### Problem Set 10h

This is the honors version of Problem Set 10.  Programming Language ISL+

Purpose This problem set concerns trees, graphs, and the design of functions with accumulators. Orderly Legos

Let’s build a different kind of lego building where all legos in the building must have distinct labels and brick labels appear in a certain order in the building.

Here is the data definition for our ordered lego buildings:
 (define-struct lego (label color width)) ; A Lego is a structure: ;    (make-lego Number Symbol Number) ; interpretation: (make-lego l c w) is the lego brick ; with label l, color c, and width w (in pixels). ; (define-struct bigger (lego left right)) ; An OrdLegoBldg (ordered lego building) is one of: ; - 'none ; - Lego ; - (make-bigger Lego OrdLegoBldg OrdLegoBldg) ; where each lego brick in the building has a unique label and ; where each (make-bigger l lft rgt) has the property that ;  – the label of l is bigger than all the labels in lft ;  – the label of l is smaller than all the labels in rgt

All functions that you design below must exploit/maintain the distinct-label and label-ordering properties of OrdLegoBldgs.

Problem 1 Design a function, ord-lb-contains?, that takes an ordered lego building and a label and determines whether the building contains a lego with that label.

Problem 2 Design a function, colors-in-order, that takes an ordered lego building and produces a list of colors. The output list must be ordered so that it starts with the color of the lego with the smallest label in the building and ends with the color of the lego with the highest label.

You may use append to concatenate lists of colors. Your function should not traverse/examine parts of the building more than once.

Problem 3 Design a function, grow-ord-lb, that takes an ordered lego building and a lego brick and adds the brick to the building. If the new brick’s label is the same as the label of a brick already in the building, return an error: Label already in building. The new building must be an OrdLegoBldg that is, it must satisfy the distinct-label and label-ordering properties of OrdLegoBldg.

Problem 4 Design a function, build-ord-lb, that takes a list of legos and produces an ordered lego building built using these legos. The building must be an OrdLegoBldg (i.e., it must satisfy the distinct-label and label-ordering properties of OrdLegoBldg). If the input list contains multiple legos with the same label, report an error. Graphs

For this set of problems, use the following data definition for graphs.

 (define-struct graph (nodes neighbors node=?)) ; A [Graph X] is a structure: ; (make-graph [Listof X] [X -> [Listof X]] [X X -> Boolean])

Problem 5 Design the function find-paths.

 ; find-paths : [Graph X] X X -> [Listof [Listof X]] ; Find all of the paths in the graph from src to dest (define (find-paths g src dest) ...)

Note that input graphs may contain cycles. Make sure that your code can handle cycles in the graph and doesn’t loop forever. Below are some tests to clarify what we expect.

 (define G1 (make-graph '(A B C D E F G) (lambda (n) (cond [(symbol=? n 'A) '(B E)] [(symbol=? n 'B) '(E F)] [(symbol=? n 'C) '(D)] [(symbol=? n 'D) '()] [(symbol=? n 'E) '(C F A)] [(symbol=? n 'F) '(D G)] [(symbol=? n 'G) '()])) symbol=?)) (check-expect (find-paths G1 'C 'C) (list (list 'C))) ; src = dest (check-expect (find-paths G1 'C 'G) empty) ; no paths from 'C to 'G (check-expect (find-paths G1 'A 'B) (list (list 'A 'B))) (check-expect (find-paths G1 'E 'G) (list (list 'E 'F 'G) (list 'E 'A 'B 'F 'G))) (check-expect (find-paths G1 'B 'G) (list (list 'B 'E 'F 'G) (list 'B 'F 'G))) (check-expect (find-paths G1 'A 'G) (list (list 'A 'B 'E 'F 'G) (list 'A 'B 'F 'G) (list 'A 'E 'F 'G)))

Problem 6 Design the function same-graph?

 ; same-graph? : [Graph X] [Graph X] -> Boolean ; Determine whether g1 and g2 have the same nodes, ; and each node in g1 has the same neighbors as that node in g2. ; Assume that both graphs have the same node equality function ; and that this node equality function is symmetric. (define (same-graph? g1 g2) ...) ; Examples (same-graph? (make-graph '() (lambda (x) '()) symbol=?) (make-graph '() (lambda (x) '()) symbol=?)) --> true (same-graph? (make-graph '(a) (lambda (x) '()) symbol=?) (make-graph '() (lambda (x) '()) symbol=?)) --> false (same-graph? (make-graph '(a) (lambda (x) '()) symbol=?) (make-graph '(a) (lambda (x) '()) symbol=?)) --> true (same-graph? (make-graph '(b) (lambda (x) '()) symbol=?) (make-graph '(a) (lambda (x) '()) symbol=?)) --> false (same-graph? (make-graph '(a b) (lambda (x) '()) symbol=?) (make-graph '(b a) (lambda (x) '()) symbol=?)) --> true (same-graph? (make-graph '(a b) (lambda (x) (cond [(symbol=? x 'a) '(b)] [(symbol=? x 'b) '()])) symbol=?) (make-graph '(a b) (lambda (x) (cond [(symbol=? x 'b) '(a)] [(symbol=? x 'a) '()])) symbol=?)) --> false (same-graph? (make-graph '(a b) (lambda (x) (cond [(symbol=? x 'a) '(a b)] [(symbol=? x 'b) '()])) symbol=?) (make-graph '(a b) (lambda (x) (cond [(symbol=? x 'a) '(b a)] [(symbol=? x 'b) '()])) symbol=?)) --> true

Problem 7 Design the following function and use same-graph? to test it.

 ; reverse-edge-graph : [Graph X] -> [Graph X] ; Build a graph with the same nodes as g, but with reversed edges. ; That is, if g has an edge from a to b then the result graph will ; have an edge from b to a. (define (reverse-edge-graph g) ...)

Problem 8 Design the following function.
 ; undirected? : [Graph X] -> Boolean ; Determine if each edge in g has a matching edge going the opposite direction (define (undirected? g) ...)