🚀 Backtracking

Backtracking is a powerful algorithmic technique used for solving problems that require searching through all possible solutions. It is particularly useful when the problem involves combinatorial search or decision-making with constraints.

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🔹 Categories of Backtracking Problems

Backtracking problems generally fall into the following categories:

1️⃣ Subset and Subsequence Generation

• Generate all subsets or subsequences of a given set.
• Often involves including or excluding elements recursively.

📌 Examples:

• 78. Subsets
• 90. Subsets II (with Duplicates)
• 131. Palindrome Partitioning

💡 Pattern:

• Choose an element, make a recursive call including it, then backtrack by removing it.

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2️⃣ Permutations and Arrangements

• Generate all possible orderings of a given set of elements.
• May involve duplicates or additional constraints.

📌 Examples:

• 46. Permutations
• 47. Permutations II (with Duplicates)
• 60. Permutation Sequence

💡 Pattern:

• Swap elements in-place to generate permutations.
• Use a boolean array or set to track used elements.

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3️⃣ Combination Problems

• Generate combinations of elements where order does not matter.
• Useful for choosing k elements from n.

📌 Examples:

• 77. Combinations
• 216. Combination Sum III
• 39. Combination Sum

💡 Pattern:

• Recursively pick elements while ensuring uniqueness.
• Maintain a current combination list and backtrack when needed.

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4️⃣ Constraint Satisfaction Problems (CSP)

• Problems where a solution must satisfy constraints, often involving grid-based solutions.

📌 Examples:

• 37. Sudoku Solver
• 51. N-Queens
• 52. N-Queens II

💡 Pattern:

• Use recursion with constraints to limit the search space.
• Maintain additional data structures (e.g., boolean arrays for rows/columns).

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5️⃣ Path-Finding in Grids (Maze, Rat in a Maze)

• Problems where you need to explore all possible paths in a grid.

📌 Examples:

• 79. Word Search
• 130. Surrounded Regions
• 980. Unique Paths III

💡 Pattern:

• Use DFS with backtracking to explore possible paths.
• Mark visited cells and unmark when backtracking.

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6️⃣ Word and String Problems

• Problems that involve constructing or searching for words using recursive backtracking.

📌 Examples:

• 17. Letter Combinations of a Phone Number
• 79. Word Search
• 212. Word Search II (Trie + Backtracking)

💡 Pattern:

• Use recursive DFS to explore character sequences.
• Often combined with Trie data structures.

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7️⃣ Mathematical and Number Problems

• Problems that involve numbers, sequences, or generating numeric solutions.

📌 Examples:

• 60. Permutation Sequence
• 526. Beautiful Arrangement
• 679. 24 Game

💡 Pattern:

• Use recursive choices with constraints on numbers.
• May involve pruning to optimize the search.

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🔹 Key Techniques in Backtracking

✅ Recursion with Decision Trees: Explore all possible choices and backtrack when invalid.
✅ Pruning (Branch Cutting): Avoid unnecessary recursive calls to improve efficiency.
✅ Bitmasking & Hashing: Track state efficiently in some cases (e.g., N-Queens).
✅ Memoization (Hybrid with DP): Store previously computed results to avoid recomputation.

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🚀 Summary Table

Category Examples Pattern

Subsets & Subsequences	Subsets, Palindrome Partitioning	Include/exclude elements recursively
Permutations	Permutations, Permutation Sequence	Swap elements, track visited states
Combinations	Combinations, Combination Sum	Recursively pick k elements
Constraint Satisfaction	Sudoku Solver, N-Queens	Recursively place elements with constraints
Path-Finding	Word Search, Unique Paths III	DFS with backtracking in a grid
Word/String Problems	Letter Combinations, Word Search	Recursive character exploration
Math & Number Problems	Beautiful Arrangement, 24 Game	Recursive numerical choices

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🛠 How to Get Better at Backtracking?

1️⃣ Solve Problems from Each Category – Start with simple ones like "Subsets" and then move to harder ones like "N-Queens".
2️⃣ Draw Decision Trees – Understand how recursion unfolds.
3️⃣ Practice Writing Constraints – This helps prune the search space.
4️⃣ Combine with Other Techniques – Hybridize with DFS, Dynamic Programming, and Bitmasking for advanced problems.

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💡 Would you like a step-by-step explanation of any of these categories? 🚀