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
aA.
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 in 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.
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Example
i j 0
1 T
2 T
3 T
4 T
5 T
1
T
T
T
T
T
2
F
F
F
F
F
3
F
F
F
T
T
4
F
F
F
T
T
5
F
F
T
T
T
6
F
F
T
T
T
7
F
F
F
F
F
8
F
F
F
T
T
9
F
T
T
T
T
10
F
T
T
T
T
11
F
F
F
F
T
12
F
F
F
T
T
13
F
F
F
T
T
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.
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Backtracking
B  0.5aA s(a)
i=n; W=B;
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); print “ai” i=i-1; }
}
}
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Time complexity
• The algorithm takes at most O(nB) time to fill in the
table, where B  0.5aA s(a) (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 ).
So B is at least O(2n). The input size is O(n2) 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.)
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0-1 version
5
6
7
8
9
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Change-making problem:
Given an amount n and unlimited quantities of coins
of each of the denominations
d1, d2, …, dm,
find the smallest number of coins that add up to n or
indicate that the problem does not have a solution.
Solution:
Let d(i) be the minimum number of coins used for
amount i.
Initial values: d(0)=0, d(i)=.
equation: d(i) = min d(i-dk)+1.
k=1, 2, …, m.
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Change-making problem:
Initial values: d(0)=0, d(i)=.
equation: d(i) = min d(i-dk)+1
k=1, 2, …, m & i-d ≥0
k
d1=2 and d2=5. i=7.
i= 0, 1, 2, 3, 4, 5, 6, 7
d(i): 0, , 1, , 2, 1, 3, 2.
Backtracking: $5, $2.
(how do you get d(7)=2? d(7)=d(2)+5. Print $5 and
goto d(2). How do you get d(2)? D(2)=2+d(0). Print
out $2 and goto d(0). Whenever reach d(0), we stop. )
for i=3, since d(3)= , there is no solution.
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More On Dynamic programming
Algorithms
Shortest path with edge constraint:
Let G=(V, E) be a directed graph with weighted edges. Let s and v be
two vertices in V. Find a shortest path from s to u with exactly k
edges. Here kn-1 is part of the input.
Solution:
Define d(i, v) be the length of the shortest path from s to v with exactly i
edges.
d(i, v)=min {c(w, v)+d(i-1, w)}
wV.
Initial values: d(i, s)=0, for i=0, d(i,s)=  for i=1, 2, …, d(0, v)=;
d(k, v) will give the length of the shortest path. A backtracking process can
give the path.
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u
5
v
8
8
-2
6
z 0
-3
8
7
-4
2
8
8
7
9
x
y
i
z,
u,
x,
v,
y
0
0




1 
6z
7z
2 

14u
4x
3
2v

9y
4y


2u
23x.
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• 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.
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A
A
A
C
Figure 1
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Assignment 4.
(Due on Friday of Week 13. Drop it in Mail Box
71 or 72)
This time, Sze Man Yuen and I can explain the
questions, but we will NOT tell you how to solve
the problems.
Question 1. (30 points) Give a polynomial time
algorithm to find the longest monotonically
increasing subsequence of a sequence of n numbers.
Your algorithm should use linear space. (10 points
for linear space) (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.
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Assignment 4.
• Question 2. (30 points) 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.
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Assignment 4.
Question 3. (40 points). Suppose you have one
machine and a set of n jobs a1, a2, …, an to process
on that machine. Each job aj has an integer
processing time tj, a profit pj and an integer deadline
dj. The machine can process only one job at a time,
and job aj must run uninterruptedly for tj
consecutive time units. If job aj is completed by its
deadline dj, you receive a profit pj, but if it is
completed after its deadline, you receive a profit of
0. Give a dynamic programming algorithm to find
the schedule that obtains the maximum amount of
profit. What is the running time of your algorithm?
(Let d be the biggest deadline, the running time can be
related to d. )
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