最长公共子序列

/**
 * Longest Common Subsequence
 * 
 * f(i, j) = max{f(i-1, j-1), f(i-1, j-1)+1, f(i-1, j), f(i, j-1)}
 * 
 * 主要在空间优化，逐行扫描，下一行只依赖与上一行和本行，可以只需要定义2个数组，更好的办法只需要一个数组
 * 每次计算出一个新值，先不更新，等下一个求出来之后，再更新前一个。
 * O(min{m, n}+1)
 *
 */
public class LCS {

	public static int lcs(int[] a, int[] b){
		assert(true);
		int[][] v = new int[a.length+1][b.length+1];
		for(int i=1; i<=a.length; i++){
			for(int j=1; j<=b.length; j++){
				int max = v[i-1][j-1];
				if(a[i-1] == b[j-1]){
					max++;
				}
				int max2 = v[i-1][j] > v[i][j-1] ? v[i-1][j] : v[i][j-1];
				v[i][j] = max > max2 ? max : max2;
			}
		}
		return v[a.length][b.length];
	}
	
	public static int lcs2(int[] a, int[] b){
		int longLength = a.length;
		int shortLength = b.length;
		int[] longArray = a;
		int[] shortArray = b;
		if(a.length < b.length){
			longLength = b.length;
			shortLength = a.length;
			longArray = b;
			shortArray = a;
		}
		
		int[] v = new int[shortLength+1];
		for(int i=1; i<=longLength; i++){
			int pre = 0;
			for(int j=1; j<=shortLength; j++){
				int max = v[j-1];
				if(longArray[i-1] == shortArray[j-1]){
					max++;
				}
				int max2 = pre>v[j] ? pre : v[j];
				v[j-1] = pre;
				pre = max>max2 ? max : max2;
			}
			v[shortLength] = pre;
		}
		return v[shortLength];
	}
	
	/**
	 * @param args
	 */
	public static void main(String[] args) {
		int[] a = {1, 3, 9, 6, 2,4,7,5};
		int[] b = {7,2,4,8,3,5,1,9,0,6};
		
		int max = LCS.lcs(a, b);
		int max2 = LCS.lcs2(a, b);
		System.out.println(max);
		System.out.println(max2);

	}

}