# CS 2110 Lab 11 Generating an Element in a Group

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## Description

Consider the group G = (Z≥0, +) where + is the usual integer addition. The set {0, 1} is a
generating set for G, but you can also consider other generators. Note that 0 must be present in
every generating set for G. Therefore, we do not need to explicitly state 0 — we can implicitly
understand 0 to be in every generating set. Given a generating set S ⊆ Z≥0 and an i ∈ Z≥0, we
can ask: how efficiently can we generate i using the elements in S. For example, let S = {1, 5}
1
and i = 11. You can generate 11 by the sequence 1 → 2 → 3 → 4 → · · · → 10. More efficiently,
you can generate 11 by 1 → 2 → 4 → 8 → 10 → 11. Optimally (i.e., most efficiently) you can
generate 11 by 5 → 10 → 11 or 1 → 6 → 11. In general, you can define the efficiency of your
generation by the number of times you use the + operator. Thus, the problem is:
Given a set S ⊆ Z≥0 and an element i ∈ Z≥0, produce a sequence of elements
generated from S such that you reach i most efficiently (i.e., fewest number of
times you use the + operator).
(In this lab exercise, bulk of your grade is for getting your program to work when S = {1}.
Therefore, first focus on that simple case. This will also help you get an intuition for how to
solve the more general case when S can be an arbitrary subset of Z≥0.)
0.1 Command Line Arguments
Your command line argument for this assignment is a single input file name.
1Recall that 0 is implicitly in S
1
0.2 The Input File Format
Each input file should solve one instance of the problem. The format is as follows. First you
must have an integer n which represents the number of elements of the generating set S. Then,
you provide the elements in S one after another. Finally, you provide your value of i.
You should allow your input to be in free format. Confer Lab 10 for what I mean by free
format.
You may assume that i and the largest element in S are never more than 100 and the
cardinality of S is at most 10.
0.3 Output
The output should contain the following:
1. A sequence that leads to i.
2. The number of times you needed to use the + operator to reach the value i.
As an example, consider the input:
3 1 7 13 42
Then, the output format is:
13+7=20, 20+1 = 21, 21+21 = 42
Number of times we used the + operator = 3
The output format is not strict, but it should be very easy for the human reader to get the
above two pieces of information.
There will be NO viva for this exercise. You must upload your source file by Friday (Nov 9)
night (11:55PM). Getting your program to work for S = {1} constitutes 80% of your marks.
The rest 20% will be for getting your program to work for any S.
Since there is no viva component, we will check for plagiarism very carefully. You are
allowed to discuss ideas with each other, but you must write your own program. And, more
importantly, you must not share your program with others.