Young tableaus An m n Young tableau is an m n matrix such that the entries of each row are in sorted order from left to right and the entries of each column are in sorted order from top to bottom. Some of the entries of a Young tableau may be 1, which we treat as nonexistent elements. Thus, a Young tableau can be used to hold r mn finite numbers. a. Draw a 4 4 Young tableau containing the elements f9; 16; 3; 2; 4; 8; 5; 14; 12g. b. Argue that an m n Young tableau Y is empty if Y 1; 1 D 1. Argue that Y is full (contains mn elements) if Y m; n < 1. 168 Chapter 6 Heapsort c. Give an algorithm to implement EXTRACT-MIN on a nonempty m n Young tableau that runs in O.m C n/ time. Your algorithm should use a recursive subroutine that solves an m n problem by recursively solving either an .m 1/ n or an m .n 1/ subproblem. (Hint: Think about MAXHEAPIFY.) Define T .p/, where p D m C n, to be the maximum running time of EXTRACT-MIN on any m n Young tableau. Give and solve a recurrence for T .p/ that yields the O.m C n/ time bound. d. Show how to insert a new element into a nonfull m n Young tableau in O.m C n/ time. e. Using no other sorting method as a subroutine, show how to use an n n Young tableau to sort n2 numbers in O.n3/ time. f. Give an O.m C n/-time algorithm to determine whether a given number is stored in a given m n Young tableau.

L11 - 2 Def. The line y = L is called a horizontal asymptote of the graph of f(x)f i How many horizontal asymptotes can a graph have Consider the following functions: x ex. f(x)= x ex. f(x)=1 − e ✻ ✻ ✛ ✲ ✛ ✲ ❄ ❄ −1 ex. f(x)=t n (x) ✻ ✛ ✲ ❄ −1 −1 x→∞m tan (x)=