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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 3 l! ~% R5 P" i: l  K4 h

4 B7 k2 N7 w5 k, P# D3 M; ?/ K6 ?4 h; t解释的不错
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' g" D6 h. S, E; j, ]6 x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。; m% H& z1 Q3 M4 t0 E" `

. Y, B0 n/ o$ X/ h  i  ] 关键要素
3 H# A* ~1 Q, `3 n1 d" @1. **基线条件(Base Case)**
9 e3 z0 n5 Q" r- B, R   - 递归终止的条件,防止无限循环
$ V9 l! K. ]( \5 {   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1% i* p7 Y, ?" \" h# x* s- @6 T

8 P: P9 X/ U6 ]2. **递归条件(Recursive Case)**3 Z" a/ ]+ V7 W" p: n
   - 将原问题分解为更小的子问题
% l3 e7 q: h$ I! J4 h   - 例如:n! = n × (n-1)!
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经典示例:计算阶乘
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def factorial(n):
' \, U( u# L9 y! H9 D    if n == 0:        # 基线条件- Y: H0 ?: x! n& n6 c) d! C1 T
        return 1
+ g. U1 s3 w( O% u$ Y& k  ], H    else:             # 递归条件, Q& X: _- k) x4 h9 F
        return n * factorial(n-1)7 T+ F0 y. E1 ?( T& G
执行过程(以计算 3! 为例):& Y# W' t- r$ `/ ?% R- e3 r
factorial(3)
, f; X: d1 j) F- d- h3 * factorial(2)9 }  e( K+ G' Y7 G
3 * (2 * factorial(1))+ ^* f5 K- |* e& K
3 * (2 * (1 * factorial(0)))
; H" K, a* L* A* G( `3 G3 * (2 * (1 * 1)) = 6! O* Q# Y% ~: m. L0 R1 G& V
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递归思维要点
5 y2 w0 h4 Y$ u& T' B1. **信任递归**:假设子问题已经解决,专注当前层逻辑& ^9 B! q5 J1 r! t% }* d
2. **栈结构**:每次调用都会创建新的栈帧(内存空间)0 f" ?. I9 `" c. `$ u( X* v9 s
3. **递推过程**:不断向下分解问题(递)7 [- Q4 }5 {; F
4. **回溯过程**:组合子问题结果返回(归)& ^' ~% q* ]# r# g4 Z" p* G* z% G
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注意事项
/ ^! N" Y# ~( P% f& `$ g必须要有终止条件( @2 X) ^$ S3 ^4 Z, c' f" Q* v
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)5 {1 O0 r( N  ~8 e$ q+ A: d- I
某些问题用递归更直观(如树遍历),但效率可能不如迭代( M# e' y1 J* q( I. V
尾递归优化可以提升效率(但Python不支持)9 F. e) e) n7 i% |$ s

4 |: T0 b4 w) E' {. [5 j 递归 vs 迭代
% j: O1 k/ c/ J|          | 递归                          | 迭代               |/ z7 D4 `2 j8 n; {
|----------|-----------------------------|------------------|
: W. ~6 z6 ~9 ^| 实现方式    | 函数自调用                        | 循环结构            |
, T2 y( X& }+ v) s| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |* k6 `# t$ z! U. `
| 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         |
" Q' Y! M+ Z2 o$ O; T" s| 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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8 n% j; u# R) p* M" L; @9 j 经典递归应用场景$ u. S) i. M) O* L2 a2 @
1. 文件系统遍历(目录树结构)- F. F2 v9 g0 O0 i6 y) a6 M" X# s
2. 快速排序/归并排序算法
( l: w+ \8 z9 w2 G  C' M5 c- v7 r, D( J3. 汉诺塔问题1 ~: y9 ?7 ^. `+ a  h  S6 a4 R
4. 二叉树遍历(前序/中序/后序)! d& m5 {3 E/ e6 u1 X
5. 生成所有可能的组合(回溯算法)/ ]" W% I, i1 \2 O
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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,
4 {0 K5 ~  q3 s我推理机的核心算法应该是二叉树遍历的变种。
, v/ K( j0 g  o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:% p4 @) c9 R& j4 ?# m+ L* _
Key Idea of Recursion& X+ S" G8 d# Z" d2 k& L1 U

; }: f# F9 b2 t8 CA recursive function solves a problem by:
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    Breaking the problem into smaller instances of the same problem.6 B1 n( w" B$ A' `

1 V  S* G& |8 @  T3 ]    Solving the smallest instance directly (base case).! L1 \" j* y! H# W, }% O$ O& ~

+ W3 v6 \0 C5 Q$ k* x6 x% P    Combining the results of smaller instances to solve the larger problem.# M# z6 H4 u2 E- L2 t
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Components of a Recursive Function
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    Base Case:
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- l) N  c( B5 X  |1 V& W        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.9 c5 o3 ^5 d7 x  j3 f: \5 H# H
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        It acts as the stopping condition to prevent infinite recursion.* K3 i; m* ?6 M& V( Y3 {

/ ^, r3 v# k% |9 `7 n  _4 N        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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( r* ^' v- @- g, e    Recursive Case:
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        This is where the function calls itself with a smaller or simpler version of the problem.3 A7 Y4 t: |" @8 h! \1 x

* n* ?6 I. F7 I2 G2 ]        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).2 L1 {2 Q8 Q. O/ L: v  O) X& w

3 L6 y) d' R2 M. S+ x  a: P3 n8 \Example: Factorial Calculation( J( B, t8 ?) g2 c3 l0 g
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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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* K$ {7 b6 ?) g1 K    Recursive case: n! = n * (n-1)!7 x% u2 X/ M( j! e( Y6 |3 I

" R2 U. s" w! ^. eHere’s how it looks in code (Python):
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/ B1 p6 }4 C9 ?& W6 `8 N/ @, Ddef factorial(n):
# K5 P+ u( U  ?  Y, Y: n( E/ Q9 d" M    # Base case
* o1 ^/ o/ ~  Q+ R6 ~% v    if n == 0:" J! H) f3 @! K$ C
        return 1
! ]5 O9 E" g( x0 S& G/ K    # Recursive case
- z' t7 y1 }) ~    else:4 `! K! [9 \+ G6 u0 v. p- R% m
        return n * factorial(n - 1)# F6 M7 p. N! }3 y8 F# u

% l. g: G& |* |5 d6 G) X6 F: A) w# Example usage
. k- s; s, b* H6 w# V6 Y, dprint(factorial(5))  # Output: 1202 f. o8 x" K: K" \3 K% o

4 f- `, ]( J+ UHow Recursion Works
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    The function keeps calling itself with smaller inputs until it reaches the base case." ~& C" m- K- f: j3 S
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    Once the base case is reached, the function starts returning values back up the call stack.# _- c5 c; M- Y$ G( M& ?' M( y
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    These returned values are combined to produce the final result.  i9 f% p1 {9 v1 |

; ]' V7 D! H, ]$ b" r/ M6 W9 NFor factorial(5):9 \' E- ]' n, o, ^! E
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factorial(5) = 5 * factorial(4)8 A& f; D: T6 |7 Q6 ?4 H0 u2 B
factorial(4) = 4 * factorial(3)
/ H0 k& K. V" d1 h" W+ t. Ofactorial(3) = 3 * factorial(2)
1 F/ k' f% F6 W0 M0 \/ W  s  lfactorial(2) = 2 * factorial(1). J9 `+ p9 Z+ c9 P9 S
factorial(1) = 1 * factorial(0)
9 N/ J: x) z3 P/ v( T  Mfactorial(0) = 1  # Base case
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Then, the results are combined:
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) a0 b! P0 }' o) cfactorial(1) = 1 * 1 = 1
) i; U+ y/ s+ a: I3 f8 \9 g0 tfactorial(2) = 2 * 1 = 2
: b5 F6 T( C- z5 B( efactorial(3) = 3 * 2 = 67 [2 e% P% {) L- K3 Y
factorial(4) = 4 * 6 = 241 o5 M7 }, |( A( F( s! p* n
factorial(5) = 5 * 24 = 120
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4 m% ?3 p! Z( k; _Advantages of Recursion$ ]' C& N( N0 ?

% i1 V, _2 ?' J. [! [    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).3 t0 g9 A. J8 c2 b/ M; I4 J. l/ R. g
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    Readability: Recursive code can be more readable and concise compared to iterative solutions.8 K% |: H& ]4 D. E7 \

7 O1 {+ G# T; g* I& z( `Disadvantages of Recursion
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) v9 F2 @' L: f3 k- g2 `! T    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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    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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When to Use Recursion
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    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).# a5 K2 l7 [3 {

, \) W: k) P! I) W    Problems with a clear base case and recursive case.
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Example: Fibonacci Sequence
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; g5 e! w0 `9 A* H+ j2 yThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
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: B2 K5 c4 L" g) v) s( j6 I0 D    Base case: fib(0) = 0, fib(1) = 1
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    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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python
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  D: Y. @, F9 edef fibonacci(n):; L& _0 Q4 }7 i0 X: ]
    # Base cases
- H) u" g( B- |6 ~3 K7 l    if n == 0:
# _8 d# O! ]  H! @+ W        return 0
. m8 Z; [- `  E1 ^    elif n == 1:! X- X! n  V& f4 r, e
        return 1
) O/ ~5 k" y, ?; i7 G% V    # Recursive case# y  Y9 I: [9 W7 m
    else:
" u3 m! b- X4 H9 D( |6 @- P        return fibonacci(n - 1) + fibonacci(n - 2)
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6 `4 R: o- D  ~# Example usage; B1 E3 F' X( A7 u9 _, _0 y
print(fibonacci(6))  # Output: 8
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6 K  {4 D* J  E' g9 p- Q$ f0 gTail Recursion5 Z. P; U' t; R- n3 X8 Y

0 n: M) d* i$ C4 zTail 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).
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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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