Problem Solving with Algorithms and Data Structure - Graphs

Study slides for http://interactivepython.org/runestone/static/pythonds/index.html

Focus on Graphs and Graph Algorithms

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1. Problem Solving with Algorithms and Data Structure — Graph Bruce Tsai
2. http://interactivepython.org/runestone/ static/pythonds/Graphs/Objectives.html
3. Graph A set of objects where some pairs of objects are connected by links road map airline flights internet connection skill tree Tree is special kind of graph Vertex (node) Edge (arc) 3
4. Goal Represent problem in form of graph Use graph algorithms to solve problem 4
5. (Undirected) Graph G = (V, E) V = {v_1, v_2, v_3, v_4, v_5, v_6} E = {(v_1, v_2), (v_1, v_5), (v_2, v_3), (v_2, v_5), (v_3, v_4), (v_4, v_5), (v_4, v_6)} (v_1, v_5) == (v_5, v_1) 5
6. Directed Graph D = (V, A) V = {v_0, v_1, v_2, v_3, v_4, v_5} A = {(v_0, v_1), (v_0, v_5), (v_1, v_2), (v_2, v_3), (v_3, v_5), (v_3, v_4), (v_4, v_0), (v_5, v_2), (v_5, v_4)} 6
7. Attributed Graph Attribute Characters of graph vertices or edges weight, color, anything you want 7
8. Path and Cycle Path Sequence of vertices that are connected by edges [v_1, v_5, v_4, v_6] Cycle A path that starts and ends at same vertex [v_2, v_3, v_4, v_5, v_2] 8
9. Adjacency Matrix 9 from to
10. Discussion Question A B C D E F 7 5 1 2 7 3 2 8 1 2 4 5 6 1 8 10
11. Adjacency List 11
12. Discussion Question 1 2 3 4 5 6 10 15 5 7 7 10 7 13 5 12
13. Word Ladder Problem Make change occur gradually by changing one letter at a time FOOL -> POOL -> POLL -> POLE -> PALE -> SALE -> SAGE 13
14. Word Ladder Graph 14
16. O(V*L) O(2E) O(V+E)
18. Breadth First Search (BFS) 18
19. BFS Analysis O(V+E) 19
20. Discussion Question - BFS [1] [1, 2, 3, 6] [2, 3, 6] [3, 6, 4] [6, 4, 5] [4, 5] [5] 1 2 3 4 5 6 20
21. Knight’s Tour Problem 21
22. Legal Move Graph 22
24. Knight Graph 24
27. Knight’s Tour 27
28. Choose Better Next Vertex Visit hard-to-reach corners early Use the middle square to hop across board only when necessary 28
30. General Depth First Search (DFS) Knight’s Tour is special case of depth first search create the depth first tree General depth first search is to search as deeply as possible 30
31. O(V+E) O(V)
32. queue stack
33. Computational Complexity
34. P versus NP problem P: questions for which some algorithms can provide an answer in polynomial time NP: questions for which an answer can be verified in polynomial time Subset sum problem {-7, -3, -2, 5, 8} -> {-3, -2, 5} 34
35. decision problem set optimization problems search problems
36. NP-complete Reduction: algorithm for transforming one problem into another problem NP-complete: a problem p in NP is NP- Complete if every other problem in NP can be transformed into p in polynomial time hardest problems in NP 36
37. NP-hard Decision problem: any arbitrary yes-or-no question on an infinite set of inputs NP-hard: a decision problem H in NP-hard when for any problem L in NP, there is polynomial-time reduction from L to H at least as hard as the hardest problems in NP 37
38. Topological Sorting Linear ordering of all vertices of a directed graph (u, v) is an edge of directed graph and u is before v in ordering 38
40. Start and Finish Time
41. Implementation 1. Call dfs(g) 2. Store vertices based on decreasing order of finish time 3. Return the ordered list 41
42. Lemma A directed graph G is acyclic if and only if a depth-first search of G yields no back edges. Back edge: edge (u, v) connecting vertex u to an ancestor v in depth-first tree (v ↝ u → v) =>: back edge exists then cycle exists <=: cycle exists then back edge exists 42
43. Proof of Implementation For any pair of distinct vertices u, v ∈ V, if there is an edge in G from u to v, then f[v] < f[u]. Consider edge (u, v) explored by DFS, v cannot be gray since then that edge would be back edge. Therefore v must be white or black. If v is white, v is descendant of u, and so f[v] < f[u] If v is black, it has already been finished, so that f[v] < f[u] 43
44. Strongly Connected Components G = (V, E) and C ⊂ V such that ∀ (v_i, v_j) ∈ C, path v_i to v_j and path v_j to v_i both exist C is strongly connected components (SCC) 44
46. Implementation 1. Call dfs(g) 2. Compute transpose of g as g_t 3. Call dfs(g_t) but explore each vertex in decreasing order of finish time 4. Each tree of step 3 is a SCC original transpose 46
47. dfs(g)
48. dfs(g_t)
49. Lemma Let C and C’ be distinct SCCs in directed graph G = (V, E), let u, v ∈ C, let u’, v’ ∈ C’, and suppose that there is a path u ↝ u’ in G. Then there cannot also be a path v’ ↝ v in G. if v’ ↝ v exists then both u ↝ u’ ↝ v’ and v’ ↝ v ↝ u exist C and C’ are not distinct SCCs 49
50. Lemma Let C and C’ be distinct SCCs in directed graph G = (V, E). Suppose that there is an edge (u, v) ∈ E, where u ∈ C and v ∈ C’. Then f(C) > f(C’) from x ∈ C to w ∈ C’ from y ∈ C’ cannot reach any vertex in C 50
51. Corollary Let C and C’ be distinct SCCs in directed graph G = (V, E). Suppose that there is an edge (u, v) ∈ ET, where u ∈ C and v ∈ C’. Then f(C) < f(C’). (v, u) ∈ E then f(C’) < f(C) 51
52. Proof of Implementation
53. Shortest Path Problem Find the path with the smallest total weight along which to route any given message 53
54. Dijkstra’s Algorithm Iterative algorithm providing the shortest path from one particular starting node to all other nodes in graph Path distance, priority queue 54
59. O(V) O(logV) O(logV) O(E*logV) O(V*logV) O((V+E)*logV)
60. Broadcast Problem
61. Minimum Spanning Tree T is acyclic subset of E that connects all vertices in V The sum of weight of edges in T is minized 61
62. Prim’s Spanning Tree Algorithm Special case of generic minimum spanning tree Find shortest paths in graph 62
65. Kruskal’s Algorithm Sort edges in E into ascending order by weight For (u, v) ∈ E, if set of u not equal to set of v A ← A ∪ {(u, v)} 65
68. Reference http://www.personal.kent.edu/~rmuhamma/Algorithms/MyAlgorithms/ AproxAlgor/TSP/tsp.htm http://en.wikipedia.org/wiki/Graph_%28mathematics%29 http://en.wikipedia.org/wiki/P_versus_NP_problem http://en.wikipedia.org/wiki/NP-complete http://en.wikipedia.org/wiki/NP-hard http://en.wikipedia.org/wiki/Reduction_(complexity) http://en.wikipedia.org/wiki/Knight%27s_tour http://staff.ustc.edu.cn/~csli/graduate/algorithms/book6/chap24.htm Introduction to Algorithms, 2nd edition