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Method in computer science
In computer science , the Akra–Bazzi method , or Akra–Bazzi theorem , is used to analyze the asymptotic behavior of the mathematical recurrences that appear in the analysis of divide and conquer algorithms where the sub-problems have substantially different sizes. It is a generalization of the master theorem for divide-and-conquer recurrences , which assumes that the sub-problems have equal size. It is named after mathematicians Mohamad Akra and Louay Bazzi .[1]
Formulation [ edit ]
The Akra–Bazzi method applies to recurrence formulas of the form:[1]
T
(
x
)
=
g
(
x
)
+
∑
i
=
1
k
a
i
T
(
b
i
x
+
h
i
(
x
)
)
for
x
≥
x
0
.
{\displaystyle T(x )=g(x )+\sum _{i=1}^{k}a_{i}T(b_{i}x+h_{i}(x ))\qquad {\text{for }}x\geq x_{0}.}
The conditions for usage are:
sufficient base cases are provided
a
i
{\displaystyle a_{i}}
and
b
i
{\displaystyle b_{i}}
are constants for all
i
{\displaystyle i}
a
i
>
0
{\displaystyle a_{i}>0}
for all
i
{\displaystyle i}
0
<
b
i
<
1
{\displaystyle 0<b_{i}<1}
for all
i
{\displaystyle i}
|
g
′
(
x
)
|
∈
O
(
x
c
)
{\displaystyle \left|g'(x )\right|\in O(x^{c})}
, where c is a constant and O notates Big O notation
|
h
i
(
x
)
|
∈
O
(
x
(
log
x
)
2
)
{\displaystyle \left|h_{i}(x )\right|\in O\left({\frac {x}{(\log x)^{2}}}\right)}
for all
i
{\displaystyle i}
x
0
{\displaystyle x_{0}}
is a constant
The asymptotic behavior of
T
(
x
)
{\displaystyle T(x )}
is found by determining the value of
p
{\displaystyle p}
for which
∑
i
=
1
k
a
i
b
i
p
=
1
{\displaystyle \sum _{i=1}^{k}a_{i}b_{i}^{p}=1}
and plugging that value into the equation:[2]
T
(
x
)
∈
Θ
(
x
p
(
1
+
∫
1
x
g
(
u
)
u
p
+
1
d
u
)
)
{\displaystyle T(x )\in \Theta \left(x^{p}\left(1+\int _{1}^{x}{\frac {g(u )}{u^{p+1}}}du\right)\right)}
(see Θ ). Intuitively,
h
i
(
x
)
{\displaystyle h_{i}(x )}
represents a small perturbation in the index of
T
{\displaystyle T}
. By noting that
⌊
b
i
x
⌋
=
b
i
x
+
(
⌊
b
i
x
⌋
−
b
i
x
)
{\displaystyle \lfloor b_{i}x\rfloor =b_{i}x+(\lfloor b_{i}x\rfloor -b_{i}x)}
and that the absolute value of
⌊
b
i
x
⌋
−
b
i
x
{\displaystyle \lfloor b_{i}x\rfloor -b_{i}x}
is always between 0 and 1,
h
i
(
x
)
{\displaystyle h_{i}(x )}
can be used to ignore the floor function in the index. Similarly, one can also ignore the ceiling function . For example,
T
(
n
)
=
n
+
T
(
1
2
n
)
{\displaystyle T(n )=n+T\left({\frac {1}{2}}n\right)}
and
T
(
n
)
=
n
+
T
(
⌊
1
2
n
⌋
)
{\displaystyle T(n )=n+T\left(\left\lfloor {\frac {1}{2}}n\right\rfloor \right)}
will, as per the Akra–Bazzi theorem, have the same asymptotic behavior.
Example [ edit ]
Suppose
T
(
n
)
{\displaystyle T(n )}
is defined as 1 for integers
0
≤
n
≤
3
{\displaystyle 0\leq n\leq 3}
and
n
2
+
7
4
T
(
⌊
1
2
n
⌋
)
+
T
(
⌈
3
4
n
⌉
)
{\displaystyle n^{2}+{\frac {7}{4}}T\left(\left\lfloor {\frac {1}{2}}n\right\rfloor \right)+T\left(\left\lceil {\frac {3}{4}}n\right\rceil \right)}
for integers
n
>
3
{\displaystyle n>3}
. In applying the Akra–Bazzi method, the first step is to find the value of
p
{\displaystyle p}
for which
7
4
(
1
2
)
p
+
(
3
4
)
p
=
1
{\displaystyle {\frac {7}{4}}\left({\frac {1}{2}}\right)^{p}+\left({\frac {3}{4}}\right)^{p}=1}
. In this example,
p
=
2
{\displaystyle p=2}
. Then, using the formula, the asymptotic behavior can be determined as follows:[3]
T
(
x
)
∈
Θ
(
x
p
(
1
+
∫
1
x
g
(
u
)
u
p
+
1
d
u
)
)
=
Θ
(
x
2
(
1
+
∫
1
x
u
2
u
3
d
u
)
)
=
Θ
(
x
2
(
1
+
ln
x
)
)
=
Θ
(
x
2
log
x
)
.
{\displaystyle {\begin{aligned}T(x )&\in \Theta \left(x^{p}\left(1+\int _{1}^{x}{\frac {g(u )}{u^{p+1}}}\,du\right)\right)\\&=\Theta \left(x^{2}\left(1+\int _{1}^{x}{\frac {u^{2}}{u^{3}}}\,du\right)\right)\\&=\Theta (x^{2}(1+\ln x))\\&=\Theta (x^{2}\log x).\end{aligned}}}
Significance [ edit ]
The Akra–Bazzi method is more useful than most other techniques for determining asymptotic behavior because it covers such a wide variety of cases. Its primary application is the approximation of the running time of many divide-and-conquer algorithms. For example, in the merge sort , the number of comparisons required in the worst case, which is roughly proportional to its runtime, is given recursively as
T
(
1
)
=
0
{\displaystyle T(1 )=0}
and
T
(
n
)
=
T
(
⌊
1
2
n
⌋
)
+
T
(
⌈
1
2
n
⌉
)
+
n
−
1
{\displaystyle T(n )=T\left(\left\lfloor {\frac {1}{2}}n\right\rfloor \right)+T\left(\left\lceil {\frac {1}{2}}n\right\rceil \right)+n-1}
for integers
n
>
0
{\displaystyle n>0}
, and can thus be computed using the Akra–Bazzi method to be
Θ
(
n
log
n
)
{\displaystyle \Theta (n\log n)}
.
See also [ edit ]
References [ edit ]
^ a b Akra, Mohamad; Bazzi, Louay (May 1998). "On the solution of linear recurrence equations". Computational Optimization and Applications . 10 (2 ): 195–210. doi :10.1023/A:1018373005182 . S2CID 7110614 .
^ "Proof and application on few examples" (PDF) .
^ Cormen, Thomas; Leiserson, Charles; Rivest, Ronald; Stein, Clifford (2009). Introduction to Algorithms . MIT Press. ISBN 978-0262033848 .
External links [ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Akra–Bazzi_method&oldid=1222385349 "
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