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strings – If we want to map abbreviations of full-English words (e.g. map “Jan” to “January”), how can we identify abbreviations which map to multiple words?

Short Version:

How can we construct such a trie which maps abbreviations of names-of-the-month to full-month (we map the abbreviation “mar” to “march”)?

  • The set of all abbreviations is formed by:
    • keeping the first letter of the month name. (all abbreviations of “january” begin with “j“)
    • deleting 1 or more characters (“jan” deletes “uary” from “january“)

The Looonnnnggg Version:

How can we construct such a trie?
What algorithm will build the appropriate trie from the container of verbose strings.

Consider the English names for months of the year:

  • January
  • February
  • March
  • April
  • (… truncated …)
  • October
  • November
  • December

We find it useful say that the English names for months of the year are “verbose” strings.
For any “verbose” string $v$, and any string $a$ we say that $a$ is an “abbreviation” of $v$ if and only if all of the following conditions are met:

  • $a$ non-empty. $|a| geq 1$
  • $a$ can formed by deleting 1 or more characters from “verbose” string $v$
  • $a(1) = v(1)$. Assume that string indexing begins at $1$, and not $0$.

For example, “jan” is an abbreviation of “january.”

Suppose you want to write an algorithm which:

  • accepts a list of verbose strings as inputs.
  • the algorithm outputs a “trie” data-structure (information retrieval tree) $T$ such that:
    • The trie $T$ accepts any ASCII string as input.
    • An output (leaf node) of the trie should be set of strings $S$ such that:
      • every string in $S$ is a verbose string
      • the string fed as input into trie $T$ is an abbreviation of every verbose string in container $S$

Some examples of input to the trie and output of the trie are shown below:

  • Example 1

    • Input: “Ma
    • Output: ${$March“, “May$}$
  • Example 2

    • Input: “Mar
    • Output: ${$March$}$
  • Example 3

    • Input: “Decuary
    • Output: ${$ $}$ ….. the empty-set

The output from the trie should be one of:

  • the empty set
  • a set of one item
  • a set of two or more items

For months of the year, we might write javascript so that an end-user can type in any half-way reasonable date-format, instead getting an error message when they put slashes instead of dashes, etc….

If you do not like the months of the year application, a different use-case would be to write write your own Linux Shell (similar to BASH). Maybe any half-way reasonable abbreviation of “make directory” will map to “mkdir” In that case, we could have many-to-one mapping from high-level shell-commands to low-level Linux commands.

The question is:

How can we construct such a trie?
What algorithm will build the appropriate trie from the container of verbose strings.

Also, can we avoid brute-force generating a list of all aberrations before-hand? The set of all strings form-able by deleting 1 or more characters from the verbose strings is quite large. We would like to avoid combinatorial explosion, if we can.

The programming language (Java, python, C+ + ) does not matter for answering this question.

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Number of words of length n for special language

Let $Sigma$ be an alphabet and let $L$ be a language over it with the following properties:

  1. if $win L$ then there exists $vin Sigma^*$ such that $wv in L$ and for every $sin Sigma$ the word $wvs$ does not lie in $L$
  2. $wvin L$ then $vw in L$
  3. It is prefix-closed, i.e. prefix of any word is still in the language.

Note that by the definition, it is not cyclic language. I’m trying to compute its growth function, by that I mean $gamma_n:= |{win L mid |w| = n}|$. I know about my specific case that it is not regular and my hypothesis is that function $Gamma(x) = sum_{n=1}^infty gamma_nx^n$ is not rational. However, I couldn’t find any information about these functions for non-regular languages. Maybe, there’s a formula that connects entropy of language, i.e. $e(L):= limsuplimits_{ntoinfty} frac{loggamma_n}{n}$ and the $Gamma$ function. Or for such a language there’s a way to describe its growth throughout the growth of the language $operatorname{End}(L) = { win L mid forall sin Sigma ,ws text{ is not in } L }$.

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recursion – Is my recursive algorithm for Equivalent Words correct?

Here is my problem.

Problem
Given two words and a dictionary, find out whether the words are equivalent.

Input: The dictionary, D (a set of words), and two words v and w from the dictionary.

Output: A transformation of v into w by substitutions such that all intermediate words belong to D. If no transformation is possible, output “v and w are not equivalent.”

I need to write both recursive and dynamic programming algorithm. As for recursion, I came up with this algorithm. Is it correct?

 EquivalentWordsProblem(v, w, D)
 1.m <- len (v)
 2.n <-  len (w)
 3.substitutions = () #array to save substitutions 
 4.if m != n:
 5.  return "v and w are not equivalent"
 6.else
 7.for i <- m to 1 <-1 do
 8.  for j <- n to j <- 1 do
 9.    if v(i) != w(j):
 10.           substituted_word <- v(1…i-1)+v(j) #we substitute v(i) for w(j)
 11.              if substituted_word in D:
 12.                substitutions.append(substituted_word)
 13.                   return EquivalentWordsProblem(v(1…m-i), w, D) #recur on the string of length m - i
 14.   else:
         return EquivalentWordsProblem(v(1…m-1), w, D) #recur on the string decreasing its length by 1
 15.if len(substitutions) != 0:
 16.  return substitutions 
 17.else
 18.   return (“v and w are not equivalent”)