Divide-and-Conquer Optimization Algorithm

A Divide-and-Conquer Optimization Algorithm is an optimization algorithm (to solve computational tasks) that systematically breaks down complex problems into smaller subproblems (that can be solved independently and combined).

• AKA: Divide and Conquer Approach, D&C Algorithm.
• Context:
  • It can transform Complex Problem into multiple subproblems through problem decomposition.
  • It can apply Core Strategy through recursive division, independent solution, and solution combination.
  • It can ensure Solution Correctness through mathematical induction and algorithmic proof.
  • It can optimize Computational Performance through parallel processing and workload distribution.
  • It can handle Problem Scale through systematic breakdown and controlled recursion.
  • ...
  • It can often improve Algorithm Efficiency through task parallelization and memory locality.
  • It can often reduce Problem Complexity through systematic simplification and subproblem optimization.
  • It can often enhance Solution Quality through divide-and-conquer refinement and recursive improvement.
  • ...
  • It can range from being a Simple Partitioning Algorithm to being a Complex Recursive System, depending on its problem decomposition strategy.
  • It can range from being a Sequential Processor to being a Parallel Solver, depending on its execution model.
  • ...
  • It can integrate with Dynamic Programming for optimization problems.
  • It can support Parallel Computing Frameworks for distributed execution.
  • It can utilize Memory Hierarchy for performance optimization.
  • ...
• Examples:
  • Classical D&C Algorithms, such as:
    • Sorting Algorithms, such as:
      • Merge Sort for efficient sorting.
      • Quick Sort for in-place sorting.
    • Search Algorithms, such as:
      • Binary Search for sorted data search.
      • Binary Space Partitioning for spatial search.
  • Numerical D&C Algorithms, such as:
    • Matrix Operations, such as:
      • Strassen Algorithm for matrix multiplication.
      • Fast Fourier Transform for frequency analysis.
  • Geometric D&C Algorithms, such as:
    • Computational Geometrys, such as:
      • Closest Pair Algorithm for point set analysis.
      • Convex Hull Algorithm for boundary computation.
  • Divide and Conquer Learning Algorithm, ...
  • ...
• Counter-Examples:
  • Greedy Algorithms, which make local optimal choices without problem division.
  • Dynamic Programming Algorithms, which solve overlapping subproblems through tabulation.
  • Linear Algorithms, which process input data sequentially without recursion.
• See: Algorithm Design Pattern, Recursive Algorithm, Problem Decomposition Strategy, Algorithmic Complexity, Parallel Algorithm, Recursive Partitioning Algorithm.

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References

2025-01

• LLM

Algorithm: OPTIMIZE(problem, constraints)
Input: Optimization problem and constraints
Output: Best solution found

1. If IS_SMALL_ENOUGH(problem) Then
   Return SOLVE_DIRECTLY(problem)

2. // Divide
  subproblems = SPLIT_PROBLEM(problem)
  solutions = []
 
3. // Conquer
  For each sub in subproblems:
      If IS_FEASIBLE(sub, constraints):
          solution = OPTIMIZE(sub, constraints)
          solutions.append(solution)

4. // Combine
  result = MERGE_SOLUTIONS(solutions)

5. Return LOCAL_OPTIMIZE(result)

SPLIT_PROBLEM(problem):
   // Divide into smaller subproblems
   mid = problem.size / 2
   return [problem[0:mid], problem[mid:end]]

MERGE_SOLUTIONS(solutions):
   // Combine subsolutions into complete solution
   result = INITIALIZE_SOLUTION()
   For each s in solutions:
       result = MERGE(result, s)
   Return result

LOCAL_OPTIMIZE(solution):
   // Improve combined solution
   current = solution
   While CAN_IMPROVE(current):
       current = IMPROVE_SOLUTION(current)
   Return current

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