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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