Chapter 1 Introduction Mathematics Review

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Chapter 1IntroductionMathematics Review
Sections 1.1, 1.2
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Outline

• Why does program execution time for large inputs
  matter?
• Basic mathematical background

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Selection Problem

• Find the kth largest number from a group of N
  numbers
• How would you solve this?
• Algorithm 1
• Sort the N numbers and pick the kth one.
• Easy to implement, but
• Sorting requires many comparisons
• Much work comparing elements having no chance to
  be at position K
• To sort N elements, need N log2(N)
  comparisons/swaps in general
• An unsorted data set with 10,000,000 elements and
  1,000 swaps/sec will take around 2 hours

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Selection Problem (Contd)

• Algorithm 2 (Better)
• Sort first K elements in the array.
• Then insert elements (K1) to N, discarding the
  smallest element each time.
• Then pick the kth element.
• What if N10 million and K5,000,000?
• How long do you think this would take?
• Both algorithms are impractical.
• A better algorithm can solve this in a second!

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Mathematics Review

• Exponents
• Logarithms
• Series
• Modular arithmetic
• Proof techniques

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Exponents
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Logarithms

• All logarithms are to the base 2, unless
  otherwise specified
• Definition 1.1
• XA B if and only if logxB A

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Properties of logarithms

• Theorem 1.1 (base-change)
• A, B, C gt
  0, A ? 1
• Proof Let X logCB, Y logCA, and Z logAB
• Then CX B, CY A, and AZ B
• Implies B CX (CY)Z CYZ
• Hence X YZ
• And Z X/Y

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Logarithms (contd.)

• Theorem 1.2
• log(AB) log A log B A, B gt 0
• Proof (Assume base 2)
• Let X log A, Y log B, and Z log AB
• Hence 2X A, 2Y B, and 2Z AB
• Combining the above 2Z AB 2X2Y 2(XY)
• Implies X Y Z

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Logarithms (contd.)

• More results
• log(A/B) log A log B
• log(AB) B log A
• log X lt X for all X gt 0
• log 1 0
• log2 2 1
• log2 1024 10
• log2 1,048,576 20

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Geometric Series

• If 0 lt A lt 1,
• If N -gt , we have
• How??

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Arithmetic Series

• How about 258(3k-1) ?

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Modular Arithmetic

• We say that A is congruent to B modulo N
• Written as (mod N)
• If N divides (A-B)
• In other words, the remainder is the same if
  either A or B are divided by N
• E.g.
• (mod 10)
• Similar to equality, if (mod N), then
• (mod N), and
  (mod N)

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Proof techniques

• Two common proof techniques in data structure and
  algorithm analysis (and in CS, in general)
• Proof by induction
• Proof by contradiction
• Another common technique
• Proof a statement false with a counterexample

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Proof by Induction

• Given a theorem
• First prove a base case
• Show the theorem is true for some small
  degenerate values
• Next assume an inductive hypothesis
• Assume the theorem is true for all cases up to
  some limit k
• Then prove that the theorem holds for the next
  value (k1)

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Proof by Induction - example

• Fibonacci Series
• F0 1, F1 1, Fi F(i-1) F(i-2), for igt1
• Show that
• Fi lt (5/3)i,for igt0
• Base case
• F1 1 lt 5/3
• F2 2 lt (5/3)225/9
• Inductive Hypothesis
• Assuming Fi lt (5/3) i , i 1, 2, ..., k

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Proof by Induction - example (contd.)

• Now prove that Fk1 lt (5/3)k1
• From definition of Fibonacci Sequence
• Fk1 Fk Fk-1
• Using inductive hypothesis

Fk1 lt (5/3) k (5/3) k-1 (5/3) k1 3/5
(3/5)2 (5/3) k1 24/25 lt (5/3)
k1
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Other types of proofs

• Disprove with a counter-example
• The statement is false in the
  Fibonacci series
• Proof F11 144 gt 112 121
• Proof by contradiction
• Initially assume that the theorem is false
• Then show that some known property would be false
  as well.
• Example There is an infinite number of prime
  numbers
• Proof
• Assume the theorem is false (so there are only
  finite prime)
• Let P1, P2, ..., Pk be all the primes in
  increasing order.
• Let N P1?P2 ? ??? ?Pk 1,N is gt Pk , so it
  is not a prime
• But it is also not divisible by any of the listed
  primes, contradicting the factorization of
  integers into primes.
• We reach a contradiction