Merge sort is a classic sorting algorithm. Its core idea is to divide the array into two sub-arrays, sort them separately, and then combine the sorted two sub-arrays into an ordered array. This divide-and-conquer strategy makes merge sorting highly efficient when dealing with big data. In terms of time complexity, the time complexity of merge sort is O (n log n), where n is the amount of data to be sorted. This is because merge sort divides the data into two parts, each part is inserted once, and then the two sorted parts are merged into an ordered array. Therefore, the time complexity of merge sort is O (n log n). In order to optimize the efficiency of merge sort, we can use the "tail recursion" technique to reduce the depth of the function call stack, thereby improving the performance of the algorithm. Tail recursion is a recursive method that allows us to use recursive calls directly inside the function body without using additional parameters outside the function to save state. This reduces the depth of the function call stack, thereby reducing memory consumption and improving running speed.

In the process of building the Raspberry Pi NAS storage server, we realize it through the cooperation of the Samba service and the external hard disk. First, we need to install the Samba service on the Raspberry Pi and configure its user permissions. Next, we connect the external hard disk to the Raspberry Pi and specify it as a shared directory in the configuration file of the Samba service. Finally, we can access the shared directory on other computers in the network for the purpose of data storage and backup.

Binary search is an efficient lookup algorithm that determines the location of the target value in the array by comparing the target value with the middle element of the array. This method has a time complexity of O (log n) and is more efficient than linear search (O (n)). The following is an example of a simple sorted array binary search using a Python implementation: ```python def binary_search(arr, target): left, right = 0, len(arr) - 1 while left <= right: mid = (left + right) // 2 if arr[mid] == target: return mid elif arr[mid] < target: left = mid + 1 else: right = mid - 1 return -1 ``` In this example, we define a function called `binary _ search`, which accepts an ordered array `arr` and a target value `targe` as input. We initialize two pointers, `lef` and `righ`, to the beginning and end positions of the array, respectively. Then, we enter a loop until `left` is greater than `righ`. In the loop, we calculate the index `mid` of the intermediate element and update the value of `left` or `righ` according to the relationship between the target value and the intermediate element. When the target value is found, the function returns its index; otherwise, returning -1 means that the target value was not found.

Self-Attention Mechanism (Self-Attention Mechanism) is the core component of the Transformer model, which allows the model to capture global dependencies when processing sequence data. This mechanism works by calculating the relevance of each element in the input sequence to the entire sequence, rather than simply treating the sequence as a block of fixed size. This greatly enhances the model's ability to understand and utilize contextual information. In a simple Transformer model, we first define an encoder layer that receives an input sequence and outputs a fixed-length encoding vector. Then, we use the decoder layer, which receives the encoded vector as input and outputs the predicted value of the sequence. Between these two layers, we insert a self-attention layer, which is used to calculate the correlation between each element in the input sequence and the entire sequence. The calculation process of the self-attention layer is as follows: 1. For each element in the input sequence, calculate its relevance score to the entire sequence. This is usually achieved by calculating the cosine similarity or dot product of the elements. 2. According to the correlation score, select the other elements most relevant to the current element and calculate the weighted sum of these elements. Weights are usually determined based on their correlation scores. 3. Multiply the weighted sum with the original value of the current element to get the new element value. 4. Combine the new element value with the current element to form a new element vector and pass it to the next time step. In this way, the self-attention mechanism can capture the global dependencies of the sequence data, so that the Transformer model performs well when dealing with complex tasks.

The KMP algorithm is an efficient string matching algorithm that reduces repeated matching steps through the prefix table. The prefix table is a pre-calculated array of strings that stores the length of the prefix substring for each position and its corresponding longest common prefix. During the matching process, the KMP algorithm first checks whether the current character matches a character in the prefix table. If the match is successful, it will continue to match backwards; if it does not match, it will move the prefix table forward one bit and match again. In this way, the KMP algorithm can skip the repeated matching steps without backtracking, thereby improving the efficiency of the algorithm.

Merge sort is a classic sorting algorithm, which divides the array into two halves, sorts the two halves separately, and then merges the two sorted subarrays into an ordered array. The time complexity of this algorithm is O (nlogn), where n is the length of the array. In merge sort, the idea of divide and conquer is widely used. First, divide the array into two halves, and then recursively sort the two subarrays. When both subarrays are sorted, they are combined into an ordered array. Time complexity: The time complexity of merge sort is O (nlogn), because each recursive call will halve the problem size, so logn recursions are required to complete the entire sorting process.

Build a Raspberry Pi NAS storage server, through Samba service and external hard disk, can realize centralized management and remote access of data. First, make sure your Raspberry Pi has the necessary packages installed, including Samba, Python, etc. Then, configure Samba to allow external access, and set the mount point of the external hard disk. Next, write a simple Python script to manage file uploads and downloads. Finally, test the network connection to make sure everything is normal. In this way, you can easily manage the Raspberry Pi NAS through the Web interface or command line tools.

The "0-1 backpack problem" is a classic combinatorial optimization problem that requires selecting a group of items from a limited resource to maximize the total value. Dynamic programming is a common method to solve such problems, which is efficiently solved by constructing state transition equations. Recursive implementations usually start by calculating the value of individual elements and then gradually expand to the value of the entire backpack. This method is intuitive and easy to understand, but may be inefficient due to double counting. Iterative implementation does not use recursion, but processes elements one by one, calculating the weight and value of each element. This strategy avoids double counting and improves the execution efficiency of the algorithm. No matter which implementation is adopted, the key is to understand the state transition equation and the properties of the optimal substructure, which helps us to effectively utilize the memory space and ensure the correctness of the algorithm.

The maze problem is a classic problem that requires us to find the shortest path from the starting point to the ending point in a maze of walls and doors. In this problem, we usually use the breadth-first search (BFS) algorithm to solve this problem. BFS is an algorithm used to traverse or search trees or graphs. In the maze problem, we can think of the maze as a graph in which each position is a node, and each node has a list of adjacent nodes. By using queues to implement BFS, we can access the next adjacent node in each iteration until the end point is found. When implementing this solution, we need to first create a queue and add the starting point to the queue. Then, we start the iteration, and in each iteration, we take a node from the queue and add all its adjacent nodes that have not been accessed to the queue. In this way, we can ensure that we always follow the shortest path. Finally, we add the endpoint to the queue and continue to iterate until the queue is empty. At this point, we have found the shortest path from the start to the end.

LSTM (Long Short Term Memory Network) model is a deep learning technology, especially suitable for processing time series data. It predicts future values by capturing long-term dependencies in the data. In the fields of stock market forecasting and weather forecasting, the LSTM model can accurately predict future price trends or weather changes based on historical data and real-time information. For example, in stock forecasting, LSTM can analyze historical stock price data to identify price trends and potential market turning points. In meteorological forecasting, LSTM can predict future meteorological indicators such as temperature and precipitation based on past weather patterns and current environmental conditions. Although the LSTM model is powerful, factors such as data preprocessing, feature engineering and model tuning need to be considered in practical applications to improve prediction accuracy.