Matrix Rotation
June 6, 2017
The first program creates a new matrix of the appropriate dimensions, then copies each item from the old matrix to the new matrix:
(define (rotate source)
(let* ((n (matrix-rows source))
(m (matrix-cols source))
(target (make-matrix m n)))
(for (i 0 m)
(for (j 0 n)
(matrix-set! target i j
(matrix-ref source (- n j 1) i))))
target))
> (define m '#( #(a b c) #(d e f) #(g h i) #(j k l) #(m n o)))
> (rotate m)
#(#(m j g d a) #(n k h e b) #(o l i f c))
The second program uses a subroutine to swap four matrix items at once, at the four corners of a rectangle that successively visits each matrix item (except the center item in an odd-sized matrix):
(define (rotate! m)
(define (rot4 m topleft topright botleft botright)
(let ((t (matrix-ref m (car botright) (cdr botright))))
(matrix-set! m (car botright) (cdr botright)
(matrix-ref m (car botleft) (cdr botleft)))
(matrix-set! m (car botleft) (cdr botleft)
(matrix-ref m (car topright) (cdr topright)))
(matrix-set! m (car topright) (cdr topright)
(matrix-ref m (car topleft) (cdr topleft)))
(matrix-set! m (car topleft) (car topright) t)))
(let ((n (matrix-rows m)))
(for (i 0 (quotient n 2))
(let ((top i) (bottom (- n i 1))
(left i) (right (- n i 1)))
(for (j 0 (- n i i 1))
(rot4 m (cons top (+ left j))
(cons (+ top j) right)
(cons bottom (- right j))
(cons (- bottom j) left)))))
m))
> (define m '#( #(a b c d e) #(f g h i j) #(k l m n o) #(p q r s t) #(u v w x y))
> (rotate! m)
#(#(u p k f a) #(v q l g b) #(w r m h c) #(x s n i d) #(y t o j e))
> (rotate! m)
#(#(y x w v u) #(t s r q p) #(o n m l k) #(j i h g f) #(e d c b a))
> (rotate! m)
#(#(e j o t y) #(d i n s x) #(c h m r w) #(b g l q v) #(a f k p u))
> (rotate! m)
#(#(a b c d e) #(f g h i j) #(k l m n o) #(p q r s t) #(u v w x y))
Four rotations restore the original matrix. It is remarkably easy to get mixed up between rows and columns while writing these functions. I did; don’t get discouraged if you do, too.
You can run the program at http://ideone.com/UoKFU2.
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;;; Implement a matrix as a vector in row-major order, with width. (define imp::matrix-contents cdr) (define matrix-width car) (define (matrix-height m) (/ (vector-length (imp::matrix-contents m)) (matrix-width m))) (define (make-matrix h w . o) (let ((o (if (null? o) #f (car o)))) (cons w (make-vector (* h w) o)))) (define (matrix-ref m r k) (vector-ref (imp::matrix-contents m) (+ (* (matrix-width m) r) k))) (define (matrix-set! m r k o) (vector-set! (imp::matrix-contents m) (+ (* (matrix-width m) r) k) o)) ;;; Some of the points appear to go from (r, k) to (k, h - r - 1), so ;;; guess they all do. By linearity or something. (define (rot m) (let* ((h (matrix-height m)) (w (matrix-width m)) (m1 (make-matrix w h))) (do ((r 0 (+ r 1))) ((= r h)) (do ((k 0 (+ k 1))) ((= k w)) (matrix-set! m1 k (- h r 1) (matrix-ref m r k)))) m1)) ;;; At each (r, k) apply the rotating transformation to the four ;;; corresponding points at once. But this rotates the matrix four ;;; times! So only do half of the rows and half of the columns to ;;; rotate only once. Rounding both ways seems to work already. (define (rot! m) (let ((hw (matrix-height m))) (do ((r 0 (+ r 1))) ((= r (floor (/ hw 2)))) (do ((k 0 (+ k 1))) ((= k (ceiling (/ hw 2)))) (let* ((nr r) (nk k) (n (matrix-ref m nr nk)) (er nk) (ek (- hw nr 1)) (e (matrix-ref m er ek)) (sr ek) (sk (- hw er 1)) (s (matrix-ref m sr sk)) (wr sk) (wk (- hw sr 1)) (w (matrix-ref m wr wk))) (matrix-set! m er ek n) (matrix-set! m sr sk e) (matrix-set! m wr wk s) (matrix-set! m nr nk w)))))) ;;; Nice display for ocular inspection. (define (lay m) (let* ((w (matrix-width m)) (h (matrix-height m))) (do ((r 0 (+ r 1))) ((= r h)) (do ((k 0 (+ k 1))) ((= k w) (newline)) (write-char #\space) (write (matrix-ref m r k)))))) ;;; Test some. Caught a few thinkoes and a typo. (define A (cons 3 '#(a b c d e f g h i j k l m n o))) (define B (cons 5 (apply vector '(a b c d e f g h i j k l m n o p q r s t u v w x y)))) (define C (cons 4 (apply vector '(a b c d e f g h i j k l m n o p)))) (lay A) (newline) (lay (rot A)) (newline) (lay B) (newline) (lay (rot B)) (newline) (rot! B) (lay B) (newline) (lay C) (newline) (lay (rot C)) (newline) (rot! C) (lay C) (newline)The first part of the exercise in Python
def rot_right(matrix): return list(zip(*reversed(matrix)))Both parts of the exercise in Julia. Just a single generic method covering all possibilities.
function rotate{T <: Any}(X::Matrix{T})
n = size(X, 1)
n1 = n + 1
Y = X'
for i = 1:div(n, 2)
j = n1 – i
Y[:,i], Y[:,j] = Y[:,j], Y[:,i]
end
return Y
end
The second part in Python. The rotation is done by swapping two matrix elements at the time.
def rot_right_in_place(matrix): n = len(matrix) c = n-1 size = (n+1)//2 def rot(row, col): col0, row0, col, row = col, row, row, c - col matrix[row0][col0], matrix[row][col] = matrix[row][col], matrix[row0][col0] # swap return row, col for i in range(size): for j in range(size if n % 2 == 0 else size-1): rot(*rot(*rot(i, j)))A better version of the second part. This version sets the matrix elements only once.
def rot_right_in_place(matrix): n = len(matrix) c = n-1 size = (n+1)//2 def rot(row, col): save = matrix[row][col] for _ in range(3): col0, row0, col, row = col, row, row, c - col matrix[row0][col0] = matrix[row][col] matrix[row][col] = save for i in range(size): for j in range(size if n % 2 == 0 else size-1): rot(i, j)Klong version
:" assign array to a" a::[[:a :b :c] [:d :e :f] [:g :h :i] [:j :k :l] [:m :n :o]] [[:a :b :c] [:d :e :f] [:g :h :i] [:j :k :l] [:m :n :o]] |'+a :"transpose the array and reverse each row" [[:m :j :g :d :a] [:n :k :h :e :b] [:o :l :i :f :c]] :" assign array to b" b::[[:a :b :c :d :e] [:f :g :h :i :j] [:k :l :m :n :o] [:p :q :r :s :t] [:u :v :w :x :y]] [[:a :b :c :d :e] [:f :g :h :i :j] [:k :l :m :n :o] [:p :q :r :s :t] [:u :v :w :x :y]] |'+b :"transpose the array and reverse each row" [[:u :p :k :f :a] [:v :q :l :g :b] [:w :r :m :h :c] [:x :s :n :i :d] [:y :t :o :j :e]]Python 3.6.
rot() is basically the same as Paul’s.
rot_in_place() moves four corresponding elements at a time. ‘i’ can be thought of as the layer or ring, 0 being the outermost ring.
‘i’, ‘ir’, ‘j’, and ‘jr’ are the row or column indices of the elements being moved.
rot = lambda m:[*zip(*m[::-1])]
def rot_in_place(a):
n = len(a)
for i in range((n+1) // 2):
ir = -i-1
for j in range(i, n-i-1):
jr = -j-1
(a[i][j], a[j][ir],
a[jr][i],a[ir][jr]) = (a[jr][i], a[i][j],
a[ir][jr],a[j][ir])