HINTS FOR HW 5

Question 1, correctness of TopSort
Question 6, quicksort
Programming Question

QUESTION 1. Correctness of Topological Sort on p.205.

Here is one method of proof. The first thing to do is to give a FORMAL statement of what it means to be CORRECT.

Here it is: let G be a DAG on n nodes. If there is an edge from node u to node v, we call v a "successor" of u, and call u a "predescessor" of v. Calling TopSort(G) will print a list of all the nodes in G.

	CORRECTNESS of TopSort(G) MEANS THIS:


	If 
		L = (u₁, u₂, ..., u_n)

	is the list of nodes printed by TopSort(G), then

		(P1) Every node in G appears in the list L

		(P2) If u_i is a successor of
			u_j, then i< j.

Now, property (P1) is obvious from the algorithm (why?). To prove property (P2), let us use induction on i to show that when u_i is printed, then all its predecessors have been printed.

BASE CASE: Show that u₁ is printed correctly.
INDUCTION: Show that if i>1, then u_i is printed correctly.

We need some properties about TopHelp(G,v): Consider the moment just before v is printed. Since TopHelp(G,v) is a recursive algorithm, there is a recursive stack at this moment. Question: is it true that every node that is VISITED is already printed? HINT: think about this stack. Your argument should make use of the fact that the graph G is actually a DAG.

QUESTION 6: QUICKSORT

Basically, we want to make sure that after we partition the problem into two, we push the 2 subproblems onto the stack in such a way that the BIGGER subproblem is pushed first.

The property you need is this. Suppose that the list of subproblems on the stack has sizes

s₁, s₂, s₃, ... s_k. I.e., there are k subproblems on the stack, with s₁ at the bottom of the stack and with s_k at the top of the stack. Then (P3) s₁ >= s₂ >= s₃ >= ... >= s_k, and for i=2,...,k-1, we have (P4) s_i-1 + s_i-1 <= s_i-1. Prove properties (P3) and (P4). Conclude from this that the stack has depth at most 2 lg n. [Note: actually, this implies a better bound of log_c n where c=1.618... (the Golden Ratio).

PROGRAMMING QUESTION.

Most of you execute each multiplication for some N number of times (where N is some arbitrary integer like 10 or 50). You then add up the individual times to get the TOTAL TIME. Finally, you divide the TOTAL TIME by N to get the AVERAGE TIME per multiplication.
HINT: I suggest that you DO NOT divide by N. Just use the TOTAL TIME for your time. All that this does is to change your constant C in T(n) = C n^e by N. But when you compute the exponent, this factor is automatically removed! The advantage of this hint is that you will not get a time of 0 (which may happen if you take the average, and the number of bits is too small).