Final answer:
For O(1) support of minimum, maximum, successor, and predecessor functions in an augmented order-statistic tree, pointers to the minimum and maximum elements are kept at the root, and parent pointers in nodes enable fast navigation, without affecting the O(log n) performance of other operations.
Step-by-step explanation:
The question asks how to support the minimum, maximum, successor, and predecessor operations in O(1) time on an augmented order-statistic tree. An order-statistic tree is a type of self-balancing binary search tree such as a Red-Black Tree that maintains additional information in its nodes. Specifically, each node holds an extra piece of data indicating the size of the subtree for which it is the root.
To achieve O(1) performance for the minimum and maximum functions, we can augment the tree structure to keep pointers to the minimum and maximum elements at the root of the tree. Whenever an insertion or deletion operation changes the minimum or maximum, we update these pointers accordingly, which can be done in constant time during the rebalancing operations.
The successor and predecessor operations can also be carried out in O(1) time by maintaining additional parent pointers in each node, which allows us to navigate the tree upwards. With parent pointers, it is straightforward to move to the next higher or lower element according to the binary search tree properties.
These modifications do not affect the O(log n) performance of other tree operations, such as insertion, deletion, and searching, because the additional work done is still within the bounds of constant time.