A few weeks ago I mentioned completing Part 1 of the online Coursera/Stanford “Algorithms: Design and Analysis” course. Part 2 of Algorithms: Design and Analysis isn’t due to start again until next year, but I didn’t want to wait, so I enrolled in the archived version of the course to watch the videos and do the assignments. I should be ready to just reuse my work when Part 2 starts again for real.
Part 2 was where things got really interesting. The assignments required implementing these algorithms, though the course covered others too:
- A Greedy Algorithm for job scheduling.
- Prim’s and Kruskal’s minimum spanning tree algorithms. (Both O(m log n) but Prims does better on dense graphs, with more edges.)
- Modifying a minimum spanning tree to identify clusters.
- The Knapsack problem (Dynamic Programming).
- Shortest Path with the Belmann-Ford SSSP (Single-Source Shortest Path) algorithm (O(mn) and works with negative paths, but fails with negative cycles) as an alternative to Dijkstra’s Shortest Path algorithm (O(m log n) and works only with positive paths).
- All Pairs Shortest Path with the Floyd Warshall algorithm (dynamic programming) (O(n3) and works with negative paths, though it fails when it detects negative cycles).
- All Pairs Shortest path with Johnson’s algorithm, via one call to Belmann-Ford on a modified graph and repeated calls to Dijkstra on a reweighted graph. (O(mn log n) and works with negative paths, but fails with negative cycles.)
- A dynamic programming algorithm for the Traveling Salesman Problem, both bottom up and recursive.
- Local search with the 2Sat problem, using Papadimitriou’s Algorithm.
I particularly enjoyed exploring “dynamic programming”, which is really just avoiding unnecessary repeated work after you’ve identified the appropriate sub-problems. It’s identifying the sub problems that is really hard. I enjoyed playing with bottom-up dynamic programming (filling in an array as you go, often discarding the n-2th set of results as you go), and top-down dynamic programming, also known as memoization (usually recursing into sub problems and ideally not doing as many sub problems as you’d do working bottom up).
While implementing a dynamic programming solution for the Traveling Salesman Problem, I learned about Gosper’s Hack for iterating over subsets. It’s now a personal favorite in my toolbox.
As with part 1 of the course, I am not allowed to publish the code of my homework solutions. But I did create a public version of the knapsack problem for solving the Make Change problem without a canonical currency (not a real world set of coins), using dynamic programming, though you shouldn’t use that as a first way to understand the classic knapsack problem. I also implemented a simple greedy algorithm for the Make Change problem with a canonical currency (real world set of coins).
The Make Change problem was interesting because I’ve read that people can and should learn to recognize NP-Complete problems, such as the traveling salesman problem. However, it is not obvious which sets of coins would cause the Make Change problem to be solvable with a greedy algorithm and which would need dynamic programming, though it might at first seem like a minor detail. (I haven’t actually read that paper yet.)
I’ve also been reading through Steven Skiena’s The Algorithm Design Manual book, which I can highly recommend. It’s more practical and enjoyable than the classic Introduction to Algorithms book by Cormen et al.