Ask Question, Ask an Expert


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

The web server on the dmz web server system renames

The Web server on the DMZ Web server system renames temporary files used to record transactions. The name has the form trns followed by the integer representation of the date and time, followed by one or more digits. Why ...

Design a help-desk priority manager store help requests

Design a help-desk priority manager. Store help requests stored in a text file with the following structure: priority, id of requesting party, time of request The priority is an integer in the range 1-5 with 1 being the ...

1 every country participating in the computer products

1. Every country participating in the computer products security evaluation has a list of evaluated products. Find out how to find this list. Does the ISO keep a global list of evaluated products? 2. Why is the product r ...

Al was an attorney in a small town and was very well

Al was an attorney in a small town and was very well respected. A couple retained Al to represent them on the purchase of a home. Before closing, Al informed the couple of payments that they were required to make by chec ...

1 compare and contrast distance-vector and link-state

1. Compare and contrast distance-vector and link-state routing. If you were designing a new routing protocol, which would you use and why? 2. Analyze the routing table on your computer. In Windows you can view it with a ...

This chapter covered network management whereas the

This chapter covered network management, whereas the previous chapter covered network security. Note that network design tasks are often interwoven, however, and shouldn't be considered discrete just because a book is di ...

An inverted page table can be used to further optimize

An inverted page table can be used to further optimize space and time. How many PTEs are needed to store the page table? Assuming a hash table implementation, what are the common case and worst case numbers of memory ref ...

Answer each question in 8 paragraphs 4 pages-1 should there

Answer each question in 8 paragraphs (4 pages)- 1) Should there be a single programming language for all programming domains? Please use the concepts that you learned in this class to justify your answer. 2) Suppose that ...

Deterministic and random behavior it is human nature to

Deterministic and random behavior It is human nature to assume that there is a causal force behind everything that we observe, but some times this is not the case. However, even when we have a perfect understanding of a ...

A searching method that uses linear interpolation can give

A searching method that uses linear interpolation can give fast retrieval when the ordered data set is relatively evenly distributed over its range of possible values. With this method. the magnitude of the value x we ar ...

  • 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

WalMart Identification of theory and critical discussion

Drawing on the prescribed text and/or relevant academic literature, produce a paper which discusses the nature of group

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