Dynamic Set Data Structures and their Implementations

Key Insight

Dynamic sets are core to computer science, representing mutable collections that can grow, shrink, or change over time, unlike static mathematical sets. Algorithms often require operations like inserting, deleting, and testing membership, defining a structure known as a dictionary. Elements in a dynamic set are objects typically containing an identifying key and optional 'satellite data'; if keys are from a 'totally ordered set,' additional operations like finding the minimum element or successor are possible.

Operations on dynamic sets are categorized into queries, which return information, and modifying operations, which change the set. Queries include SEARCH(S, k) to find an element by key, MINIMUM(S) and MAXIMUM(S) for ordered sets, and SUCCESSOR(S, x) or PREDECESSOR(S, x) to navigate elements. Modifying operations are INSERT(S, x) and DELETE(S, x). The time efficiency of these operations is generally measured relative to the set's size 'n', with some advanced data structures supporting all these operations in O(lg n) time in the worst case.

Elementary dynamic sets like stacks and queues are commonly implemented using arrays, offering O(1) time complexity for their primary operations. Stacks operate on a Last-In, First-Out (LIFO) principle, using PUSH for insertion and POP for deleting the most recently added element. Queues adhere to a First-In, First-Out (FIFO) policy, with ENQUEUE for adding elements to the tail and DEQUEUE for removing elements from the head. Linked lists provide a flexible representation using pointers (next and pre attributes) to define linear order, supporting variations like singly/doubly linked, sorted/unsorted, and circular lists. 'Sentinels' (dummy objects) can simplify boundary conditions in linked list operations. Objects and pointers can be synthesized from arrays in languages without native support, managing unused objects via a 'free list' that acts like a stack for allocation and deallocation.