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
computational problems.
for
solving
• 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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MEASURING COMPLEXITY (cont.)
• 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.
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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 for
describing 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.)
In stage 1
• 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.
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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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BIG-O AND SMALL-O NOTATION
(cont.)
• 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.
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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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COMPLEXITY RELATIONSHIPS AMONG
MODELS (cont.)
• 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))
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SPACE COMPLEXITY
INTRACTABILITY
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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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INTRACTABILITY
Pages 335 - 338
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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 .
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