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Pseudo-polynomial time algorithm (The concept and the terminology are important). Partition Problem: Input: Finite set A=(a 1 , a 2 , …, a n } and a size s(a) (integer) for each a A. Question: Is there a subset A’A such that a A’ s(a) = a A –A’ s(a)?

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## Pseudo-polynomial time algorithm (The concept and the terminology are important)

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**Pseudo-polynomial time algorithm(The concept and the**terminology are important) Partition Problem: Input: Finite set A=(a1, a2, …, an} and a size s(a) (integer) for each aA. Question: Is there a subset A’A such that a A’ s(a) = a A –A’ s(a)? Theorem: Partition problem is NP-complete (Karp, 1972). An dynamic algorithm: For in and j 0.5 a A s(a) , define t(i, j) to be true if and only if there is a subset Ai of {a1, a2, …, ai} such that a Ai s(a)=j. Formula: T(i,j)=true if and only if t(i-1, j)=true or t(i-1, j-s(ai))=true.**Example**j Figure 4.8 Table of t(i,j) for the instance of PARTITION for which A={a1,a2,a3,a4,a5}, s(a1)=1, s(a2)=9, s(a3)=5, s(a4)=3, and s(a5)=8. The answer for this instance is "yes", since t(5,13)=T, reflecting the fact that s(a1)+s(a2)+s(a4)=13=26/2.**Backtracking**if t(n, W) is not true then print “ no such partition” and stop; i=n; w=W; if ( t(n, W)== false) then stop; While (i> 0 ) do { if (t(i, w) == true) { if (t(i-1, W)== true) then i=i-1; else { W=W-s(ai); i=i-1; print “ai”} } }**Time complexity**The algorithm takes at most O(nB) time to fill in the table. (Each cell needs constant time to compute). Do we have a polynomial time algorithm to solve the Partition Problem and thus all NP-complete problems? No. O(nb) is not polynomial in terms of the input size. S(ai)=2n=10000…0 . (binary number of n+1 bits , n 0’s). So B is at least O(2n). The input size is O(n) if there some ai with S(ai)=2n. B is not polynomial in terms of n (input size) in general. However, if any upper bound is imposed on B, (e.g., B is Polynomial), the problem can be solved in polynomial time for this special case. (This is called pseudo-polynomial.)**Exercise: Let T be a rooted binary tree, where each**internal node in the tree has two children and every node (except the root) in T has a parent. Each leaf in the tree is assigned a letter in ={A, C, G, T}. Figure 1 gives an example. Consider an edge e in T. Assume that every end of e is assigned a letter. The cost of e is 0 if the two letters are identical and the cost is 1 if the two letters are not identical. The problem here is to assign a letter in to each internal node of T such that the cost of the tree is minimized, where the cost of the tree is the total cost of all edges in the tree. Design a polynomial-time dynamic programming algorithm to solve the problem.**A**A C Figure 1 A**Assignment 4. (Due on May 2, 2007. Drop it in Mail Box 59)**This time, Helena and I can explain the questions, but we will NOT tell you how to solve the problems. Question 1. Give a polynomial time algorithm to find the longest monotonically increasing subsequence of a sequence of n numbers. (Assume that each integer appears once in the input sequence of n numbers) Example: Consider sequence 1,8, 2,9, 3,10, 4, 5. Both subsequences 1, 2, 3, 4, 5 and 1, 8, 9, 10 are monotonically increasing subsequences. However, 1,2,3, 4, 5 is the longest.**Assignment 4. (Due on May 2, 2007)**• Question 2. Given an integer d and a sequence of integers s=s1s2…sn. Design a polynomial time algorithm to find the longest monotonically increasing subsequence of s such that the difference between any two consecutive numbers in the subsequence is at least d. • Example: Consider the input sequence 1,7,8, 2,9, 3,10, 4, 5. The subsequence 1, 2, 3, 4, 5 is a monotonically increasing subsequence such that the difference between any two consecutive numbers in the subsequence is at least 1. • 1, 3, 5 is a monotonically increasing subsequence such that the difference between any two consecutive numbers in the subsequence is at least 2.**Assignment 4. (Due on May 2, 2007)**Question 3. Suppose that there are n sequences s1, s2, …, snon alphabet ={a1, a2, …, am }. Every sequence si is of length m and every letter in appears exactly once in each si. Design a polynomial time algorithm to compute the LCS of the n sequences. What is the time complexity of your algorithm?

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