Ask Question, Ask an Expert

+1-415-315-9853

info@mywordsolution.com

Ask Computer Engineering Expert

problem 1: This problem involves the problem of computing change for a given coin system. A coin system is defined to be a sequence of coin values v1 < v2 < . . . < vn, such that v1 = 1. For ex, in the U.S. coin system we have six coins with values h1, 5, 10, 25, 50, 100i. The problem is what is the best way to make change for a given integer amount A.

a) Let c ≥ 2 be an integer constant. Suppose that you have a coin system where there are n types of coins of integer values v1 < v2 < . . . < vn, such that v1 = 1 and, for 1 < i ≤ n, vi = c · vi−1. (For ex, for c = 3 and n = 4, an ex would be h1, 3, 9, 27i.) Describe an algorithm which given n, c, and an initial amount A, outputs an n-element vector that indicates the minimum number of coins in this system that sums up to this amount. (Hint: Use a greedy approach.)

b) Given an initial amount A ≥ 0, let hm1, . . . ,mni be the number of coins output by your algorithm.

Prove that the algorithm is correct. In particular, prove the following:

i) For 1 ≤ i ≤ n, mi ≥ 0
ii) Pni=1mi · vi = A
iii) The number of coins used is as small as possible.

Prove that your algorithm is optimal (in the sense that of generating the minimum number of coins) for any such currency system.

c) Give an ex of a coin system (either occurring in history, or one of your own invention) for which the greedy algorithm may fail to produce the minimum number of coins for some amount. Your coin system must have a 1-cent coin.

problem 2: You are given an undirected graph G = (V,E) in which the edge weights are highly restricted.

In particular, each edge has a positive integer weight of either {1, 2, . . . ,W}, where W is a constant (independent of the number of edges or vertices). Show that it is possible to compute the single-source shortest paths in such a graph in O(n + m) time, where n = |V | and m = |E|. (Hint: Because W is a constant, a running time of O(W(n + m)) is as good as O(n + m).)
 
Requirement: Algorithm running time needs to be in DJIKstra’s running time or better.

problem 3: You are the mayor of a beautiful city by the ocean, and your city is connected to the mainland by a set of k bridges. Your city manager tells you that it is necessary to come up with an evacuation plan in the event of a hurricane. Your idea is to add a sign at each intersection pointing the direction of the route to the closest of the k bridges. You realize that this can be modeled as a graph problem, where the street intersections are nodes, the roads are edges, and the edge weights give the driving time between two adjacent intersections. Note that some of the roads in your city are one-way roads.

a) describe how to solve the evacuation-route problem in O(mlog n) time, where n is the number of intersections and m is the number of streets connecting two adjacent intersections. Note that the number of bridges k is not a constant, that is, it may depend on n and m. Therefore, a running time of O(k · mlog n) is not an acceptable solution. The output of your algorithm will be a labeling of the intersections, with an arrow pointing to the road to take to the closest bridge.

b) Some of the bridges can hold more capacity than others. For 1 ≤ i ≤ k, we associate a positive numeric weight ci with each bridge. To encourage people to use bridges with higher capacity, we treat distances to different bridges differently. In particular, if δ is the actual distance from some intersection to bridge i, we assign it a capacity-weighted distance of δ/ci. Therefore, the higher the bridge capacity is, the lower the weighted distance. Present an O(mlog n) time algorithm to solve the evacuation-route problem using capacity-weighted distances.
 
Additional info: Some of the bridges will be higher capacity. Their weights will be function of (distance/number of lanes) → this will give us shorter weight and we can treat it as a better route to take even though the distance is farther away.

Computer Engineering, Engineering

  • Category:- Computer Engineering
  • Reference No.:- M91939

Have any Question? 


Related Questions in Computer Engineering

1 the ieee 80211 task group i tgi is developing new wlan

1. The IEEE 802.11 Task Group i (TGi) is developing new WLAN security protocols named TKIP and CCMP. CCMP is envisioned to supersede WEP and TKIP. Research and study these efforts and comment on the progress. 2. It has b ...

1 use keplers law to check the accuracy of a given period

1. Use Kepler's law to check the accuracy of a given period and altitude for an Iridium satellite. 2. Use Kepler's law to check the accuracy of a given period and altitude for a Globalstar satellite. 3. Find the efficien ...

1 modify the expressiontree class from the chapter to

1. Modify the ExpressionTree class from the chapter to handle the unary operator - and unary mathematical function n!. 2. Implement the general Priority Queue ADT using the min-heap implemented as an array. Instead of ha ...

1 do you foresee a fusion between lan man and wan

1. Do you foresee a fusion between LAN, MAN, and WAN technologies in the future? Support your response. 2. Network technology is in transition. Discuss the direction of network technology

Show that the real and imaginary parts of the unitary dff

Show that the real and imaginary parts of the unitary DFf matrix are not orthogonal matrices in general. Show that the N x N cosine transform matrix C is orthogonal. Verify your proof for the case N = 4. Show that the N ...

1 suppose the stock of capital and the workforce are both

1. Suppose the stock of capital and the workforce are both increasing at 3 percent annually in the country of Wholand. At the same time, real output is growing at 6 percent. How is that possible in the short run and in t ...

Keeping the iv secret in ofb mode does not make an

Keeping the IV secret in OFB mode does not make an exhaustive key search more complex. Describe how we can perform a brute-force attack with unknown IV. What are the requirements regarding plaintext and ciphertext  Descr ...

1 why are vulnerabilities difficult to predict2 discuss the

1. Why are vulnerabilities difficult to predict? 2. Discuss the sources of system vulnerabilities. 3. Is it possible to locate all vulnerabilities in a network? In other words, can one make an authoritative list of those ...

Disscussion questionsneed a paragraphs on each

Disscussion Questions Need a Paragraphs on each question Choosing a Power Supply Choosing a Motherboard Name the 9 major parts of the motherboard and describe it purpose Your group has been asked to present to a high sch ...

Explain the relationship between mps due forecast quantity

Explain the relationship between MPS, due, forecast quantity, and customer orders.

  • 4,153,160 Questions Asked
  • 13,132 Experts
  • 2,558,936 Questions Answered

Ask Experts for help!!

Looking for Assignment Help?

Start excelling in your Courses, Get help with Assignment

Write us your full requirement for evaluation and you will receive response within 20 minutes turnaround time.

Ask Now Help with Problems, Get a Best Answer

Section onea in an atwood machine suppose two objects of

SECTION ONE (a) In an Atwood Machine, suppose two objects of unequal mass are hung vertically over a frictionless

Part 1you work in hr for a company that operates a factory

Part 1: You work in HR for a company that operates a factory manufacturing fiberglass. There are several hundred empl

Details on advanced accounting paperthis paper is intended

DETAILS ON ADVANCED ACCOUNTING PAPER This paper is intended for students to apply the theoretical knowledge around ac

Create a provider database and related reports and queries

Create a provider database and related reports and queries to capture contact information for potential PC component pro

Describe what you learned about the impact of economic

Describe what you learned about the impact of economic, social, and demographic trends affecting the US labor environmen