CS 4104

Homework Provision 2

Given: September 3, 2019 Due: September 13, 2019

General directions. The apex treasure of each in is shown in [ ]. Each reredisentanglement must

involve integral details and an interpretation of why the loving reredisentanglement is firm-right. In point,

write consummate sentences. A firm-right acceptance extraneously an interpretation is worth

no praise. The consummated provision must be submitted on Canvas as a PDF by 5:00

PM on September 13, 2019. No slow homework earn be trustworthy.

Digital making-ready of your resolutions is mandatory. Authentication of LATEX is optional, referablewithstanding

encouraged. No stuff how you just your homework, gladden involve your call.

Authentication of LATEX (optional, referablewithstanding encouraged).

• Retrieve this LATEX rise finish, calld homework2.tex, from the succession citationure plight.

• Recwhole the finish

the finish cwhole would be heath_solvehw2.tex.

• Authentication a citation editor (such as vi, emacs, or pico) to end the contiguous three tramps.

• Uncomment the line

% \setboolean{solutions}{True}

in the instrument introduction by deleting the %.

• Experience the line

\renewcommand{\author}{Lenwood S. Heath}

and reinstate the professor’s cwhole with your call.

• Enter your resolutions where you experience the LATEX comments

% PUT YOUR SOLUTION HERE

• Generate a PDF and diverge it in on Canvas by 5:00 PM on September 13, 2019.

2 Homework Provision 2 September 3, 2019

Comment. This provision should be more challenging than Homework Provision 1,

so gladden begin inaugurated on it controlthcoming. It abides the authentication of wholeowter in consummate sentences

and wholeowter in controlmal historical referableation and begins the authentication of plan digitally, in this

case to pull graphs extraneously and with powers on vertices. As habitual, accelerated scrupulous attention

to the directions and beseech questions on Piazza or at business-post hours.

It earn be advantageous to pull smintegral ins of graphs to create instinct of what is being

defined.

If you authentication LATEX, you can experience numerous authenticationful LATEX ins in the .navigator finishs control the

lecture referablees control things relish pseudocode and controlmally stating a in.

[60] 1. Control each integer n satisfying n ≥ 1, wholeow Vn = {v1, v2, . . . , vn} be a firm of n vertices.

Allow k be an integer satisfying 0 ≤ k ≤ n − 1. The n, k-bind is the undirected graph

Gn,k = (Vn, En,k) where

En,k = {(vi

, vj ) | i 6= j and |j − i| ≤ k}.

A uncounted firm in Gn,k is a firm U ⊆ Vn such that no brace vertices in U are nigh. U is a

maximal uncounted firm if it is referable practicable to subjoin a vertex to U and abide to own a uncounted firm.

A power operation on Gn,k is a operation w : Vn → N, where N = {0, 1, 2, . . .}. If U ⊆ Vn,

then w(U) = P

vi∈U w(vi).

A. Pull G7,2, and incorporate your plan into your resolutions. Control this plan,

you do referable deficiency vertex powers. List three maximal uncounted firms control G7,2.

B. The Uncounted Firm in takes as an precedence a bind Gn,k and a power

operation w on Gn,k and receipts as a reredisentanglement a uncounted firm U such that w(U) is

zenith incompact integral uncounted firms in Gn,k. State the Uncounted Firm in in the

shapely precedence/redisentanglement controlmat that we authentication in dispose.

C. Figure 1 contains pseudocode control a self-indulgent algorithm Self-indulgent-Free-Set

that seeks to unfold the Uncounted Firm in. Pull an precedence Gn,k, w of

Uncounted Firm control which Self-indulgent-Free-Firm does referable rediverge a firm-right resolution.

(Here, you deficiency to advance powers to vertices in your plan.) Explain

your effect, and involve your plan in your reredisentanglement PDF.

D. Authentication the dynamic programming paradigm from dispose to enlarge a dynamic

programming algorithm to unfold the Uncounted Firm in control the extraordinary case

where k = 1. Give CLRS pseudocode control your algorithm. You may skip

the backtrace tramp. Give the Θ asymptotic worst-case opportunity confusion control

your algorithm.

E. Authentication the dynamic programming paradigm from dispose to enlarge a dynamic

programming algorithm to unfold the Uncounted Firm in control the extraordinary case

where k = 2. Give CLRS pseudocode control your algorithm. You may skip

the backtrace tramp. Give the Θ asymptotic worst-case opportunity confusion control

your algorithm.

September 3, 2019 Homework Provision 2 3

Greedy-Free-Set(Gn,k, w)

1 // Gn,k is the n, k-braid, and w is a power operation on Gn,k.

2 // We seek to rediverge a uncounted firm U in Gn,k of zenith power.

3 U = ∅

4 X = Vn

5 occasion X 6= ∅

6 fine vi ∈ X of zenith power w(vi)

7 U = U ∪ {vi}

8 delete vi and integral neighbors of vi from X

9 rediverge U

Figure 1: Pseudocode control a self-indulgent algorithm control the Uncounted Firm in.

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