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

Does bmw have a guided missile corporate culture and

Does BMW have a guided missile corporate culture, and incubator corporate culture, a family corporate culture, or an Eiffel tower corporate culture?

Rebecca borrows 10000 at 18 compounded annually she pays

Rebecca borrows $10,000 at 18% compounded annually. She pays off the loan over a 5-year period with annual payments, starting at year 1. Each successive payment is $700 greater than the previous payment. (a) How much was ...

Jeff decides to start saving some money from this upcoming

Jeff decides to start saving some money from this upcoming month onwards. He decides to save only $500 at first, but each month he will increase the amount invested by $100. He will do it for 60 months (including the fir ...

Suppose you make 30 annual investments in a fund that pays

Suppose you make 30 annual investments in a fund that pays 6% compounded annually. If your first deposit is $7,500 and each successive deposit is 6% greater than the preceding deposit, how much will be in the fund immedi ...

Question -under what circumstances is it ethical if ever to

Question :- Under what circumstances is it ethical, if ever, to use consumer information in marketing research? Explain why you consider it ethical or unethical.

What are the differences between four types of economics

What are the differences between four types of economics evaluations and their differences with other two (budget impact analysis (BIA) and cost of illness (COI) studies)?

What type of economic system does norway have explain some

What type of economic system does Norway have? Explain some of the benefits of this system to the country and some of the drawbacks,

Among the who imf and wto which of these governmental

Among the WHO, IMF, and WTO, which of these governmental institutions do you feel has most profoundly shaped healthcare outcomes in low-income countries and why? Please support your reasons with examples and research/doc ...

A real estate developer will build two different types of

A real estate developer will build two different types of apartments in a residential area: one- bedroom apartments and two-bedroom apartments. In addition, the developer will build either a swimming pool or a tennis cou ...

Question what some of the reasons that evolutionary models

Question : What some of the reasons that evolutionary models are considered by many to be the best approach to software development. The response must be typed, single spaced, must be in times new roman font (size 12) an ...

  • 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

Why might a bank avoid the use of interest rate swaps even

Why might a bank avoid the use of interest rate swaps, even when the institution is exposed to significant interest rate

Describe the difference between zero coupon bonds and

Describe the difference between zero coupon bonds and coupon bonds. Under what conditions will a coupon bond sell at a p

Compute the present value of an annuity of 880 per year

Compute the present value of an annuity of $ 880 per year for 16 years, given a discount rate of 6 percent per annum. As

Compute the present value of an 1150 payment made in ten

Compute the present value of an $1,150 payment made in ten years when the discount rate is 12 percent. (Do not round int

Compute the present value of an annuity of 699 per year

Compute the present value of an annuity of $ 699 per year for 19 years, given a discount rate of 6 percent per annum. As