Dynamic hashing is a technique used in computer science and database systems to handle the efficient storage and retrieval of data in hash tables. In traditional static hashing, the number of hash buckets is fixed, which can lead to performance issues when the number of items stored in the hash table increases.

Dynamic hashing addresses this limitation by allowing the hash table to grow or shrink dynamically based on the number of items stored in it. The main idea behind dynamic hashing is to use an additional level of indirection to access the hash buckets.

Here’s a high-level overview of how dynamic hashing works:

- Initially, a dynamic hash table starts with a small number of hash buckets. Each bucket can hold a fixed number of items.
- When an item needs to be inserted into the hash table, the hash function is applied to the item’s key to determine its hash value.
- The hash value is used to index into a directory, which contains pointers to the actual hash buckets.
- If the targeted hash bucket has space available, the item is inserted directly into the bucket.
- If the targeted hash bucket is full, it undergoes a split operation. During the split operation, the bucket is divided into two new buckets, and the items in the original bucket are redistributed between the two new buckets based on their hash values.
- The directory is updated to point to the new buckets, and the original bucket is no longer used.
- The split operation can also cause the directory to grow if it becomes too crowded. In this case, a new directory with a larger size is created, and the pointers from the old directory are redistributed to the new one.
- When the number of items decreases or certain conditions are met, dynamic hashing supports a merge operation. Merging combines two adjacent buckets into one, reducing the number of buckets and updating the directory accordingly.

Dynamic hashing provides several benefits:

- It allows the hash table to grow or shrink dynamically, adapting to the number of items stored and improving performance.
- It reduces the possibility of collisions because the buckets are split and merged as needed, spreading the items more evenly.
- It minimizes the impact of changes in data distribution, as the hash table can adjust itself dynamically.

Overall, dynamic hashing is a technique that enables efficient storage and retrieval of data in hash tables by dynamically adjusting the size and structure of the hash table based on the number of items and the distribution of data.

### How to search a key:

To search for a key in a hash table, including in a dynamically hashed table, you can follow these steps:

- Calculate the hash value of the key using the hash function specific to the hash table implementation. The hash function should distribute the keys uniformly across the hash table buckets.
- Use the hash value to locate the bucket or index in the hash table where the key is expected to be stored.
- If the hash table is dynamically hashed, ensure that the directory or structure holding the bucket pointers is consulted to find the actual bucket corresponding to the hash value. This step may involve additional indirection.
- Once you have located the bucket, search for the key within that bucket. The specific search algorithm will depend on the collision resolution method employed within the bucket. Some common techniques for searching within a bucket include linear probing, chaining with linked lists, or open addressing.
- Linear Probing: In linear probing, the keys are stored directly within the bucket array, and if a collision occurs, the algorithm linearly searches for the next available slot in the bucket array until it finds the key or an empty slot.
- Chaining: In chaining, each bucket contains a linked list or another data structure that holds all the keys hashed to that bucket. To search for a key, you traverse the linked list or data structure within the bucket and compare the search key with the keys stored in each node until a match is found or the end of the list is reached.
- Open Addressing: In open addressing, the keys are stored directly within the bucket array, and if a collision occurs, the algorithm uses a predefined probing sequence to search for the key in alternative locations within the hash table.

- If the key is found, the search is successful, and you can retrieve the associated data or perform any other desired operations.
- If the key is not found within the bucket, it means that the key does not exist in the hash table.

Remember to consider the specifics of the hash table implementation you are working with, including collision resolution methods and any additional data structures used for dynamic hashing.

### How to insert a new record:

To insert a new record into a hash table, including a dynamically hashed table, you can follow these steps:

- Calculate the hash value of the key for the new record using the hash function specific to the hash table implementation. The hash function should distribute the keys uniformly across the hash table buckets.
- Use the hash value to determine the bucket or index in the hash table where the new record should be inserted.
- If the hash table is dynamically hashed, ensure that the directory or structure holding the bucket pointers is consulted to find the actual bucket corresponding to the hash value. This step may involve additional indirection.
- Once you have located the bucket, consider the collision resolution method employed within the bucket to handle any potential collisions. The specific method will depend on the hash table implementation. Common collision resolution techniques include:
- Linear Probing: If a collision occurs and the bucket is already occupied, you can linearly probe for the next available slot in the bucket array until you find an empty slot to insert the new record.
- Chaining: If the bucket uses chaining, you can add the new record to the linked list or data structure within the bucket. This involves creating a new node or element and appending it to the existing list.
- Open Addressing: If the bucket uses open addressing, you need to follow the predefined probing sequence to search for an empty slot in the hash table. Once an empty slot is found, you can insert the new record in that slot.

- Insert the new record into the appropriate location determined by the collision resolution method. Ensure that you store both the key and the associated data in the hash table.
- If necessary, update any metadata or data structures used to track the number of items in the hash table or to support dynamic hashing.

By following these steps, you can successfully insert a new record into a hash table, taking into account the specific collision resolution method and any dynamic hashing mechanisms in place.

### Insert key 9 with hash address 10001 into the above structure:

Based on the provided hash address (10001), let’s assume that it corresponds to the binary representation of the address. For simplicity, I’ll assume a dynamic hash table with a directory and linear probing as the collision resolution method.

- Calculate the hash value: Convert the hash address (10001) to its decimal equivalent, which is 17. This will be the hash value for the key.
- Use the hash value to locate the bucket or index in the hash table. In this case, we’ll assume that the hash table is an array of buckets.
- Assuming the hash table is dynamically hashed, consult the directory or structure holding the bucket pointers to find the actual bucket corresponding to the hash value.
- Once you have located the bucket, check if the bucket is already occupied or not.
- If the bucket is empty, you can directly insert the key 9 into that bucket.
- If the bucket is occupied, you need to perform linear probing to find the next available slot in the bucket array. Linear probing involves incrementing the index by one until an empty slot is found or the end of the bucket array is reached.
Let’s say the bucket is occupied until index 20. Starting from index 17 (the hash value), you would increment the index until you find an empty slot. Assuming index 18, 19, and 20 are occupied, you continue probing until you find an empty slot.

- Once you find an empty slot (let’s say index 21), you can insert the key 9 into that slot. Now the bucket at index 21 contains the key 9.
- If necessary, update any metadata or data structures used to track the number of items in the hash table or to support dynamic hashing.

Note that the above steps assume a simplified representation of a hash table. In practice, hash tables can have variations in implementation, collision resolution techniques, and dynamic hashing mechanisms, so the exact steps may vary.

### Advantages of dynamic hashing:

Dynamic hashing offers several advantages over static hashing:

- Efficient Space Utilization: Dynamic hashing allows the hash table to grow or shrink dynamically based on the number of items stored. This flexibility ensures that the hash table can adapt to the actual number of items, avoiding excessive memory allocation or wasted space.
- Improved Performance: With dynamic hashing, the hash table can maintain a balanced load factor, which helps distribute the items evenly across the buckets. This balanced distribution reduces the likelihood of collisions and improves the efficiency of search, insertion, and deletion operations.
- Scalability: Dynamic hashing enables the hash table to scale gracefully as the number of items increases. It can accommodate a larger number of items without requiring a complete restructuring of the hash table. This scalability is particularly beneficial for applications where the number of items fluctuates over time.
- Flexibility in Data Distribution: Dynamic hashing can handle changes in data distribution more effectively than static hashing. When data distribution changes, such as when certain keys become more or less frequent, dynamic hashing can redistribute the items by splitting or merging buckets, ensuring a more even distribution and minimizing the impact on performance.
- Reduced Collisions: By adjusting the size and structure of the hash table dynamically, dynamic hashing aims to minimize collisions. Collisions occur when multiple items hash to the same bucket, and they can degrade performance. The ability to split and merge buckets allows for a more even distribution of items, reducing the probability of collisions.
- Lower Memory Overhead: Dynamic hashing typically requires less memory overhead compared to static hashing, as it only allocates space for the current number of items in the hash table. This can result in more efficient memory utilization, especially when dealing with large hash tables or varying numbers of items.

Overall, dynamic hashing provides flexibility, scalability, and improved performance compared to static hashing. It addresses the limitations of fixed-size hash tables and adapts to changing data characteristics, resulting in more efficient and effective storage and retrieval of data.

### Disadvantages of dynamic hashing:

While dynamic hashing offers several advantages, it also has some potential disadvantages:

- Increased Complexity: Dynamic hashing introduces additional complexity compared to static hashing. The need to maintain a directory or structure to track the bucket locations and handle splits and merges adds complexity to the implementation. This complexity can make the code harder to understand, debug, and maintain.
- Overhead of Directory or Structure: Dynamic hashing requires a directory or structure to map the hash values to the actual bucket locations. This overhead increases memory usage and can impact the overall performance of the hash table. The directory needs to be updated and accessed for every hash operation, adding extra computational overhead.
- Insertion and Deletion Overhead: The dynamic nature of the hash table can introduce additional overhead during insertions and deletions. When a bucket split or merge occurs, the redistribution of items and updating of the directory may involve multiple operations, affecting the performance of these operations.
- Potential for Fragmentation: As the hash table grows and shrinks dynamically, fragmentation can occur. When buckets are split and merged, it can result in smaller gaps of unused space within the hash table. This fragmentation may reduce space utilization efficiency and may require periodic reorganization or compaction to reclaim wasted space.
- Increased Memory Usage: Dynamic hashing may require additional memory compared to static hashing. The overhead of maintaining the directory and adapting the hash table structure dynamically can consume extra memory, especially when dealing with large hash tables or a high number of items.
- Slower Initialization: The initialization of a dynamically hashed table can be slower compared to a statically hashed table. Creating and initializing the directory or structure to support dynamic hashing may involve additional steps and memory allocations, contributing to increased initialization time.

It’s important to consider these potential disadvantages when deciding whether to use dynamic hashing. The choice between dynamic hashing and static hashing depends on the specific requirements and characteristics of the application, including the expected number of items, the nature of data distribution, and the desired trade-offs between performance and complexity.