ICS H21 • UC IRVINE • DAVID G. KAY • FALL 2009

Seventh Homework

This assignment is due at the start of lab on Monday, November 9.

(1) Take a look back at the discussion of the function reduce in section 21.5. (Reduce is sometimes called accumulate or foldr, sometimes with the arguments in a different order. In DrScheme, the predefined function is named foldr.) Using foldr/reduce, map, and filter, you can define many powerful operations very compactly, without explicit recursion. For example, suppose we have a list of restaurant (rrant) structures as we've used in class before; call that list RL. To produce a list of the names of the cheap Thai restaurants in RL, we only need to say

(map rrant-name (filter cheap? (filter Thai? RL))).

To calculate the average price of the cheap Thai restaurants, we can say

(local ((define cheap-thai-restaurant-prices
                (map rrant-price (filter cheap? (filter Thai? RL)))))
  (/ (foldr + 0 cheap-thai-restaurant-prices)
     (length cheap-thai-restaurant-prices)))

(In the above example, note that using the local expression saves us from computing the list of prices two separate times.) If you have trouble figuring out how these expressions work, first make sure you understand map, filter, and foldr individually (look at their "contracts") and then look at what each part of the expression returns, starting from the inside out.

Do each of the following problems. Be aware, also, that problems like these are likely to show up on exams.

(1.1) Write the function convert-to-1 that takes one argument (of any type) and returns the number 1, no matter what the argument is. Next, use map and convert-to-1 to define the function list-of-ones that takes a list of items and returns a list of 1s that's the same length as its argument. Finally, use foldr and list-of-ones to rewrite the last line of the average-price code above without using length. We did this in class; now you should be able to do it, too.

(1.2) What is the result of evaluating each of these expressions?

•   (foldr + 0 '(1 2 3 4 5))
•   (foldr (lambda (a b) (+ b (if (even? a) a 0))) 0 '(1 2 3 4 5))
•   (foldr cons '() '(Huey Dewey Louie))
•   (foldr max -1 '(1953 1956 1949 1991 1964))

(1.3) Assume you have a function (interval a b) that returns a list of all the integers between a and b, inclusive (so that (interval 5 10) would return (5 6 7 8 9 10)). (Re-)write the function factorial using foldr (and interval), without any explicit recursion.

(1.4) Now, think back to the restaurant collection program and assume we have a list (called RL) of the restaurant objects as described there. For each of the following expressions, describe in one English sentence what value it returns. Don't just say, "It does a foldr of plus and zero to a map of ... ;" give a description of what the expression means, something you could put in a software catalog so that a prospective buyer could find what he or she wanted.

•   (foldr + 0 (map (lambda (R) 1) RL))
•   (filter (lambda (R) (equal? 'Ethiopian (rrant-cuisine R))) RL)
•   (/ (foldr + 0 (map (lambda (R) (rrant-price R)) RL))
     (foldr + 0 (map (lambda (R) 1) RL)))
•   (local ((define PRL (filter (lambda (R) (equal? 'pizza (rrant-dish R))) RL)))
     (/ (foldr + 0 (map rrant-price PRL))
        (foldr + 0 (map (lambda (R) 1) PRL))))

(1.5) Using map, filter, and foldr, write an expression to return each of the following values without using explicit recursion:

• a list of all the French and Italian restaurants in RL
• a list of all the (best) dishes served at the French and Italian restaurants in RL; it's okay for this list to have duplicates in it, though you may attempt to remove them if you'd like (as it can be done without explicit recursion)
• a list of all the restaurants in RL whose best dish costs between $10.00 and $20.00 (inclusive)
• the name of the lowest-priced French restaurant in RL
• a list of all the restaurants in RL, where every French restaurant whose best dish's price is less than the average (price of best dishes at French restaurants) has its price changed to that average price

(2) Chapter 25 talks about recursive algorithms that don't follow a conventional template or formula. Chapters 26, 27, and 28 continue this theme. You can certainly skip sections 27.3, 27.4, and 27.5. The discussion of backtracking in Chapter 28 is something you'll see in ICS H23.

(3) Intermezzo 5 (Chapter 29) of the HtDP text talks about O-notation and vectors. For a preview of topics in ICS H22, look at exercises 29.2.1, 29.2.2, and 29.2.3; you're not responsible for those exercises this quarter, though.

Based in part on ICS H21assignments by David G. Kay from Fall 2001; modified by David G. Kay, Fall 2004–Fall 2009.