Describe an efficient algorithm that, given a set {x1, } of points on the real line, determines the smallest set of unit-length closed intervals that contains all of the given points. (a unit length interval just means any closed interval with length 1. i. e., an interval [a, b] where b − a = 1.)
a. briefly describe a greedy algorithm for the unit length interval problem.
b. state and prove a "swapping lemma" for your greedy algorithm.
c. write a proof that uses your swapping lemma to show that your greedy algorithm does indeed produce a set of intervals that contain all of the points {x1, } with the fewest number of intervals.

Write a function balancechemical to balance chemical reactions by solving a linear set of equations. the inputs arguments are: reagents: symbols of reagents in string row array products: symbols of products in string row array elements: elements in the reaction in string row array elcmpreag: elemental composition of reactants in two dimensional numeric array elcmpprdcts: elemental composition of prducts in two dimensional numeric array hint: the first part of the problem is setting up the set of linear equations that should be solve. the second part of the problem is to find the integers from the solution. one way to do this is to mulitiply the rational basis for the nullspace by increasing larger integers until both the left-and right-side integers exist. for example, for the reaction that involves reacting with to produce and : reagents=["ch4", "o2"]; products =["co2", "h2o"]; elements =["c","h", "o"] elcmpreag=[1,4,0;