The Stony Brook Algorithm Repository
cs.sunysb.edu
cs.sunysb.edu
There are hundreds of problems on the site, roughly graded by difficulty, easiest first. In the beginning you get to solve a few easy problems. As you solve each problem, more problems become available for you to try.
They are "a platform for learning bioinformatics through problem solving" so many of the problems are about strings, but there are also problems on graphs and other structures.
They also have a new section called "Algorithmic Heights", which is "A collection of exercises in introductory algorithms to accompany "Algorithms", the popular textbook by Dasgupta, Papadimitriou, and Vazirani." I have submitted this for discussion here: https://news.ycombinator.com/item?id=7456390
(Edit: Minor fixes.)
[0] https://www.cs.cmu.edu/afs/cs.cmu.edu/project/pscico-guyb/re...