/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 54 Let \(T_{1}\) and \(T_{2}\) be s... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 54

Let \(T_{1}\) and \(T_{2}\) be spanning trees of a graph. The distance between \(T_{1}\) and \(T_{2}\) is the number of edges in \(T_{1}\) and \(T_{2}\) that are not common to \(T_{1}\) and \(T_{2}\) . $$ \begin{array}{l}{\text { Suppose that } T_{1}, T_{2}, \text { and } T_{3} \text { are spanning trees of the sim- }} \\ {\text { ple graph } G . \text { Show that the distance between } T_{1} \text { and } T_{3}} \\ {\text { does not exceed the sum of the distance between } T_{1} \text { and }} \\ {T_{2} \text { and the distance between } T_{2} \text { and } T_{3} .}\end{array} $$

Short Answer

Expert verified
The distance between \(T_1\) and \(T_3\) is less than or equal to the sum of the distances between \(T_1\) and \(T_2\), and \(T_2\) and \(T_3\).

Step by step solution

01

Define the Distance Between Trees

The distance between two spanning trees, say \(T_{1}\) and \(T_{2}\) is given by the number of edges that are not common to both trees. Formally, if \(E(T_1)\) and \(E(T_2)\) are the edge sets of \(T_1\) and \(T_2\), then the distance is defined as \[ d(T_1, T_2) = |E(T_1) \Delta E(T_2)| \] where \(\Delta\) denotes the symmetric difference.
02

Apply the Triangle Inequality Concept

Consider three spanning trees \(T_1\), \(T_2\), and \(T_3\). To show that \[ d(T_1, T_3) \le d(T_1, T_2) + d(T_2, T_3) \], use the concept of triangle inequality within the metric space formed by the distances.
03

Symmetric Difference and Cardinality

The symmetric difference between the edge sets of the trees can be visualized: \[ d(T_1, T_3) = |E(T_1) \Delta E(T_3)| \]. By definition, the symmetric difference \( A \Delta C\) can be decomposed as: \[ (A \Delta B) \cup (B \Delta C) \] with overlaps counted twice being discarded.
04

Apply the Union Bound

According to union bound principles, the cardinality of the symmetric difference will always be less than or equal to the sum of individual differences: \[ |A \Delta C| \le |A \Delta B| + |B \Delta C| \]. Substitute \(A = E(T_1)\), \(B = E(T_2)\), and \(C = E(T_3)\): \[ d(T_1, T_3) \le d(T_1, T_2) + d(T_2, T_3). \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph Theory
Graph theory is a branch of mathematics that studies the properties and applications of graphs. A graph is a collection of vertices (or nodes) and edges (or arcs) that connect pairs of vertices. Graphs can be used to model many real-world situations, such as social networks, computer networks, and transportation systems.

In this exercise, we are dealing with spanning trees. A spanning tree of a graph is a subgraph that is a tree and includes all the vertices of the original graph. Trees are a special type of graph with no cycles and a unique path between any two vertices. Spanning trees are useful for optimally connecting different parts of a network.
Symmetric Difference
The symmetric difference between two sets is a new set containing elements that are in either of the original sets but not in their intersection.

Mathematically, the symmetric difference of two sets, A and B, is denoted by \(A \Delta B\), and defined as:
\[A \Delta B = (A - B) \cup (B - A)\]

For the exercise, the distance between two spanning trees, T1 and T2, is defined as the number of edges not common to both trees. In other words, \[d(T_1, T_2) = |E(T_1) \Delta E(T_2)|\]

Here, \(d(T_1, T_3) \Delta d(T_2, T_3)\) captures the edges either in T1 or T3 but not in both.
Triangle Inequality
The triangle inequality is a fundamental concept in mathematics, especially in metric spaces. It states that for any three points (or objects) A, B, and C, the distance between A and C is never greater than the sum of the distances between A to B and B to C.

Formally, for distances d(A, B), d(B, C), and d(A, C), the inequality is written as:
\[d(A, C) \leq d(A, B) + d(B, C)\]

In the context of our exercise, we are using this property to show that the distance between two spanning trees T1 and T3 is never greater than the sum of the distances between T1 and T2, and T2 and T3. This is expressed as:
\[d(T_1, T_3) \leq d(T_1, T_2) + d(T_2, T_3)\]
Metric Space
A metric space is a set where a notion of distance (called a metric) is defined between elements of the set. This distance measures how far apart elements are from each other.

A metric must satisfy certain properties:
  • Non-negativity: d(x, y) 鈮� 0 for all x, y, and d(x, x)=0
  • Symmetry: d(x, y) = d(y, x) for all x, y
  • Triangle inequality: d(x, z) 鈮� d(x, y) + d(y, z) for all x, y, z


In the exercise, the edge sets of spanning trees form a metric space, with the distance function being the symmetric difference of their edge sets. The triangle inequality ensures that the distance between any two spanning trees within this space adheres to the core properties of a metric.

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Most popular questions from this chapter

A chain letter starts with a person sending a letter out to 10 others. Each person is asked to send the letter out to 10 others, and each letter contains a list of the previous six people in the chain. Unless there are fewer than six names in the list, each person sends one dollar to the first person in this list, removes the name of this person from the list, moves up each of the other five names one position, and inserts his or her name at the end of this list. If no person breaks the chain and no one receives more than one letter, how much money will a person in the chain ultimately receive?

A rooted spanning tree of a directed graph is a rooted tree containing edges of the graph such that every vertex of the graph is an endpoint of one of the edges in the tree. $$ \begin{array}{l}{\text { Give an algorithm to build a rooted spanning tree for con- }} \\ {\text { nected directed graphs in which each vertex has the same }} \\ {\text { in-degree and out-degree. }}\end{array} $$

Explain how breadth-first search or depth-first search can be used to order the vertices of a connected graph.

a) Draw the complete binary tree with 15 vertices that represents a tree- connected network of 15 processors. b) Show how 16 numbers can be added using the 15 processors in part (a) using four steps.

What does each of these represent in a tree that represents the structure of an organization? a) the parent of a vertex b) a child of a vertex c) a sibling of a vertex d) a sibling of a vertex e) the ancestors of a vertex f) the level of a vertex g the height of the tree
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