Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 22.2, Problem 5E
Program Plan Intro
To argue the value of
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Check out a sample textbook solutionStudents have asked these similar questions
Consider the following undirected binary tree T with 17 vertices.
a
lo
Starting with the root vertex a, we can use Breadth-First Search (BFS) or Depth-First Search
(DFS) to pass through all of the vertices in this tree.
Whenever we have more than one option, we always pick the vertex that appears earlier in the
alphabet. For example, from vertex a, we go to b instead of c.
Clearly explain the difference between Breadth-First Search and Depth-First Search, and deter-
mine the order in which the 17 vertices are reached using each algorithm.
In the following G, each adjacency list is sorted in increasing alphabetical order.
B
H
G
Do depth-first search in G, considering vertices in increasing alphabetical or-
der. Show the final result, with vertices labeled with their starting and finishing times,
and edges labeled with their type (T/B/F/C). Here T, B, F and C refer to tree, back,
forward, and cross edges, respectively. For your convenience, the graph is reproduced
enlarged below.
B
D
E
H
G
F
4. Consider the graph G, shown below, which is simply a copy of K5.
02
V3
5
V1
24
V5
How many distinct spanning trees does G have? (Hint: Break up your search by the
isomorphism type of the tree, as discovered on the previous page. So for example,
start by counting the paths of length 5 in G. Then proceed to the next type of tree
with 5 vertices. The total number of trees is 125, but please use this answer only to
check that your solution is complete!)
Chapter 22 Solutions
Introduction to Algorithms
Ch. 22.1 - Prob. 1ECh. 22.1 - Prob. 2ECh. 22.1 - Prob. 3ECh. 22.1 - Prob. 4ECh. 22.1 - Prob. 5ECh. 22.1 - Prob. 6ECh. 22.1 - Prob. 7ECh. 22.1 - Prob. 8ECh. 22.2 - Prob. 1ECh. 22.2 - Prob. 2E
Ch. 22.2 - Prob. 3ECh. 22.2 - Prob. 4ECh. 22.2 - Prob. 5ECh. 22.2 - Prob. 6ECh. 22.2 - Prob. 7ECh. 22.2 - Prob. 8ECh. 22.2 - Prob. 9ECh. 22.3 - Prob. 1ECh. 22.3 - Prob. 2ECh. 22.3 - Prob. 3ECh. 22.3 - Prob. 4ECh. 22.3 - Prob. 5ECh. 22.3 - Prob. 6ECh. 22.3 - Prob. 7ECh. 22.3 - Prob. 8ECh. 22.3 - Prob. 9ECh. 22.3 - Prob. 10ECh. 22.3 - Prob. 11ECh. 22.3 - Prob. 12ECh. 22.3 - Prob. 13ECh. 22.4 - Prob. 1ECh. 22.4 - Prob. 2ECh. 22.4 - Prob. 3ECh. 22.4 - Prob. 4ECh. 22.4 - Prob. 5ECh. 22.5 - Prob. 1ECh. 22.5 - Prob. 2ECh. 22.5 - Prob. 3ECh. 22.5 - Prob. 4ECh. 22.5 - Prob. 5ECh. 22.5 - Prob. 6ECh. 22.5 - Prob. 7ECh. 22 - Prob. 1PCh. 22 - Prob. 2PCh. 22 - Prob. 3PCh. 22 - Prob. 4P
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Similar questions
- 3. Consider the following algorithm for the vertex cover problem for trees. Pick an edge (u, v), where u is a leaf. Include v in the cover, delete all edges incident to v, and repeat until there are no edges left. Show that this algorithm finds a smallest cover. Hint: you may argue using matchings as in the analysis of the 2-approximation algorithm.arrow_forward3. From the graph above determine the vertex sequence of the shortest path connecting the following pairs of vertex and give each length: a. V & W b. U & Y c. U & X d. S & V e. S & Z 4. For each pair of vertex in no. 3 give the vertex sequence of the longest path connecting them that repeat no edges. Is there a longest path connecting them?arrow_forwardFor the following graph: 4 (a) Determine the order of vertices traversed by depth-first search (with lower index as priority in case of ambiguity) and show how the stack would look when running it. (b) Compute the discovery and finishing times (DFS numbers) for each vertex. (c) What edges make up the DFS tree or forest? Classify the rest of the edges. (d) Determine whether there is a cycle or not in this graph. If there is not, give a possible topological ordering.arrow_forward
- Given the following adjacency list representation of an undirected graph, give the visited node order forDepth-First Search, starting with v1. The format of the solution: add number of node space node e.g. 1 2 34 5678 Adj[1]->4->6->7 Adj[2]->3->4->7 Adj[3]-> 2->5->6 Adj[4]->1->2->5->6 Adj[5]->3->4 Adj[6]->1->3->4->7 Adj[7]->1->2->6 467321arrow_forward4. Please apply Kruskal's spanning tree algorithm in the graph below and find the minimum spanning tree (MST). Edge weights are in the adjacency matrix table. (you can list the edges in MST or draw the tree below) d b a e h f The adjacency matrix for the undirected weighted graph is mentioned below: a d e g h a 4 8 4 8 11 8. 7 4 2 d 7 14 e 9. 10 f 4 14 10 2 1 6. 8. 11 1 7 7arrow_forwardGiven the graph below, list the vertex visitation order of a depth-first search (DFS) beginning at vertex A. [follow alphabetical order] * B D E F H. O A>B>C>E>D>F>H>l>G O A>B>C>F>D>E>H>G>l O A>B>C>F>D>E>l>H>G A>B>C>E>F>G>H>l>D Other:arrow_forward
- For the following graph, perform Breadth First Search (starting v1, tie breaker rule, choose node with smaller index first) explain step by step with priority queue.arrow_forwardUse depth-first search starting at vertex SS to construct a spanning tree of the graph below. If we tie-break alphabetically, what is the order of edges we construct our spanning tree? a. {S,T}, {T,X}, {W,X}, {W,Z}, {V,Z}, {X,Y}, {U,Y} b. {S,T}, {S,V}, {V,Z}, {W,Z}, {W,X}, {T,X}, {X,Y}, {U,Y} c. {S,T}, {T,X}, {W,X}, {S,V}, {V,Z}, {Y,Z}, {U,Y} d. {S,T}, {T,X}, {W,X}, {W,Z}, {V,Z}, {Z,Y}, {U,Y}arrow_forwardKruskal's algorithm, which incorporates the union-find data structure, is as follows.algorithm Pre-cond KruskalMST (G): G is an undirected graph.post-cond: The result is a minimum spanning tree.arrow_forward
- For the graph in Figure 1 below, construct a spanning tree using the breadth-first algorithm. Show all working (including data structures and all steps). Decide first whether the arc weights are needed for this problem. Then, process vertices by lowest-numbered first and alphabetical order where appropriate. 15 3 4 V2 7 6 V3 8 7 Figure 1 Co V5 V6 5 7 Zarrow_forwardAS Minimum Spanning Tree - Problem Use both the Kruskal's algorithm and the Prim's algorithm to find the maximum spanning tree for the following graph. Indicate the order of edge selection. (A maximum spanning tree's total weight is maximized. When use the Prim's, start with vertex a (shaded black).) 8 7 9. d 9. 1 4 14 a e 7. 10 8 1 2 2.arrow_forwardQuestion 4. Given the following directed graph: B F E Perform a Depth-First Search (DFS) on the given graph, using vertex G as the source. (a) Suppose that neighbors of a vertex is accessed in alphabetical order. List the vertices in the discovered order of DFS and show for each vertex v, its discovery time d[v] and finish time f[v]. (b) Draw the depth-first tree obtained.arrow_forward
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