ICS%20241

1
ICS 241

• Discrete Mathematics II
• William Albritton, Information and Computer
  Sciences Department at University of Hawaii at
  Manoa
• For use with Kenneth H. Rosens Discrete
  Mathematics Its Applications (5th Edition)
• Based on slides originally created by
• Dr. Michael P. Frank, Department of Computer
  Information Science Engineering at University
  of Florida

2
6.3 Divide Conquer R.R.s

• Many types of problems are solvable by reducing a
  problem of size n into some number a of
  independent subproblems, each of size ??n/b?,
  where a?1 and bgt1.
• The time complexity to solve such problems is
  given by a recurrence relation
• T(n) aT(?n/b?) g(n)

3
Divide Conquer Example

• Binary search
• Reduce problem to 1 sub-problem (namely,
  searching half of the list) (thus, a1) of size
  ??n/2? (thus, b2).
• Use two comparisons to implement reduction (g(n)
  2)
• Which half of list
• Whether any terms remain
• So T(n) T(?n/2?)2

4
Divide Conquer Example

• Find min max of a sequence
• If n1, then a1 is max min
• If ngt1, then split into two sequences (so a2)
• Each new sequence is one-half original (so b2)
• Use two comparisons to find max min of two
  sequences (so g(n) 2)
• So T(n) 2T(?n/2?)2

5
Divide Conquer Example

• Merge sort
• Break list of length n into 2 sublists (a2),
• Each of size ??n/2? (so b2),
• Sort them, then merge, in g(n) T(n) time (fewer
  than n comparisons)
• So T(n) 2T(?n/2?) n

6
Class Exercise

• Section 6.3, Exercise 7.a.b.c. (p. 433)
• Each pair of students should use only one sheet
  of paper while solving the class exercises

7
Fast Multiplication Example

• The ordinary grade-school algorithm takes T(n2)
  steps to multiply two n-digit numbers.
• Seems like too much work!
• Lets find an asymptotically faster algorithm!
• To find the product cd of two 2n-digit base-b
  numbers, c(c2n-1c2n-2c0)b and
  d(d2n-1d2n-2d0)b first, we break c and d in
  half cbnC1C0, dbnD1D0, and then...
  (see next slide)

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Derivation of Fast Multiplication

(Multiply out polynomials)
(Factor last term)
9
Recurrence Rel. for Fast Mult.

• Notice that the time complexity T(n) of the fast
  multiplication algorithm obeys the recurrence
• T(2n)3T(n)?(n)
• T(n)3T(n/2)Cn
• So a3, b2.

Time to do the needed adds subtracts of
n-digit and 2n-digit numbers
10
Theorem 1

• Consider increasing function f(n) that satisfies
  the recurrence relation
• f(n) af(n/b) c, with a1, integer bgt1, real
  cgt0. Then
• f(n) is ?(nlog b a) if a gt 1
• f(n) is ?(log n) if a 1
• When n bk, with k positive integer
• f(n) C1 n log b a C2
• C1 f(1) c/(a-1) C2 -c/(a-1)

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Theorem 1 Example

• Let f(n) 5f(n/2)3, f(1) 7, n2k. Find f(n)
  estimate f(n) if f is an increasing function
• a5, b2, c3, nbk 2k
• f(n) C1 n log b a C2
• n log b a nlog 2 5
• C1 f(1) c/(a-1) C2 -c/(a-1)
• C1 7 3/4 31/4 C2 -3/4
• f(n) (31/4) nlog 2 5 3/4
• f(n) is ?(nlog b a) if a gt 1, so ?(nlog25)

12
Class Exercise

• Section 6.3, Exercise 12 13 (p. 434)
• Work together in pairs using one sheet of paper
  to solve the class exercises
• Use Theorem 1 to solve these

13
The Master Theorem

• Consider a function f(n) that, for all nbk for
  all k?Z,,satisfies the recurrence relation
• f(n) af(n/b) cnd
• with a1, integer bgt1, real cgt0, d0. Then

14
Master Theorem Example

• Recall that complexity of fast multiply was
• T(n) 3T(n/2) Cn
• Thus, a3, b2, d1. So a gt bd, so case 3 of the
  master theorem applies, so
• which is O(n1.58), so the new algorithm is
  strictly faster than ordinary T(n2) multiply!

15
Class Exercise

• Exercise 21.a.b. (p. 434)
• Work together in pairs using one sheet of paper
  to solve the class exercises