Divide-and-Conquer Optimization Algorithm: Difference between revisions
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(Created page with "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). * <B>AKA:</B> Divide and Conquer Approach, D&C Algorithm. * <B>Context:</B> ** It can transform Complex Problem into multiple subproblems through problem decomposition. ** It can apply Core Strategy throu...") |
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** [[Linear Algorithm]]s, which process [[input data]] sequentially without recursion. | ** [[Linear Algorithm]]s, which process [[input data]] sequentially without recursion. | ||
* <B>See:</B> [[Algorithm Design Pattern]], [[Recursive Algorithm]], [[Problem Decomposition Strategy]], [[Algorithmic Complexity]], [[Parallel Algorithm]], [[Recursive Partitioning Algorithm]]. | * <B>See:</B> [[Algorithm Design Pattern]], [[Recursive Algorithm]], [[Problem Decomposition Strategy]], [[Algorithmic Complexity]], [[Parallel Algorithm]], [[Recursive Partitioning Algorithm]]. | ||
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---- | |||
== References == | |||
=== 2025-01 === | |||
* LLM | |||
<code> | |||
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 | |||
</code>< | |||
---- | ---- |
Latest revision as of 22:22, 29 January 2025
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:
- Search Algorithms, such as:
- Numerical D&C Algorithms, such as:
- Matrix Operations, such as:
- Geometric D&C Algorithms, such as:
- Divide and Conquer Learning Algorithm, ...
- ...
- Classical D&C Algorithms, such as:
- 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.
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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