1 / 13

CSE 390B: Graph Algorithms

CSE 390B: Graph Algorithms. Based on CSE 373 slides by Jessica Miller, Ruth Anderson. A Graph: Course Prerequisites. 461. 322. 373. 143. 321. 326. 142. 415. 410. 370. 341. 413. 417. 378. 421. Nodes = courses Directed edge = prerequisite. 401. Depth-First Search (DFS).

brooklyn
Download Presentation

CSE 390B: Graph Algorithms

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. CSE 390B: Graph Algorithms Based on CSE 373 slidesby Jessica Miller, Ruth Anderson

  2. A Graph: Course Prerequisites 461 322 373 143 321 326 142 415 410 370 341 413 417 378 421 Nodes = courses Directed edge = prerequisite 401

  3. Depth-First Search (DFS) • depth-first search (DFS): find path between two vertices by exploring each path as many steps as possible before backtracking • often implemented recursively with a stack for the path in progress • always finds a path, but not necessarily the shortest one • easy to reconstruct the path once you have found it (just unroll the calls) • DFS path search order from A to others (assumes ABC edge order): • A • A  B • A  B  D • A  B  F • A  B  F E • A  C • A  C  G

  4. DFS pseudocode dfs(v1, v2): path = new Stack(). dfs(v1, v2, path) dfs(v1, v2, path): path.push(v1). mark v1 as visited. if v1 = v2: path is found. for each unvisited neighbor vi of v1 with edge (v1  vi): if dfs(vi, v2, path) finds a path, path is found. path.pop(). // path is not found.

  5. Breadth-First Search (BFS) breadth-first search (DFS): find path between two nodes by taking one step down all paths and then immediately backtracking often implemented with a queue of next vertices to visit always finds the shortest path (fewest edges); optimal for unweighted graphs harder to reconstruct the path once you have found it BFS path search order from A to others: A A  B A  C A E A  B  D A  B  F A  C  G 5

  6. BFS pseudocode bfs(v1, v2): Q = {v1}. mark v1 as visited. while Q not empty: v = Q.dequeue(). // remove from front if v is v2: path is found. for each unvisited neighbor vi of v1 with edge (v1  vi): mark vi as visited. Q.enqueue(vi). // add at end path is not found.

  7. Implementation: Adjacency Matrix an n × n 2D array where M[a][b] = edges from va to vb Sometimes implemented as a Map<V, Map<V, E>> Good for quickly asking, "is there an edge from vertex i to j?" How do we figure out the degree of a vertex? 2 1 3 1 2 3 4 5 6 7 1 2 3 4 5 6 7 0 1 0 0 1 1 0 7 1 0 1 0 0 0 1 0 1 0 1 0 0 0 4 0 0 1 0 1 0 1 1 0 0 1 0 1 1 6 1 0 0 0 1 0 0 5 0 1 0 1 1 0 0 7

  8. 2 1 3 7 1 2 3 4 5 6 7 2 5 6 1 3 7 4 2 4 6 3 5 7 5 1 4 6 7 1 5 2 4 5 Implementation: Adjacency Lists • n lists of neighbors; L[a] = all edges out from va • Sometimes implemented as a Map<V, List<V>> • Good for processing all-neighbors, sparse graphs (few edges) • How do we figure out the degree of a vertex?

  9. Dijkstra's algorithm Dijkstra's algorithm: finds shortest (min weight) path between a pair of vertices in a weighted directed graph with nonnegative edges solves the "one vertex, shortest path" problem basic algorithm concept: create a table of information about the currently known best way to reach each vertex (distance, previous vertex) and improve it until it reaches the best solution 2 A B 1 3 10 4 2 2 C D E 5 8 4 6 1 F G 9

  10. Dijkstra pseudocode Dijkstra(v1, v2): for each vertex v: // Initialize state v's distance := infinity. v's previous := none. v1's distance := 0. Q := {all vertices}. while Q is not empty: v := remove Q's vertex with min distance. mark v as known. for each unknown neighbor n of v: dist := v's distance + edge (v, n)'s weight. if dist is smaller than n's distance: n's distance := dist. n's previous := v. reconstruct path from v2 back to v1, following previous pointers.  0 2 A B 1 3 10 4   2 2 C D E  5 8 4 6 1 F G   examine A: update B(2),D(1)examine D: update C(3),E(3),F(9),G(5)examine B,E: update noneexamine C: update F(8)examine G: update F(6)examine F: update none 10

  11. Floyd-Warshall algorithm Floyd-Warshall algorithm: finds shortest (min weight) path between all pairs of vertices in a weighted directed graph solves the "all pairs, shortest paths" problem (demo) idea: repeatedly find best path using only vertices 1..k inclusive floydWarshall(): int path[n][n]. for each (i, j) from (0, 0) to (n, n): path[i][j] = edge_weight[i][j]. for k = 0 to n: for i = 0 to n: for j = 0 to n: path[i][j] = min(path[i][j], path[i][k] + path[k][j]). 11

  12. Topological Sort Topological sort: finds a total ordering of vertices such thatfor any edge (v, w) in E, v precedes w in the ordering e.g. find an ordering in which all UW CSE courses can be taken B C A D E F A D E F B C 12

  13. B C A D E F Topological Sort pseudocode V = {all vertices}. E = {all edges}. L = []. while V is not empty: for each vertex v in V: if v has no incoming edges: V.remove(v). L.append(v). for each edge e (v  n): E.remove(e). return L. examine A,Dexamine Bexamine Cexamine E examine F

More Related