Consider the following problem. INPUT: Positive integers r1, ··· , rn and c1, ··· , cn . OUTPUT: An n by n matrix A with 0/1 entries such that for all i the sum of the ith row in A is ri and the sum of the ith column in A is ci , if such a matrix exists. Think of the problem this way. You want to put pawns on an n by n chessboard so that the ith row has ri pawns and the ith column has ci pawns. Consider the following greedy algorithm that constructs A row by row. Assume that the first i-1 rows have been constructed. Let aj be the number of 10 s in the jth column in the first i-1 rows. Now the ri columns with with maximum cj -aj are assigned 1’s in row i, and the rest of the columns are assigned 00 s. That is, the columns that still needs the most 1’s are given 1’s. Formally prove that this algorithm is correct using an exchange argument.

Consider the following problem. INPUT: Positive integers r1, ··· , rn and c1, ··· , cn . OUTPUT: An n by n matrix A with 0/1 entries such that for all i the sum of the ith row in A is ri and the sum of the ith column in A is ci , if such a matrix exists. Think of the problem this way. You want to put pawns on an n by n chessboard so that the ith row has ri pawns and the ith column has ci pawns. Consider the following greedy algorithm that constructs A row by row. Assume that the first i-1 rows have been constructed. Let aj be the number of 10 s in the jth column in the first i-1 rows. Now the ri columns with with maximum cj -aj are assigned 1’s in row i, and the rest of the columns are assigned 00 s. That is, the columns that still needs the most 1’s are given 1’s. Formally prove that this algorithm is correct using an exchange argument.