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(',');
  }