学号：16340008

题目：127. Word Ladder

Question:

Given two words (beginWord and endWord), and a dictionary's word list, find the length of shortest transformation sequence from beginWord to endWord, such that:

1. Only one letter can be changed at a time.
2. Each transformed word must exist in the word list. Note that beginWord is not a transformed word.

Note:

• Return 0 if there is no such transformation sequence.
• All words have the same length.
• All words contain only lowercase alphabetic characters.
• You may assume no duplicates in the word list.
• You may assume beginWord and endWord are non-empty and are not the same.

Example 1:

Input:
beginWord = "hit",
endWord = "cog",
wordList = ["hot","dot","dog","lot","log","cog"]

Output: 5

Explanation: As one shortest transformation is "hit" -> "hot" -> "dot" -> "dog" -> "cog",
return its length 5.

Example 2:

Input:
beginWord = "hit"
endWord = "cog"
wordList = ["hot","dot","dog","lot","log"]

Output: 0

Explanation: The endWord "cog" is not in wordList, therefore no possible transformation.

Answer:

题目的意思如下：题目给出一个单词beginWord，一个单词列表wordList，一个单词endWord，要求我们从beginWord开始，每次只能变单词中的一个字母，使其成为wordList中的一个单词，重复若干次上述步骤直到变化成endWord，求最少的变化次数（若能成功变化成endWord）。

分析题目我们可以得到以下方案：

1. 对beginWord，以wordList作为候选池，进行BFS，得到其下一层A
2. 层A中的所有单词从候选池（wordList）中删去，避免搜索路径树出现环。而且其他将来可能经过该层单词的路径都将比层A发散出去的路径更长
3. 将层A中的单词推进队列
4. 队列头弹出一个单词，对其BFS，所有下一层的单词都从wordList中删去，并将该层单词推进队列。（从树根到队列头单词的下一层某单词K的步数肯定小于等于从树根到队列中其他单词到单词K的步数）
5. 重复第四步，直到某队列头的下一层单词中有endWord（有解）或队列为空（无解）

在函数开始，先检测了endWord是否在wordList中，避免无谓的路径搜索。

声明了类State，有属性单词和属性步数，作为搜索的节点记录数据，相当于树的节点。方法中将State对象推进或弹出队列，对搜索到的下一层单词创建State对象记录其单词和步数（上一层+1）。

从wordList删除搜索到的下一层单词是为了后续搜索时能搜索更少的节点。

定义了方法check用于校验两个单词是否只差一个字母。

然后得到初步的代码如下（Python3）：

import time
import queue
import copy

class State:
    def __init__(self, on, times):
        self.now = on
        self.times = times

class Solution:
    def ladderLength(self, beginWord, endWord, wordList):
        """
        :type beginWord: str
        :type endWord: str
        :type wordList: List[str]
        :rtype: int
        """
        self.q = queue.Queue()
        if endWord not in wordList:
            return 0
        
        self.list = wordList
        self.end = endWord
        begin = State(beginWord, 1)
        self.q.put(begin)
        
        while not self.q.empty():
            temp = self.q.get()
            i = 0
            while i < len(self.list):
                word = self.list[i]
                if self.check(word, temp.now):
                    if word == self.end:
                        return temp.times + 1
                    if word != temp.now:
                        self.q.put(State(word, temp.times+1))
                    self.list.remove(word)
                else: i += 1
        return 0


    def check(self, wordA, wordB):
        flag = True
        for i in range(len(wordA)):
            if wordA[i] != wordB[i] and flag == True:
                flag = False
            elif wordA[i] != wordB[i] and flag == False:
                return False
        return True

 然而当wordList中元素足够多（如以下代码中的29/39次test的2855个数据）时，提交反馈Time Limit Exceeded。这是因为搜索的效率低下。对于一个弹出的队列头单词A，搜索与他相似（指只相差一个字母）的单词需要遍历整个wordList，复杂度为O(n)，同时remove一个相似单词需要对所有后续元素进行移动。若用一个标识（如0）取代remove，在另一方面由加重了遍历的工作量（实际上通过time计时发现时间更长了）。由于python3中list的数据结构相当于一个动态数组，因此假如不改变算法，这将无可避免。曾考虑使用链表，然而只能优化remove而无法改变搜索需要的遍历。

以下代码为测试代码，其中wordList有2855个数据，用以上算法在VScode调试下计时得6-7s，步数为11。

test = Solution()
beginWord = "sand"
endWord = "acne"
wordList = ["slit","bunk","wars","ping","viva","wynn","wows","irks","gang","pool","mock","fort","heel","send","ship","cols","alec","foal","nabs","gaze","giza","mays","dogs","karo","cums","jedi","webb","lend","mire","jose","catt","grow","toss","magi","leis","bead","kara","hoof","than","ires","baas","vein","kari","riga","oars","gags","thug","yawn","wive","view","germ","flab","july","tuck","rory","bean","feed","rhee","jeez","gobs","lath","desk","yoko","cute","zeus","thus","dims","link","dirt","mara","disc","limy","lewd","maud","duly","elsa","hart","rays","rues","camp","lack","okra","tome","math","plug","monk","orly","friz","hogs","yoda","poop","tick","plod","cloy","pees","imps","lead","pope","mall","frey","been","plea","poll","male","teak","soho","glob","bell","mary","hail","scan","yips","like","mull","kory","odor","byte","kaye","word","honk","asks","slid","hopi","toke","gore","flew","tins","mown","oise","hall","vega","sing","fool","boat","bobs","lain","soft","hard","rots","sees","apex","chan","told","woos","unit","scow","gilt","beef","jars","tyre","imus","neon","soap","dabs","rein","ovid","hose","husk","loll","asia","cope","tail","hazy","clad","lash","sags","moll","eddy","fuel","lift","flog","land","sigh","saks","sail","hook","visa","tier","maws","roeg","gila","eyes","noah","hypo","tore","eggs","rove","chap","room","wait","lurk","race","host","dada","lola","gabs","sobs","joel","keck","axed","mead","gust","laid","ends","oort","nose","peer","kept","abet","iran","mick","dead","hags","tens","gown","sick","odis","miro","bill","fawn","sumo","kilt","huge","ores","oran","flag","tost","seth","sift","poet","reds","pips","cape","togo","wale","limn","toll","ploy","inns","snag","hoes","jerk","flux","fido","zane","arab","gamy","raze","lank","hurt","rail","hind","hoot","dogy","away","pest","hoed","pose","lose","pole","alva","dino","kind","clan","dips","soup","veto","edna","damp","gush","amen","wits","pubs","fuzz","cash","pine","trod","gunk","nude","lost","rite","cory","walt","mica","cart","avow","wind","book","leon","life","bang","draw","leek","skis","dram","ripe","mine","urea","tiff","over","gale","weir","defy","norm","tull","whiz","gill","ward","crag","when","mill","firs","sans","flue","reid","ekes","jain","mutt","hems","laps","piss","pall","rowe","prey","cull","knew","size","wets","hurl","wont","suva","girt","prys","prow","warn","naps","gong","thru","livy","boar","sade","amok","vice","slat","emir","jade","karl","loyd","cerf","bess","loss","rums","lats","bode","subs","muss","maim","kits","thin","york","punt","gays","alpo","aids","drag","eras","mats","pyre","clot","step","oath","lout","wary","carp","hums","tang","pout","whip","fled","omar","such","kano","jake","stan","loop","fuss","mini","byrd","exit","fizz","lire","emil","prop","noes","awed","gift","soli","sale","gage","orin","slur","limp","saar","arks","mast","gnat","port","into","geed","pave","awls","cent","cunt","full","dint","hank","mate","coin","tars","scud","veer","coax","bops","uris","loom","shod","crib","lids","drys","fish","edit","dick","erna","else","hahs","alga","moho","wire","fora","tums","ruth","bets","duns","mold","mush","swop","ruby","bolt","nave","kite","ahem","brad","tern","nips","whew","bait","ooze","gino","yuck","drum","shoe","lobe","dusk","cult","paws","anew","dado","nook","half","lams","rich","cato","java","kemp","vain","fees","sham","auks","gish","fire","elam","salt","sour","loth","whit","yogi","shes","scam","yous","lucy","inez","geld","whig","thee","kelp","loaf","harm","tomb","ever","airs","page","laud","stun","paid","goop","cobs","judy","grab","doha","crew","item","fogs","tong","blip","vest","bran","wend","bawl","feel","jets","mixt","tell","dire","devi","milo","deng","yews","weak","mark","doug","fare","rigs","poke","hies","sian","suez","quip","kens","lass","zips","elva","brat","cosy","teri","hull","spun","russ","pupa","weed","pulp","main","grim","hone","cord","barf","olav","gaps","rote","wilt","lars","roll","balm","jana","give","eire","faun","suck","kegs","nita","weer","tush","spry","loge","nays","heir","dope","roar","peep","nags","ates","bane","seas","sign","fred","they","lien","kiev","fops","said","lawn","lind","miff","mass","trig","sins","furl","ruin","sent","cray","maya","clog","puns","silk","axis","grog","jots","dyer","mope","rand","vend","keen","chou","dose","rain","eats","sped","maui","evan","time","todd","skit","lief","sops","outs","moot","faze","biro","gook","fill","oval","skew","veil","born","slob","hyde","twin","eloy","beat","ergs","sure","kobe","eggo","hens","jive","flax","mons","dunk","yest","begs","dial","lodz","burp","pile","much","dock","rene","sago","racy","have","yalu","glow","move","peps","hods","kins","salk","hand","cons","dare","myra","sega","type","mari","pelt","hula","gulf","jugs","flay","fest","spat","toms","zeno","taps","deny","swag","afro","baud","jabs","smut","egos","lara","toes","song","fray","luis","brut","olen","mere","ruff","slum","glad","buds","silt","rued","gelt","hive","teem","ides","sink","ands","wisp","omen","lyre","yuks","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time_start=time.time()
print(test.ladderLength(beginWord, endWord, wordList))
time_end=time.time()
print('time cost',time_end-time_start,'s')

 最后参考题目的Discuss得到，可以通过对单词字母的广度优先搜索来避免对wordList的遍历。一个长度为m的单词，其所有相似单词共有25*m个。由于我们已经得到单词可能的搜索结果，只要查找其是否在wordList即可避免遍历。这样我们可以使用一个可哈希的数据结构存储wordList，这里使用了set。这样检测一个单词是否在wordList的耗时将缩减到最小。此时搜索的复杂度为O(1)。同样在搜索到一个相似单词时在set中删除它，这样可以避免维护一个容器来储存搜索过的单词。然后将代码优化如下：

import queue
import string

class State:
    def __init__(self, on, times):
        self.now = on
        self.times = times

class Solution:
    def ladderLength(self, beginWord, endWord, wordList):
        """
        :type beginWord: str
        :type endWord: str
        :type wordList: List[str]
        :rtype: int
        """
        self.q = queue.Queue()
        
        self.list = set(wordList)

        if endWord not in self.list:
            return 0

        self.end = endWord
        begin = State(beginWord, 1)
        self.q.put(begin)

        while not self.q.empty():
            temp = self.q.get()
            for index in range(len(temp.now)):
                for ch in string.ascii_lowercase:
                    if ch == temp.now[index]:
                        continue
                    else:
                        nextWord = temp.now[:index] + str(ch) + temp.now[index+1:]
                        if nextWord in self.list:
                            if nextWord == self.end:
                                return temp.times + 1
                            else:
                                self.q.put(State(nextWord, temp.times+1))
                                self.list.discard(nextWord)

        return 0

这次提交的Runtime为1064 ms - 1804ms 间浮动（实际上主要在1600ms-1800ms间浮动）。VScode调试计时为0.3s+。

假如进一步优化，将queue换成collections.deque，VScode调试计时将减小为0.2s+，LeetCode的Runtime为952ms - 1120ms

优化后如下：

import time
import string
import collections

class Solution:
    def ladderLength(self, beginWord, endWord, wordList):
        """
        :type beginWord: str
        :type endWord: str
        :type wordList: List[str]
        :rtype: int
        """
        
        self.list = set(wordList)

        if endWord not in self.list:
            return 0

        self.end = endWord
        self.q = collections.deque([(beginWord, 1)])

        while self.q:
            (w,t) = self.q.popleft()
            for index in range(len(w)):
                for ch in string.ascii_lowercase:
                    if ch == w[index]:
                        continue
                    else:
                        nextWord = w[:index] + str(ch) + w[index+1:]
                        if nextWord in self.list:
                            if nextWord == self.end:
                                return t + 1
                            else:
                                self.q.append((nextWord, t+1))
                                self.list.discard(nextWord)

        return 0

VScode调试计时： 

 LeetCode下的Runtime：

最后

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