求最长公共子序列-DP问题
Longest common subsequence problem
The longest common subsequence (LCS) problem is the problem of finding the longest subsequence common to all sequences in a set of sequences (often just two sequences). It differs from the longest common substring problem: unlike substrings, subsequences are not required to occupy consecutive positions within the original sequences.
Example
- LCS for input Sequences
ABCDGHandAEDFHRisADHof length 3. - LCS for input Sequences
AGGTABandGXTXAYBisGTABof length 4
以s1={1,3,4,5,6,7,7,8},s2={3,5,7,4,8,6,7,8,2}为例。
1.构建二维矩阵A[n+1][n+1] s1为列，s2为行
（1）A[0][any]=0,A[any][0]=0
（2）if(s1[columnIndex-1]==s2[rowIndex-1])则A[rowIndex][columnIndex]=A[rowIndex-1][columnIndex-1]+1;
else A[rowIndex][columnIndex]=max(A[rowIndex][columnIndex-1],A[rowIndex-1][columnIndex])
（3）由A[8][9]可知最大子序列长度。
图如下：
2.针对构建的二维矩阵求最长子序列
（1）当columnIndex>0或者rowIndex>0时。
（2）如果s1[columnIndex-1]==s2[rowIndex-1]则longestSequence.unshift(s1[columnIndex-1]).rowIndex--;columnIndex--.
（3）如果不等，
则1.如果A[rowIndex][columnIndex]==A[rowIndex][columnIndex-1];columnIndex--;//向左
2.否则 rowIndex--;
代码如下：
/**
* @param {string[]} set1
* @param {string[]} set2
* @return {string[]}
*/
export default function longestCommonSubsequence(set1, set2) {
// Init LCS matrix.
const lcsMatrix = Array(set2.length + ).fill(null).map(() => Array(set1.length + ).fill(null)); // Fill first row with zeros.
for (let columnIndex = ; columnIndex <= set1.length; columnIndex += ) {
lcsMatrix[][columnIndex] = ;
} // Fill first column with zeros.
for (let rowIndex = ; rowIndex <= set2.length; rowIndex += ) {
lcsMatrix[rowIndex][] = ;
} // Fill rest of the column that correspond to each of two strings.
for (let rowIndex = ; rowIndex <= set2.length; rowIndex += ) {
for (let columnIndex = ; columnIndex <= set1.length; columnIndex += ) {
if (set1[columnIndex - ] === set2[rowIndex - ]) {
lcsMatrix[rowIndex][columnIndex] = lcsMatrix[rowIndex - ][columnIndex - ] + ;
} else {
lcsMatrix[rowIndex][columnIndex] = Math.max(
lcsMatrix[rowIndex - ][columnIndex],
lcsMatrix[rowIndex][columnIndex - ],
);
}
}
} // Calculate LCS based on LCS matrix.
if (!lcsMatrix[set2.length][set1.length]) {
// If the length of largest common string is zero then return empty string.
return [''];
} const longestSequence = [];
let columnIndex = set1.length;
let rowIndex = set2.length; while (columnIndex > || rowIndex > ) {
if (set1[columnIndex - ] === set2[rowIndex - ]) {
// Move by diagonal left-top.
longestSequence.unshift(set1[columnIndex - ]);
columnIndex -= ;
rowIndex -= ;
} else if (lcsMatrix[rowIndex][columnIndex] === lcsMatrix[rowIndex][columnIndex - ]) {
// Move left.
columnIndex -= ;
} else {
// Move up.
rowIndex -= ;
}
} return longestSequence;
}
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