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标题: 突然想到让deepseek来解释一下递归 [打印本页]

作者: 密银    时间: 2025-1-29 14:16
标题: 突然想到让deepseek来解释一下递归
本帖最后由 密银 于 2025-1-29 14:19 编辑 & J, a. d7 c. y: _7 ?1 k8 r) z  o

* h+ x" T& W) z$ s2 Y解释的不错
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( i+ n# s% K$ i5 X- A" F; ]递归是一种通过将问题分解为更小的同类子问题来解决问题的方法。它的核心思想是:**函数直接或间接地调用自身**,直到满足终止条件。  J6 @& P5 a& p' m) o1 h# W

( [9 F4 J+ @: n7 ]( r( Y2 q5 n$ P5 t 关键要素6 ~5 l# n3 [7 ~$ \  k% R, ~
1. **基线条件(Base Case)**
! M1 a5 d+ u) P6 ^4 ^) G   - 递归终止的条件,防止无限循环
0 ]  k1 Z# w, m8 l   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1& v7 ^& r3 [9 e, k" f. Q7 d
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2. **递归条件(Recursive Case)**
7 s0 B. p; v& d5 T% o   - 将原问题分解为更小的子问题
2 h+ _1 M/ @4 a3 L! M% J   - 例如:n! = n × (n-1)!+ h3 ?5 y: f( }( T
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经典示例:计算阶乘4 ^! @: F0 I& a+ g5 ]* I4 {2 Z
python
: ?' @$ j7 z8 l3 edef factorial(n):
2 N$ s! W/ G" a* A' n* O7 ?    if n == 0:        # 基线条件+ ~* O% r+ q9 u( s% e- S. n- N
        return 15 C" c6 w/ c  i9 ^4 `, |
    else:             # 递归条件+ c5 r/ x1 B1 e) C# \5 X* u
        return n * factorial(n-1)
( u  f, q5 x% S6 N& s$ m执行过程(以计算 3! 为例):
0 p3 F+ z! a2 \! |factorial(3)7 U' V) v0 _$ J1 M
3 * factorial(2)4 S/ E' ^# ^" ~& u! j
3 * (2 * factorial(1))
4 B! V) ]3 B$ H9 m% h( G: G) z3 * (2 * (1 * factorial(0)))
0 [+ P) G) P% ]- V) L2 l3 * (2 * (1 * 1)) = 6$ n$ n4 I) ?& h( u. f

9 t' C  h. q. E" |/ N4 d 递归思维要点
  q, |' B  `9 ?3 R1. **信任递归**:假设子问题已经解决,专注当前层逻辑4 c3 G" P+ p* {3 b
2. **栈结构**:每次调用都会创建新的栈帧(内存空间)1 x; g" m0 \: q6 W
3. **递推过程**:不断向下分解问题(递)0 d/ O5 n* Q/ ^1 g3 z4 w3 j
4. **回溯过程**:组合子问题结果返回(归)- O5 Z* Y/ c& Q# l( `) u; C

0 r) R* L0 w  k9 ^$ |  W, O 注意事项
3 v8 A2 l, ]% Q# _; [+ L& ~必须要有终止条件4 j+ A% j" N6 u0 A$ m
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)3 U" |) V* R; F; }( ^, _& ^9 N3 C
某些问题用递归更直观(如树遍历),但效率可能不如迭代
9 M. [+ I0 M8 H尾递归优化可以提升效率(但Python不支持)
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递归 vs 迭代+ h0 c" @' O9 l* p( _1 G
|          | 递归                          | 迭代               |
4 @5 ~8 A& C. l|----------|-----------------------------|------------------|" h: ]& i& `8 N  T, U9 g
| 实现方式    | 函数自调用                        | 循环结构            |, ?: _8 x1 S( X% E
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |: ^: o0 A, [6 h3 }, z; E
| 代码可读性  | 对符合递归思维的问题更直观                | 线性流程更直接         |
& d3 L; B# ]  P  u+ z  n6 Z& r( m, Y7 N* ~| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
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经典递归应用场景
% H4 C: J0 j' z: ]1. 文件系统遍历(目录树结构)
2 w5 C/ e9 V( D0 h2. 快速排序/归并排序算法7 Y& q: R# t! N; V
3. 汉诺塔问题+ ]- Q3 b% m% P9 V$ ~& z8 J% W* L1 i
4. 二叉树遍历(前序/中序/后序)
# W; k2 j  ~( c/ e$ ]7 D5. 生成所有可能的组合(回溯算法)
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( x5 p5 O$ X9 S试着用递归思维想象:你站在一面镜子前,镜子里有无数个逐渐变小的你在照镜子,这就是递归的直观体现。但记住每个"分身"最终都要有结束的时刻,这就是基线条件的重要性。
作者: testjhy    时间: 2025-1-30 00:07
挺好,递归思维要点与我能够回忆起来我当时写递归程序的思路很一致,,或者被它唤醒,/ l: z/ z7 }0 l8 H, k" k1 O; U
我推理机的核心算法应该是二叉树遍历的变种。5 _: b) P  l. O. r2 T" e
另外知识系统的推理机搜索深度(递归深度)并不长,没有超过10层的,如果输入变量多的话,搜索宽度很大,但对那时的286-386DOS系统,计算压力也不算大。
作者: nanimarcus    时间: 2025-2-2 00:45
Recursion in programming is a technique where a function calls itself in order to solve a problem. It is a powerful concept that allows you to break down complex problems into smaller, more manageable subproblems. Here's a detailed explanation:
" m, X% j- |5 g/ r* A! RKey Idea of Recursion" c5 [5 D3 t) u

& h8 Z( ~- J  y3 a( G) B3 uA recursive function solves a problem by:
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2 G; W0 P0 v' P7 V    Breaking the problem into smaller instances of the same problem.8 n( G5 b/ t9 _0 m3 W5 B" x0 p* S

4 d9 [5 a% F+ F/ O# N5 A    Solving the smallest instance directly (base case).6 g0 e( a; u4 q6 R2 g* H
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    Combining the results of smaller instances to solve the larger problem., A$ h8 O- F. \$ D9 W* r0 i
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Components of a Recursive Function
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; Y) x$ R8 m, X* O$ @    Base Case:0 K9 m# \6 J( \: i. ]$ G) B$ @" I

2 m0 e: _* V& H9 {& M' m6 h) W        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.: Y- a0 E/ `- P: s6 }

& [* T1 A' N, Z: f- L1 t& V        It acts as the stopping condition to prevent infinite recursion.9 E( D0 z3 g! a

- i* |# l; a5 i: y        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    Recursive Case:* f7 m4 T3 u3 N9 X
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        This is where the function calls itself with a smaller or simpler version of the problem.
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- n7 u1 }0 w6 J, \8 w+ r+ |        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).: Y) ~5 Q0 u( ?

8 ^3 H" j$ D. S- q2 b0 qExample: Factorial Calculation
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The factorial of a number n (denoted as n!) is the product of all positive integers less than or equal to n. It can be defined recursively as:
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    Base case: 0! = 1
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    Recursive case: n! = n * (n-1)!
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Here’s how it looks in code (Python):
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def factorial(n):
0 c3 |' [! ~& l5 M: A    # Base case
$ n2 y! U4 H% F# Q+ ?    if n == 0:
  _+ N; e- t1 S( J2 r4 E1 Q        return 16 V( T  v) c2 ~0 P3 {1 L
    # Recursive case
+ a; Y1 ~5 S* \4 D; J( A7 ]    else:
  m* }2 m/ J; O# h        return n * factorial(n - 1)
# U+ I2 H3 O2 F0 o% I& b) `$ {0 v0 u' z6 V9 _; z, z
# Example usage
( H/ s/ m& _7 ?3 ]: F7 Fprint(factorial(5))  # Output: 120
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How Recursion Works
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( `% O/ R2 A, O2 b0 a) _, p    The function keeps calling itself with smaller inputs until it reaches the base case.2 J, e, c9 u- x0 m1 ]

- x  C3 o# W6 ]! l    Once the base case is reached, the function starts returning values back up the call stack.  M# P7 z+ ^  S' r) @( @# a

( d; ^4 }8 p7 |, c8 F    These returned values are combined to produce the final result.
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$ ]4 D9 E* A! m# c% o5 e6 D5 rFor factorial(5):
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factorial(5) = 5 * factorial(4). w' b2 B( `3 y7 }& c6 [  A% s) P
factorial(4) = 4 * factorial(3)$ B0 a9 x$ ~1 ]4 \: J4 l5 _: [5 H# D
factorial(3) = 3 * factorial(2)
) A/ I) h3 ^3 F0 g- r2 Wfactorial(2) = 2 * factorial(1)
7 g, w& S, b. U1 d3 b7 efactorial(1) = 1 * factorial(0)3 u+ o3 ]" t0 @
factorial(0) = 1  # Base case' X1 b& D$ g% \/ G: X
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Then, the results are combined:
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factorial(1) = 1 * 1 = 13 z" [* l$ M$ \
factorial(2) = 2 * 1 = 2
' T; R3 L- ~$ v( n; \& w4 pfactorial(3) = 3 * 2 = 6
  q- O. M) F' \8 Efactorial(4) = 4 * 6 = 247 h2 ^, m2 P& c+ ~
factorial(5) = 5 * 24 = 120
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Advantages of Recursion
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    Simplicity: Recursive solutions are often more intuitive and easier to write for problems that have a natural recursive structure (e.g., tree traversals, divide-and-conquer algorithms).
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! q+ ]+ s4 w% `/ ~! X' j    Readability: Recursive code can be more readable and concise compared to iterative solutions.* P: ?3 B% z) t( j  m. g2 |
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Disadvantages of Recursion
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* C1 L0 n) l+ f; z6 l5 w: H    Performance Overhead: Each recursive call adds a new layer to the call stack, which can lead to high memory usage and potential stack overflow for deep recursion.
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: b# V5 X% V7 _0 O' k- ]. a    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).. b( L; J1 M9 B' |' {/ K- E

8 ?0 X" f4 U7 J6 X! T; D7 eWhen to Use Recursion: j3 h( \. P$ `# E. o3 W/ V
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    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
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4 O; b" t# n: n) V: [    Problems with a clear base case and recursive case.
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  Q1 s* q' Y. X' ]- fExample: Fibonacci Sequence
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The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
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, B6 f# C% ]. A- |0 C/ ?    Base case: fib(0) = 0, fib(1) = 1
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6 a2 M4 A5 g0 X4 g0 ?    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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python. M" b  \' v( ]# a1 |: u
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0 B& M# W/ o& x* F) C" d& [def fibonacci(n):
* h7 c. o# s0 s+ H2 i    # Base cases
$ e9 r+ n3 ]# N& V8 J2 c6 {6 H    if n == 0:
$ h+ g& J9 w5 K( L7 k        return 0
6 g$ L' R# n  D, w; R/ ~    elif n == 1:
; A2 a  Y# U) O( v- u; Y8 |        return 15 R4 O/ `; e8 H$ p5 M1 {3 @
    # Recursive case) \  a( j6 A4 t& O8 O" b) d
    else:8 G6 t6 N( O! f0 ]) r" l: i
        return fibonacci(n - 1) + fibonacci(n - 2)% j; Z$ h  \8 A0 H7 t. D6 X& N

: _5 d' Z' u+ i2 N; @7 M# Example usage$ |0 ^" J7 f3 e( w
print(fibonacci(6))  # Output: 8$ s# o' o. y- R0 l2 L

( d# B/ D2 w' |7 q# b9 rTail Recursion
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Tail recursion is a special case of recursion where the recursive call is the last operation in the function. Some programming languages optimize tail-recursive functions to avoid stack overflow, but not all languages (e.g., Python does not optimize tail recursion).4 [+ Q: @: R7 p7 C) p9 R
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In summary, recursion is a fundamental concept in programming that allows you to solve problems by breaking them into smaller, self-similar subproblems. It’s important to define a base case to avoid infinite recursion and to understand the trade-offs between recursion and iteration.
作者: nanimarcus    时间: 2025-2-2 00:47
我还让Deepseek 给我讲讲Linux Kernel Driver 现在的开发流程,让一个老同志复习复习,快忘光了。




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