In this paper we give a definition of "algorithm," "finite algorithm," "equivalent algorithms," and what it means for a single algorithm to dominate a set of algorithms. We define a derived algorithm which may have a smaller mean execution time than any of its component algorithms. We give an explicit expression for the mean execution time (when it exists) of the derived algorithm. We give several illustrative examples of derived algorithms with two component algorithms. We include mean execution time solutions for two-algorithm processors whose joint density of execution times are of several general forms. For the case in which the joint density for a two-algorithm processor is a step function, we give a maximum-likelihood estimation scheme with which to analyze empirical processing time data.
Deep Dive into Optimal Synthesis of Multiple Algorithms.
In this paper we give a definition of “algorithm,” “finite algorithm,” “equivalent algorithms,” and what it means for a single algorithm to dominate a set of algorithms. We define a derived algorithm which may have a smaller mean execution time than any of its component algorithms. We give an explicit expression for the mean execution time (when it exists) of the derived algorithm. We give several illustrative examples of derived algorithms with two component algorithms. We include mean execution time solutions for two-algorithm processors whose joint density of execution times are of several general forms. For the case in which the joint density for a two-algorithm processor is a step function, we give a maximum-likelihood estimation scheme with which to analyze empirical processing time data.
It can categorically be said that no algorithm is unique. By this we mean that for a given task, invariably more than one algorithm exists which will accomplish that task. One strategy is to select one algorithm deemed generally superior to the rest, and to use that algorithm exclusively. This paper examines an alternative strategy. We ask, given two or more equivalent algorithms, is it ever possible to create a new derived algorithm whose mean execution time is less than that of all of the original algorithms? If so, how can such an algorithm be derived? , , , N α α α be a set of equivalent algorithms. We say that n α
, , , N α α α if and only if for every ω ∈Ω , ( ) (
Now suppose we are given a set of finite equivalent algorithms { }
, , , N α α α . Suppose further that there exists a probability space over such that ( )
, , ,
, , , :
is the event consisting of the points n S ω ∈Ω on which none of the algorithms 1 2 , , , n α α α completes processing within each algorithm’s permitted run time limit. The derived algorithm is then defined to be the pair .
( ) ( )
, , ,
represents the time taken for the derived algorithm to execute when presented with the task ω , and
⎦ represents the derived algorithm’s output when presented with the task ω .
We may envision an implementation of this algorithm as follows. When presented with a task ω ∈Ω , a timer is started, and 1 α is applied.
, , ,
given by the following Theorem 1:
Proof: Recall that
Telescoping sums yield
as desired.
and
)
In this case,
Suppose the joint density of completion times for the two algorithms is given by
) , Notation: In the following, if B is a Boolean expression, then ( )
In particular we define ( )
Suppose the joint density of completion times for the two algorithms is given by The minimum occurs at 1 3 τ = . Note that if 1 3 τ = , then ( )
. In this case the derived algorithm has better mean execution time than either of the original algorithms. Its mean execution time is approximately 6% less than that of either of the original algorithms.
Suppose the joint density of completion times for the two algorithms is given by ( ) ( ) Note that for any choice of 1 τ , then ( )
In this case the derived algorithm has exactly the same mean execution time as do the original algorithms, so a derived algorithm would be of no benefit.
( )
so in this case ( )
ET α α π π τ does not exist.
( )
ax by cx dy
Notation: In the following, if is a Boolean expression, then .
( )
)
,
)
)
)
)
In particular, quadratic function. Notice that the set of points of connection of the pieces is a subset of { } This global minimum is given by
This minimum can be computed in ( )
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