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

1
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.

2
Example
j
i 0 1 2 3 4 5 6 7 8 9 10 11 12 13
1 T T F F F F F F F F F F F F
2 T T F F F F F F F T T F F F
3 T T F F F T T F F T T F F F
4 T T F T T T T F T T T F T T
5 T T F T T T T F T T T T T T
Figure 4.8 Table of t(i,j) for the instance of
PARTITION for which Aa1,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)1326/2.
3
Backtracking

• in WB
• if ( t(n, W) false) then stop
• While (igt 0 ) do
• if (t(i, W) true)
• if (t(i-1, W) true) then ii-1
• else
• WW-s(ai) print ai
  ii-1

4
Time complexity

• The algorithm takes at most O(nB) time to fill
  in the table, where (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)2n100000 . (binary number of n1 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.)

5
0-1 version
6
(No Transcript)
7
(No Transcript)
8
(No Transcript)
9
(No Transcript)
10
(No Transcript)
11

• 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.
• k1, 2, , m.

12

• Change-making problem
• Initial values d(0)0, d(i)?.
• equation d(i) min d(i-dk)1
• k1, 2, , m
  i-dk0
• d12 and d25. i7.
• 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)2d(0).
  Print out 2 and goto d(0). Whenever reach d(0),
  we stop. ) for i3, since d(3) ?, there is no
  solution.

13

• 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 i0,
  d(i,s) ? for i1, 2, , d(0, v)?
• d(k, v) will give the length of the shortest
  path. A backtracking process can give the path.

14

• i z, u, x, v, y
• 0 0 ? ? ? ?
• 1 ? 6z 7z ? ?
• ? ? 14u 4x 2u
• 3 4y 2v ? 9y 23x.

15

• 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.

16

A
17

• 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.

18

• Assignment 4.
• Question 2. (30 points) Given an integer d and
  a sequence of integers ss1s2sn. 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.

19

• 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. )