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TIME COMPLEXITY

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Pages 247 - 256

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Introduction

• Even when a problem:- is decidable and- computationally solvable

• it may not be solvable in practice if the solution requires an inordinate amount of time or memory.

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Objectives

• investigation of the:- time, - memory, or - Other resources required for solving

computational problems.

• to present the basics of time complexity theory.

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Objectives (cont.)

• First - introduce a way of measuring the time used to solve a problem.

• Then - show how to classify problems according to the amount of time required.

• After - discuss the possibility that certain decidable problems require enormous amounts of time and how to determine when you are faced with such a problem.

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MEASURING COMPLEXITY

The language

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MEASURING COMPLEXITY (cont.)

• How much time does a single-tape Turing machine need to decide A?

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• The number of steps that an algorithm uses on a particular input may depend on several parameters: (if the input is a graph)

- the number of steps may depend on :- the number of nodes, - the number of edges, and - the maximum degree of the graph, or - some combination of these and/or other

factors.

MEASURING COMPLEXITY (cont.)

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Analysis

• worst-case analysis - consider the longest running time of all inputs of a particular length.

• average-case analysis - consider the average of all the running times of inputs of a particular length.

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BIG-O AND SMALL-O NOTATION

• Exact running time of an algorithm often is a complex expression. (estimation)

• Asymptotic analysis - seek to understand the running time of the algorithm when it is run on large inputs.

The asymptotic notation or big-O notation fordescribing this relationship is

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Self study

• EXAMPLE 7.3

• EXAMPLE 7.4

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BIG-O AND SMALL-O NOTATION (cont.)

• Performing this scan uses n steps. • Typically use n to represent the length of the

input. • Repositioning the head at the left-hand end of

the tape uses another n steps. • The total used in this stage is 2n steps.

In stage 1

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BIG-O AND SMALL-O NOTATION (cont.)

In stage 4 the machine makes a single scan to decide whether to accept or reject. The time taken in this stage is at most O(n).

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• Thus the total time of M1 on an input of length n is

O(n) + O(n2 ) + O(n) or O(n2 ).

In other words, it's running time is O(n2 ), which completes the time analysis of this machine.

BIG-O AND SMALL-O NOTATION (cont.)

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Is there a machine that decides A asymptotically more quickly?

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Executed Time

• Stages 1 and 5 are executed once, taking a total of O(n) time.

• Stage 4 crosses off at least half the 0s and 1s is each time it is executed, so at most 1+log2 n.

• the total time of stages 2, 3, and 4 is (1 + log2 n)O(n), or O(n log n).

• The running time of M2 is O(n) + O (n logn) = O(n log n).

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COMPLEXITY RELATIONSHIPS AMONG MODELS

• We consider three models: - the single-tape Turing machine;- the multi-tape Turing machine; and - the nondeterministic Turing machine

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• convert any multi-tape TM into a single-tape TM that simulates it.

• Analyze that simulation to determine how much additional time it requires.

• simulating each step of the multi-tape machine uses at most O(t(n)) steps on the single-tape machine.

• the total time used is O(t2 (n)) steps.O(n) + O(t2 (n)) running time O(t2 (n))

COMPLEXITY RELATIONSHIPS AMONG MODELS (cont.)

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SPACE COMPLEXITY I N T R A C T A B I L I T Y

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Pages 303 – 308

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Objective

• Consider the complexity of computational problems in terms of the amount of space, or memory, that they require.

• Time and space are two of the most important considerations when we seek practical solutions to many computational problems.

• Space complexity shares many of the features of time complexity and serves as a further way of classifying problems according to their computational difficulty.

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Introduction

• select a model for measuring the space used by an algorithm.

• Turing machines are mathematically simple and close enough to real computers to give meaningful results.

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Estimation the space complexity

• We typically estimate the space complexity of Turing machines by using asymptotic notation.

• EXAMPLE 8.3

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EXAMPLE 8.4

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SAVITCH'S THEOREM read only

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Pages 335 - 338

I N T R A C T A B I L I T Y

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• Certain computational problems are solvable in principle, but the solutions require so much time or space that they can't be used in practice. Such problems are called intractable.

• Turing machines should be able to decide more languages in time n3 than they can in time n2. The hierarchy theorems prove that .