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Introduction to Computer Science I Topic 6: Generative Recursion

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Introduction to Computer Science ITopic 6: Generative Recursion

Prof. Dr. Max Mühlhäuser

Dr. Guido Rößling

- Introduction to generative recursion
- Sorting with generative recursion
- Guidelines for the design of generative recursive procedures
- Structural versus generative recursion
- Backtracking: Traversing graphs

- So far, we have used structural recursion to process structurally recursive data
- We have decomposed the data to their immediate structural components
- We have processed these components and combined the results

- However:
- Not all problems can be solved using structurally recursive functions
- Even if there is a structurally recursive solution for a problem, it might not be optimal

- Now we will consider a new programming style: Generative recursion

- Divide and Conqueris an important principle underlying generative recursion
- If the problem is trivial to solve, the solution for the trivial case is returned
- Otherwise:
- Divide the problem in new smaller problems (generate new sub-problems)
- Conquer the sub-problems by applying the same technique recursively
- Combine the solutions for the sub-problems into a solution for the original problem

- The design of generative recursive programs is more an ad-hoc activity as compared to the design of the structural recursive programs that needs an insight – a “Eureka!"

Task Description:

- The ball rolls at a constant speed until it drops off the edge of the table
- We can model the table as a canvas/surface with a pre-defined length and width
- The ball can be represented as a disc that moves across the canvas
- The disc movement can be represented by repeating the following steps:
- Draw the disc at the current position on the canvas
- Wait for a certain pre-defined time
- Erase the disc at the current position
- Move it to a new position

;;TeachPack: draw.ss

;; structure: (make-ball numbernumbernumbernumber)

(define-structball (x y delta-x delta-y))

;; draw-and-clear : a-ball -> true

(define (draw-and-clear a-ball)

(and

(draw-solid-disk (make-posn

(ball-x a-ball)

(ball-y a-ball)) 5 'red)

(sleep-for-a-while DELAY)

(clear-solid-disk (make-posn

(ball-x a-ball)

(ball-y a-ball)) 5 'red)))

;; move-ball : ball -> ball

(define (move-ball a-ball)

(make-ball (+ (ball-x a-ball) (ball-delta-x a-ball))

(+ (ball-y a-ball) (ball-delta-y a-ball))

(ball-delta-x a-ball)

(ball-delta-y a-ball)))

;; Dimension ofsurface

(defineWIDTH100)

(defineHEIGHT100)

;; Delay constant

(defineDELAY.1)

To move the ball multiple times we can write:

This gets tedious after a while.

We need a function that moves the ball until it is out of bounds.

(definethe-ball (make-ball10 20 -5 +17))

(and

(draw-and-clear the-ball)

(and

(draw-and-clear (move-ball the-ball))

...))

Determine whether a-ball is outside of the bounds:

Template for moving the ball until it is out of bounds:

;; out-of-bounds? : a-ball -> boolean

(define (out-of-bounds? a-ball)

(not

(and

(<=0 (ball-x a-ball) WIDTH)

(<=0 (ball-y a-ball) HEIGHT))))

The trivial case: returntrue

;; move-until-out : a-ball -> true

(define (move-until-out a-ball)

(cond

[(out-of-bounds? a-ball) ... ]

[else...]))

true

?

After drawing and moving the ball, we apply move-until-outagain, which means the function is recursive:

(define (move-until-out a-ball)

(cond

[(out-of-bounds? a-ball) true]

[else (and (draw-and-clear a-ball)

(move-until-out

(move-ball a-ball)))]))

We can now test the function as follows: The code creates a canvas of proper size and a ball that moves to the bottom left of the canvas.

(startWIDTHHEIGHT)

(move-until-out (make-ball1020-5+17))

(stop)

- The procedure move-until-out uses a new type of recursion:
- Conditions have nothing to do with the input data
- Recursive application in the body does not use part of the input
- move-until-outgenerates an entirely new and different ball structure and uses it for the recursion

We do not have a design recipe for this

(define (move-until-out a-ball)

(cond

[(out-of-bounds? a-ball) true]

[else (and (draw-and-clear a-ball)

(move-until-out

(move-ball a-ball)))]))

- Introduction to generative recursion
- Sorting with generative recursion
- Guidelines for the design of generative recursive procedures
- Structural versus generative recursion
- Backtracking: Traversing graphs

- We are once more concerned with sorting the elements of a list …
- We have seen insertion sort:
- A structurally recursive procedure

- Now we will see two other algorithms for sorting: Quicksort and Mergesort
- Classic examples of generative recursion
- Based on the idea of “divide and conquer”

- We have seen insertion sort:

;; insertion-sort: list-of-numbers -> list-of-numbers

;; creates a sortedlistofnumb. fromnumbers in alon

(define (insertion-sortalon)

(cond

[(empty? alon) empty]

[else (insert (firstalon)

(insertion-sort (restalon)))]))

unsorted

sorted

an

an

- An arbitrary intermediate step: sorting an arbitrary sub-list:L0=(list elp…elr)
- Divide: PartitionL0 in two (possibly empty) lists, L1=(list elp…elq-1) and L2=(list elq+1…elr), such that
- each element of L1 is smaller than elq, and
- the latter is in turn smaller than each element in L2

- Conquer: Apply the same procedure recursively to sort L1 and L2
- Combine: simply concatenate the sorted lists L1 and L2

- Divide: PartitionL0 in two (possibly empty) lists, L1=(list elp…elq-1) and L2=(list elq+1…elr), such that

Turningpoint

elq

<= elq

> elq

- Two open questions so far:
- How do we select the pivot element?
- We always use the first element

- When do we stop? That is: what is the trivial case of Quicksort?
- The empty list is always sorted!

- How do we select the pivot element?

- Select thefirstelementfromthelistaspivot item
- Determinethefirst sub-listwithelements <= pivot
- Sortthis sub-listrecursivelyusingQuicksort
- Determinethesecond sub-listwithelements > pivot
- Sortthis sub-listrecursivelyusingQuicksort
- Concatenatethesorted sub-liststo a newlist

Sort(list 11 8 7 14):

Select thefirstelementfrom '(11 8 7 14) aspivot item:11

Determinethefirst sub-list <= 11: '(8 7)

Sortthis sub-list

Select thefirstelementfrom '(8 7) aspivot item: 8

Determinethefirst sub-listwithelements <= 8: '(7)

Sortthis sub-list

Select thefirstelementfrom '(7) aspivot item: 7

Determinethefirst sub-listwithelements <= 7: empty

Sortthis sub-list Resultempty

Determinethesecond sub-listwithelements > 7: empty

Sortthis sub-list Resultempty

Concatenate (empty 7 empty) to a newlist -> (list 7)

Determinethesecond sub-listwithelements > 8: empty

Sortthis sub-list Resultempty

Concatenate((list 7) 8 empty) to a newlist (list 7 8)

Determine…

11

(list 14)

(list 8 7)

empty

14

empty

8

empty

(list 7)

empty

7

empty

(list 14)

(list 7)

(list 7 8)

(list 7 8 11 14)

pivot item

quick-sort distinguishes two cases:

If the input is empty, it returns empty.

Otherwise, it performs recursion.

Each sub-list is sorted separately using quick-sort

The sorted versions of the two lists are then combined using append

;; quicksort2: (listofnumber) -> (listofnumber)

(define (quicksort2 alon)

(cond

[(empty? alon) empty]

[else

(append

(quicksort2 (less-or-equal (restalon)

(firstalon)))

(list (firstalon))

(quicksort2 (greater-than (restalon)

(firstalon))))

]))

- greater-than filters out the items that are larger than threshold:
- less-than filters out the items that are smaller than threshold:

(define (greater-than alonthreshold)

(filter1 > alon threshold)))

(define (less-than alonthreshold)

(filter1 < alon threshold)))

(quick-sort (list11 8 14 7))

= (append (quick-sort (list8 7))

(list11)

(quick-sort (list14)))

= (append (append (quick-sort (list7))

(list8)

(quick-sortempty))

(list 11)

(quick-sort (list 14)))

= (append (append (append (quick-sortempty)

(list7)

(quick-sortempty))

(list 8)

(quick-sortempty))

(list 11)

(quick-sort (list 14)))

= ...

= (append (append (appendempty

(list7)

empty)

(list 8)

empty)

(list 11)

(quick-sort (list 14)))

= (append (append (list7)

(list8)

empty)

(list 11)

(quick-sort (list 14)))

= (append (list78)

(list11)

(quick-sort (list14)))

= ...

Idea:

split the list in the middle

apply the function to the sub-lists recursively

merge the two sorted lists into a new sorted list

- Given two ordered lists ls1 and ls2
- How do we merge them into a new ordered list?

ls-1

ls-2

1

2

4

7

3

5

6

8

Compareandcopythesmallerelement, thencontinue

sorted-list

1

2

3

4

5

6

7

8

(define (merge ls1 ls2)

(cond

[(empty? ls1) ls2]

[(empty? ls2) ls1]

[(<= (first ls1) (first ls2))

(cons (first ls1)

(merge (rest ls1) ls2))

]

[else (cons (first ls2)

(merge ls1 (rest ls2)))]

)

)

(define (merge-sortalon)

(local

((define (merge-stepleftright)

(cond

[(>= leftright) alon]

[else

(local(

(definemid (floor (/ (+ leftright) 2)))

(defineleft-list

(merge-sort (extractalonleftmid)))

(defineright-list

(merge-sort

(extractalon (+ mid 1) right))))

(mergeleft-listright-list)

)

]

)))

(merge-step 1 (lengthalon))))

(define (extract alonleft right)

(cond

[(empty? alon) empty]

[(> left right) empty]

[(> left 1)

(extract

(rest alon)

(- left 1)

(- right 1))]

[else (cons

(first alon)

(extract

alon

(+ left 1)

right))]))

- Introduction to generative recursion
- Sorting with generative recursion
- Guidelines for the design of generative recursive procedures
- Structural versus generative recursion
- Backtracking: Traversing graphs

- Understand the nature of the data of the procedure
- Describe the process in terms of data, creating a new structure or partitioning a list of numbers
- Distinguish between those input data
- which can be processed trivially,
- and those which cannot

- The generation of problems is the key to algorithm design
- The solutions of the generated problems must be combined

- Data analysis and design:
- Analyze and define data collections representing the problem

- Contract, purpose, header
- Specify what the function does
- Explain in general terms how it works

- Function examples:
- Illustrate how the algorithm proceeds for some given input

- Template:
- Follow a general template

- Definition:
- Answer the questions posed by the template

- Test
- Test the completed function
- Eliminate the bugs

(define (generative-recursive-funproblem)

(cond

[(trivially-solvable? problem)

(determine-solutionproblem)]

[else

(combine-solutions

... problem ...

(generative-recursive-fun

(generate-problem-1 problem))

...

(generative-recursive-fun

(generate-problem-n problem)))]))

- What is a trivially solvable problem, and the pertaining solution?
- How do we generate new problems that are easier to solve than the initial problem?
- Is there one new problem to generate, or are there more?

- Is the solution for the given problem the same as for (one of) the new problems?
- Or do we have to combine solutions to create a solution for the initial problem?
- If so, do we require parts of the data constituting the initial problem?

- So far, each function has produced an output for a valid input
Evaluation of structurally recursive procedures has always terminated.

- Important characteristic of the recipe for structurally recursive procedures:
- Each step of natural recursion uses an immediate component of the input, not the input itself

- Because data is constructed in a hierarchical manner, the input shrinks at every stage
- The function sooner or later consumes an atomic piece of data and terminates.

- This characteristic is not true for generative recursive functions.
- The internal recursions do not consume an immediate component of the input, but some new piece of data, which is generated from the input.

- This step may produce the initial input over and over again and thus prevent the evaluation from ever producing a result
- We say that the program is trapped in an infinite loop.

- What happens if we place the following three expressions at the bottom of the DrScheme Definitions window and click execute?
- Does the second expression ever produce a value so that the third expression is evaluated and the canvas disappears?

(startWIDTHHEIGHT)

(move-until-out (make-ball102000))

(stop)

The slightest mistake in process definition may cause an infinite loop:

;; less-or-equal: (list-of-numbers) number-> (list-of-numbers)

(define(less-or-equalalonthreshold)

(cond

[(empty?alon) empty]

[else (if (<= (firstalon) threshold)

(cons (firstalon)

(less-or-equalalonthreshold))

(less-or-equal(restalon) threshold))]))

Insteadof(restalon)

Quicksortdoes not terminatewiththenewfunction

(quick-sort (list 5))

= (append (quicksort (less-or-equal5 (list 5)))

(list 5)

(quicksort (greater-than5 (list 5))))

= (append (quicksort(list 5))

(list 5)

(quicksort (greater-than5 (list 5))))

- The termination argument is an additional step in the design recipe of generative recursive procedures
- Termination argument explains
- why the process produces an output for every input
- how the function implements this idea
- when the process possibly does not terminate

At each step, quick-sortpartitions the list into two sublists using less-or-equaland greater-than.

Each function produces a list that is smaller than the input (the second argument), even if the pivot element (the first argument) is an item on the list.

Hence, each recursive application of quick-sort consumes a strictly shorter listthan the given one.

Eventually, quick-sortreceives an empty list and returns empty.

Termination argument may reveal additional termination cases.

This knowledge can be added to the algorithm:

So liefert (less-or-equal N (list N))und(greater-than N (list N)) immer empty

;; quick-sort : (listofnumber) -> (listofnumber)

(define (quick-sortalon)

(cond

[(empty? alon) empty]

[(empty? (restalon)) alon]

[else

(append

(quick-sort(less-or-equal (restalon)

(firstalon)))

(list (firstalon))

(quick-sort(greater-thanalon(firstalon))))

]))

- Introduction to generative recursion
- Sorting with generative recursion
- Guidelines for the design of generative recursive procedures
- Structural versus generative recursion
- Backtracking: Traversing graphs

trivially-solvable?

empty?

generate-problem

rest

Template forgenerative recursion

(define (generative-recursive-funproblem)

(cond

[(trivially-solvable? problem)

(determine-solutionproblem)]

[else

(combine-solutions

problem

(generative-recursive-fun

(generate-problemproblem)))]))

Template forlistprocessing

(define (generative-recursive-funproblem)

(cond

[(empty?problem) (determine-solutionproblem)]

[else

(combine-solutions

problem

(generative-recursive-fun (restproblem)))]))

- Is there a difference between structural and generative recursion?
- Structurally recursive functions seem to be just special cases of generative recursion
- But this "it's all the same" attitude does not improve the understanding of the design process
- Structurally and generative recursive functions are designed using different approaches and have different consequences.

- Examples
- 6 and 25 are both numbers with several denominators:
- 6 is evenly divisible by 1, 2, 3, and 6;
- 25 is evenly divisible by 1, 5, and 25.
- The greatest common denominator of 25 and 6 is 1.

- 18 and 24 have many common denominators :
- 18 is evenly divisible by 1, 2, 3, 6, 9, and 18;
- 24 is evenly divisible by 1, 2, 3, 4, 6, 8, 12, and 24.
- The greatest common denominator is 6.

- 6 and 25 are both numbers with several denominators:

Test for every number i = [min(n,m) .. 1] whether it divides both n and m evenly and return the first such number.

;; gcd-structural : N[>= 1] N[>= 1] -> N

;; structuralrecursionusingdatadefinitionof N[>= 1]

(define (gcd-structural n m)

(local

((define (first-divisor i)

(cond

[(= i 1) 1]

[(and(= (remainder n i) 0)

(= (remainder m i) 0))

i]

[else (first-divisor (- i 1))]

)

)

)

(first-divisor (min m n))))

Inefficientfor large numbers!

- gcd-structural simply tests every number whether it divides both n and m evenly and returns the first such number
- For small natural numbers, this process works just fine

- However, for (gcd-structural 101135853 450146405) = 177 the procedure will compare 101135853 – 177 = 101135676 numbers!
- Even reasonably fast computers spend several minutes on this task.

- Enter the definition of gcd-structural into the Definitions window and evaluate the following expression… in the Interactions window … and go get a coffee
(time (gcd-structural 101135853 450146405))

- The Euclidean algorithm to determine the greatest common denominator (GCD) of two integers is one of the oldest algorithms known
- Appeared inEuclid’s Elements around 300 BC
- However, the algorithm probably was not discovered by Euclid and it may have been known up to 200 years earlier.

Insight: For two natural numbersnandm, n > m, GCD(n, m) = GCD(m, remainder(n,m))

(gcd larger smaller)

= (gcdsmaller (remainder larger smaller))

Example: GCD(18, 24) = GCD(18,remainder(24/18)) = GCD(18,6) = GCD(6,0) = 6

;; gcd-generative : N[>= 1] N[>=1] -> N

(define (gcd-generative n m)

(local

((define (clever-gcd larger smaller)

(cond

[(= smaller 0) larger]

[else (clever-gcd

smaller

(remainder larger smaller))]))

)

(clever-gcd (max m n) (min m n))))

clever-gcdis based on generative recursion:

- The trivially solvable case is when smaller is 0
- The generative step calls clever-gcd with smaller and (remainder larger smaller)

(gcd-generative101135853450146405)needsonly9 iterations!

Let n = mq + r, then any number u which dividesbothn and m (n = su and m = tu), also dividesr

r = n – qm = su – qtu = (s – qt)u

Any number v which divides both m and r(m = s’v and r = t’v ), also divides n

n = qm + r = qs´v + t´v = (s´q + t´)v

- Therefore, every common denominator of n and m is also a common denominator of m and r.
- gcd(n,m) = gcd(m,r)
- It is enough if we continue the process with m and r
- Since r is smaller in absolute value than m, we will reach r = 0 after a finite number of steps.

- Question: can we conclude that generative recursion is better than structural recursion?
- Answer: no, not automatically.
- Even a well-designed generative procedure is not always faster than structural recursion.
- For example, quick-sortwins over insertion sort only for large lists

- Structural recursion is easier to design
- Designing generative recursive procedures often requires deep mathematical insight

- Structural recursion is easier to understand
- It may be difficult to grasp the idea of the generative step.

- Even a well-designed generative procedure is not always faster than structural recursion.

Start by using structural recursion.

If it is too slow, try to design generative recursion.

Document the problem generation with good examples, give a good termination argument.

- Introduction to generative recursion
- Sorting with generative recursion
- Guidelines for the design of generative recursive procedures
- Structural versus generative recursion
- Backtracking: Traversing graphs

B

C

D

A

E

F

G

- A graph is a collection of nodes and edges.
- The edges represent one-way connections between the nodes.
- Canbe used to describe
- a plan of one-way streets in a city,
- relationships between persons,
- connections on the Internet, etc.

Scheme - list representation

(defineGraph

'((A (BE))

(B (EF))

(C (D))

(D ())

(E (C))

(F (DG))

(G ())))

D

B

C

A

E

F

G

;; find-route : node node graph -> (listof node)

;; to create a path from origin to destination in G

;; false, if there is no path,

(define (find-route origin destination G) ...)

(find-route 'C 'DGraph) = (list 'C 'D)

(find-route 'E 'DGraph) = (list 'E 'C 'D)

(find-route 'C 'GGraph) = false

A backtracking algorithm follows a specific template:

- Pursue a (possible) path to a solution until
- the solution is found (success! terminate), or
- the path cannot be continued.

- If the path cannot be continued:
- Walk back along the path unto the last branch where alternatives exist that have not yet been chosen,
- Choose such an alternative and proceed with step 1.

- If the starting point is reached again and there are no more alternatives: failure! terminate
Usage examples:

- "N Queens Problem", finding paths in graphs

C

D

B

A

E

F

G

BACKTRACK !

- Find the path from node A to G!

- If the origin is equal to the destination, the problem is trivial; the answer is (list destination).
- Otherwise, try to find route from all neighbors of the origin.

(define (find-route originationdestinationaGraph)

(cond

[(symbol=? originationdestination)

(listdestination)]

[else ...

(find-route/list

(neighborsoriginationaGraph)

destination

aGraph)

...]))

neighbors is similar to the function contains-doll?

;; neighbors : node graph -> (listof node)

;; to lookup the neighbors of node in graph

(define (neighbors node graph)

(cond

[(empty? graph)

(error 'neighbors "empty graph")]

[(symbol=? (first (first graph)) node)

(second (first graph))]

[else (neighbors node (rest graph))]))

- find-route/list
- Processes a list of nodes
- Determine, for each of them, whether a path to the destination node in this graph exists

;; find-route/list :

;; (listof node) node graph -> (listof node) or false

(define (find-route/list lo-origins destination G)

...)

- The result of find-routedepends on that of find-route/list, which can be one of these:
- a path from a neighbor node to the destination
- false, if no path from one of the neighbors could be found

(define (find-route originationdestinationaGraph)

(cond

[(symbol=? originationdestination)

(listdestination)]

[else

(local

((definepossible-route

(find-route/list

(neighborsoriginationaGraph)

destination

aGraph)))

(cond

[(boolean? possible-route) ...]

[else (cons? possible-route) ...]))]))

(define (find-route origin destination aGraph)

(cond

[(symbol=? origin destination)

(list destination)]

[else

(local ((define possible-route

(find-route/list

(neighbors origin aGraph)

destination

aGraph)))

(cond

[(boolean? possible-route) false]

[else (cons origin possible-route)]))]))

C

D

B

A

E

F

G

- Find the path from node A to G!

Neighbors of A

B

C

D

B

C

D

(define (find-route/list lo-Os destaG)

(cond

[(empty? lo-Os) false]

[else

(local

((define possible-route

(find-route (first lo-Os) destaG)))

(cond

[(boolean? possible-route)

(find-route/list (rest lo-Os) destaG)]

[else possible-route])

)

])

)

A

A

E

F

G

E

F

G

B

C

D

B, E, C is a cycle

A

E

F

G

The function fails to terminate in a graph with a cycle:

(find-route 'B 'D Cyclic-graph)

= ... (find-route/list (list'E 'F) 'D Cyclic-graph) ...

= ... (find-route 'E 'D Cyclic-graph) ...

= ... (find-route/list (list'C 'F) 'D Cyclic-graph) ...

= ... (find-route 'C 'D Cyclic-graph) ...

= ... (find-route/list (list'B 'D) 'D Cyclic-graph) ...

= ... (find-route 'B 'D Cyclic-graph) ...

= ...

- There are problems that cannot be solved, or can only be solved sub-optimally, using structural recursion
- Generative recursion is based on the principle: “divide-and-conquer”
- The design recipe for generative recursive functions has to be adapted:
- In particular, we need an argument for the termination

- Structurally recursive functions are a subset of generative recursive functions
- When both strategies are possible the choice depends on a case-by-case basis
- one cannot say that one class is better than the other