CS1800: Graph Algorithms (2nd Part) Professor Kevin Gold

Transcription

1 S1800: raph lgorithms (2nd Part) Professor Kevin old

2 Summary So ar readth-irst Search (S) and epth-irst Search (S) are two efficient algorithms for finding paths on graphs. S also finds the shortest path. S searches close to home first by using a queue (distant nodes must wait their turn) done S goes deep first with a stack that makes whatever is new, the most exciting old in new new in done ijkstra s algorithm is similar to S but works with weighted graphs, finding cheapest paths when edges have costs - uses a priority queue done old in

3 Sample xplorations rom S Queue: () S:,,,,, restart at S:,,,, backtrack and, restart at

4 Sample xplorations rom S Queue: ( ) S:,,,,, restart at S:,,,, backtrack and, restart at

5 Sample xplorations rom S Queue: ( ) S:,,,,, restart at S:,,,, backtrack and, restart at

6 Sample xplorations rom S Queue: ( ) S:,,,,, restart at S:,,,, backtrack and, restart at

7 Sample xplorations rom S Queue: ( ) S:,,,,, restart at S:,,,, backtrack and, restart at

8 Sample xplorations rom S Queue: () S:,,,,, restart at S:,,,, backtrack and, restart at

9 Sample xplorations rom S Queue: () S:,,,,, restart at S:,,,, backtrack and, restart at

10 Sample xplorations rom S Queue: () then Queue: () S:,,,,, restart at S:,,,, backtrack and, restart at

11 Sample xplorations rom S Stack: S:,,,,, restart at S:,,,, backtrack and, restart at

12 Sample xplorations rom S Stack: S:,,,,, restart at S:,,,, backtrack and, restart at

13 Sample xplorations rom S Stack: S:,,,,, restart at S:,,,, backtrack and, restart at

14 Sample xplorations rom S Stack: S:,,,,, restart at S:,,,, backtrack and, restart at

15 Sample xplorations rom S Stack: (tossing the stale s) S:,,,,, restart at S:,,,, backtrack and, restart at

16 Sample xplorations rom S Stack: S:,,,,, restart at S:,,,, backtrack and, restart at

17 Sample xplorations rom S Stack: then S:,,,,, restart at S:,,,, backtrack and, restart at

18 Running Time nalysis of S/S On exploring a new node, both S and S do some work for every neighbor of the node. heck whether the neighbor has been marked as previously seen. If it hasn t, add it to the stack/queue. This work per neighbor is done exactly once for each vertex; the algorithms marking scheme makes sure it isn t repeated. So what is the total work?

19 Loose and Tight ounds We could estimate the work done per vertex as being no worse than V, the number of vertices, because that is the maximum number of neighbors. Then the total work would be V * V or O( V 2 ). It s especially appropriate to use O here because we rounded up; we don t know if our bound is tight. tighter bound comes from the fact that we did work proportional to the sum of the degrees of all vertices, Σ v deg(v). We know from the handshaking lemma that this is 2. So a more careful analysis is that the running time is on the order of (dropping the constant). This is considered better than V 2 because a sparse graph like a tree may have more like V edges. 3 = deg(v) V v

20 onsidering the (No-)dge ases Is the running time of S/S O( )? lmost (and it s often expressed that way), but consider the case of having no edges. The algorithm still must hop from vertex to vertex and discover no neighbors for each. This and other setup makes the running times of S and S O( V + ): linear in the number of edges and vertices.

21 nother Well-Known raph Problem: Minimal Spanning Trees Suppose you want to build a network connecting a set of nodes, and you know the cost of connecting any two nodes. You want to build the network with the cheapest overall cost. minimal spanning tree algorithm solves this problem. It finds the minimum weight tree that connects all nodes in a graph.

22 Method 1: Prim s lgorithm row from * * c 7 8 a 3 9 b a,b,c in priority queue keys: a=20,b=9,c=5 (c used to be 7) There is a start node and a priority queue of nodes to try next. When a node is explored, its neighbors are added to the priority queue in order of their edge weights. Not total distance to the node, as in ijkstra s (which is otherwise almost identical) The node closest to the existing tree gets explored next.

23 Method 2: Kruskal s lgorithm Jump around & grab cheapest 1 5 c a Kruskal s algorithm sorts all the edge weights, then adds the cheapest edge that doesn t create a cycle The added edges need not be adjacent, unlike Prim s b We ve covered sorting algorithms, but efficiently avoiding cycles we defer to later courses (undies 2, lgorithms) no cycle 3 - no cycle 4 - no cycle 5 - no cycle 7 - cycle, skip 8 - cycle, skip [9 - no cycle, done]