What is the difference between a fusion tree and VEB-Tree?
Fusion tree and Van Emde Boas (VEB) tree are both data structures used for maintaining a dynamic set of integers, but they differ in their approaches and performance characteristics.

A fusion tree is a balanced tree that uses the idea of fusion to improve its performance. In a fusion tree, elements are initially sorted into a sorted list, and then merged to form a complete binary tree with perfect balance. The tree is then recursively split into two sub-trees, with one sub-tree containing the elements less than the median and the other containing the elements greater than the median. This process is repeated until the base case of a single element or an empty set is reached. A fusion tree supports operations such as insert, delete, and search in O(log n) time, where n is the size of the set.

A VEB tree is a hierarchical structure that recursively partitions a universe of size U into √U smaller universes of size √U. The root of the tree maintains a bit vector of size √U, where each bit represents the presence or absence of an element in the set. The VEB tree then recursively partitions each smaller universe into another VEB tree, with the process continuing until the base case of a universe of size 2 is reached. A VEB tree supports operations such as insert, delete, and search in O(log log U) time, which is faster than the O(log n) time of a fusion tree, but requires more memory.

In summary, the main differences between a fusion tree and a VEB tree are their approach to maintaining a dynamic set of integers and their performance characteristics. A fusion tree uses the idea of fusion and is more memory-efficient, while a VEB tree is a hierarchical structure that provides faster operations with larger memory requirements.
What do you mean by universe in this context?