A cycle in an undirected graph G that goes through all the vertices of G is call

Question

A cycle in an undirected graph G that goes through all the vertices of G is called a spanning
cycle. Assume that H is an undirected graph that has a spanning cycle and we construct
another undirected graph H? from H by adding a single vertex at a time together with some
edges between the new vertex and some of the old vertices. More precisely, we have a sequence
of graphs H = H0,H1,H2,

Explanation / Answer

We want to run Dijkstra algorithm whose edge weights are integers in the range 0,1,...,W where W is a relatively small number. a)Using a bucket implementation (also known as Dial’s implementation) Dijkstra algorithm can be made to run in O(W|V | + |E|). Idea is to ?nd an upper bound on shortest distance labels. Since the maximum edge weight is W and a vertex can be updated at most |V |-1 times, we get a bound of O(W|V |) on shortest path distances. Using the property of Dijkstra algorithm that the distance labels that are designated permanent are non-decreasing we can present the following algorithm procedure Efficient-Dijkstra1(G,l,W,s) Input: Graph G=(V, E), edge weights le, Max edge weight W, vertex s ? V Output: Shortest path distance labels for all vertices v ? V : dist(v) = 8 prev(v) = nil bucket(v) = nil Create an array B of size W|V |: B[i] keep vertex of distance label i B[0] = s,dist(s)=0,bucket(s)=0 index = 0 while index =6 W|V | Increment index: B[index] =6 Ø u = B[index] for all edges (u,v) ? E: temp = dist(v) if dist(v) > dist(u) + l(u,v): dist(v) = dist(u)+l(u,v) prev(v) = u B[dist(v)] = v if temp =6 8: Remove v from B[temp] Doubly linked list should be used for array B, which allows us to do the following operations in O(1) time: (1) Checking whether a bucket is empty or nonempty (2) Deleting an element from a bucket (3) Adding an element to the bucket (b)The algorithm is same as the algorithm shown on Page 110. We claim the following lemma

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.