Prüfer
Encoding/Decoding of a Tree in Common Lisp
Description of the problem:
Given a tree, or a simple graph, the standard way of describing it is
giving the set of vertices and edges. However, there are myriad
different ways of encoding a tree. Pruefer
encoding is one of them. Today, I am going to write two functions:
one to write the Prüfer code of a given tree as a list of edges, and
another to write a list of edges from a Prüfer code.
The algorithm
The encoding algorithm is pretty simple:
Function Encode:
Input: A tree T as a list of edges
Output: An ordered list rs of vertices with repetitions
Begin
Let rs be the empty list
While T contains two or more vertices do
Find the leaf x (vertex of degree 1) in T with the minimal label
Let y be the other end of the edge connected to x
Remove x and the edge xy connected to x from T
Append y to rs
End
Return rs
End
Let me show how this works over an example: Let T be the tree
0 -- 1 -- 2 -- 3 -- 4
| |
5 6
In the first pass we pick 0 but add 1 to the Prüfer code, and we
remove the edge 01. In the second pass, 4 is the smallest label and we
add 3 to the code before we remove 43. Next is 5, we add 1 again and
remove 15. Next is 1, we add 2 to the code and remove the edge 12. Next
is 2, we add 3 to the code and remove the edge 23. And then we stop
since we only have two vertices 3 and 6. The Prüfer code for the tree
above is 13123.
The decoding algorithm is a little more complex
Function Decode:
Input: An ordered list rs of vertices with repetitions
Output: A tree T as a list of edges
Begin
Let n be length(rs)
Do
Let x be the first element of the list rs
Let y be the vertex with the minimal label
that does not appear in rs
Add the edge xy to T
Remove x from rs
Append y to the end of rs
Until T contains n edges
Let x and y be the only vertices not in rs
Add xy to T
Return T
End
Let us decode the code we obtained above: The code was 13123 and n is
set to 5. In the first pass x is 1, and y is 0. So, we add the edge 01
to T. The code is now 31230. In the next pass x is 3, y is 4. We add 34
to T and the code becomes 12304. Next, we get x is 1, y is 5 and we add
15 to T while the code is updated to 23045. Then x is 2, y is 1, the
code becomes 30451 and we add 12 to T. In the final pass x is 3, y is 2
and we add 23 to T while the code becomes 04512. The only two vertices
that are not in the code are 3 and 6. So T is given by the list of
edges: 01, 34, 15, 12, 23, 36.
An implementation in common
lisp
Before we write the code for the encoding part, I am going to need
few utility functions for graph related calculations. I got hooked on to
the following macro from clojure:
(defmacro ->> (x &rest forms)
(dolist (f forms x)
(if (listp f)
(setf x (append f (list x)))
(setf x (list f x)))))
The macro makes the code below more readable. Next,
(defun neighbors (G x)
"Returns the set of neighbors of a vertex x in G."
(remove x
(->> G
(remove-if-not (lambda (ys) (member x ys)))
(reduce (lambda (us vs) (append us (remove x vs)))))))
(defun degree (G x)
"Returns the degree of a vertex x in G."
(length (neighbors G x)))
(defun vertices (G)
"Returns the set of vertices of a graph G."
(remove-duplicates (reduce #'append G)))
(defun delete-vertex (G x)
"Deletes a vertex x all edges adjacent to x from G."
(remove-if (lambda (edge) (member x edge)) G))
NEIGHBORS
DEGREE
VERTICES
DELETE-VERTEX
Now that we are done with generic graph operations, let us move on to
the specific function we need. First is for finding the leaf with
minimal label.
(defun min-leaf (G)
"Find the leaf in G with the minimal label."
(->> (vertices G)
(remove-if-not (lambda (y) (= 1 (degree G y))))
(reduce #'min)))
Now, we can implement the encoding function as follows:
(defun prufer-encode (G &optional xs)
"Returns the Prüfer encoding of a tree given as a list of edges."
(if (= 1 (length G))
(nreverse xs)
(let* ((x (min-leaf G))
(y (car (neighbors G x))))
(prufer-encode (delete-vertex G x) (cons y xs)))))
Let us test:
(prufer-encode '((0 1) (1 2) (2 3) (3 4) (1 5) (3 6)))
The decode function strangely is more straightforward to
implement:
(defun prufer-decode (xs)
"Returns the tree as a list of edges from a given Prüfer code."
(let* ((n (+ 2 (length xs)))
(ys (loop for i from 0 below n collect i)))
(labels ((helper (code k G)
(let ((zs (sort (set-difference ys (remove-duplicates code)) #'<)))
(if (= k 2)
(append G (list zs))
(helper (append (cdr code) (list (car zs)))
(1- k)
(append G (list (list (car code) (car zs)))))))))
(helper xs n nil))))
And we test it:
(prufer-decode '(1 3 1 2 3))
((1 0) (3 4) (1 5) (2 1) (3 2) (3 6))
Now, let me test the algorithm on a larger random tree:
(defun random-tree (n)
(loop for i from 1 below n
collect (list (random i) i)))
(defun straighten (G)
(mapcar (lambda (xs) (sort xs #'<)) G))
OK. I am going to test the algorithm on a random tree on 50 vertices:
The code below should return nil:
(defvar G (straighten (random-tree 50)))
(set-difference G (straighten (prufer-decode (prufer-encode G))) :test #'equal)
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