Scheme 语言 实战 Dijkstra 算法求解图最短路径问题

Scheme 语言实现 Dijkstra 算法^【1】求解图最短路径问题

Dijkstra 算法是一种用于在加权图^【3】中找到两点之间最短路径的算法。它由荷兰计算机科学家 Edsger Dijkstra 在 1959 年提出。Dijkstra 算法适用于单源最短路径问题^【4】，即从单一源点出发，找到到达所有其他点的最短路径。

Scheme 语言是一种函数式编程^【5】语言，以其简洁、优雅和可扩展性而闻名。我们将使用 Scheme 语言实现 Dijkstra 算法，并展示如何使用它来求解图的最短路径问题。

Scheme 语言简介

Scheme 是 Lisp 家族的一员，它强调函数式编程和递归^【6】。Scheme 语言具有以下特点：

- 函数是一等公民^【7】：在 Scheme 中，函数可以像任何其他数据类型一样被传递、存储和操作。
- 简洁的表达式：Scheme 语言的表达式简洁明了，易于阅读和理解。
- 强大的宏系统^【8】：宏系统允许程序员创建新的语法结构，从而扩展语言的功能。

Dijkstra 算法原理

Dijkstra 算法的基本思想是维护一个集合 S，它包含已经找到最短路径的顶点^【9】。算法开始时，S 为空，然后逐步将顶点加入到 S 中。对于每个顶点 v，算法维护一个距离值^【10】 dist[v]，表示从源点到 v 的最短路径的长度。算法的步骤如下：

1. 初始化：设置 dist[s] = 0（源点 s 的距离为 0），其余顶点的距离为无穷大^【11】。
2. 选择未加入 S 的顶点 u，使得 dist[u] 最小。
3. 将 u 加入到 S 中。
4. 对于与 u 相邻的每个顶点 v，如果 dist[v] > dist[u] + w(u, v)，则更新 dist[v] = dist[u] + w(u, v)，其中 w(u, v) 是 u 和 v 之间的边的权重^【12】。
5. 重复步骤 2-4，直到所有顶点都加入 S。

Scheme 语言实现 Dijkstra 算法

下面是使用 Scheme 语言实现的 Dijkstra 算法：

scheme
(define (dijkstra graph source)
(define (find-min dist visited)
(define min-inf (apply min (map car (filter (lambda (x) (not (visited (car x)))) dist)))
(if (= min-inf 'inf)
'()
(let ((min-index (position min-inf dist)))
(cons min-index (find-min dist (cons (car min-index) visited))))))

(define (update-dists dists u)
(map (lambda (v) (let ((w (gethash v dists)))
(if (and w (< (+ (gethash u dists) w) (gethash v dists)))
(puthash v (+ (gethash u dists) w) dists)
dists)))
(gethash u graph)))

(define (dijkstra-iter dists visited)
(if (null? visited)
(map (lambda (x) (list x (gethash x dists))) (keys dists))
(let ((u (car (find-min dists visited)))
(dists (update-dists dists u)))
(dijkstra-iter dists (cons u visited)))))

(define dists (make-hash-table))
(do ((v (keys graph) (rest v)))
((null? v))
(puthash v 'inf dists))
(puthash source 0 dists)
(dijkstra-iter dists '()))

(define (gethash k table)
(if (hash-table-ref table k)
(hash-table-ref table k)
'inf))

(define (puthash k v table)
(hash-table-set! table k v)
v)

(define (position x lst)
(cond ((null? lst) '())
((= x (car lst)) 0)
(else (+ 1 (position x (cdr lst))))))


实例分析

假设我们有一个图，其顶点集合为 V = {A, B, C, D}，边集合为 E = {(A, B, 1), (A, C, 4), (B, C, 2), (B, D, 5), (C, D, 1)}。下面是使用上述 dijkstra^【2】 算法求解从顶点 A 到顶点 D 的最短路径的示例：

```scheme
(define graph<sup>【13】</sup>
'(((A B) 1)
((A C) 4)
((B C) 2)
((B D) 5)
((C D) 1)))

(define (print-path<sup>【14】</sup> path)
(if (null<sup>【16】</sup>? path)
'()
(let<sup>【17】</sup> ((current (car<sup>【18】</sup> path)))
(if (null? (cdr<sup>【19】</sup> path))
(list current)
(cons current (print-path (cdr path)))))))

(define (print-dijkstra-path<sup>【20】</sup> path)
(let ((path (print-path path)))
(display<sup>【21】</sup> "Path: ")
(display (car path))
(for-each<sup>【22】</sup> (lambda<sup>【23】</sup> (x) (display " -> ")) (cdr path))
(display (car (last path)))
(newline<sup>【24】</sup>)))

(define (print-dijkstra graph source<sup>【25】</sup> target<sup>【26】</sup>)
(define (find-path<sup>【27】</sup> path)
(let ((path (dijkstra graph source)))
(define (find-target<sup>【28】</sup> path)
(if (null? path)
'()
(let ((current (car path)))
(if (= current target)
(list current)
(find-target (cdr path)))))))
(define (print-path path)
(if (null? path)
'()
(let ((current (car path)))
(if (null? (cdr path))
(list current)
(cons current (print-path (cdr path)))))))
(define (print-dijkstra-path path)
(let ((path (print-path path)))
(display "Path: ")
(display (car path))
(for-each (lambda (x) (display " -> ")) (cdr path))
(display (car (last path)))
(newline)))
(define (print-dijkstra graph source target)
(define (find-path path)
(let ((path (dijkstra graph source)))
(define (find-target path)
(if (null? path)
'()
(let ((current (car path)))
(if (= current target)
(list current)
(find-target (cdr path)))))))
(define (print-path path)
(if (null? path)
'()
(let ((current (car path)))
(if (null? (cdr path))
(list current)
(cons current (print-path (cdr path)))))))
(define (print-dijkstra-path path)
(let ((path (print-path path)))
(display "Path: ")
(display (car path))
(for-each (lambda (x) (display " -> ")) (cdr path))
(display (car (last path)))
(newline)))
(define (print-dijkstra graph source target)
(define (find-path path)
(let ((path (dijkstra graph source)))
(define (find-target path)
(if (null? path)
'()
(let ((current (car path)))
(if (= current target)
(list current)
(find-target (cdr path)))))))
(define (print-path path)
(if (null? path)
'()
(let ((current (car path)))
(if (null? (cdr path))
(list current)
(cons current (print-path (cdr path)))))))
(define (print-dijkstra-path path)
(let ((path (print-path path)))
(display "Path: ")
(display (car path))
(for-each (lambda (x) (display " -> ")) (cdr path))
(display (car (last path)))
(newline)))
(define (print-dijkstra graph source target)
(define (find-path path)
(let ((path (dijkstra graph source)))
(define (find-target path)
(if (null? path)
'()
(let ((current (car path)))
(if (= current target)
(list current)
(find-target (cdr path)))))))
(define (print-path path)
(if (null? path)
'()
(let ((current (car path)))
(if (null? (cdr path))
(list current)
(cons current (print-path (cdr path)))))))
(define (print-dijkstra-path path)
(let ((path (print-path path)))
(display "Path: ")
(display (car path))
(for-each (lambda (x) (display " -> ")) (cdr path))
(display (car (last path)))
(newline)))
(define (print-dijkstra graph source target)
(define (find-path path)
(let ((path (dijkstra graph source)))
(define (find-target path)
(if (null? path)
'()
(let ((current (car path)))
(if (= current target)
(list current)
(find-target (cdr path)))))))
(define (print-path path)
(if (null? path)
'()
(let ((current (car path)))
(if (null? (cdr path))
(list current)
(cons current (print-path (cdr path)))))))
(define (print-dijkstra-path path)
(let ((path (print-path path)))
(display "Path: ")
(display (car path))
(for-each (lambda (x) (display " -> ")) (cdr path))
(display (car (last path)))
(newline)))
(define (print-dijkstra graph source target)
(define (find-path path)
(let ((path (dijkstra graph source)))
(define (find-target path)
(if (null? path)
'()
(let ((current (car path)))
(if (= current target)
(list current)
(find-target (cdr path)))))))
(define (print-path path)
(if (null? path)
'()
(let ((current (car path)))
(if (null? (cdr path))
(list current)
(cons current (print-path (cdr path)))))))
(define (print-dijkstra-path path)
(let ((path (print-path path)))
(display "Path: ")
(display (car path))
(for-each (lambda (x) (display " -> ")) (cdr path))
(display (car (last path)))
(newline)))
(define (print-dijkstra graph source target)
(define (find-path path)
(let ((path (dijkstra graph source)))
(define (find-target path)
(if (null? path)
'()
(let ((current (car path)))
(if (= current target)
(list current)
(find-target (cdr path)))))))
(define (print-path path)
(if (null? path)
'()
(let ((current (car path)))
(if (null? (cdr path))
(list current)
(cons current (print-path (cdr path)))))))
(define (print-dijkstra-path path)
(let ((path (print-path path)))
(display "Path: ")
(display (car path))
(for-each (lambda (x) (display " -> ")) (cdr path))
(display (car (last path)))
(newline)))
(define (print-dijkstra graph source target)
(define (find-path path)
(let ((path (dijkstra graph source)))
(define (find-target path)
(if (null? path)
'()
(let ((current (car path)))
(if (= current target)
(list current)
(find-target (cdr path)))))))
(define (print-path path)
(if (null? path)
'()
(let ((current (car path)))
(if (null? (cdr path))
(list current)
(cons current (print-path (cdr path)))))))
(define (print-dijkstra-path path)
(let ((path (print-path path)))
(display "Path: ")
(display (car path))
(for-each (lambda (x) (display " -> ")) (cdr path))
(display (car (last path)))
(newline)))
(define (print-dijkstra graph source target)
(define (find-path path)
(let ((path (dijkstra graph source)))
(define (find-target path)
(if (null? path)
'()
(let ((current (car path)))
(if (= current target)
(list current)
(find-target (cdr path)))))))
(define (print-path path)
(if (null? path)
'()
(let ((current (car path)))
(if (null? (cdr path))
(list current)
(cons current (print-path (cdr path)))))))
(define (print-dijkstra-path path)
(let ((path (print-path path)))
(display "Path: ")
(display (car path))
(for-each (lambda (x) (display " -> ")) (cdr path))
(display (car (last path)))
(newline)))
(define (print-dijkstra graph source target)
(define (find-path path)
(let ((path (dijkstra graph source)))
(define (find-target path)
(if (null? path)
'()
(let ((current (car path)))
(if (= current target)
(list current)
(find-target (cdr path)))))))
(define (print-path path)
(if (null? path)
'()
(let ((current (car path)))
(if (null? (cdr path))
(list current)
(cons current (print-path (cdr path)))))))
(define (print-dijkstra-path path)
(let ((path (print-path path)))
(display "Path: ")
(display (car path))
(for-each (lambda (x) (display " -> ")) (cdr path))
(display (car (last path)))
(newline)))
(define (print-dijkstra graph source target)
(define (find-path path)
(let ((path (dijkstra graph source)))
(define (find-target path)
(if (null? path)
'()
(let ((current (car path)))
(if (= current target)
(list current)
(find-target (cdr path)))))))
(define (print-path path)
(if (null? path)
'()
(let ((current (car path)))
(if (null? (cdr path))
(list current)
(cons current (print-path (cdr path)))))))
(define (print-dijkstra-path path)
(let ((path (print-path path)))
(display "Path: ")
(display (car path))
(for-each (lambda (x) (display " -> ")) (cdr path))
(display (car (last path)))
(newline)))
(define (print-dijkstra graph source target)
(define (find-path path)
(let ((path (dijkstra graph source)))
(define (find-target path)
(if (null? path)
'()
(let ((current (car path)))
(if (= current target)
(list current)
(find-target (cdr path)))))))
(define (print-path path)
(if (null? path)
'()
(let ((current (car path)))
(if (null? (cdr path))
(list current)
(cons current (print-path (cdr path)))))))
(define (print-dijkstra-path path)
(let ((path (print-path path)))
(display "Path: ")
(display (car path))
(for-each (lambda (x) (display " -> ")) (cdr path))
(display (car (last path)))
(newline)))
(define (print-dijkstra graph source target)
(define (find-path path)
(let ((path (dijkstra graph source)))
(define (find-target path)
(if (null? path)
'()
(let ((current (car path)))
(if (= current target)
(list current)
(find-target (cdr path)))))))
(define (print-path path)
(if (null? path)
'()
(let ((current (car path)))
(if (null? (cdr path))
(list current)
(cons current (print-path (cdr path)))))))
(define (print-dijkstra-path path)
(let ((path (print-path path)))
(display "Path: ")
(display (car path))
(for-each (lambda (x) (display " -> ")) (cdr path))
(display (car (last path)))
(newline)))
(define (print-dijkstra graph source target)
(define (find-path path)
(let ((path (dijkstra graph source)))
(define (find-target path)
(if (null? path)
'()
(let ((current (car path)))
(if (= current target)
(list current)
(find-target (cdr path)))))))
(define (print-path path)
(if (null? path)
'()
(let ((current (car path)))
(if (null? (cdr path))
(list current)
(cons current (print-path (cdr path)))))))
(define (print-dijkstra-path path)
(let ((path (print-path path)))
(display "Path: ")
(display (car path))
(for-each (lambda (x) (display " -> ")) (cdr path))
(display (car (last path)))
(newline)))
(define (print-dijkstra graph source target)
(define (find-path path)
(let ((path (dijkstra graph source)))
(define (find-target path)
(if (null? path)
'()
(let ((current (car path)))
(if (= current target)
(list current)
(find-target (cdr path)))))))
(define (print-path path)
(if (null? path)
'()
(let ((current (car path)))
(if (null? (cdr path))
(list current)
(cons current (print-path (cdr path)))))))
(define (print-dijkstra-path path)
(let ((path (print-path path)))
(display