Improving algorithmic efficiency without altering its structure can be achieved through several techniques:

1. Code Optimization: Refactoring the code to eliminate unnecessary operations, using more efficient functions and methods, and optimizing data handling can reduce the overhead and improve performance.
2. Using Better Compiler/Interpreter Options: Modern compilers and interpreters often have optimization options that can significantly improve the performance of the existing code.
3. Choosing Efficient Data Types: Using data types that consume less memory or allow faster operations for the same tasks can enhance efficiency.
4. Parallel Execution: Distributing the workload across multiple processors or cores can significantly reduce execution time, particularly for algorithms that can be parallelized.
5. Caching and Memorization: Storing the results of expensive function calls and reusing them instead of recalculating can improve efficiency, particularly in algorithms with many redundant calculations.
6. Efficient Memory Management: Reducing memory footprint and efficient memory allocation strategies can decrease the time and space complexities. While these techniques do not change the algorithm's fundamental structure, they optimize its implementation, making it more efficient in practical applications.

Loop unrolling is a technique used to increase an algorithm's efficiency by reducing the number of iterations and, consequently, the overhead of loop control. In loop unrolling, multiple iterations of the loop are combined into a single iteration, reducing the loop's overall execution count. This process decreases the overhead caused by the loop control mechanisms (like incrementing the counter and checking the loop condition).

For example, in a loop that increments by one, you could increase the increment to two, processing two elements per iteration instead of one. This technique is especially beneficial in loops with small bodies where the loop control overhead is significant compared to the execution time of the loop's body. However, it's crucial to balance the extent of unrolling because overly unrolled loops can lead to increased code size (code bloat), which might negatively impact the instruction cache efficiency and, thus, the overall performance. Also, too much unrolling can decrease readability and maintainability of code. Therefore, loop unrolling is a trade-off and needs to be applied judiciously, considering the algorithm's nature and the hardware on which it runs.

A while loop is more efficient than a for loop in scenarios where the termination condition is dependent on something other than a straightforward counter. For instance, consider a program that reads data from a stream or a file until it encounters a specific marker or end-of-file (EOF). Since the number of iterations isn't known beforehand and is determined by when the condition (finding the marker or EOF) is met, a while loop is a more natural and efficient choice. It directly associates the loop's continuation with the desired condition, avoiding the need for an arbitrary counter or checking a condition within the loop body (which might be necessary with a for loop). This directness can lead to clearer, more maintainable code and can prevent unnecessary iterations that might occur if a for loop's counter doesn't align precisely with the condition being met.

The choice of data structures significantly impacts algorithm efficiency, particularly in terms of time and space complexity. Different data structures are optimised for different operations. For instance, arrays allow fast access (O(1)) to elements if the index is known, making them excellent for problems where direct access is frequent. However, adding or removing elements from an array, especially at the beginning or in the middle, is less efficient (O(n)) because it may require shifting elements. In contrast, linked lists offer faster insertion and deletion (O(1) if the node is known; O(n) for searching the node), but slower access times (O(n)) as elements must be accessed sequentially. Data structures like hash tables provide fast access, addition, and deletion times (average and best-case O(1), worst-case O(n)) but at the cost of higher space complexity and less order. Trees, particularly balanced ones like AVL or Red-Black trees, provide O(log n) access, addition, and deletion times, making them suitable for dynamic dataset problems where order and quick operations are essential. Thus, choosing an appropriate data structure based on the specific needs and operations of an algorithm is crucial for optimising its overall efficiency.

Considering the best, worst, and average cases in algorithm efficiency analysis is crucial because it provides a comprehensive understanding of how the algorithm will perform under different circumstances. The worst-case analysis is essential for understanding the maximum time or space an algorithm will require, ensuring that it meets performance requirements under all scenarios. The best-case analysis, while often less critical, can give insight into the optimal performance and scenarios where the algorithm is most efficient. The average-case analysis, arguably the most practical, provides a realistic expectation of performance in typical usage. Some algorithms might have excellent best-case performance but poor worst-case performance (e.g., QuickSort), which might be acceptable in some contexts but not in others, like real-time systems where predictability is key. Thus, a thorough analysis helps in choosing the right algorithm based on specific application needs and understanding potential performance bottlenecks.