Algorithms Homework Assignment #9

1. AlgorithmsHomework Assignment #9 Jing-Jie Lin

2. HW#9 No.2 (a) Give an example of a weighted connected undirected graph G = (V, E) and a vertex v, such that the minimum-cost spanning tree of G is the same as the shortest-path tree rooted at v. (10pt) (b) Give an example (5pt) of a weighted connected undirected graph G = (V, E) and a vertex v, such that the minimum-cost spanning tree of G is very different from the shortest path tree rooted at v. Can the two trees be completely disjoint?(5pt)

3. HW#9 No.2 (b) No, it is impossible that the two trees are completely disjointed. Let Tm denote the minimum cost spanning tree of G Ts be the shortest path tree rooted at v. • v has only one edge The only edge must be both in Tm and in Ts. Otherwise v will be disconnected from Tm and from T • v has more than oneedge Let (v, u) be the minimum weighted edge among all edges of v. Then (v, u) must belong to both Tm and Ts. • If (v, u) is not in Tm, Tm must include some other edge (v, w). But by replacing (v, w) with (v, u) in Tm=> a new tree with lower weight • If (v, u) is not in Ts, the shortest path π from v to u must include some other edge (v, w), but the edge (v, u) is shorter than π

4. HW#9 No.3 (a) Run the strongly connected components algorithm on the directed graph shown in Figure 1. (10pt) • When traversing the graph, the algorithm should follow the given DFS numbers. • Show the High values as computed by the algorithm in each step. 1 (b) Add the edge (4, 2) to the graph and discuss the changes this makes to the execution of the algorithm. (10pt) 4 9 7 2 6 8 5 3

5. (a) 1 4 9 7 2 6 8 8 5 3

6. (a) 8 1 8 7 7 6 6 5 5 4 4 3 3 2 2 1 4 9 max 7 2 6 8 5 max 3 max max max max 9 1 max 3 9 max 9 8 max

7. (b) 1 4 9 7 2 6 8 5 3

8. HW#9 No.4 • Let G = (V, E) be a connected weighted undirected graph and T be a minimum-cost spanning tree (MCST) of G. • Suppose that the cost of one edge {u, v} in G is changed (increased or decreased) • {u, v} may or may not belong to T. • Design an algorithm (10pt) to either find a new MCST or to determine that T is still an MCST. • The more efficient your algorithm is, the more points you will be credited for this problem. • Explain why your algorithm is correct(5pt) and analyze its time complexity (5pt).

9. {u, v} HW#9 No.4 T1 T2 f Lemma: Adding additional edge k to a minimal cost spanning tree T, it will create a cycle . In the cycle, k is the maximum cost edge.

10. Input: G(V, E): undirected graph, T: MCST, e={u, v} 2 G • Output: T’: new MCST or print “No Change” • 17else if e T and e is decrease then 18 T’ := T + e; • 19 C := Find_Cycle(T, e); • 20for all edge k in C do • 21ifk.weight> w then • 22w := k.weight; 23new_edge:=k; 24 T’ := T’ – new_edge; 25if new_edge = e then 26 print “No Change” 27 else 28return T’; 29 else • 30 print “No Change” AlgorithmnewMCST (G, T, e); 1 begin 2w := e.weight; 3new_edge := e; • 4ife2 T and e is increase then 5T’ := T - e; 6 T1 := DFS_Tree(T’, u); 7 T2 := DFS_Tree(T’, v); 8 for all edges k connecting T1 and T2 do 9ifk.weight < w then 10 w := k.weight; 11new_edge := k; 12 if k =e then 13print “No Change” • 14 else • 15 T’ := T’ + new_edge; • 16 return T’; O(|V|+|E|) O(|V|) O(|V1|+|E1|) O(|V2|+|E2|) O(|E|) Time Complexity: O(|V|+|E|)