动态规划DynamicProgrammingDP-天津大学计算机科学与技术学院

* * * * * * * * * * * * * * * * * * * * * yMax=min(w[n]-1,c) * * * * TraceBack For i=1 to n-1 If xi from P(i+1) then xi=1 else xi=0 * * * * * * * * * * * * * * * * * * * * * * * * * * * Constructing an LCS PrintLCS(b,X,i,j) 1. i=m 2. j=n; 3. if i==0 or j==0 then exit; 4. if b[i,j]==1 then { i=i-1; j=j-1; print “xi”; } 5. if b[i,j]==2 i=i-1 6. if b[i,j]==3 j=j-1 7. Goto Step 3. The time complexity: O(nm). 0 1 2 3 4 i Xi A B C Yj B B A C D 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 2 1 1 1 1 2 1 2 2 1 1 2 2 3 B B C B LCS (reversed order): B C B LCS (straight order): Hidden Markov Model HMM λ = (ΩI , ΩO, A, B, π ) ΩI= {i1,....iN}：set of state ΩO = {v1,...,vM}：set of observation A = {aij}，aij = p(It+1 = ij |It = ii): transition probability B = {bik}，bik = p(Ot = vk | It = ii): symbol probability π = {πi}， πi = p(I1 = ii): init state distribution Where: I=< i1,...iT > is the state sequence O= is the output sequence Three Problems of HMM Evaluation: Given model λ and output sequence O, computing p(O| λ) Decoding: Given model λ and output sequence O, to find the state sequence X such that: maximize {p(O, I| λ)} Learning Given output sequence O, to find the model λ such that: maximize {p(O, I| λ)} 复习要求 根据优化原理列递归式 设计实现递归式的迭代算法(列表) 应用 0/1背包问题 矩阵乘法链 求各对点之间的最短路 MNS 要求会做实例；分析算法的复杂度 课堂练习(1) 0/1背包问题: n=4,c=20,w=(10,15,6,9) p=(2,5,8,1) 产生元组集合P(1),P(2),P(3)和该背包问 实例的解 证明当重量和效益值均为整数时动态规划算法的时间复杂度为 O(min{2n,nΣ1≤i≤npi ,nc}) 提示:|P(i)|≤ 1+Σi≤j≤npj 课堂练习(2) 子集和数问题:设S={s1,s2,...,sn} 为n个正数的集合,试找出和数不超过M且最大的S的子集 该问题是NP-难度问题,试用动态规划法设计一算法 练习(3) r=(10,20,50,1,100),给出优化的乘法顺序和元素乘法数目 练习(4),习题19 T(i,j)=任务i 按第j种方式所需时间(j=1,2) C(i,j)=任务i 按第j种方式所需成本(j=1,2) 任务是拓扑排序的，必须先完成任务1才能完成任务2… 要求在时间t之前完成所有任务且成本最小 cost(i,j)＝任务1到i能在j时间内完成的最小成本 cost(1,j)=∞ j