Chapter 7: Computation

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7.1 An introduction to algorithms

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The if statement and the if-else statement

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The for loop

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The while loop

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7.2 Asymptotic growth of functions

O notation

Definition 7.2.1: O notation.

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Proving that f is O(g)

Proof.

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Proving that f is not O(g)

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Ω notation

Definition 7.2.2: Ω Notation.

Proving that f is \Omega(g)

\Theta notation and polynomials

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Asymptotic growth of logarithm functions with different bases

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The growth rate of common functions in analysis of algorithms

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Rules about asymptotic growth

Figure 7.2.3: Rules for the asymptotic growth of functions.

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7.3 Analysis of algorithms

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ComputeSum

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ComputeSum.

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7.3.2: Number of atomic operations performed by ComputeSum.

Asymptotic complexity

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FindSmallest

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FindSmallest.

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SequenceProduct

AverageOfEnds

Worst-case complexity

SearchSequence

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CountDuplicatePairs

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Proof.

FindDuplicate

Additional information: 7.3.1: Efficient algorithms.

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CountValuesLessThanT

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MysteryAlgorithm

7.4 Finite state machines

COIN, COIN, PUSH, COIN, PUSH.

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INPUT: COIN, COIN, PUSH, PUSH.

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BUY/GUMBALL

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MESSAGE”.

NONE”.

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Finite state machines that recognize properties

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Recognizing valid passwords

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FSM 1

FSM 2

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7.5 Turing machines

The definition of a Turing machine

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A simple Turing machine

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A more complex Turing machine

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7.6 Decision problems and languages

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Additional information 7.6.1: Uncomputable problems.

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