Final review Depth-first search vs. Breadth-first search Depth-first search. Put unvisited vertices on a stack. Breadth-first search. Put unvisited vertices on a queue. source set of marked vertices s Connected component: maximal set of connected vertices. DFS marks all vertices connected to s in time v proportional to the sum of their degrees. no such edge set of can exist unmarked vertices x w BFS finds path from s to t that uses fewest number of edges. Breadth-frst Intuition.maze exploration BFS examines vertices in increasing distance from s. 2 ‣ directed graphs 3 Depth-first search in digraphs Same method as for undirected graphs. • Every undirected graph is a digraph (with edges in both directions). • DFS is a digraph algorithm. DFS (to visit a vertex v) Mark v as visited. Recursively visit all unmarked vertices w adjacent from v. 4 Breadth-first search in digraphs Same method as for undirected graphs. • Every undirected graph is a digraph (with edges in both directions). • BFS is a digraph algorithm. s BFS (from source vertex s) w Put s onto a FIFO queue, and mark s as visited. Repeat until the queue is empty: - remove the least recently added vertex v - for each unmarked vertex adjacent from v: add to queue and mark as visited.. v Is w reachable from v in this digraph? Proposition. BFS computes shortest paths (fewest number of edges). 5 Topological sort DAG. Directed acyclic graph. Topological sort. Redraw DAG so all edges point up. 0→5 0→2 0 0→1 3→6 3→5 3→4 2 5 1 5→4 6→4 6→0 3→2 1→4 3 4 6 directed edges DAG Solution. Reverse DFS Postorder! topological order 6 Reverse DFS postorder in a DAG marked[] reversePost 0 dfs(0) 1 0 0 0 0 0 0 - dfs(1) 1 1 0 0 0 0 0 - dfs(4) 1 1 0 0 1 0 0 - 2 5 1 4 done 1 1 0 0 1 0 0 4 1 done 1 1 0 0 1 0 0 4 1 3 4 dfs(2) 1 1 1 0 1 0 0 4 1 2 done 1 1 1 0 1 0 0 4 1 2 6 dfs(5) 1 1 1 0 1 1 0 4 1 2 check 2 1 1 1 0 1 1 0 4 1 2 5 done 1 1 1 0 1 1 0 4 1 2 5 0→5 0 done 1 1 1 0 1 1 0 4 1 2 5 0 0→2 check 1 1 1 1 0 1 1 0 4 1 2 5 0 check 2 1 1 1 0 1 1 0 4 1 2 5 0 0→1 dfs(3) 1 1 1 1 1 1 0 4 1 2 5 0 3→6 check 2 1 1 1 1 1 1 0 4 1 2 5 0 3→5 check 4 1 1 1 1 1 1 0 4 1 2 5 0 3→4 check 5 1 1 1 1 1 1 0 4 1 2 5 0 dfs(6) 1 1 1 1 1 1 1 4 1 2 5 0 5→4 6 done 1 1 1 1 1 1 1 4 1 2 5 0 6 6→4 3 done 1 1 1 1 1 1 1 4 1 2 5 0 6 3 6→0 check 4 1 1 1 1 1 1 0 4 1 2 5 0 6 3 3→2 check 5 1 1 1 1 1 1 0 4 1 2 5 0 6 3 check 6 1 1 1 1 1 1 0 4 1 2 5 0 6 3 reverse DFS 1→4 done 1 1 1 1 1 1 1 4 1 2 5 0 6 3 postorder is a topological order! 7 Strongly-connected components Def. Vertices v and w are strongly connected if there is a directed path from v to w and a directed path from w to v. Key property. Strong connectivity is an equivalence relation: • v is strongly connected to v. • If v is strongly connected to w, then w is strongly connected to v. • If v is strongly connected to w and w to x, then v is strongly connected to x. Def. A strong component is a maximal subset of strongly-connected vertices. A graph and its connected components A digraph and its strong components 8 Kosaraju's algorithm Simple (but mysterious) algorithm for computing strong components. • Run DFS on GR to compute reverse postorder. • Run DFS on G, considering vertices in order given by first DFS. DFS in reverse digraph (ReversePost) DFS in original digraph G dfs(1) dfs(0) dfs(11) dfs(6) dfs(7) 1 done dfs(5) check 4 check 9 check 6 dfs(12) check 4 dfs(8) check unmarked vertices in the order check unmarked vertices in the order dfs(4) dfs(3) dfs(9) check 0 check 7 0 1 2 3 4 5 6 7 8 9 10 11 12 1 0 2 4 5 3 11 9 12 10 6 7 8 check 5 check 11 6 done check 9 dfs(2) dfs(10) 8 done dfs(0) dfs(1) check 0 check 12 7 done check 8 dfs(6) 1 done check 3 10 done 9 done dfs(7) dfs(0) 2 done 3 done 12 done dfs(8) dfs(5) check 2 11 done check 7 dfs(4) 4 done check 9 8 done dfs(3) 5 done check 12 7 done check 5 check 1 check 10 6 done dfs(2) 0 done check 2 dfs(2) check 0 check 4 dfs(4) check 3 check 5 dfs(11) 2 done check 3 dfs(9) 3 done dfs(12) Second check 2 DFS gives strong components. (!!) check 11 4 done dfs(10) 5 done strong check 9 check 1 components 10 done 0 done 9 12 done check 2 check 8 check 4 check 6 check 5 9 done check 3 11 done dfs(11) check 6 check 4 dfs(5) dfs(12) dfs(3) dfs(9) check 4 check 11 check 2 dfs(10) 3 done check 12 check 0 10 done 5 done 9 done 4 done 12 done check 3 11 done 2 done check 9 0 done check 12 dfs(1) check 10 check 0 dfs(6) 1 done check 9 check 2 check 4 check 3 check 0 check 4 reverse 6 done check 5 postorder dfs(7) check 6 for use check 6 check 7 in second dfs(8) check 8 dfs() check 7 check 9 (read up) check 9 check 10 8 done check 11 7 done check 12 check 8 Kosaraju’s algorithm for fnding strong components in digraphs ‣ minimum spanning trees 10 Minimum spanning tree Given. Undirected graph G with positive edge weights (connected). Def. A spanning tree of G is a subgraph T that is connected and acyclic. Goal. Find a min weight spanning tree. 24 4 23 6 9 18 5 11 16 8 7 10 14 21 spanning tree T: cost = 50 = 4 + 6 + 8 + 5 + 11 + 9 + 7 Brute force. Try all spanning trees? 11 Kruskal's algorithm Kruskal's algorithm. [Kruskal 1956] Consider edges in ascending order of weight. Add the next edge to the tree T unless doing so would create a cycle. next MST edge is red graph edges sorted by weight MST edge (black) 0-7 0.16 2-3 0.17 1-7 0.19 0-2 0.26 5-7 0.28 1-3 0.29 1-5 0.32 2-7 0.34 4-5 0.35 1-2 0.36 4-7 0.37 0-4 0.38 6-2 0.40 3-6 0.52 6-0 0.58 6-4 0.93 obsolete edge (gray) grey vertices are a cut defined by the vertices connected to one of the red edge’s vertices 12 Trace of Kruskal’s algorithm Kruskal's algorithm: implementation challenge Challenge. Would adding edge v–w to tree T create a cycle? If not, add it. Efficient solution. Use the union-find data structure. • Maintain a set for each connected component in T. • If v and w are in same set, then adding v–w would create a cycle. • To add v–w to T, merge sets containing v and w. w v w v Case 1: adding v–w creates a cycle Case 2: add v–w to T and merge sets containing v and w 13 Prim's algorithm Prim's algorithm. [Jarník 1930, Dijkstra 1957, Prim 1959] Start with vertex 0 and greedily grow tree T. At each step, add to T the min weight edge with exactly one endpoint in T. edges with exactly one endpoint in T (sorted by weight) 0-7 0.16 1-7 0.19 0-2 0.26 0-2 0.26 0-2 0.26 5-7 0.28 0-4 0.38 5-7 0.28 1-3 0.29 6-0 0.58 2-7 0.34 1-5 0.32 4-7 0.37 2-7 0.34 0-4 0.38 1-2 0.36 6-0 0.58 4-7 0.37 0-4 0.38 0-6 0.58 2-3 0.17 5-7 0.28 4-5 0.35 5-7 0.28 1-5 0.32 4-7 0.37 1-3 0.29 4-7 0.37 0-4 0.38 1-5 0.32 0-4 0.38 6-2 0.40 4-7 0.37 6-2 0.40 3-6 0.52 0-4 0.38 3-6 0.52 6-0 0.58 6-2 0.40 6-0 0.58 6-0 0.58 6-2 0.40 3-6 0.52 6-0 0.58 6-4 0.93 Trace of Prim’s algorithm 14 Prim's algorithm: lazy implementation Challenge.