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Algorithm Design and Analysis

L ECTURE 7 Greedy Graph Algorithms Topological sort Shortest paths. Algorithm Design and Analysis. CSE 565. Adam Smith. The (Algorithm) Design Process. Work out the answer for some examples Look for a general principle Does it work on *all* your examples? Write pseudocode

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Algorithm Design and Analysis

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  1. LECTURE 7 • Greedy Graph Algorithms • Topological sort • Shortest paths Algorithm Design and Analysis CSE 565 Adam Smith A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  2. The (Algorithm) Design Process • Work out the answer for some examples • Look for a general principle • Does it work on *all* your examples? • Write pseudocode • Test your algorithm by hand or computer • Does it work on *all* your examples? • Prove your algorithm is always correct • Check running time Be prepared to go back to step 1! A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  3. Writing algorithms • Clear and unambiguous • Test: You should be able to hand it to any student in the class, and have them convert it into working code. • Homework pitfalls: • remember to specify data structures • writing recursive algorithms: don’t confuse the recursive subroutine with the first call • label global variables clearly A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  4. Writing proofs • Purpose • Determine for yourself that algorithm is correct • Convince reader • Who is your audience? • Yourself • Your classmate • Not the TA/grader • Main goal: Find your own mistakes A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  5. Exploring a graph Classic problem: Given vertices s,t ∈ V, is there a path from s to t? Idea: explore all vertices reachable from s Two basic techniques: • Breadth-first search (BFS) • Explore children in order of distance to start node • Depth-first search (DFS) • Recursively explore vertex’s children before exploring siblings How to convert these descriptions to precise algorithms? A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  6. 1 if (i, j) ∈ E, 0 if (i, j) E. A[i, j] = A 1 2 3 4 1 0 1 1 0 2 1 • Storage: ϴ(V 2) • Good for dense graphs. • Lookup: O(1) time • List all neighbors: O(|V|) 2 0 0 1 0 3 0 0 0 0 3 4 4 0 0 1 0 Adjacency-matrix representation The adjacency matrix of a graph G = (V, E), where V = {1, 2, …, n}, is the matrix A[1 . . n, 1 . . n] given by A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  7. 2 1 3 4 Adjacency list representation • An adjacency list of a vertex v ∈ V is the list Adj[v] of vertices adjacent tov. Typical notation:n = |V| = # verticesm = |E| = # edges Adj[1] = {2, 3} Adj[2] = {3} Adj[3] = {} Adj[4] = {3} For undirected graphs,|Adj[v] | = degree(v). For digraphs,|Adj[v] | = out-degree(v). How many entries in lists? 2|E| (“handshaking lemma”) Total ϴ(V + E) storage — good for sparse graphs. A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  8. DFS pseudocode • Maintain a global counter time • Maintain for each vertex v • Two timestamps: • v.d = time first discovered • v.f= time when finished • “color”: v.color • white = unexplored • gray = in process • black = finished • Parent v.π in DFS tree A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  9. DFS pseudocode • Note: recursive function different from first call… • Exercise: change code to set “parent” pointer? A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  10. DFS example • T = tree edge (drawn in red) • F = forward edge (to a descendant in DFS forest) • B= back edge (to an ancestor in DFS forest) • C = cross edge (goes to a vertex that is neither ancestor nor descendant) A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  11. DFS with adjacency lists • Outer code runs once, takes time O(n) (not counting time to execute recursive calls) • Recursive calls: • Run once per vertex • time = O(degree(v)) • Total: O(m+n) A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  12. Toplogical Sort A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  13. Directed Acyclic Graphs • Def. An DAG is a directed graph that contains no directed cycles. • Ex. Precedence constraints: edge (vi, vj) means vi must precede vj. • Def. A topological order of a directed graph G = (V, E) is an ordering of its nodes as v1, v2, …, vn so that for every edge (vi, vj) we have i < j. v2 v3 v6 v5 v4 v1 v2 v3 v4 v5 v6 v7 v7 v1 a DAG a topological ordering

  14. Precedence Constraints • Precedence constraints. Edge (vi, vj) means task vi must occur before vj. • Applications. • Course prerequisite graph: course vi must be taken before vj. • Compilation: module vi must be compiled before vj. Pipeline of computing jobs: output of job vi needed to determine input of job vj. • Getting dressed shirt mittens socks boots jacket underwear pants

  15. Recall from book • Every DAG has a topological order • If G graph has a topological order, then G is a DAG. A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  16. Review • Suppose your run DFS on a DAG G=(V,E) • True or false? • Sorting by discovery time gives a topological order • Sorting by finish time gives a topological order A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  17. Shortest Paths A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  18. Shortest Path Problem • Input: • Directed graph G = (V, E). • Source node s, destination node t. • for each edge e, length (e) = length of e. • length path = sum of edge lengths • Find: shortest directed path from s to t. 23 2 3 9 s 18 14 6 2 6 4 19 Length of path (s,2,3,5,t)is 9 + 23 + 2 + 16 = 50. 30 11 5 15 5 6 16 20 t 7 44 A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  19. Rough algorithm (Dijkstra) • Maintain a set of explored nodes S whose shortest path distance d(u) from s to u is known. • Initialize S = {s}, d(s) = 0. • Repeatedly choose unexplored node v which minimizes • add v to S, and set d(v) = (v). shortest path to some u in explored part, followed by a single edge (u, v) (e) v d(u) u S s A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  20. Rough algorithm (Dijkstra) • Maintain a set of explored nodes S whose shortest path distance d(u) from s to u is known. • Initialize S = {s}, d(s) = 0. • Repeatedly choose unexplored node v which minimizes • add v to S, and set d(v) = (v). Invariant: d(u) is known for all vertices in S shortest path to some u in explored part, followed by a single edge (u, v) BFS withweighted edges (e) v d(u) u S s A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  21. Demo of Dijkstra’s Algorithm 2 Graph with nonnegative edge lengths: B D 10 8 A 1 4 7 9 3 C E 2 A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  22. Demo of Dijkstra’s Algorithm ¥ ¥ Initialize: 2 B D 10 8 A 0 1 4 7 9 3 C E Q: A B C D E 2 ¥ ¥ ¥ ¥ 0 ¥ ¥ S: {} A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  23. Demo of Dijkstra’s Algorithm ¥ ¥ “A”EXTRACT-MIN(Q): 2 B D 10 8 A 0 1 4 7 9 3 C E Q: A B C D E 2 ¥ ¥ ¥ ¥ 0 ¥ ¥ S: { A } A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  24. Demo of Dijkstra’s Algorithm ¥ 10 Explore edges leaving A: 2 B D 10 8 A 0 1 4 7 9 3 C E Q: A B C D E 2 ¥ ¥ ¥ ¥ 0 ¥ 3 3 ¥ ¥ 10 S: { A } A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  25. Demo of Dijkstra’s Algorithm ¥ 10 “C”EXTRACT-MIN(Q): 2 B D 10 8 A 0 1 4 7 9 3 C E Q: A B C D E 2 ¥ ¥ ¥ ¥ 0 ¥ 3 3 ¥ ¥ 10 S: { A, C } A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  26. Demo of Dijkstra’s Algorithm 7 11 Explore edges leaving C: 2 B D 10 8 A 0 1 4 7 9 3 C E Q: A B C D E 2 ¥ ¥ ¥ ¥ 0 3 5 3 ¥ ¥ 10 7 11 5 S: { A, C } A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  27. Demo of Dijkstra’s Algorithm 7 11 “E”EXTRACT-MIN(Q): 2 B D 10 8 A 0 1 4 7 9 3 C E Q: A B C D E 2 ¥ ¥ ¥ ¥ 0 3 5 3 ¥ ¥ 10 7 11 5 S: { A, C, E } A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  28. Demo of Dijkstra’s Algorithm 7 11 Explore edges leaving E: 2 B D 10 8 A 0 1 4 7 9 3 C E Q: A B C D E 2 ¥ ¥ ¥ ¥ 0 3 5 3 ¥ ¥ 10 7 11 5 S: { A, C, E } 7 11 A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  29. Demo of Dijkstra’s Algorithm 7 11 “B”EXTRACT-MIN(Q): 2 B D 10 8 A 0 1 4 7 9 3 C E Q: A B C D E 2 ¥ ¥ ¥ ¥ 0 3 5 3 ¥ ¥ 10 7 11 5 S: { A, C, E, B } 7 11 A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  30. Demo of Dijkstra’s Algorithm 7 9 Explore edges leaving B: 2 B D 10 8 A 0 1 4 7 9 3 C E Q: A B C D E 2 ¥ ¥ ¥ ¥ 0 3 5 3 ¥ ¥ 10 7 11 5 S: { A, C, E, B } 7 11 9 A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  31. Demo of Dijkstra’s Algorithm 7 9 “D”EXTRACT-MIN(Q): 2 B D 10 8 A 0 1 4 7 9 3 C E Q: A B C D E 2 ¥ ¥ ¥ ¥ 0 3 5 3 ¥ ¥ 10 7 11 5 S: { A, C, E, B, D } 7 11 9 A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  32. Proof of Correctness (Greedy Stays Ahead) Invariant. For each node u S, d(u) is the length of the shortest path from s to u. Proof:(by induction on |S|) • Base case: |S| = 1 is trivial. • Inductive hypothesis: Assume for |S| = k 1. • Let v be next node added to S, and let (u,v) be the chosen edge. • The shortest s-u path plus (u,v) is an s-v path of length (v). • Consider any s-v path P. We'll see that it's no shorter than (v). • Let (x,y) be the first edge in P that leaves S,and let P' be the subpath to x. • P + (x,y) has length ·d(x)+ (x,y)·(y)·(v) P y x P' s S u v A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  33. Review Question • Is Dijsktra’s algorithm correct with negative edge weights? A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  34. Implementation • For unexplored nodes, maintain • Next node to explore = node with minimum (v). • When exploring v, for each edge e = (v,w), update • Efficient implementation: Maintain a priority queue of unexplored nodes, prioritized by (v). Priority Queue A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  35. Priority queues • Maintain a set of items with priorities (= “keys”) • Example: jobs to be performed • Operations: • Insert • Increase key • Decrease key • Extract-min: find and remove item with least key • Common data structure: heap • Time: O(logn) per operation A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  36. Pseudocode for Dijkstra(G, ) • d[s]  0 • for each vÎV – {s} • dod[v]  ¥; p[v]  ¥ • S   • Q  V ⊳Q is a priority queue maintainingV – S,keyed on [v] • whileQ ¹ • dou  EXTRACT-MIN(Q) • S  SÈ {u}; d[u]  p[u] • for each vÎAdjacency-list[u] • doif[v] > [u] + (u, v) • then [v]  d[u] + (u, v) explore edges leaving v Implicit DECREASE-KEY A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  37. n times degree(u) times Handshaking Lemma  ·mimplicit DECREASE-KEY’s. Analysis of Dijkstra • whileQ ¹ • dou  EXTRACT-MIN(Q) • S  SÈ {u} • for each vÎAdj[u] • doif d[v] > d[u] + w(u, v) • then d[v]  d[u] + w(u, v) PQ Operation Dijkstra Array Binary heap d-way Heap Fib heap † ExtractMin n n log n HW3 log n DecreaseKey m 1 log n HW3 1 Total n2 m log n m log m/n n m + n log n † Individual ops are amortized bounds A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

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