Question

I was trying to write an algorithm for given problem:

we are given a set of numbers- {n1,n2,n3,n4,n5……}
and we have to check that can we derive a number(Say X) using addition and subtraction by given numbers. X will always be less than all elements of the given set.

Eg.

Set: {2,3,4,6,9}
given number: 1, Result: Yes
9-4-4 =1

Set: {3,4,6,9}
given number: 2, Result: Yes
6-4 = 2

Thanks in advance.

Answer 1

Effectively you are looking for the ideal generated by the numbers in your set. The intergers form a principal ideal domain, which means every ideal is generated by a single integer. All you have to do is find this single integer — say g — and check whether X can be devided by g. Finding g is also easy — it’s the greatest common divisor of all elements in your set, which can be found using the Euclidean algorithm.

You example sets can generate every integer by addition and substraction, since the can generate 1. For example for the set {3,4,6,9} you have 1=4-3, and any integer n can be written as n times the sum of 4-3.