Code-VTGraph/network.rkt

;; Copyright 2019 Edward L. Platt <elplatt@alum.mit.edu>
;;
;; This module implements a family of graphs based on permutations of
;; sylvester-radix numbers.

#lang racket

(provide
 
 ;; Generate the list of graph nodes.
 ;;
 ;; Params:
 ;; m: Dimensionality of sylvester-radix numbers used as vertices.
 ;; 
 ;; Example:
 ;; > (sylvester-nodes 2)
 ;; '((0 0) (1 0) (0 1) (1 1) (0 2) (1 2))
 sylvester-nodes

 ;; Generate the list of directed graph edges.
 ;;
 ;; Params:
 ;; m: Dimensionality of sylvester-radix numbers used as vertices.
 ;;
 ;; Example:
 ;; > (sylvester-edges 1)
 ;; '(((0) (1)) ((1) (0)))
 sylvester-edges

 ;; Create cyclic permutation operator for one digit of sylvester-radix number.
 ;;
 ;; Params:
 ;; i: Index of digit to permute.
 ;; n: Number of times to cycle digit.
 ;;
 ;; Example:
 ;; > ((cycle-sylvester-radix 2 1) '(0 0 0 0))
 ;; '(0 0 1 0)
 cycle-sylvester-radix

 ;; Create skip operator.
 ;; The skip operator is a composition of single-digit cyclic permutations.
 ;; Skip operators are used to generate the graph edges.
 ;;
 ;; Params:
 ;; i: Results in an operator acting on digits 0 to i.
 ;;
 ;; Example:
 ;; > ((skip-sylvester-radix 2) '(0 0 0))
 ;; '(1 2 6)
 skip-sylvester-radix)

(require "sylvester.rkt")

(define (sylvester-nodes m)
  (map (compose (extend-sylvester-radix m)
                make-sylvester-radix)
       (range (- (sylvester m) 1))))

(define (sylvester-edges m)
  (foldr
   append
   '()
   (map (lambda (k) (map (sylvester-color-edge k) (sylvester-nodes m)))
        (range m))))

(define (sylvester-color-edge k)
  (lambda (v) (list v ((skip-sylvester-radix k) v))))

(define (cycle-sylvester-radix i n)
  (lambda (v)
    (let* ([v ((extend-sylvester-radix (+ i 1)) v)]
           [head (take v i)]
           [tail (if (< i (- (length v) 1))
                     (drop v (+ i 1))
                     '())]
           [element (list-ref v i)])
      (append
       head
       (cons (modulo (+ element n) (sylvester i))
             tail)))))

(define (skip-sylvester-radix i)
  (lambda (v)
    (if (= i 0)
        ((cycle-sylvester-radix 0 1) v)
        (let* ([prev (skip-sylvester-radix (- i 1))]
               [partial (take v (min i (length v)))]
               [value (sylvester-radix-value (prev partial))])
          ((compose (cycle-sylvester-radix i (+ 1 value)) prev) v)))))