在演算法分析 中，支配理论 （英語：master theorem ）提供了用渐近符号（大O符号 ）表示许多由分治法 得到的递推关系式 的方法。这种方法最初由Jon Bentlery Dorothea Haken James B. Saxe Cormen Leiserson Rivest Stein 算法导论 》 (introduction to algorithm) 推广而为人熟知。
不过，并非所有递推关系式都可应用支配理论。该定理的推广形式包括Akra-Bazzi定理 。
支配理论 假设有递归关系式
T
(
n
)
=
a
T
(
n
b
)
+
f
(
n
)
{\displaystyle T(n)=a\;T\!\left({\frac {n}{b}}\right)+f(n)}
a
≥
1
,
b
>
1
{\displaystyle a\geq 1{\mbox{, }}b>1}
其中，
n
{\displaystyle n}
a
{\displaystyle a}
n
b
{\displaystyle {\frac {n}{b}}}
f
(
n
)
{\displaystyle f(n)}
情形一 如果存在常数
ϵ
>
0
{\displaystyle \epsilon >0}
f
(
n
)
=
O
(
n
log
b
(
a
)
−
ϵ
)
{\displaystyle f(n)=O\left(n^{\log _{b}(a)-\epsilon }\right)}
则
T
(
n
)
=
Θ
(
n
log
b
a
)
{\displaystyle T(n)=\Theta \left(n^{\log _{b}a}\right)}
情形二 如果存在常数
ϵ
≥
0
{\displaystyle \epsilon \geq 0}
f
(
n
)
=
Θ
(
n
log
b
a
log
ϵ
n
)
{\displaystyle f(n)=\Theta \left(n^{\log _{b}a}\log ^{\epsilon }n\right)}
则
T
(
n
)
=
Θ
(
n
log
b
a
log
ϵ
+
1
n
)
{\displaystyle T(n)=\Theta \left(n^{\log _{b}a}\log ^{\epsilon +1}n\right)}
情形三 如果存在常数
ϵ
>
0
{\displaystyle \epsilon >0}
f
(
n
)
=
Ω
(
n
log
b
(
a
)
+
ϵ
)
{\displaystyle f(n)=\Omega \left(n^{\log _{b}(a)+\epsilon }\right)}
同时存在常数
c
<
1
{\displaystyle c<1}
n
{\displaystyle n}
a
f
(
n
b
)
≤
c
f
(
n
)
{\displaystyle af\left({\frac {n}{b}}\right)\leq cf(n)}
则
T
(
n
)
=
Θ
(
f
(
n
)
)
{\displaystyle T\left(n\right)=\Theta \left(f\left(n\right)\right)}
常用演算法中的应用
演算法
递回关系式
运算时间
备注
二分搜尋演算法
T
(
n
)
=
T
(
n
2
)
+
Θ
(
1
)
{\displaystyle T(n)=T\left({\frac {n}{2}}\right)+\Theta (1)}
Θ
(
log
n
)
{\displaystyle \Theta (\log n)}
情形二（
k
=
0
{\displaystyle k=0}
二叉树 遍历
T
(
n
)
=
2
T
(
n
2
)
+
Θ
(
1
)
{\displaystyle T(n)=2T\left({\frac {n}{2}}\right)+\Theta (1)}
Θ
(
n
)
{\displaystyle \Theta (n)}
情形一
最佳排序矩阵搜索(已排好序的二维矩阵)
T
(
n
)
=
2
T
(
n
2
)
+
O
(
log
n
)
{\displaystyle T(n)=2T\left({\frac {n}{2}}\right)+O(\log n)}
Θ
(
n
)
{\displaystyle \Theta (n)}
合并排序
T
(
n
)
=
2
T
(
n
2
)
+
Θ
(
n
)
{\displaystyle T(n)=2T\left({\frac {n}{2}}\right)+\Theta (n)}
Θ
(
n
log
n
)
{\displaystyle \Theta (n\log n)}
情形二（
k
=
0
{\displaystyle k=0}
参考文献 Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms , Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7 . Sections 4.3 (The master method) and 4.4 (Proof of the master theorem), pp. 73–90.
Michael T. Goodrich and Roberto Tamassia. Algorithm Design: Foundation, Analysis, and Internet Examples . Wiley, 2002. ISBN 0-471-38365-1 . The master theorem (including the version of Case 2 included here, which is stronger than the one from CLRS) is on pp. 268–270. 
