A good set of algorithms to get started with is sorting algorithms. There are many different options, from easy (selection sort, bubble sort), to medium (shell sort, mergesort, quicksort) to hard (heapsort, timsort, mergesort with linear space complexity).
A different field with many options is graph algorithms. There are easy options such as depth first search and breadth first search. Problems such as topological sorting (using depth-first search) and minimum spanning tree (Kruskal's algorithm) are of medium difficulty. There are also many algorithms related to pathfinding, the most fundamental being Dijkstra's algorithm. A* is an algorithm that extends Dijkstra by a heuristic that makes it faster if assumptions can be made about the graph. Flow networks are also a field with many interesting algorithms such as the Ford–Fulkerson algorithm or the Edmonds–Karp algorithm.
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