最长公共子序列问题的改进算法

不同于动态规划法的一种新的求解最长公共子序列问题的方法,该算法主要是把求解公共字符串问题转化为求解矩阵L(p,m)的问题，在利用定理求解矩阵的元素过程中（1）while(i<k),L(k,i)=null，
                  （2）while(L(k,i)=k),L(k,i+1)=L(k,i+2)=…L(k,m)=k；

    求出每列元素，一直到发现第p+1 行都为null 时退出循环，得出矩阵L(k,m)后，B[L(1,m-p+1)]B[L(2,m-p+2)]…B[L(p,m)]即为A 和B 的LCS，其中p 为LCS 的长度。

public function construct_LCS_Matrix(str1:String,str2:String):Array{const INT_MAX:int = 999999;var matrix:Array=[[]];var m:int = str1.length;var n:int = str2.length;var k:int;for(k=0;k<=m;k++){matrix[0][k]=-1;}for(k=1;k<=m;k++){matrix[k]=[];for(var i:int=0;i<=m;i++){matrix[k][i] = INT_MAX;if(i<k){continue;}if(matrix[k][i-1]==k){matrix[k][i] = k;continue;}for(var j:int=0;j<n;j++){if(str1.charAt(i-1)==str2.charAt(j)&& j > matrix[k-1][i-1]){if(matrix[k][i-1]>-1){matrix[k][i] = (j<matrix[k][i-1] ? j : matrix[k][i-1]);}else{matrix[k][i] = j;}break;}}}if(matrix[k][m]==INT_MAX){break;}}return matrix;}

var a:String = 'gawegweew';var b:String = 'egwaeweewwfgewa';var arr:Array = ac.construct_LCS_Matrix(a,b);var indexes:Array=[];var subLen:int = arr[0].length-1;for(i=0;i<arr.length-2;i++){indexes.push(b.charAt(arr[arr.length-2-i][subLen-i]));}indexes.reverse();trace(indexes.join(''));

参考Url：http://blog.csdn.net/v_july_v/article/details/6695482

javascript 版本

function LCS(A, B) {  var m = A.length;  var n = B.length;  var L = [[]];  var maxkk = [];  for (var i=0; i<=m; i++) {    L[0][i] = 0;    if (L[i] == null) {      L[i] = [];    }    L[i][0] = 0;    maxkk[i] = 0;  }  var k=1;  for (k=1; k<=m; k++) {    for (var i=1; i<=m; i++) {      L[k][i] = L[k][i-1];      if (i < k) {        L[k][i] = 0; // 无效值 //i<k 时,L(k,i)=null，N 代表无穷大            continue;      }      if (L[k][i-1]==k && i <= m) {        L[k][i]=k; //L(k,i)=k 时,L(k,i+1)=L(k,i+2)=…L(k,m)=k          maxkk[i]=k;        continue;      }      if (i > k && L[k][k]==0) {        continue;      }      for (var j = 1; j <= n; j ++) {        //定理4 的实现   递增的方式         if (A[i-1]==B[j-1]) {          // console.log("ding ling4: " + j + " " + maxkk[i])          if (j > maxkk[i]) {            if (L[k][i-1] > 0) {              L[k][i] = j < L[k][i-1] ? j : L[k][i-1];            } else {              L[k][i] = j;            }            maxkk[i] = L[k][i];            break;          }        }       } // for j       if (L[k][i] < L[k-1][i-1]) {        L[k][i] = 0;      }    } // for i    // console.log(L[k].join(","));  } // for k  // console.log("LCS matrix getted " + L.length);  var L2 = [];  for (var i=0; i<L.length; i++) {    if (i > 0) {      L[i].splice(0,1);      // console.log(L[i].join(","));      var sum = 0;      for (var j=0; j<L[i].length; j++) {        sum += L[i][j];      }      if (sum > 0) {        L2.push(L[i]);        // console.log(L[i].join(","));      }          }  }function getMaximumSequence(userInput, answer) {    var userInputArr = userInput.split(" ");    var answerArr = answer.split(" ");    var matrix = LCS(userInputArr, answerArr);    if (matrix[0] == null) {      return "";    }    var subLen = matrix[0].length - 1;    var maximumSubs = [];    for (var j=0; j<matrix.length; j++) {      var zeroAppeared = false;      maximumSubs=[];      for(var i=0; i<matrix.length; i++) {        var line = matrix.length-1-i-j;        if (line < 0) {          break;        }        var index = matrix[line][subLen-i];        if (index == 0) {          zeroAppeared = true;          // console.log("zeroAppeared");          break;        }        maximumSubs.push(answerArr[index - 1]);      }      if (!zeroAppeared) {        break;      }    }        maximumSubs.reverse();    return maximumSubs.join(" ");  }

相关函数

var MISSING = 1;  var INCORRECT = 2;  var NEEDLESS = 3;  function verifyUserInput(userInput, answer) {    resultBox.innerHTML = answer;    var input = removeSymbols(userInput);    var rightOne = removeSymbols(answer);    var intersect = getMaximumSequence(input, rightOne);    console.log("intersect:"+intersect);    var inputArr = input.split(' ');    var rightArr = rightOne.split(' ');    var intersectArr = intersect.split(' ');    var len = inputArr.length;    if (len < rightArr.length) {      len = rightArr.length;    }    var errorObjMap = {};    var errorMapA = getErrorMap(inputArr, intersectArr);    var errorMapB = getErrorMap(rightArr, intersectArr);    for (var key in errorMapA) {      var error = {};      errorObjMap[key] = error;      error.incorrect = errorMapA[key];      if (errorMapB[key] == null) {        error.type = NEEDLESS;      } else {        error.correct = errorMapB[key];        error.type = INCORRECT;      }    }    for (var key in errorMapB) {      if (errorMapA[key] == null) {        var error = {};        error.correct = errorMapB[key];        error.type = MISSING;        errorObjMap[key] =  error;      }    }    var retStr="";    var errorStatistics = [[],[],[],[]];    for (var i=0; i<=intersectArr.length; i++) {      var error = errorObjMap[i];      if (error != null) {        switch (error.type) {          case MISSING:            retStr += " <b><font color='#009900'>" + error.correct + "</font></b> ";            errorStatistics[MISSING].push(error.correct);            break;          case INCORRECT:            retStr += " <font color='#990000'><del>" + error.incorrect + "</del></font color='#990000'> " +            "<b><font color='#009900'>" + error.correct + "</font></b> ";            errorStatistics[INCORRECT].push(error.correct);            break;          case NEEDLESS:            retStr += " <b><font color='#999900'><del>" + error.incorrect + "</del></font></b> ";            errorStatistics[NEEDLESS].push(error.incorrect);            break;        }      }      renderStatistics(errorStatistics);      var correct = intersectArr[i];      if (correct != null) {        retStr += " " + correct;      }    }    inputResult.innerHTML = retStr;  }  function test() {    var a = removeSymbols("OK, let's all heave it up at the same time. ");    var b = removeSymbols("ok let's   all  heave it up at the same time");    // console.log(getMaximumSequence("A B C A D F A G B", "A B H A I J K A G B"));    // console.log(getMaximumSequence("a b c d f g h i j k m", "a b c e f g h i j k n"));    // console.log(getMaximumSequence(a, b));    verifyUserInput(b, a);  }

function getErrorMap(strArr, intersectArr) {    var errorMap = {};    var index = 0;    for (var i=0; i<strArr.length; i++) {      if (index < intersectArr.length) {        if (intersectArr[index] != strArr[i]) {          if (errorMap[index] != null) {            errorMap[index] = errorMap[index] + " " + strArr[i];          } else {            errorMap[index] = strArr[i];          }        } else {          index++;        }      } else {        if (errorMap[index] != null) {          errorMap[index] = errorMap[index] + " " + strArr[i];        } else {          errorMap[index] = strArr[i];        }      }    }    return errorMap;  }    function removeSymbols(input) {    var str = input.trim();    str = str.replace(/\s+/g, ' '); //多个连续空格替换为一个空格    var arr = [];    for (var i=0; i<str.length; i++) {      var chr = str.charAt(i);      if (IsAlpha(chr)) {        arr.push(chr.toLocaleLowerCase());      }      if (chr == ' ' || chr == '-' || chr == '"' || chr == "'") {        arr.push(chr);      }    }    return arr.join('');  }  function IsAlpha(cCheck) {     return ((('a'<=cCheck) && (cCheck<='z')) || (('A'<=cCheck) && (cCheck<='Z')));  }  function getMaximumSequence(userInput, answer) {    var userInputArr = userInput.split(" ");    var answerArr = answer.split(" ");    var matrix = LCS(userInputArr, answerArr);    if (matrix[0] == null) {      return "";    }    var subLen = matrix[0].length - 1;    var maximumSubs = [];    for (var j=0; j<matrix.length; j++) {      var zeroAppeared = false;      maximumSubs=[];      for(var i=0; i<matrix.length; i++) {        var line = matrix.length-1-i-j;        if (line < 0) {          break;        }        var index = matrix[line][subLen-i];        if (index == 0) {          zeroAppeared = true;          // console.log("zeroAppeared");          break;        }        maximumSubs.push(answerArr[index - 1]);      }      if (!zeroAppeared) {        break;      }    }        maximumSubs.reverse();    return maximumSubs.join(" ");  }  function renderStatistics(errorData) {    wrongSpellSp.innerHTML= errorData[INCORRECT].length + " " + errorData[INCORRECT].join(',');    missSpellSp.innerHTML= errorData[MISSING].length + " " + errorData[MISSING].join(',');    needlessSpellSp.innerHTML= errorData[NEEDLESS].length + " " + errorData[NEEDLESS].join(',');  }