Optimal Resizable Arrays

SIAM Journal on Computing, Volume 53, Issue 5, Page 1354-1380, October 2024.
Abstract. A resizable array is an array that can grow and shrink by the addition or removal of items from its end, or both its ends, while still supporting constant-time access to each item stored in the array given its index. Since the size of an array, i.e., the number of items in it, varies over time, space-efficient maintenance of a resizable array requires dynamic memory management. A standard doubling technique allows the maintenance of an array of size [math] using only [math] space, with [math] amortized time, or even [math] worst-case time, per operation. Sitarski, and (apparently independently) Brodnik, Carlsson, Demaine, Munro, and Sedgewick describe much better solutions that maintain a resizable array of size [math] using only [math] space, still with [math] time per operation. Brodnik et al. give a simple proof that this is best possible. We distinguish between the space needed for storing a resizable array, and accessing its items, and the temporary space that may be needed while growing or shrinking the array. For every integer [math], we show that [math] space is sufficient for storing and accessing an array of size [math], if [math] space can be used briefly during grow and shrink operations. Accessing an item by index takes [math] worst-case time, while grow and shrink operations take [math] amortized time. Using an exact analysis of a growth game, we show that for any data structure from a wide class of data structures that uses only [math] space to store the array, the amortized cost of grow is [math], even if only grow and access operations are allowed. The time for grow and shrink operations cannot be made worst-case unless [math].