Advanced Algorithms

Advanced algorithms are sophisticated techniques used to solve complex computational problems efficiently. These algorithms often combine multiple paradigms and data structures to achieve optimal solutions.

Greedy Algorithms

Greedy algorithms make locally optimal choices at each step with the hope of finding a global optimum.

Examples:

• Huffman Coding
• Dijkstra's Algorithm
• Minimum Spanning Tree (Prim's and Kruskal's)
• Activity Selection

Divide and Conquer

Divide and Conquer algorithms break down problems into smaller, independent subproblems, solve them recursively, and combine their solutions.

Examples:

• Merge Sort
• Quick Sort
• Strassen's Matrix Multiplication
• Closest Pair of Points

Backtracking

Backtracking is an algorithmic technique for solving problems recursively by trying to build a solution incrementally, abandoning a path as soon as it determines that it cannot lead to a valid solution.

Examples:

• N-Queens Problem
• Sudoku Solver
• Hamiltonian Path
• Subset Sum

Network Flow

Network flow algorithms solve problems related to flows in networks, where a network is represented as a directed graph with capacities on edges.

Examples:

• Maximum Flow (Ford-Fulkerson, Edmonds-Karp)
• Minimum Cut
• Bipartite Matching
• Circulation with Demands

Advanced Data Structures

• Segment Trees: Used for range queries and updates
• Fenwick Trees (Binary Indexed Trees): Efficient prefix sums
• Disjoint Set Union (Union-Find): Tracks a set of elements partitioned into disjoint subsets
• Trie: Efficient string storage and retrieval
• Suffix Array/Tree: Advanced string processing

Common Problems

1. Word Search II (Backtracking + Trie)
2. Shortest Path with Alternating Colors (Graph Theory)
3. Maximum Profit in Job Scheduling (Dynamic Programming)
4. Sliding Window Maximum (Deque)
5. Critical Connections in a Network (Graph Theory)