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Advanced Backtracking

Complex backtracking problems and advanced applications.

Graph coloring​

  • Problem: Color graph vertices with minimum colors so no adjacent vertices have same color
  • Constraints: No two adjacent vertices can have same color
  • Backtracking: Try each color for vertex, backtrack if conflict
  • Optimization: Use color classes and constraint propagation

Hamiltonian cycle​

  • Problem: Find cycle that visits each vertex exactly once
  • Constraints: Each vertex visited exactly once, cycle must be closed
  • Backtracking: Build path incrementally, backtrack if dead end
  • Optimization: Use pruning based on remaining vertices

Knight's tour​

  • Problem: Find sequence of moves for knight to visit every square on chessboard
  • Constraints: Knight must move in L-shape, visit each square once
  • Backtracking: Try each possible move, backtrack if stuck
  • Optimization: Use Warnsdorff's rule for better performance

Partition problems​

  • Problem: Partition set into subsets with specific properties
  • Constraints: Each element in exactly one subset, subsets satisfy conditions
  • Backtracking: Try assigning each element to different subsets
  • Optimization: Use dynamic programming for overlapping subproblems

Constraint optimization​

  • Problem: Find solution that satisfies constraints and optimizes objective
  • Constraints: Must satisfy all given constraints
  • Optimization: Minimize or maximize objective function
  • Backtracking: Explore solution space, prune based on objective bounds