Lecture 34 Fall 2018 Wednesday November 28

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1 Greedy Algorithms Oliver W. Layton CS231: Data Structures and Algorithms Lecture 34 Fall 2018 Wednesday November 28

2 Plan Friday office hours: 3-4pm instead of 1-2pm Dijkstra's algorithm example Minimum spanning trees Kruskal's algorithm Prim's algorithm Heap sort

3 Spanning tree A spanning tree is a subgraph that: Contains ALL ver5ces in the original graph (connected subgraph) Needs to be a tree...which means...? No cycles and connected. Let's draw some examples.

4 Number of spanning trees If the graph is complete, then Cayley s formula specifies the total number of spanning trees, which is for ver;ces. For example, for a complete 4 vertex graph, there are spanning trees:

5 Minimum spanning trees (MSTs) Assumes a weighted graph. Minimum spanning tree: a spanning tree with the smallest total sum of weight (cost) along all the edges (there may be more than one).

6 Example: Minimum spanning tree (1/2) Distance between Chicago suburbs: A telephone company wants to install internet fiber cabling, but wants to minimum the total cost, and hence the total amount of fiber cable used.

7 Example: Minimum spanning tree (2/2) A minimum spanning tree tells the company how to lay the cable to cover every town and save the most money:

8 Example: Hunt the Wumpus Random room extension: Implement MST algorithms for a set of randomly posi9oned rooms will allow you to create a valid ini9al maze.

9 Kruskal s algorithm Greedy algorithm (what's this again?) to compute a graph's minimum spanning tree. General strategy Make single root vertex trees (for ver3ces). Join together trees connec3ng the next smallest edges, as long as they don t create cycles (which spanning tree condi3on would this invalidate?).

10 Anima&on 1 of Kruskal s algorithm in ac&on 1 Source: Wikipedia

11 Kruskal s algorithm: Pseudocode 1. Ini&alize a set of trees (Set<Tree> treeset), one for each of the ver&ces in the graph. Each tree just has a single root node. So a graph of ver&ces should have a set data structure with singlenode trees in it. 2. Create a list of all the edges in the original graph, sorted by weight (min-tomax). 3. Loop through the edge list If the edge joins two trees together, union the trees together (what does this opera&on do again?)

12 Kruskal s algorithm: Example Let's work out an example on the board

13 Kruskal s algorithm analysis What makes the algorithm greedy? Why are sets used as the data structure to hold the trees?

14 Another approach to MSTs: Prim s algorithm Key difference: Builds up a single large tree rather than merge smaller mul7-node trees. General strategy Ini%alize the distance from the (the current tree) to all other ver%ces as, except gets distance 0. This is just like Dijkstra. While we don t have all the ver%ces added to the tree: Add the vertex with the smallest distance to the current tree. Update distance to the neighboring ver%ces outside the tree, to see if the added vertex reduced the distance.

15 Prim's algorithm: Anima1on

16 Prim's algorithm: Pseudocode 1. Ini&alize an empty tree (e.g. represented by a set of ver&ces). This will eventually be the MST. 2. Ini&alize a min-heap. Add all the ver&ces in the graph with Inf priority, except, which gets priority While the heap isn t empty: Remove the next vertex from the heap. Add to the tree. For every neighbor of : Update the priority of if it s in the heap and < u.getpriority(): heap.updatepriority(u, dist(u, v)); What makes the algorithm greedy?

17 Prim's algorithm: Example Let's work out an example on the board

18 Heap Sort Topic next week is sor0ng algorithms. e.g. {3, 1, 2} {1, 2, 3} One common method, heap sort, leverages min-heaps, so makes sense to introduce now. Let's write the pseudocode on the board and analyze the 0me complexity.

Algorithm Analysis Graph algorithm. Chung-Ang University, Jaesung Lee

Algorithm Analysis Graph algorithm Chung-Ang University, Jaesung Lee Basic definitions Graph = (, ) where is a set of vertices and is a set of edges Directed graph = where consists of ordered pairs