# Lecture 17 - Sequences

We have discussed the importance of parallelism several times thus far in the course, and even conducted mathematical analysis of time complexity in the presence of parallelism, but so far given a limited selection of parallel-friendly data structures. Our go-to data structure for parallelism purposes is the tree, as the list is an inherently sequential data structure, but it doesn't need to be the only one.

In this lecture, we introduced sequences, which are essentially immutable arrays. They are fixed-size lists that admit constant-time access to any given element, which is a very strong property that allows us to have nice parallel operations. For instance, mapping a function onto a sequence can be done faster than $O(n)$ in the length of the sequence, as each cell can be done at the same time, whereas on a list it is bottlenecked to be at least linear, due to needing to traverse to the end of the list.

We walked through the sequence library, which provided many familiar list functions, albeit on sequences. We also discussed the theory of cost graphs, which are a mathematical tool that we can use to analyze the time complexity of nested sequence operations.

While sequences are useful, they are generally better for bulk operations when the number of elements is known ahead of time. For serial operations involving singular elements, lists tend to be better. Sequences are overall an interesting application of a module with an abstract interface, as well as a nice case study of how we can obtain the benefits of imperative structures like arrays, without compromising immutability.