When I have a number in my list that is greater than 9 I want to separate the
digits and add them to the running sum.
The code I have is giving me and error in my sum-list definition.
(define sum-list (lst)
(if (null lst)
0
(if (>9 car lst?)
(cons ((mod (car lst) 10)) + (* (remainder (/car lst 10) 10))))
(if (>9 cdr lst?)
(cons ((mod (cdr lst)10)) + (* (remainder (/cdr lst 10) 10))))
(+ (car lst) (sum-list (cdr lst)))))
I am getting an error"Expected only one expression after the name sum-list but found one extra part.
I wrote this now in mit-scheme. I split the problem in 2 subproblems -- to conver the number to the list of digits and then to sum the digits in the resulting list.
(define n->l
(lambda (n return)
((lambda (s) (s s n return))
(lambda (s n col)
(if (zero? n)
(col '())
(s s
(quotient n 10)
(lambda (rest)
(col (cons (remainder n 10) rest)))))))))
(define sum-digits
(lambda (n)
(n->l n (lambda (l) (fold-left + 0 l)))))
(sum-digits 100)
(sum-digits 123)
I am trying to build a 6-tuple store on top of wiredtiger. The tuples can be described as follow:
(graph, subject, predicate, object, alive, transaction)
Every tuple stored in the database is unique.
Queries are like regular SPARQL queries except that the database store 6 tuples.
Zero of more elements of a tuple can be variable. Here is an example query that allows to retrieve all changes introduces by a particular transaction P4X432:
SELECT ?graph ?subject ?predicate ?object ?alive
WHERE
{
?graph ?subject ?predicate ?object ?alive "P4X432"
}
Considering all possible patterns ends up with considering all combinations of:
(graph, subject, predicate, object, alive, transaction)
That is given by the following function:
def combinations(tab):
out = []
for i in range(1, len(tab) + 1):
out.extend(x for x in itertools.combinations(tab, i))
assert len(out) == 2**len(tab) - 1
return out
Where:
print(len(combinations(('graph', 'subject', 'predicate', 'object', 'alive', 'transaction'))))
Display:
63
That is there 63 combinations of the 6-tuples. I can complete each indices with the missing tuple item, e.g. the following combination:
('graph', 'predicate', 'transaction')
Will be associated with the following index:
('graph', 'predicate', 'transaction', 'subject', 'alive', 'object')
But I know there is a smaller subset of all permutations of the 6-tuple that has the following property:
A set of n-permutations of {1, 2, ..., n} where all combinations of {1, 2, ..., n} are prefix-permutation of at least one element of the set.
Otherwise said, all combinations have a permutation that is prefix of one element of the set.
I found using a brute force algorithm a set of size 25 (inferior to 63) that has that property:
((5 0 1 2 3 4) (4 5 0 1 2 3) (3 4 5 0 1 2) (2 3 4 5 0 1) (1 2 3 4 5 0) (0 1 2 3 4 5) (0 2 1 3 4 5) (0 3 2 1 5 4) (0 4 3 1 5 2) (0 4 2 3 1 5) (2 1 5 3 0 4) (3 2 1 5 0 4) (3 1 4 5 0 2) (3 1 5 4 2 0) (3 0 1 4 2 5) (3 5 2 0 1 4) (4 3 1 0 2 5) (4 2 1 5 3 0) (4 1 0 2 5 3) (4 5 2 1 0 3) (5 4 1 2 3 0) (5 3 0 1 4 2) (5 2 1 3 4 0) (5 1 2 4 0 3) (5 0 2 4 3 1))
Here is the r7rs scheme program I use to compute that solution:
(define-library (indices)
(export indices)
(export permutations)
(export combination)
(export combinations)
(export run)
(import (only (chezscheme) trace-define trace-lambda random trace-let))
(import (scheme base))
(import (scheme list))
(import (scheme comparator))
(import (scheme hash-table))
(import (scheme process-context))
(import (scheme write))
(begin
(define (combination k lst)
(cond
((= k 0) '(()))
((null? lst) '())
(else
(let ((head (car lst))
(tail (cdr lst)))
(append (map (lambda (y) (cons head y)) (combination (- k 1) tail))
(combination k tail))))))
(define (factorial n)
(let loop ((n n)
(out 1))
(if (= n 0)
out
(loop (- n 1) (* n out)))))
(define (%binomial-coefficient n k)
;; https://en.wikipedia.org/wiki/Binomial_coefficient#Multiplicative_formula
(let loop ((i 1)
(out 1))
(if (= i (+ k 1))
out
(loop (+ i 1) (* out (/ (- (+ n 1) i) i))))))
(define (memo proc)
(let ((m (make-hash-table (make-equal-comparator))))
(lambda args
(if (hash-table-contains? m args)
(hash-table-ref m args)
(let ((v (apply proc args)))
(hash-table-set! m args v)
v)))))
(define binomial-coefficient
(memo
(lambda (n k)
(cond
((= n k) 1)
((= k 0) 1)
(else (%binomial-coefficient n k))))))
;; k-combination ranking and unranking procedures according to
;; https://en.wikipedia.org/wiki/Combinatorial_number_system
(define (ranking lst)
(let loop ((lst (sort < lst)) ;; increasing sequence
(k 1)
(out 0))
(if (null? lst)
out
(loop (cdr lst) (+ k 1) (+ out (binomial-coefficient (car lst) k))))))
(define (%unranking k N)
(let loop ((n (- k 1)))
(if (< N (binomial-coefficient (+ n 1) k))
n
(loop (+ n 1)))))
(define (unranking k N)
(let loop ((k k)
(N N)
(out '()))
(if (= k 0)
out
(let ((m (%unranking k N)))
(loop (- k 1) (- N (binomial-coefficient m k)) (cons m out))))))
(define fresh-random
(let ((memo (make-hash-table (make-eqv-comparator))))
(lambda (n)
(when (= (hash-table-size memo) n)
(error 'oops "no more fresh number" n
))
(let loop ()
(let ((r (random n)))
(if (hash-table-contains? memo r)
(loop)
(begin (hash-table-set! memo r #t) r)))))))
(define (random-k-combination k n)
(unranking k (fresh-random (binomial-coefficient n k))))
(define (combinations lst)
(if (null? lst) '(())
(let* ((head (car lst))
(tail (cdr lst))
(s (combinations tail))
(v (map (lambda (x) (cons head x)) s)))
(append s v))))
;; (define (combinations lst)
;; (append-map (lambda (k) (combination k lst)) (iota (length lst))))
(define (permutations s)
;; http://rosettacode.org/wiki/Permutations#Scheme
(cond
((null? s) '(()))
((null? (cdr s)) (list s))
(else ;; extract each item in list in turn and permutations the rest
(let splice ((l '()) (m (car s)) (r (cdr s)))
(append
(map (lambda (x) (cons m x)) (permutations (append l r)))
(if (null? r) '()
(splice (cons m l) (car r) (cdr r))))))))
(define (shift lst index)
(append (drop lst index) (take lst index)))
(define (rotations lst)
(reverse! (map (lambda (index) (shift lst index)) (iota (length lst)))))
(define (prefix? lst other)
"Return #t if LST is prefix of OTHER"
(let prefix ((lst lst)
(other other))
(if (null? lst)
#t
(if (= (car lst) (car other))
(prefix (cdr lst) (cdr other))
#f))))
(define (indices lst)
(let ((candidates (permutations lst)))
(let loop ((out (rotations lst)) ;; all rotations are solutions
(combinations (combinations lst)))
(if (null? combinations)
(reverse! out)
(let ((permutations (permutations (car combinations))))
(if (any (lambda (s) (any (lambda (p) (prefix? p s)) permutations)) out)
;; there is an existing "solution" for the
;; permutations of COMBINATION move to the next
;; combination
(loop out (cdr combinations))
(loop (cons (find (lambda (c) (if (member c out)
#f
(any (lambda (p) (prefix? p c)) permutations)))
candidates)
out)
(cdr combinations))))))))
(define (permutation-prefix? c o)
(any (lambda (p) (prefix? p o)) (permutations c)))
(define (ok? combinations candidate)
(every (lambda (c) (any (lambda (p) (permutation-prefix? c p)) candidate)) combinations))
(define (run)
(let* ((n (string->number (cadr (command-line))))
(N (iota n))
(solution (indices N))
(min (length solution))
(rotations (rotations N))
(R (length rotations))
;; other stuff
(cx (combinations N))
(px (filter (lambda (x) (not (member x rotations))) (permutations N)))
;; other other stuff
(pn (length px))
(PN (iota pn)))
(display "(length solution) => ") (display (length solution))
(display "\n")
(display "(length rotations) => ") (display R)
(display "\n")
(let try ((x (- (length solution) 1)))
(let ((count (binomial-coefficient pn (- x R))))
(let loop ((index 0)
(cxx (map (lambda (x) (list-ref px x)) (random-k-combination (- x R) pn))))
(when (= (modulo index (expt 10 5)) 0)
(display "n=") (display n) (display " x=") (display x)
(display " ")
(display index) (display "/") (display count) (display "\n"))
(let ((candidate (append rotations cxx)))
(let ((continue? (not (ok? cx candidate))))
(if continue?
(loop (+ index 1)
(map (lambda (x) (list-ref px x)) (random-k-combination (- x R) pn)))
(begin (display "new solution n=") (display n)
(display " length=") (display x)
(display " ") (display candidate)
(display "\n")
(try (- x 1)))))))))))
))
With that list of permutations I can query any pattern.
I am wondering if there is a smaller set and whether there is definitive algorithm to compute that kind of set.
Based on this answer https://math.stackexchange.com/a/3146793/23663
The following program yields a solution that is a minimal solution according to math ™:
import itertools
import math
f = math.factorial
bc = lambda n, k: f(n) // f(k) // f(n-k) if k<n else 0
def pk(*args):
print(*args)
return args[-1]
def stringify(iterable):
return ''.join(str(x) for x in iterable)
def combinations(tab):
out = []
for i in range(1, len(tab) + 1):
out.extend(stringify(x) for x in itertools.combinations(tab, i))
assert len(out) == 2**len(tab) - 1
return out
def ok(solutions, tab):
cx = combinations(tab)
px = [stringify(x) for x in itertools.permutations(tab)]
for combination in cx:
pcx = [''.join(x) for x in itertools.permutations(combination)]
# check for existing solution
for solution in solutions:
if any(solution.startswith(p) for p in pcx):
# yeah, there is an existing solution
break
else:
print('failed with combination={}'.format(combination))
break
else:
return True
return False
def run(n):
tab = list(range(n))
cx = list(itertools.combinations(tab, n//2))
for c in cx:
L = [(i, i in c) for i in tab]
A = []
B = []
while True:
for i in range(len(L) - 1):
if (not L[i][1]) and L[i + 1][1]:
A.append(L[i + 1][0])
B.append(L[i][0])
L.remove((L[i + 1][0], True))
L.remove((L[i][0], False))
break
else:
break
l = [i for (i, _) in L]
yield A + l + B
for i in range(7):
tab = stringify(range(i))
solutions = [stringify(x) for x in run(i)]
assert ok(solutions, tab)
print("n={}, size={}, solutions={}".format(i, len(solutions), solutions))
The above program output is:
n=0, size=1, solutions=['']
n=1, size=1, solutions=['0']
n=2, size=2, solutions=['01', '10']
n=3, size=3, solutions=['012', '120', '201']
n=4, size=6, solutions=['0123', '2031', '3012', '1230', '1302', '2310']
n=5, size=10, solutions=['01234', '20341', '30142', '40123', '12340', '13402', '14203', '23410', '24013', '34021']
n=6, size=20, solutions=['012345', '301452', '401253', '501234', '203451', '240513', '250314', '340521', '350124', '450132', '123450', '142503', '152304', '134502', '135024', '145032', '234510', '235104', '245130', '345210']
I am following the cs61a spring 2015 class.
One of the problem in the scheme project is:
Implement the list-partitions procedure, which lists all of the ways to
partition a positive integer total without using consecutive integers. The
contents of each partition must be listed in decreasing order.
Hint: Define a helper procedure to construct partitions. The built-in append
procedure creates a list containing all the elements of two argument lists.
The cons-all procedure in questions.scm adds a first element to each list in a list of lists.
The number 5 has 4 partitions that do not contain consecutive integers:
5
4, 1
3, 1, 1
1, 1, 1, 1, 1
The following partitions of 5 are not included because of consecutive
integers:
3, 2
2, 2, 1
2, 1, 1, 1
I found one solution but cannot understand it
;; List all ways to partition TOTAL without using consecutive numbers.
(define (apply-to-all proc items)
(if (null? items)
'()
(cons (proc (car items))
(apply-to-all proc (cdr items)))))
(define (cons-all first rests)
(apply-to-all (lambda (rest) (cons first rest)) rests))
(define (caar x) (car (car x)))
(define (cadr x) (car (cdr x)))
(define (cddr x) (cdr (cdr x)))
(define (cadar x) (car (cdr (car x))))
(define (cdar x) (cdr (car x)))
(define (partitions-r a b)
(if (= a 0) nil
(append (cons-all a (list-partitions b))
(cons-f (partitions-r (- a 1) (+ b 1))
))
))
(define (cons-f lst)
(cond
((eq? lst nil) nil)
((eq? (cdar lst) nil) lst)
((< (caar lst) (cadar lst)) (cons-f (cdr lst)))
((= (caar lst) (+ 1 (cadar lst))) (cons-f (cdr lst)))
(else (cons (car lst) (cons-f (cdr lst))))
))
(define (list-partitions total)
(cond ((= total 1) '((1)) )
((= total 0) '(()) )
(else (append nil (partitions-r total 0)))
))
; For these two tests, any permutation of the right answer will be accepted.
(list-partitions 5)
; expect ((5) (4 1) (3 1 1) (1 1 1 1 1))
(list-partitions 7)
; expect ((7) (6 1) (5 2) (5 1 1) (4 1 1 1) (3 3 1) (3 1 1 1 1) (1 1 1 1 1 1 1))
What does the function partitions-r and cons-f do? Thank you very much!
Don't know Scheme, but recursive generation in pseudocode might look like:
function Partitions(N, LastValue, list):
if N = 0
print list
else
for i from Min(LastValue, N) downto 1
if (i != LastValue - 1) //reject consecutive number
Partitions(N - i, i, list + [i]);
I'm trying to create a function to count all the elements in a list, including the elements of its sublists. initially, to get started, i came up with a basic function myList:
(define myLength
(lambda (L)
(cond
((null? L) 0)
(else (+ 1 (myLength (cdr L)))))))
However, it doesn't help me account for function calls like:
(numAtoms '()) "...should be 0"
(numAtoms '(())) "...should be 0"
(numAtoms '(1 1)) "...should be 2"
(numAtoms '(1 (1 1) 1)) "...should be 4"
(numAtoms '(1 (1 (1 1)) 1)) "...should be 5"
I'm trying to use basic functions like length, null?, and list?.
I think the trick here is to imagine how you can transform your input into the code that you'd want to use to compute the sum. Let's write each of your inputs in the fully expanded form, in terms of cons and '() and whatever other atoms appear in your data:
'() == '()
'(()) == (cons '() '())
'(1 1) == (cons 1 (cons 1 '()))
'(1 (1 1) 1) == (cons 1 (cons 1 (cons 1 '())) (cons 1 '()))
'(1 (1 (1 1)) 1) == ...
Now, look what would happen if you replaced each occurrence of cons with +, and each occurrence of '() with 0, and each occurrence of something that's not '() with 1. You'd have:
'() => 0 == 0
(cons '() '()) => (+ 0 0) == 0
(cons 1 (cons 1 '())) => (+ 1 (+ 1 0)) == 2
(cons 1 (cons 1 (cons 1 '())) (cons 1 '())) => (+ 1 (+ 1 (+ 1 0)) (+ 1 0)) == 4
... => ... == ...
Notice that those sums are exactly the values that you want! Based on this, it seems like you might not want to treat your input as a list so much as a tree built from cons cells. In general, you can map over a tree by specifying a function to apply to the recursive results of processing a pair, and a function to process the atoms of the tree:
(define (treeduce pair-fn atom-fn tree)
(if (pair? tree)
(pair-fn (treeduce pair-fn atom-fn (car tree))
(treeduce pair-fn atom-fn (cdr tree)))
(atom-fn tree)))
You could then implement that mapping of cons to + and everything else to 1 if it's a list and 0 if it's not by:
(define (non-null-atoms tree)
(treeduce +
(lambda (atom)
(if (not (null? atom))
1
0))
tree))
This yields the kinds of results you'd expect:
(non-null-atoms '()) ;=> 0
(non-null-atoms '(())) ;=> 0
(non-null-atoms '(1 1)) ;=> 2
(non-null-atoms '(1 (1 1) 1)) ;=> 4
(non-null-atoms '(1 (1 (1 1)) 1)) ;=> 5
Here is a recursive template you can use:
(define (num-atoms lst)
(cond ((pair? lst) (+ (num-atoms <??>)
(num-atoms <??>)))
((null? lst) <??>) ; not an atom
(else <??>))) ; an atom
This next example uses a helper that has the accumulated value (num) as an argument.
(define (num-atoms lst)
;; locally defined helper
(define (helper num lst)
(cond ((pair? lst) (helper (helper <??> <??>) <??>)) ; recurse with the sum of elements from car
((null? lst) <??>) ; return accumulated value
(else (helper <??> <??>)))) ; recurse with add1 to num
;; procedure starts here
(helper 0 lst))
Hope it helps
Make my-length work for any argument type, list or 'atom'; then the recursive algorithm becomes almost trivial:
(define (my-length l)
(cond ((null? l) 0)
((list? l) (+ (my-length (car l)) (my-length (cdr l))))
(else 1))) ; atom
> (my-length '(1 (1 (1 1)) 1)))
5
I don't fully understand what the append-map command does in racket, nor do I understand how to use it and I'm having a pretty hard time finding some decently understandable documentation online for it. Could someone possibly demonstrate what exactly the command does and how it works?
The append-map procedure is useful for creating a single list out of a list of sublists after applying a procedure to each sublist. In other words, this code:
(append-map proc lst)
... Is semantically equivalent to this:
(apply append (map proc lst))
... Or this:
(append* (map proc lst))
The applying-append-to-a-list-of-sublists idiom is sometimes known as flattening a list of sublists. Let's look at some examples, this one is right here in the documentation:
(append-map vector->list '(#(1) #(2 3) #(4)))
'(1 2 3 4)
For a more interesting example, take a look at this code from Rosetta Code for finding all permutations of a list:
(define (insert l n e)
(if (= 0 n)
(cons e l)
(cons (car l)
(insert (cdr l) (- n 1) e))))
(define (seq start end)
(if (= start end)
(list end)
(cons start (seq (+ start 1) end))))
(define (permute l)
(if (null? l)
'(())
(apply append (map (lambda (p)
(map (lambda (n)
(insert p n (car l)))
(seq 0 (length p))))
(permute (cdr l))))))
The last procedure can be expressed more concisely by using append-map:
(define (permute l)
(if (null? l)
'(())
(append-map (lambda (p)
(map (lambda (n)
(insert p n (car l)))
(seq 0 (length p))))
(permute (cdr l)))))
Either way, the result is as expected:
(permute '(1 2 3))
=> '((1 2 3) (2 1 3) (2 3 1) (1 3 2) (3 1 2) (3 2 1))
In Common Lisp, the function is named "mapcan" and it is sometimes used to combine filtering with mapping:
* (mapcan (lambda (n) (if (oddp n) (list (* n n)) '()))
'(0 1 2 3 4 5 6 7))
(1 9 25 49)
In Racket that would be:
> (append-map (lambda (n) (if (odd? n) (list (* n n)) '()))
(range 8))
'(1 9 25 49)
But it's better to do it this way:
> (filter-map (lambda (n) (and (odd? n) (* n n))) (range 8))
'(1 9 25 49)