Last modified: Jan 31 2026 at 10:09 PM • 3 mins read

Hard Problems

Table of contents

  1. Overview
  2. Problem Categories
  3. Solved Problems
  4. Key Patterns for Hard Problems
  5. Tips for Hard Problems
  6. Preparation Strategy

Overview

Hard problems represent the most challenging algorithmic puzzles, requiring:

  • Advanced algorithm design
  • Complex optimization techniques
  • Multiple algorithmic concepts combined
  • System design considerations
  • Mathematical insights

Problem Categories

Advanced Dynamic Programming

  • Multi-dimensional DP
  • DP with complex state transitions
  • Optimization with constraints

Complex Graph Algorithms

  • Shortest path algorithms
  • Network flow problems
  • Advanced graph traversals

Mathematical & Computational Geometry

  • Number theory problems
  • Geometric algorithms
  • Mathematical modeling

System Design & Data Structures

  • Custom data structure design
  • Concurrent programming concepts
  • Memory-efficient solutions

Solved Problems

Problems will be added as they are solved

Key Patterns for Hard Problems

1. Advanced Dynamic Programming

def advanced_dp_example(matrix):
    m, n = len(matrix), len(matrix[0])
    dp = [[float('inf')] * n for _ in range(m)]
    dp[0][0] = matrix[0][0]
    
    for i in range(m):
        for j in range(n):
            if i > 0:
                dp[i][j] = min(dp[i][j], dp[i-1][j] + matrix[i][j])
            if j > 0:
                dp[i][j] = min(dp[i][j], dp[i][j-1] + matrix[i][j])
    
    return dp[m-1][n-1]

2. Graph Algorithms

def dijkstra_example(graph, start):
    import heapq
    
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    pq = [(0, start)]
    
    while pq:
        current_distance, current = heapq.heappop(pq)
        
        if current_distance > distances[current]:
            continue
        
        for neighbor, weight in graph[current].items():
            distance = current_distance + weight
            
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(pq, (distance, neighbor))
    
    return distances

3. Custom Data Structures

class TrieNode:
    def __init__(self):
        self.children = {}
        self.is_end = False

class Trie:
    def __init__(self):
        self.root = TrieNode()
    
    def insert(self, word):
        node = self.root
        for char in word:
            if char not in node.children:
                node.children[char] = TrieNode()
            node = node.children[char]
        node.is_end = True
    
    def search(self, word):
        node = self.root
        for char in word:
            if char not in node.children:
                return False
            node = node.children[char]
        return node.is_end

Tips for Hard Problems

  1. Break down complexity - Divide into smaller, manageable subproblems
  2. Mathematical insight - Often require mathematical or logical breakthroughs
  3. Multiple techniques - Combine several algorithmic approaches
  4. Optimization focus - Usually require optimal or near-optimal solutions
  5. System thinking - Consider scalability and real-world constraints
  6. Pattern synthesis - Build upon patterns from easier problems

Preparation Strategy

  1. Master fundamentals - Ensure solid understanding of medium problems first
  2. Study algorithms - Learn classic algorithms (Dijkstra, Floyd-Warshall, etc.)
  3. Practice regularly - Consistent practice with increasingly difficult problems
  4. Analyze solutions - Study multiple approaches and their trade-offs
  5. Time management - Hard problems often have tight time constraints in interviews

Solutions will be added as problems are solved and documented.