Table of contents
3. Steps of divide and conquer algorithm:
4. Example: merge sort algorithm
Five. Advantages and disadvantages:
introduction:
In computer science, the divide-and-conquer algorithm is an effective strategy often used in solving complex problems. The algorithm decomposes the problem into smaller, easier-to-solve sub-problems, and then obtains the solution to the original problem by merging the solutions to the sub-problems. This blog will introduce the basic principles, applicable scenarios, advantages and disadvantages of the divide and conquer algorithm, hoping to provide readers with a more comprehensive understanding.
1. Basic principles:
When we face a complex problem, the basic principle of the divide-and-conquer algorithm is to divide the problem into smaller, easier-to-solve sub-problems, and then obtain the solution to the original problem by merging the solutions of the sub-problems.
Specifically, the steps of the divide and conquer algorithm are as follows:
1. Divide: Divide the original problem into several smaller and independent sub-problems. This step is usually done recursively. Break down the problem gradually until the problem is simple enough that no further decomposition into subproblems is necessary.
2. Conquer: Solve each subproblem recursively. At this point, the original problem has been decomposed into multiple smaller sub-problems, and we can solve each sub-problem independently to obtain their solutions. This step is usually implemented by recursively calling the same divide-and-conquer algorithm.
3. Combine: Combine the solutions of the subproblems into the solution of the original problem. In this step, we combine the solutions of the subproblems according to certain rules to obtain the solution of the original problem. This merging process is usually linear, using the solutions to the subproblems to construct the solution to the original problem.
It is worth noting that the premise that the divide and conquer algorithm can work is that the solutions of the subproblems can be combined into the solution of the original problem. If the solutions to the subproblems cannot be merged directly, then we may need an additional step to perform the merge operation.
The core idea of the divide and conquer algorithm is to divide a complex problem into solvable small problems, and then solve these small problems recursively, and finally combine the results into the solution of the original problem. The time complexity of the algorithm is usually related to the number of decomposed sub-problems and the size of each sub-problem.
Summary:
Divide and conquer algorithms provide an efficient strategy for solving complex problems by breaking the problem into smaller, more solvable subproblems and gradually merging the solutions to the subproblems. It is the basis of many important algorithms and data structures, such as merge sort, quick sort and binary tree.
2. Applicable scenarios:
The divide and conquer algorithm is suitable for the following scenarios:
1. The problem can be divided into independent sub-problems: The divide-and-conquer algorithm is suitable for those problems that can be divided into independent sub-problems. This means that the original problem can be solved by recursively decomposing it into multiple smaller sub-problems.
2. The solutions of the sub-problems are easy to merge into the solution of the original problem: the divide-and-conquer algorithm requires that the solutions of the sub-problems can be combined to obtain the solution of the original problem. This means that solutions to subproblems should be composable or merged to form a solution to the original problem.
3. The scale of sub-problems can be further reduced: the divide-and-conquer algorithm divides the original problem into multiple smaller-scale sub-problems in each recursive call. Therefore, problems suitable for divide-and-conquer algorithms need to satisfy the condition that the size of sub-problems can be further reduced to avoid infinite recursion or unnecessary calculations.
Divide and conquer algorithms are used in many fields, the following are some common applicable scenarios:
1. Sorting Algorithms: Sorting algorithms like merge sort and quick sort utilize the idea of divide and conquer algorithms. They decompose the sequence to be sorted into smaller subsequences, then recursively sort the subsequences, and finally merge the sorted subsequences to obtain the final sorting result.
2. Search algorithm: Binary search is a classic search algorithm using divide and conquer algorithm. It divides the problem into two smaller sub-problems (left sub-array and right sub-array), and then determines which sub-array to continue searching by comparing the size relationship between the target value and the middle element, thereby narrowing the search range.
3. Graph Algorithms: Many problems on graphs can also be solved using divide-and-conquer algorithms, such as minimum spanning tree algorithms (such as Krus' algorithm and Prim's algorithm) and shortest path algorithms (such as Dijkstra's algorithm). These algorithms divide the graph into multiple subproblems, and then obtain the solution to the original problem by merging the solutions of the subproblems.
4. Polynomial multiplication: Polynomial multiplication is a common problem in algebra. The divide-and-conquer algorithm can be used to transform polynomial multiplication into smaller-scale sub-problems, and then obtain the solution of the original problem by merging the solutions of the sub-problems to improve computational efficiency.
Summary:
The divide-and-conquer algorithm is suitable for scenarios that can be divided into problems that are independent of each other and can be combined for solutions, and the size of the sub-problems can be further reduced. It is widely used in sorting, searching, graph algorithms and many other fields. By rationally decomposing and solving sub-problems, divide-and-conquer algorithms can effectively solve complex problems.
3. Steps of divide and conquer algorithm:
The divide-and-conquer algorithm can be broken down into three steps: split, solve, and merge. Below I will introduce the specific content of each step in detail.
1. Decomposition (Divide):
Decomposition is the process of dividing the original problem into several smaller and independent sub-problems. This step is usually done recursively. Break down the problem gradually until the problem is simple enough that no further decomposition into subproblems is necessary.
Specifically, when we face a complex problem, we need to consider how to decompose it into smaller sub-problems. This usually requires finding a reproducible structure for the problem, and an appropriate partitioning method. Each subproblem should be as similar as possible and can be solved by the same algorithm.
2. Solving (Conquer):
The solving step is the process of recursively solving each sub-problem. In this step, the original problem has been decomposed into several smaller sub-problems. We can solve each subproblem independently and obtain their solutions.
For each subproblem, we continue to apply the divide-and-conquer algorithm, continuing to decompose into smaller subproblems until we reach the base case, where the subproblems are simple enough to be solved directly. This step is usually implemented by recursively calling the same divide-and-conquer algorithm.
3. Combine:
The combining step is the process of merging the solutions of the sub-problems into the solution of the original problem. After solving all the sub-problems, we need to combine their solutions according to certain rules to get the solution of the original problem.
The merge operation is usually linear, and it uses the solutions of the subproblems to construct the solution of the original problem. The exact way you combine them depends on the specific problem being solved. Some problems have a very simple merging step, requiring only a simple combination of solutions to subproblems. Other problems may require more complex merging strategies.
These three steps together constitute the basic flow of the divide and conquer algorithm. We can efficiently solve complex problems by decomposing the original problem into subproblems, solving the subproblems recursively, and then combining the solutions of the subproblems to obtain the solution to the original problem.
It is worth noting that the divide and conquer algorithm requires that the solutions of the subproblems can be combined into the solution of the original problem. If the solutions to the subproblems cannot be merged directly, then we may need an additional step to perform the merge operation.
Summary:
The steps of the divide and conquer algorithm include decomposition, solving and merging. We can efficiently solve complex problems by dividing the original problem into smaller subproblems, solving each subproblem recursively, and combining the solutions of the subproblems into the solution of the original problem. These three steps together form the basic principle of the divide and conquer algorithm.
4. Example: merge sort algorithm
Merge Sort (Merge Sort) is a commonly used sorting algorithm, which uses the idea of divide and conquer algorithm to divide an array to be sorted into two sub-arrays of equal size, then recursively sort the sub-arrays, and finally combine the two Merge sorted subarrays into one sorted array.
The following are the detailed steps of the merge sort algorithm:
1. Divide:
Recursively decompose the array to be sorted into two smaller sub-arrays. First, find the position of the midpoint of the array, and divide the array into left and right subarrays until it can no longer be divided. This process is achieved by recursion.
2. Solve (Conquer):
Recursively apply the merge sort algorithm to each sub-array until the length of the sub-array is 1, that is, the sub-array cannot be divided. This is the base case of recursion and the solution to the subproblem.
3. Combine:
Merge two sorted sub-arrays into one sorted array. During the merging process, we use two pointers to traverse the two sub-arrays respectively, and put the elements into a temporary array in order of size until both sub-arrays are traversed. Finally, the elements in the temporary array are copied back to their corresponding positions in the original array.
The above steps are repeated recursively until the final sorting is completed.
The time complexity of merge sort is O(nlogn), where n is the length of the array to be sorted. This is due to the fact that the time complexity of splitting an array is O(logn) each time, while the time complexity of merging two sorted subarrays is O(n). Therefore, the overall time complexity of merge sort is O(nlogn).
The advantages of merge sort include stability (the order of equal elements does not change) and generalizability for various data types. But its disadvantage is that it needs additional storage space to store the temporary array, so the space complexity is high.
Summary:
Merge sort is a common sorting algorithm. Using the idea of divide and conquer algorithm, the array is decomposed into sub-arrays, the sub-arrays are sorted recursively, and then the ordered sub-arrays are merged into an ordered array. Merge sort efficiently sorts an array through the steps of decomposing, resolving, and merging. It has a time complexity of O(nlogn), but requires additional storage space.
Five. Advantages and disadvantages:
Divide and conquer algorithm is a commonly used algorithm design strategy, which divides the problem into multiple smaller and independent sub-problems, then solves these sub-problems separately, and combines their solutions into the solution of the original problem. The following is a detailed description of the advantages and disadvantages of the divide and conquer algorithm:
advantage:
1. Solve complex problems: Divide and conquer algorithms make the solution of problems simpler and clearer by decomposing complex problems into independent sub-problems. Breaking down a large problem into smaller ones can make the problem easier to understand and solve.
2. Parallelizable: Since the divide-and-conquer algorithm divides the problem into multiple sub-problems, and each sub-problem is independent of each other, these sub-problems can be easily solved in parallel. This makes the divide-and-conquer algorithm suitable for parallel computing scenarios, and can make full use of multi-core processors and distributed computing resources.
3. Improve algorithm efficiency: For some problems, the time complexity of divide and conquer algorithm is lower than other algorithms. By decomposing the problem into smaller sub-problems and recursively solving these sub-problems, the size of the original problem can be reduced, thereby improving the efficiency of the algorithm.
4. Modularity and reusability: The idea of divide and conquer algorithm makes the algorithm have good modularity and reusability. By dividing the problem into mutually independent sub-problems, each sub-problem can be designed and implemented individually, which enables algorithms to be organized and reused in a modular manner.
shortcoming:
1. Additional space is required: In the divide-and-conquer algorithm, in order to store the solutions of the subproblems and the temporary data generated during the merging process, additional storage space is often required. This may lead to a high space complexity of the algorithm.
2. The overhead of recursion: Divide and conquer algorithms usually use recursion to solve subproblems. The recursive call process involves operations such as function calls, creation and destruction of local variables, which will increase additional overhead and may cause problems such as stack overflow.
3. The overhead of sub-problem division: The divide-and-conquer algorithm needs to divide the original problem into sub-problems at each level of recursion, and this division process requires a certain amount of overhead. For some problems, partitioning into subproblems can be a time-consuming operation.
Summarize:
Divide and conquer algorithm is a powerful algorithm design strategy, which can solve complex problems, and has the advantages of parallelization, improved algorithm efficiency, modularization and reusability. However, it also requires additional space to store temporary data, recursion adds additional overhead, and subproblem partitioning may also cause overhead. Therefore, when designing and using the divide-and-conquer algorithm, factors such as the characteristics of the problem, data size, and time and space constraints need to be considered comprehensively.