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Chapter 5: Q8E (page 162)

Suppose you are given a weighted graph G=(V,E) with a distinguished vertex s and where all edge weights are positive and distinct. Is it possible for a tree of shortest paths from s and a minimum spanning tree in G to not share any edges? If so, give an example. If not, give a reason.

Short Answer

Expert verified

Every vertex of both the weighted graphG=V,Eis"s."Every one of the edge weights remains positive and unique.

Step by step solution

01

Directed Graph

If somehow the weighted graph G is directed, a tree with shortest routes from "s " can exist, and the lowest spanning tree in graph "G " will just not share any edge weights regard as the following scenario. Moreover, judge the following counter-directed graph:

鈥� Using Kruskal's approach, find the least spanning tree. The following is the minimal spanning tree for something like a directed graph:

鈥� Therefore least expense of road 3鈫�1is"1"cost for path 3鈫�2is"2,"and cost of path 3鈫�4is"3"of the presented graph 1+2+3=6.Consider the counter directed graph is given below

02

Shortest path Tree

鈥� Utilising Dijkstra's method, find the optimum from the vertex " 1." The shortest path tree for a directed graph is then shown as follows:


鈥� Our smallest path's edges being, as well as the least cost of path 1鈫�2is"6"path 1鈫�2is"6"and path 4鈫�3is"5"of the given graph 6+4+5=15 .Therefore, the directed graph is possible to find the minimum spanning tree without sharing any edge weights.

03

Undirected Graph

Assuming G seems to be an undirected weighted graph. Then, without expressing any edge weights, finding its minimum spanning tree is impossible. To determine the shortest path tree, MST must share at least one edge weight.

04

Conclusion

鈥� If the weighted graph G is undirected, then use cut property to find the minimum cut edge for shortest paths from 鈥渟鈥� and minimum spanning tree in graph 鈥淕鈥�. Because, MST is unique and edge weights are distinct.

鈥� According the Dijkstra鈥檚 algorithm, the minimum cut edge is needed to find the shortest path tree.

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Most popular questions from this chapter

A prefix-free encoding of a finite alphabet 螕 assigns each symbol in 螕 a binary codeword, such that no codeword is a prefix of another codeword. A prefix-free encoding is minimal if it is not possible to arrive at another prefix-free encoding (of the same symbols) by contracting some of the keywords. For instance, the encoding {0,101} is not minimal since the codeword 101 can be contracted to 1 while still maintaining the prefix-free property.

Show that a minimal prefix-free encoding can be represented by a full binary tree in which each leaf corresponds to a unique element of 螕, whose codeword is generated by the path from the root to that leaf (interpreting a left branch as 0 and a right branch as 1 ).

Show that for any integer n that is a power of 2 , there is an instance of the set cover problem (Section 5.4) with the following properties:

  1. There are n elements in the base set.
  2. The optimal cover uses just two sets.
  3. The greedy algorithm picks at least log n sets.

Thus the approximation ratio we derived in the chapter is tight.

Graphs with prescribed degree sequences. Given a list of n positive integers d1,d2,鈥�,dn, we want to efficiently determine whether there exists an undirected graphG=(V,E) whose nodes have degrees preciselyd1,d2,鈥�,dn . That is, if V={v1,鈥�,vn}, then the degree of vi should be exactly di. We call (d1,鈥�,dn) the degree sequence of G. This graph G should not contain self-loops (edges with both endpoints equal to the same node) or multiple edges between the same pair of nodes.

(a) Give an example of d1,d2,d3,d4 where all the di鈮�3 and d1+d2+d3+d4 is even, but for which no graph with degree sequence(d1,d2,d3,d4) exists.

(b) Suppose that d1鈮�d2鈮�d3鈮�dn and that there exists a graph G=(V,E) with degree sequence (d1,鈥�,dn). We want to show that there must exist a graph that has this degree sequence and where in addition the neighbors of v1 are v2,v3,鈥�,vdi+1 . The idea is to gradually transform G into a graph with the desired additional property.

i. Suppose the neighbors ofv1 in Gare not v2,v3,鈥�,vdi+1. Show that there exists i<j鈮�n and u鈭�V and such that {v1,vi},{u,vj}鈭�Eand {v1,vj},{u,vi}鈭�E

ii. Specify the changes you would make to G to obtain a new graph G'=(V,E') with the same degree sequence as G and where (v1,vi)鈭�E'.

iii. Now show that there must be a graph with the given degree sequence but in which v1 has neighbors v2,v3,鈥�,vdi+1.

c) Using the result from part (b), describe an algorithm that on input d1,鈥�,dn (not necessarily sorted) decides whether there exists a graph with this degree sequence. Your algorithm should run in time polynomial in n and in m=鈭�i=1ndi .

Show that if an undirected graph with n vertices has k connected components, then it has at least n - k edges.

Consider an undirected graph G=(V,E)with nonnegative edge weights role="math" localid="1658915178951" we鈮�0. Suppose that you have computed a minimum spanning tree of G, and that you have also computed shortest paths to all nodes from a particular node role="math" localid="1658915296891" s鈭�V. Now suppose each edge weight is increased by 1: the new weights are w0e=we+1.

(a) Does the minimum spanning tree change? Give an example where it changes or prove it cannot change.

(b) Do the shortest paths change? Give an example where they change or prove they cannot change.
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