寻找所有“字符相等”字符串的有效算法？

Question

如何编写输出输入字符串的"同型等价物“的有效函数？

示例1 (伪代码)：

homoglyphs_list = [
                     ["o", "0"], // "o" and "0" are homoglyphs
                     ["i", "l", "1"] // "i" and "l" and "1" are homoglyphs
                  ]

input_string    = "someinput"
output          = [
                   "someinput", "s0meinput", "somelnput",
                   "s0melnput", "some1nput", "s0me1nput"
                  ]

示例2

homoglyphs_list = [
                     ["rn", "m", "nn"],
                  ]

input_string    = "rnn"
output          = ["rnn", "rm", "mn", "rrn", "nnn", "nm", "nrn"]

示例3

homoglyphs_list = [
                     ["d", "ci", "a"], // "o" and "0" are homoglyphs
                     ["i", "l", "1"] // "i" and "l" and "1" are homoglyphs
                  ]
                  /*
                     notice that with the two rules above,
                     we can infer "d" = "ci" = "a" = "cl" = "c1"
                  */

input_string    = "pacerier"
output          = [
                   "pacerier",  "pacerler",  "pacer1er",  "pcicerier",
                   "pcicerler", "pcicer1er", "pclcerier", "pc1cerier",
                   "pclcerler", "pc1cerler", "pclcer1er", "pc1cer1er",
                   "pdcerier",  "pdcerler",  "pdcer1er"
                  ]

注意:输出数组中成员的顺序并不重要，我们可以假设给定的同型类型映射是适当的(输入不会给我们一个“无限循环”)。

我现在的算法很好用，但是它使用的是原始的蛮力，性能很差。例如，用同型"mmmmm"输入["rn", "m", "nn"]需要38秒才能运行：

// This is php code (non-pseudo just so we could test the running time), 
// but the question remains language-agnostic

public function Func($in, Array $mappings){
    $out_table = array();
    $out_table[$in] = null;
    while(true){
        $number_of_entries_so_far = count($out_table);
        foreach(array_keys($out_table) as $key){
            foreach($mappings as $mapping){
                foreach($mapping as $value){
                    for($a=0, $aa=strlen($key); $a<$aa; ++$a){
                        $pos = strpos($key, $value, $a);
                        if($pos === false){
                            continue;
                        }
                        foreach($mapping as $equal_value){
                            if($value === $equal_value){
                                continue;
                            }
                            $out_table[substr($key, 0, $pos) . $equal_value . substr($key, $pos + strlen($value))] = null;
                        }
                    }

                }
            }
        }
        if($number_of_entries_so_far === count($out_table)){
            // it means we have tried bruteforcing but added no additional entries,
            // so we can quit now
            break;
        }
    }
    return array_keys($out_table);
}

如何实现高效(快速)同型扩展算法？

Answer 1

您应该从递归实现中获得一些性能，虽然我还没有对实际的性能进行过多的考虑。这将避免多次遍历输出字符串并计算每次迭代的输出数。此外，我还添加了一个小的“缓存”，以防止计算相同的同型类型两次，为了简单起见，它不检查映射，但您可以将它实现到，或者在每次调用更改映射之前清除缓存(但这很难看，很容易引入bug)。

代码看起来如下：

cache = {}
def homoglyph(input_string, mappings):
    output = []
    character = input_string[0]
    if input_string in cache:
        return cache[input_string]
    elif not input_string:
        return []
    for charset in mappings:
        if character in charset:
            temp = input_string
            sub_homoglyphs = homoglyph(input_string[1:], mappings)
            for char in charset:
                temp[0] = char
                output.append(temp)
                #adding the homoglyph character to the rest of the string
                for subhomoglyph in sub_homoglyphs:
                    output.append(char+subhomoglyph)
    cache[input_string] = output
    return output

(这是用python编写的，因为我不熟悉php语法，所以我还没有真正运行它，所以可能会出现错误，但逻辑的要点是存在的)