Objectives
The objectives of this assignment are:
To demonstrate the ability to implement algorithms using basic data structures and operations on
To gain experience in designing an algorithm for a given problem description and implementing that algorithm in Python.
To demonstrate an understanding of complexity, and to the ability to implement algorithms of a given complexity.
Task 1: Complexity (4 Marks)
Create a Python module called complexity.py. Within this module implement the following task. For this module you may not use the Python built-in function pow(x) and you may not use the power operator **. You may not import any other libraries or modules.
Part A: Peaks (2 Marks)
Sometimes in life we care more about finding peaks than we care about finding the maximum. For example, during semester you know that the week containing the maximum amount of assessment is (probably) during exam block. But you will also have weeks in which that particular week contains more assessment than the week directly before it, or directly after it. As such, if you are planning on going to the movies with friends, you should probably do it in a week that is not an assessment peak.
Write a function local peak(lst) that takes in a list of numbers, and returns the index of a local peak.
Input: a list lst containing two or more numbers, where no number is the same as its neighbour.
Output: an integer indicating the index of a local peak. If there is more than one peak in the given list, the index of any peak may be returned. A peak is defined as a number that is greater than its neighbours. A peak may be the first or last number in the list, in which case it must be greater than its only neighbour.
For full marks, this function must have a complexity of O(log(N )), where N is the length of lst. Be aware that some Python functions have a higher complexity than you might think. One example is slicing, which has a complexity of O(k) where k is the length of the slice.
Examples
Calling local peak([0,2,4,6,5,2]) returns 3.
Calling local peak([-7,-6,-2,-10,-5,-11]) returns either 2 or 4.
Calling local peak([8.8,8.6,8.1,8.2,8.1,8.01]) returns either 0 or 3.
Part B: Powers (2 Marks)
Write a function power(n, p) that takes a number and a power and returns the number raised to the given power.
Input: a number n and a non-negative integer p.
Output: a number that is the evaluation of np
For full marks, this function must have a complexity of O(log(p)).
Examples
Calling power(2,8) returns 256.
Marking Guide (total 4 marks)
Marks are given for the correct behavior of the different functions:
1 mark for the correct behaviour of local peak, and 1 mark for implementing the function with a complexity of O(log(N )). You must be able to explain why the function has a complexity of O(log(N )) to receive the full
1 mark for the correct behaviour of power, and 1 mark for implementing the function with a complexity of O(log(p)). You must be able to explain why the function has a complexity of O(log(p)) to receive the full
Note: marks will be deducted for including print statements or function calls in submitted module.
Task 2: Birthday Card (6 Marks)
Your good friend, Xanthe, who also lives in Melbourne, is turning 21 in a few weeks. To celebrate the occassion you would like to give Xanthe a birthday card that has been signed by all of her friends. The only problem is Xanthe is incredibly well travelled, and has friends all through the country and the world.
To figure out the best way to get the card to all of these friends before Xanthe’s birthday, you have turned to graph theory. Every city and town in which a friend of Xanthe lives has been allocated a number between 1 and the number of friends (minus one), and Melbourne has been allocated the number 0. You have recorded postage times between locations as edge weights. You have found that all pairs of locations are such that the postage time from A to B is the same as the postage time from B to A, so your graph is undirected. You have also found that there is a postage connection between every pair of locations, so your graph is connected and complete. You have implemented this graph as an adjacency matrix, where each cell represents the weight of an edge. All postage times are given in integers, representing how many days the card would take to travel between the two locations. (I.e. an edge weight of 2 between A and B means it would take two days to get the card between location A and location B.)
Create a Python module called card.py. Within this module implement the following three tasks. You are encouraged to decompose the given tasks into additional functions. You may not import any other libraries or modules.
Part A: Time (1 Mark)
Write a function post time(graph, path) that returns how long it will take Xanthe’s card to travel the given path.
Input: a nested list graph that represents a graph as an adjacency matrix, that models the postage routes and times between Xanthe’s friends; and a list of integers path that represents a proposed path in the graph. You may assume the given path exists in the graph.
Output: an integer that is the number of days it would take Xanthe’s card to travel the given path.
Examples
postage1 = [ [0,1,1,3,2],
[1,0,4,5,1],
[1,4,0,1,3],
[3,5,1,0,1],
[2,1,3,1,0] ]
postage2 = [ [0,2,2,1,2],
[2,0,1,1,2],
[2,1,0,1,4],
[1,1,1,0,1],
[2,2,4,1,0] ]
The example graphs postage1 and postage2 are provided for illustrative purpose. Your implemented function must be able to handle arbitrary graphs in matrix form.
Calling post time(postage1, [0,1,4,3,2,0]) returns 5.
Calling post time(postage1, [0,3,1,2,3]) returns 13.
Part B: Where to post? (2.5 Marks)
Write a function post route(graph) that uses a greedy approach to find a fast (but not necessarily the fastest) route by which to send the card. The metric for the greedy approach is: send the card to the location that takes the shortest time to get to that the card has not yet been visited; or, if all locations have been visited, send the card back to Melbourne. If there are multiple locations to chose from, all with the same distance, choose the location that has been allocated the smaller number.
Input: a nested list graph that represents a graph as an adjacency matrix, that models the postage routes and times between Xanthe’s friends.
Output: a list of integers, where each integer is a location, ordered such that visiting each location in the order given creates a cycle that begins and ends in Melbourne. The only number that can appear twice is 0.
Examples
Calling post route(postage1) returns [0,1,4,3,2,0].
Calling post route(postage2) returns [0,3,1,2,4,0].
Part C: Can it be done? (2.5 Marks)
Write a function on time(graph,days) that determines whether there exists a postage route that can be completed before Xanthe’s birthday.
Input: a nested list graph that represents a graph as an adjacency matrix, that models the postage routes and times between Xanthe’s friends; and a non-negative integer days, which is the number of days until Xanthe’s birthday.
Output: a boolean, True if there exists a cycle beginning and ending at Melbourne, that visits every location, and that has a weight equal or less than days; otherwise False.
You must cite any lecture code you use.
Examples
Calling on time(postage1,5) returns True, because there is a cycle in graph postage1 with weight 5 that fits the criteria: [0,1,4,3,2,0].
Calling on time(postage2,8) returns True, because there is a cycle in graph postage2 with weight 8 that fits the criteria: [0,1,2,3,4,0].
Calling on time(postage2,5) returns False, because there is no cycle in graph postage2 with weight 5 that fits the
Marking Guide (total 6 marks)
Marks are given for the correct behavior of the different functions:
1 mark for post time.
5 marks for post route.
5 marks for on time.
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