This trace follows the actual state transitions behind the companion Arrays and Linked Lists: Cost Follows Access Pattern. It describes a common execution path; implementation details can vary, so keep the contract separate from the mechanism.
Step 1: Choose contiguous or linked storage
An array stores elements in contiguous indexed positions. This makes locating element i a direct address calculation and often keeps nearby values close in cache. A linked list stores each value alongside a reference to another node, which makes traversal follow pointers instead of predictable offsets.
Step 2: Locate the target position
Inserting into the middle of a packed array shifts later values, while inserting a node into a linked list can update a constant number of references once the insertion point is already known. Finding that point still requires traversal. Dynamic arrays balance capacity and memory use by occasionally allocating a larger block and copying elements.
Step 3: Read or change the element
An array index computes an offset directly, while a linked update follows references to a known node and rewires its neighbors; locating that node can dominate the operation.
At this point, record the state that changed and check the invariant before advancing. If the operation repeats, make clear which values persist and which are recomputed.
Step 4: Repair affected positions
Constant-time indexing does not imply constant-time insertion, and constant-time pointer rewiring does not imply constant-time list operations overall. Linked structures add allocation and pointer-chasing costs. Arrays may reserve unused capacity, but their locality often makes them faster for scans than a theoretically similar pointer-based structure.
Step 5: Return with the invariant intact
Choose a representation for a queue with frequent append, front removal, and occasional iteration. State which operations dominate and what guarantees your implementation needs.
The trace is complete when the result satisfies the stated contract. Compare this model with the concrete runtime or system you are studying before making a performance claim.