1. CSCE 411 Design and Analysis of Algorithms Set 9: More Graph Algorithms Prof. Jennifer Welch Spring 2012 CSCE 411, Spring 2012: Set 9 1

2. 2 Generic MST Algorithm input: weighted undirected graph G = (V,E,w) T := empty set while T is not yet a spanning tree of G find an edge e in E s.t. T U {e} is a subgraph of some MST of G add e to T return T (as MST of G)

3. CSCE 411, Spring 2012: Set 93 Kruskal's Algorithm as a Special Case of Generic Algorithm Consider edges in increasing order of weight Add the next edge iff it doesn't cause a cycle At any point, T is a forest (set of trees); eventually T is a single tree

4. CSCE 411, Spring 2012: Set 94 Idea of Prim's Algorithm Instead of growing the MST as possibly multiple trees that eventually all merge, grow the MST from a single vertex, so that there is only one tree at any point. Also a special case of the generic algorithm: at each step, add the minimum weight edge that goes out from the tree constructed so far.

5. CSCE 411, Spring 2012: Set 95 Prim's Algorithm input: weighted undirected graph G = (V,E,w) T := empty set S := {any vertex in V} while |T| < |V| - 1 do let (u,v) be a min wt. outgoing edge (u in S, v not in S) add (u,v) to T add v to S return (S,T) (as MST of G)

6. CSCE 411, Spring 2012: Set 96 Prim's Algorithm Example a b h i c g d f e 4 87 8 11 2 76 12 4 14 9 10

7. CSCE 411, Spring 2012: Set 97 Correctness of Prim's Algorithm Let T i be the tree represented by (S,T) at the end of iteration i. Show by induction on i that T i is a subtree of some MST of G. Basis: i = 0 (before first iteration). T 0 contains just a single vertex, and thus is a subtree of every MST of G.

8. CSCE 411, Spring 2012: Set 98 Correctness of Prim's Algorithm Induction: Assume T i is a subtree of some MST M. We must show T i+1 is a subtree of some MST. Let (u,v) be the edge added in iteration i+1. u v TiTi T i+1 Case 1: (u,v) is in M. Then T i+1 is also a subtree of M.

9. CSCE 411, Spring 2012: Set 99 Correctness of Prim's Algorithm Case 2: (u,v) is not in M. There is a path P in M from u to v, since M spans G. Let (x,y) be the first edge in P with one endpoint in T i and the other not in T i. u v TiTi x y P

10. CSCE 411, Spring 2012: Set 910 Correctness of Prim's Algorithm Let M' = M - {(x,y)} U {(u,v)} M' is also a spanning tree of G. w(M') = w(M) - w(x,y) + w(u,v) ≤ w(M) since (u,v) is min wt outgoing edge So M' is also an MST and T i+1 is a subtree of M' u v TiTi x y T i+1

11. CSCE 411, Spring 2012: Set 911 Implementing Prim's Algorithm How do we find minimum weight outgoing edge? First cut: scan all adjacency lists at each iteration. Results in O(VE) time. Try to do better.

12. CSCE 411, Spring 2012: Set 912 Implementing Prim's Algorithm Idea: have each vertex not yet in the tree keep track of its best (cheapest) edge to the tree constructed so far. To find min wt. outgoing edge, find minimum among these values use a priority queue to store the best edge info (insert and extract-min operations)

13. CSCE 411, Spring 2012: Set 913 Implementing Prim's Algorithm When a vertex v is added to T, some other vertices might have their best edges affected, but only neighbors of v add decrease-key operation to the priority queue u v TiTi T i+1 w w's best edge to T i check if this edge is cheaper for w x x's best edge to T i v's best edge to T i

14. CSCE 411, Spring 2012: Set 914 Details on Prim's Algorithm Associate with each vertex v two fields: best-wt[v] : if v is not yet in the tree, then it holds the min. wt. of all edges from v to a vertex in the tree. Initially infinity. best-node[v] : if v is not yet in the tree, then it holds the name of the vertex (node) u in the tree s.t. w(v,u) is v's best-wt. Initially nil.

15. CSCE 411, Spring 2012: Set 915 Details on Prim's Algorithm input: G = (V,E,w) // initialization initialize priority queue Q to contain all vertices, using best-wt values as keys let v 0 be any vertex in V decrease-key(Q,v 0,0) // last line means change best-wt[v 0 ] to 0 and adjust Q accordingly

16. CSCE 411, Spring 2012: Set 916 Details on Prim's Algorithm while Q is not empty do u := extract-min(Q) // vertex w/ smallest best-wt if u is not v 0 then add (u,best-node[u]) to T for each neighbor v of u do if v is in Q and w(u,v) < best-wt[v] then  best-node[v] := u  decrease-key(Q,v,w(u,v)) return (V,T) // as MST of G

17. CSCE 411, Spring 2012: Set 917 Running Time of Prim's Algorithm Depends on priority queue implementation. Let T ins be time for insert T dec be time for decrease-key T ex be time for extract-min Then we have |V| inserts and one decrease-key in the initialization: O(V  T ins +T dec ) |V| iterations of while one extract-min per iteration: O(V  T ex ) total

18. CSCE 411, Spring 2012: Set 918 Running Time of Prim's Algorithm Each iteration of while includes a for loop. Number of iterations of for loop varies, depending on how many neighbors the current vertex has Total number of iterations of for loop is O(E). Each iteration of for loop: one decrease key, so O(E  T dec ) total

19. CSCE 411, Spring 2012: Set 919 Running Time of Prim's Algorithm O(V(T ins + T ex ) + E  T dec ) If priority queue is implemented with a binary heap, then T ins = T ex = T dec = O(log V) total time is O(E log V) (Think about how to implement decrease-key in O(log V) time.)

20. CSCE 411, Spring 2012: Set 920 Shortest Paths in a Graph Let's review two important single-source shortest path algorithms: Dijkstra's algorithm Bellman-Ford algorithm

21. CSCE 411, Spring 2012: Set 921 Single Source Shortest Path Problem Given: directed or undirected graph G = (V,E,w) and source vertex s in V Find: For each t in V, a path in G from s to t with minimum weight Warning! Negative weights are a problem: s t 4 55

22. CSCE 411, Spring 2012: Set 922 Shortest Path Tree Result of a SSSP algorithm can be viewed as a tree rooted at the source Why not use breadth-first search? Works fine if all weights are the same: weight of each path is (a multiple of) the number of edges in the path Doesn't work when weights are different

23. CSCE 411, Spring 2012: Set 923 Dijkstra's SSSP Algorithm Assumes all edge weights are nonnegative Similar to Prim's MST algorithm Start with source vertex s and iteratively construct a tree rooted at s Each vertex keeps track of tree vertex that provides cheapest path from s (not just cheapest path from any tree vertex) At each iteration, include the vertex whose cheapest path from s is the overall cheapest

24. CSCE 411, Spring 2012: Set 924 Prim's vs. Dijkstra's s 5 4 1 6 Prim's MST s 5 4 1 6 Dijkstra's SSSP

25. CSCE 411, Spring 2012: Set 925 Implementing Dijkstra's Alg. How can each vertex u keep track of its best path from s? Keep an estimate, d[u], of shortest path distance from s to u Use d as a key in a priority queue When u is added to the tree, check each of u's neighbors v to see if u provides v with a cheaper path from s: compare d[v] to d[u] + w(u,v)

26. CSCE 411, Spring 2012: Set 926 Dijkstra's Algorithm input: G = (V,E,w) and source vertex s // initialization d[s] := 0 d[v] := infinity for all other vertices v initialize priority queue Q to contain all vertices using d values as keys

27. CSCE 411, Spring 2012: Set 927 Dijkstra's Algorithm while Q is not empty do u := extract-min(Q) for each neighbor v of u do if d[u] + w(u,v) < d[v] then  d[v] := d[u] + w(u,v)  decrease-key(Q,v,d[v])  parent(v) := u

28. CSCE 411, Spring 2012: Set 928 Dijkstra's Algorithm Example ab de c 2 8 4 9 2 4 12 10 63 source is vertex a 012345 QabcdebcdecdedededØ d[a]000000 d[b] ∞ 22222 d[c] ∞ 1210 d[d] ∞∞∞ 1613 d[e] ∞∞ 11 iteration

29. CSCE 411, Spring 2012: Set 929 Correctness of Dijkstra's Alg. Let T i be the tree constructed after i-th iteration of while loop: vertices not in Q edges indicated by parent variables Show by induction on i that the path in T i from s to u is a shortest path and has distance d[u], for all u in T i (i.e., show that T i is a correct shortest path tree). Basis: i = 1. s is the only vertex in T 1 and d[s] = 0.

30. CSCE 411, Spring 2012: Set 930 Correctness of Dijkstra's Alg. Induction: Assume T i is a correct shortest path tree. Show that T i+1 is a correct shortest path tree. Let u be the vertex added in iteration i. Let x = parent(u). s x TiTi u T i+1 Need to show path in T i+1 from s to u is a shortest path, and has distance d[u]

31. CSCE 411, Spring 2012: Set 931 Correctness of Dijkstra's Alg s x TiTi u T i+1 P, path in T i+1 from s to u (a,b) is first edge in P' that leaves T i a b P', another path from s to u

32. CSCE 411, Spring 2012: Set 932 Correctness of Dijkstra's Alg s x TiTi u T i+1 a b P' P Let P1 be part of P' before (a,b). Let P2 be part of P' after (a,b). w(P') = w(P1) + w(a,b) + w(P2) ≥ w(P1) + w(a,b) (nonneg wts) ≥ w(s->a path in T i ) + w(a,b) (inductive hypothesis) ≥ w(s->x path in T i ) + w(x,u) (alg chose u in iteration i and d-values are accurate, by inductive hypothesis) = w(P). So P is a shortest path, and d[u] is accurate after iteration i+1.

33. CSCE 411, Spring 2012: Set 933 Running Time of Dijkstra's Alg. initialization: insert each vertex once O(V T ins ) O(V) iterations of while loop one extract-min per iteration => O(V T ex ) for loop inside while loop has variable number of iterations… For loop has O(E) iterations total one decrease-key per iteration => O(E T dec ) Total is O(V (T ins + T ex ) + E T dec )

34. CSCE 411, Spring 2012: Set 934 Using Different Heap Implementations O(V(T ins + T ex ) + E  T dec ) If priority queue is implemented with a binary heap, then T ins = T ex = T dec = O(log V) total time is O(E log V) There are fancier implementations of the priority queue, such as Fibonacci heap: T ins = O(1), T ex = O(log V), T dec = O(1) (amortized) total time is O(V log V + E)

35. CSCE 411, Spring 2012: Set 935 Using Simpler Heap Implementations O(V(T ins + T ex ) + E  T dec ) If graph is dense, so that |E| =  (V 2 ), then it doesn't help to make T ins and T ex to be at most O(V). Instead, focus on making T dec be small, say constant. Implement priority queue with an unsorted array: T ins = O(1), T ex = O(V), T dec = O(1) total is O(V 2 )

36. CSCE 411, Spring 2012: Set 936 What About Negative Edge Weights? Dijkstra's SSSP algorithm requires all edge weights to be nonnegative even more restrictive than outlawing negative weight cycles Bellman-Ford SSSP algorithm can handle negative edge weights even "handles" negative weight cycles by reporting they exist

37. CSCE 411, Spring 2012: Set 937 Bellman-Ford Idea Consider each edge (u,v) and see if u offers v a cheaper path from s compare d[v] to d[u] + w(u,v) Repeat this process |V| - 1 times to ensure that accurate information propgates from s, no matter what order the edges are considered in

38. CSCE 411, Spring 2012: Set 938 Bellman-Ford SSSP Algorithm input: directed or undirected graph G = (V,E,w) //initialization initialize d[v] to infinity and parent[v] to nil for all v in V other than the source initialize d[s] to 0 and parent[s] to s // main body for i := 1 to |V| - 1 do for each (u,v) in E do // consider in arbitrary order if d[u] + w(u,v) < d[v] then d[v] := d[u] + w(u,v) parent[v] := u

39. CSCE 411, Spring 2012: Set 939 Bellman-Ford SSSP Algorithm // check for negative weight cycles for each (u,v) in E do if d[u] + w(u,v) < d[v] then output "negative weight cycle exists"

40. CSCE 411, Spring 2012: Set 940 Running Time of Bellman-Ford O(V) iterations of outer for loop O(E) iterations of inner for loop O(VE) time total

41. CSCE 411, Spring 2012: Set 941 Bellman-Ford Example s c a b 3 —4 4 2 1 process edges in order (c,b) (a,b) (c,a) (s,a) (s,c)

42. CSCE 411, Spring 2012: Set 942 Correctness of Bellman-Ford Assume no negative-weight cycles. Lemma: d[v] is never an underestimate of the actual shortest path distance from s to v. Lemma: If there is a shortest s-to-v path containing at most i edges, then after iteration i of the outer for loop, d[v] is at most the actual shortest path distance from s to v. Theorem: Bellman-Ford is correct. Proof: Follows from these 2 lemmas and fact that every shortest path has at most |V| - 1 edges.

43. CSCE 411, Spring 2012: Set 943 Correctness of Bellman-Ford Suppose there is a negative weight cycle. Then the distance will decrease even after iteration |V| - 1 shortest path distance is negative infinity This is what the last part of the code checks for.

44. CSCE 411, Spring 2012: Set 944 Speeding Up Bellman-Ford The previous example would have converged faster if we had considered the edges in a different order in the for loop move outward from s If the graph is a DAG (no cycles), we can fully exploit this idea to speed up the running time

45. CSCE 411, Spring 2012: Set 945 DAG Shortest Path Algorithm input: directed graph G = (V,E,w) and source vertex s in V topologically sort G d[v] := infinity for all v in V d[s] := 0 for each u in V in topological sort order do for each neighbor v of u do d[v] := min{d[v],d[u] + w(u,v)}

46. CSCE 411, Spring 2012: Set 946 DAG Shortest Path Algorithm Running Time is O(V + E). Example: