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[资源] 【分享】How to think about algorithms.Jeff Edmonds.Cambridge.2008[New]

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How to think about algorithms.

Authors
Jeff Edmonds
Publisher: 2008 Cambridge University Press
Pages: 462
ISBN 13:

ISBN-13 978-0-511-41370-4 eBook (EBL)
ISBN-13 978-0-521-84931-9 hardback
ISBN-13 978-0-521-61410-8 paperback

Introduction
From determining the cheapest way to make a hot dog to monitoring the workings
of a factory, there are many complex computational problems to be solved. Before
executable code can be produced, computer scientists need to be able to design the
algorithms that lie behind the code, be able to understand and describe such algorithms
abstractly, and be confident that they work correctly and efficiently. These are
the goals of computer scientists.
A Computational Problem: A specification of a computational problem uses preconditions
and postconditions to describe for each legal input instance that the computation
might receive, what the required output or actions are. This may be a function
mapping each input instance to the required output. It may be an optimization
problem which requires a solution to be outputted that is “optimal” from among a
huge set of possible solutions for the given input instance. It may also be an ongoing
system or data structure that responds appropriately to a constant stream of input.
Example: The sorting problem is defined as follows:
Preconditions: The input is a list of n values, including possible repetitions.
Postconditions: The output is a list consisting of the same n values in nondecreasing
order.
An Algorithm: An algorithm is a step-by-step procedure which, starting with an input
instance, produces a suitable output. It is described at the level of detail and abstraction
best suited to the human audience that must understand it. In contrast,
code is an implementation of an algorithm that can be executed by a computer. Pseudocode
lies between these two.
An Abstract Data Type: Computers use zeros and ones, ANDs and ORs, IFs and
GOTOs. This does not mean that we have to. The description of an algorithm may
talk of abstract objects such as integers, reals, strings, sets, stacks, graphs, and trees;
abstract operations such as “sort the list,” “pop the stack,” or “trace a path”; and abstract
relationships such as greater than, prefix, subset, connected, and child. To be
useful, the nature of these objects and the effect of these operations need to be understood.
However, in order to hide details that are tedious or irrelevant, the precise
implementations of these data structure and algorithms do not need to be specified.
Formore on this see Chapter 3.
Correctness: An algorithm for the problem is correct if for every legal input instance,
the required output is produced. Though a certain amount of logical thinking is requireds,
the goal of this text is to teach how to think about, develop, and describe
algorithms in such way that their correctness is transparent. See Chapter 28 for the
formal steps required to prove correctness, and Chapter 22 for a discussion of forall
and exist statements that are essential formaking formal statements.
Running Time: It is not enough for a computation to eventually get the correct
answer. It must also do so using a reasonable amount of time and memory space.
The running time of an algorithm is a function from the size n of the input instance
given to a bound on the number of operations the computation must do. (See
Chapter 23.) The algorithm is said to be feasible if this function is a polynomial like
Time(n) = (n2), and is said to be infeasible if this function is an exponential like
Time(n) = (2n). (See Chapters 24 and 25 formore on the asymptotics of functions.)
To be able to compute the running time, one needs to be able to add up the times
taken in each iteration of a loop and to solve the recurrence relation defining the
time of a recursive program. (See Chapter 26 for an understanding of
n
i=1 i = (n2),
and Chapter 27 for an understanding of T(n) = 2T(n2
) +n = (n logn).)
Meta-algorithms: Most algorithms are best described as being either iterative or
recursive. An iterative algorithm (Part One) takes one step at a time, ensuring that
each step makes progress whilemaintaining the loop invariant. A recursive algorithm
(Part Two) breaks its instance into smaller instances, which it gets a friend to solve,
and then combines their solutions into one of its own.
Optimization problems (Part Three) form an important class of computational
problems. The key algorithms for them are the following. Greedy algorithms (Chapter
16) keep grabbing the next object that looks best. Recursive backtracking algorithms
(Chapter 17) try things and, if they don’t work, backtrack and try something
else. Dynamic programming (Chapter 18) solves a sequence of larger and larger instances,
reusing the previously saved solutions for the smaller instances, until a solution
is obtained for the given instance. Reductions (Chapter 20) use an algorithm for
one problem to solve another. Randomized algorithms (Chapter 21) flip coins to help
them decide what actions to take. Finally, lower bounds (Chapter 7) prove that there
are no faster algorithms.

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How to think about algorithms.Jeff Edmonds.Cambridge.2008.pdf
How to think about algorithms.Jeff Edmonds.Cambridge.2008.pdf
How to think about algorithms.Jeff Edmonds.Cambridge.2008.pdf
How to think about algorithms.Jeff Edmonds.Cambridge.2008.pdf
How to think about algorithms.Jeff Edmonds.Cambridge.2008.pdf
How to think about algorithms.Jeff Edmonds.Cambridge.2008.pdf
How to think about algorithms.Jeff Edmonds.Cambridge.2008.pdf
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