動态規劃：PAT 1045 Favorite Color Stripe

ADS的第7周作業，是一個動态規劃算法題。

問題重制

Title: Favorite Color Stripe

Description:

Eva is trying to make her own color stripe out of a given one. She would like to keep only her favorite colors in her favorite order by cutting off those unwanted pieces and sewing the remaining parts together to form her favorite color stripe.

It is said that a normal human eye can distinguish about less than 200 different colors, so Eva’s favorite colors are limited. However the original stripe could be very long, and Eva would like to have the remaining favorite stripe with the maximum length. So she needs your help to find her the best result.

Note that the solution might not be unique, but you only have to tell her the maximum length. For example, given a stripe of colors {2 2 4 1 5 5 6 3 1 1 5 6}. If Eva’s favorite colors are given in her favorite order as {2 3 1 5 6}, then she has 4 possible best solutions {2 2 1 1 1 5 6}, {2 2 1 5 5 5 6}, {2 2 1 5 5 6 6}, and {2 2 3 1 1 5 6}.

Input Specification:

Each input file contains one test case. For each case, the first line contains a positive integer N ( N≤200) which is the total number of colors involved (and hence the colors are numbered from 1 to N ). Then the next line starts with a positive integer M ( M≤200 ) followed by MM Eva’s favorite color numbers given in her favorite order. Finally the third line starts with a positive integer L (L≤104​​ ) which is the length of the given stripe, followed by L colors on the stripe. All the numbers in a line a separated by a space.

Output Specification:

For each test case, simply print in a line the maximum length of Eva’s favorite stripe.

Sample Input:

6
5 2 3 1 5 6
12 2 2 4 1 5 5 6 3 1 1 5 6
           

Sample Output:

7
           

解法說明

設Eva喜歡的序列為M[0...m)，給出條紋的序列為 L[0...l) 。

經過裁減，符合條件序列的最大長度（最大公共序列長度）的解為 f(m,l) 。

對于狀态空間 {(m,l)|m∈M,l∈L} ，

平凡态：

f(0,l)=f(m,0)=0

有狀态轉移方程：

f(m,l)={f(m,l−1)+1,max{f(m−1,l),f(m,l−1)},Mm−1=Ll−1Mm−1≠Ll−1

平凡态比價容易了解：空序列的長度為0。

現在考慮當Eva喜歡的序列不變時，每當給出的條紋多了一個，即 Ll−1 ，考慮它是否與Eva喜歡的序列的最後一個( Mm−1 )相同，即它是否可以拓展，如果可以，它可以直接加到 f(m,l−1) 上，這個情況也比較好了解。

最後考慮當這個新來的條紋不比對時，它，或者Eva喜歡的最後一個顔色，兩者之一總有一個沒有卵用的：當新來的條紋不存在( f(m,l−1) )或者Eva喜歡的最後一個顔色不存在( f(m−1,l) )，比較一下哪個比較大就取哪個即可。

代碼實作

• 遞歸記憶化搜尋版本
• 空間優化并使用疊代的版本
• 半線上算法進一步優化空間版本