-level problem-solving abilities in your GCA.

We strongly encourage you to click through and study the resource links provided under each bullet point for any algorithm topics you're not fully comfortable with. These carefully selected resources cover advanced graph algorithms, multidimensional dynamic programming, space-time tradeoffs, and algorithmic design patterns that form the foundation for expert-level technical interviews. Taking the time to review these materials and researching more on your own now will significantly strengthen your understanding and improve your problem-solving capabilities during the most challenging coding assessments.

Advanced Graph Algorithms

Building on the graph algorithms introduced in Module 2, we'll explore more complex graph problems and their solutions:

def topological_sort(graph):
    """
    Perform topological sort on a directed acyclic graph.
    
    Args:
        graph: A dictionary representing an adjacency list
               {node: [list of neighbors]}
    
    Returns:
        A list of nodes in topological order
    """
    # Track visited nodes and the result order
    visited = set()
    temp_mark = set()  # For cycle detection
    result = []
    
    def visit(node):
        # Detect cycles (not a DAG)
        if node in temp_mark:
            raise ValueError("Graph has cycles, topological sort not possible")
        
        # Skip if already visited
        if node in visited:
            return
        
        # Mark node as temporarily visited
        temp_mark.add(node)
        
        # Visit all neighbors first
        for neighbor in graph.get(node, []):
            visit(neighbor)
        
        # Mark as permanently visited and add to result
        visited.add(node)
        temp_mark.remove(node)
        result.append(node)
    
    # Visit all nodes
    for node in graph:
        if node not in visited:
            visit(node)
    
    # Return in reverse order (stack representation)
    return result[::-1]

# Example usage
course_prerequisites = {
    'algorithms': ['data_structures'],
    'programming': [],
    'data_structures': ['programming'],
    'databases': ['data_structures'],
    'distributed_systems': ['databases', 'algorithms']
}

print(topological_sort(course_prerequisites))
# Output: ['programming', 'data_structures', 'algorithms', 'databases', 'distributed_systems']
                    

Advanced Dynamic Programming

We'll explore complex dynamic programming problems that require careful state representation and transition functions:

def edit_distance(word1, word2):
    """
    Calculate the minimum number of operations required to convert word1 to word2.
    Operations allowed: insert, delete, replace a character.
    
    Args:
        word1: First string
        word2: Second string
    
    Returns:
        Minimum number of operations required
    """
    m, n = len(word1), len(word2)
    
    # Create a DP table
    dp = [[0] * (n + 1) for _ in range(m + 1)]
    
    # Initialize the DP table
    for i in range(m + 1):
        dp[i][0] = i  # Delete all characters from word1
    
    for j in range(n + 1):
        dp[0][j] = j  # Insert all characters from word2
    
    # Fill the DP table
    for i in range(1, m + 1):
        for j in range(1, n + 1):
            if word1[i-1] == word2[j-1]:
                dp[i][j] = dp[i-1][j-1]  # No operation needed
            else:
                # Take the minimum of delete, insert, or replace
                dp[i][j] = 1 + min(dp[i-1][j],    # Delete
                                   dp[i][j-1],    # Insert
                                   dp[i-1][j-1])  # Replace
    
    return dp[m][n]

# Example
print(edit_distance("kitten", "sitting"))  # Output: 3
                    

Space-Time Tradeoffs

Understanding when and how to trade memory for speed (or vice versa) is crucial for optimizing algorithms:

def coin_change_optimized(coins, amount):
    """
    Find the minimum number of coins needed to make up the given amount.
    Space-optimized version that uses O(amount) space instead of O(n*amount).
    
    Args:
        coins: List of coin denominations
        amount: Target amount
    
    Returns:
        Minimum number of coins needed, or -1 if impossible
    """
    # Initialize DP array
    dp = [float('inf')] * (amount + 1)
    dp[0] = 0
    
    # Fill DP array
    for coin in coins:
        for x in range(coin, amount + 1):
            dp[x] = min(dp[x], dp[x - coin] + 1)
    
    return dp[amount] if dp[amount] != float('inf') else -1

# Example
print(coin_change_optimized([1, 2, 5], 11))  # Output: 3 (5 + 5 + 1)
                    

Algorithmic Paradigms and Design Patterns

Recognizing problem patterns and applying appropriate paradigms can significantly improve your approach:

import random


def quick_select(arr, k):
    """
    Find the kth smallest element in an unsorted array using Quick Select.
    
    Args:
        arr: Unsorted array
        k: Index of the element to find (0-indexed)
    
    Returns:
        The kth smallest element
    """
    if not arr or k < 0 or k >= len(arr):
        return None
    
    def partition(left, right, pivot_index):
        pivot = arr[pivot_index]
        # Move pivot to end
        arr[pivot_index], arr[right] = arr[right], arr[pivot_index]
        
        # Move elements smaller than pivot to the left
        store_index = left
        for i in range(left, right):
            if arr[i] < pivot:
                arr[store_index], arr[i] = arr[i], arr[store_index]
                store_index += 1
        
        # Move pivot to its final place
        arr[right], arr[store_index] = arr[store_index], arr[right]
        
        return store_index
    
    def select(left, right, k):
        if left == right:
            return arr[left]
        
        # Choose random pivot to avoid worst-case performance
        pivot_index = random.randint(left, right)
        
        # Partition around the pivot
        pivot_index = partition(left, right, pivot_index)
        
        # Check if we found the kth element
        if k == pivot_index:
            return arr[k]
        elif k < pivot_index:
            return select(left, pivot_index - 1, k)
        else:
            return select(pivot_index + 1, right, k)
    
    return select(0, len(arr) - 1, k)

# Example
arr = [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5]
print(quick_select(arr, 5))  # 6th smallest element (0-indexed)
                    

Assignment Overview

For this assignment, you'll tackle expert-level algorithm problems that combine multiple concepts and require sophisticated optimization techniques. This assignment will push your problem-solving abilities to prepare you for the hardest technical interview questions.

Requirements

1. Complete the following problems on LeetCode:
   Longest Increasing Path in a Matrix Find the longest increasing path in a matrix.
   Binary Tree Level Order Traversal Traverse a binary tree level by level.
   Trapping Rain Water Calculate the trapped water in an elevation map.
   Word Ladder II Find all shortest transformation sequences from beginWord to endWord.
   Burst Balloons Maximize coins earned by bursting balloons in an optimal order.
2. For each problem:
   • Document your approach using the REACTO framework
   • Analyze the time and space complexity of your solution
   • Identify the key data structures and algorithms used
   • Submit screenshots of your solution code

Advanced Algorithm References

Complex Data Structures

Competitive Programming Resources