Given N strings s1, s2, ..., sn, sort them in some way, then connect them one by one, we obtain a long string. Now the task is to find an arrangement of the strings such that the long string we obtain is lexicographically least among all possible long strings.
There is a simple solution to this problem. For any two strings A and B, we define that A is prior to B, if and only if A+B<B+A (here the symbol '+' means connection, '<' means 'lexicographically less than'). Then we can use any of the sorting algorithms to sort the strings by their priority and obtain the arrangement required.
But there is a problem: base on the comparison rule defined above, how to prove that the strings can be sorted correctly? That is, how to prove that the strings are sortable by the rule?
Clearly, any two strings are comparable under the rule. So the problem is whether the rule would produce a circle. If there is a circle, we failed; otherwise we can sort them in topological order.
Suppose that the rule is transitive, that is to say, if A is prior to B and B is prior to C, then A must be prior to C. If the transitivity holds, there will be no circles, and the strings are sortable under the rule.
Let la, lb and lc denote the length of string A, B and C respectively. We know that any string S can be expressed by a number of radix Z, where Z can be the size of the alphabet (e.g. 26). Let Val(S) denote the number representing string S, then string A+B can be expressed as:
Val(A+B)=Val(A)*Z^lb+Val(B)
Similarly,
Val(B+A)=Val(B)*Z^la+Val(A)
Val(B+C)=Val(B)*Z^lc+Val(C)
Val(C+B)=Val(C)*Z^lb+Val(B)
Val(A+C)=Val(A)*Z^lc+Val(C)
Val(C+A)=Val(C)*Z^la+Val(A)
Since Length(A+B) is equal to Length(B+A), the relation A+B<B+A is equivalent to the following expression in algebra:
Val(A)*Z^lb+Val(B)<Val(B)*Z^la+Val(A) ······ *
Similarly, B+C<C+B is equivalent to:
Val(B)*Z^lc+Val(C)<Val(C)*Z^lb+Val(B) ······ #
From (*) we can obtain:
Val(A)*(Z^lb-1)<Val(B)*(Z^la-1)
Recognizing that la>0 and Z>1, we get Z^la>=Z^1>1, so Z^la-1>0. Then we obtain:
Val(A)*(Z^lb-1)/(Z^la-1)<Val(B) ······ $
Similarly, we can obtain from (#) that:
Val(B)<Val(C)*(Z^lb-1)/(Z^lc-1) ······ $$
At last, we get from ($) and ($$) that:
Val(A)*(Z^lb-1)/(Z^la-1)<Val(C)*(Z^lb-1)/(Z^lc-1)
Then we can easily obtain: Val(A)*Z^lc+Val(C)<Val(C)*Z^la+Val(A), which is equivalent to A+C<C+A.
As proved above, the strings can be sorted under the rule. And after sorting, according to the quality of sorting algorithms, we can prove that the first string in the arrangement is prior to any others, so it makes the long string lexicographically less than any long strings starting with other strings. For the second and other strings in the arrangement, we can prove the optimality similarly.
