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[科技前沿] 突然想到让deepseek来解释一下递归

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  • TA的每日心情
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    2025-9-8 05:08
  • 签到天数: 3 天

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     楼主| 发表于 2025-1-29 14:16:55 | 只看该作者 回帖奖励 |倒序浏览 |阅读模式
    本帖最后由 密银 于 2025-1-29 14:19 编辑
    : O# L2 q& ?: p8 z: t
    0 t( O1 R  Y. m! o( s解释的不错
    ! B! Y+ e' ]8 [6 V, I* r( x
    - U6 h# S) d: W1 c* i; Q' ^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。
    - Q) ~: h% R' c4 |. W8 R7 P4 h3 S0 n" o* r7 P- w
    关键要素
    6 Y# d* }% C4 A  K- C  g0 q' N1. **基线条件(Base Case)**% i: I3 A  ~" o8 |- _: }# |/ @+ h
       - 递归终止的条件,防止无限循环
    - |( O+ Z; V/ l# f8 H' Y   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1& i( _! H, Q8 |

    . n) L  ~% c% x5 m7 W2. **递归条件(Recursive Case)**. {- V  r! O+ a9 ?/ w/ w* c
       - 将原问题分解为更小的子问题
    - f2 E4 g# R6 P- t3 _2 S# ]   - 例如:n! = n × (n-1)!) E* V- Z. L$ w$ \  Y3 g! ]
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    经典示例:计算阶乘# a( A6 K  H9 Q9 F+ ], ~
    python
      v  G5 m6 c  k* V# ?9 j, Tdef factorial(n):
    ! Y7 w4 l7 p9 @+ y  l+ C. t    if n == 0:        # 基线条件7 a& I2 W4 s) O% _# }  o. S, T
            return 1, Q3 f5 F  x; n0 S! R
        else:             # 递归条件5 V- h# G6 y3 v  |. b/ g3 {+ P
            return n * factorial(n-1)5 v7 b/ _- A  L! b
    执行过程(以计算 3! 为例):
    % r1 T- h% v5 @+ cfactorial(3)* g; y* A+ |3 r/ d" J% Q( `1 i
    3 * factorial(2)
    6 r6 [; E1 q" L  L' h3 * (2 * factorial(1))
    % E; X8 U7 U( i: `5 o( [3 * (2 * (1 * factorial(0)))7 |2 D( z  O4 _2 B* T0 W
    3 * (2 * (1 * 1)) = 6: i: |; C/ D0 ?

    " e4 c* k2 X+ C% H 递归思维要点+ g* n6 F( F1 A; Y2 b3 [2 ]
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    6 `" t5 u: {- I% l  |) V* o2. **栈结构**:每次调用都会创建新的栈帧(内存空间)& \7 Y& w2 r$ [
    3. **递推过程**:不断向下分解问题(递)
    * d6 X/ q$ Q( H/ |  m4. **回溯过程**:组合子问题结果返回(归)3 f5 g' b$ q+ K

    ( A7 ~5 u" e% p 注意事项) x1 {5 A2 F: ~* u8 l
    必须要有终止条件
    , a' n  N0 y4 q# s% c+ P" R2 j递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    & _& E. f3 M8 C某些问题用递归更直观(如树遍历),但效率可能不如迭代
    * `( Q6 Z$ G" t* \& R+ W尾递归优化可以提升效率(但Python不支持)
      K# d* _% q. \
    2 X  q' n$ a+ ] 递归 vs 迭代
    5 d# D, ?- k" T5 W6 a|          | 递归                          | 迭代               |
    - j4 e  B% K6 `8 o: G" r|----------|-----------------------------|------------------|
    , ]2 x/ X. [' C& v| 实现方式    | 函数自调用                        | 循环结构            |
    / u0 Z9 s0 E% h0 Q| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    3 G: q0 c0 u0 [% {; Q/ m| 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         |
    3 O; Z: y. f4 w6 v" a- c9 X+ t| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ' @- u" J  L# M* }0 d8 c- n5 F- j/ Q( }( C' ^
    经典递归应用场景' D# G5 }  |/ \- V: W& w2 s1 o+ c3 g
    1. 文件系统遍历(目录树结构)& F& [: ?/ p8 n8 R; _) X# C
    2. 快速排序/归并排序算法
    $ m/ W' H  _' o3. 汉诺塔问题
    & {9 `5 m9 |2 J: o1 i4. 二叉树遍历(前序/中序/后序)3 n5 Q% O+ {: _! S9 \1 u
    5. 生成所有可能的组合(回溯算法)$ L+ r8 {8 _9 K

    6 T# f* r" `2 k1 c: y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。

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  • TA的每日心情

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    沙发
    发表于 2025-1-30 00:07:50 | 只看该作者
    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,% Y  }! [* J# R. F4 L
    我推理机的核心算法应该是二叉树遍历的变种。! d& ~( p/ \# Y% H; @
    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。
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    板凳
    发表于 2025-2-2 00:45:59 | 只看该作者
    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:
    2 r/ X( \6 n) [* a* eKey Idea of Recursion
    , |, {! P- V) {* ^" W+ B- j. F
    9 G( V: d, a' tA recursive function solves a problem by:) ^8 H1 f0 Y" c
    4 @& t: N7 B* }5 f9 p2 N' a
        Breaking the problem into smaller instances of the same problem.1 J+ x. S: l5 A2 ^! Y

    9 x2 d) c" \) Y' G0 [/ _. {    Solving the smallest instance directly (base case).( m7 v5 e  Y, O/ _+ h
    ! T( Y7 u) [" c, Y* l/ H6 Z
        Combining the results of smaller instances to solve the larger problem.
    . x+ o2 M0 Q4 P1 v# d$ i. U3 K
    , h8 l( X, S9 M: j  i0 KComponents of a Recursive Function
    , Q7 H5 `0 x7 Y) K
    ) O: a/ \  A% Y' K8 t0 K1 Z    Base Case:& h6 O% J0 b: \* A( L. s

    ) I: x: f2 X+ m. |! k: G        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.* H' k: |" ~6 Y) ?; M, v
    % s( o- m9 o# V5 D1 P9 N5 L2 B8 J
            It acts as the stopping condition to prevent infinite recursion.
    2 M, g% H( q% \: D1 |( v* A, G0 H# {* `+ ?! t9 X3 K7 i% i
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.0 Y: j4 Q9 y' w% O% b) [

    , o/ G: f6 i& F1 b# l  k6 n5 a    Recursive Case:' N' y1 R) ]7 @) C+ A2 U

    + j/ i' {) x5 {0 F) i( ]        This is where the function calls itself with a smaller or simpler version of the problem.
    % ]1 x" Y' Q% F: ~
    " _, g6 ~8 e* l1 ^9 X        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    9 e! w$ k/ a% f# C! A8 E5 w+ m& h3 i. {. Q6 O1 G
    Example: Factorial Calculation3 A* l: _; z& h3 M- k

    ! ?# |! I  E. E$ ?8 L9 M! g" HThe 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:
    2 M5 _1 u, m0 u9 {4 s
    # ?6 Y- F9 i* b3 z    Base case: 0! = 1' s0 D: T* a/ \- h9 m% [# U$ @

    - r4 n' W. Y" U$ K% P    Recursive case: n! = n * (n-1)!& I  ]: y2 a1 H
    : S( q8 I  I; ]9 R2 H& U
    Here’s how it looks in code (Python):) j7 z, A) k6 N
    python
    ! m5 u$ z: o6 C. t( y( s) x  ]0 A* Z! e. E1 J
    7 {& T1 w9 L! h3 L
    def factorial(n):
    6 T0 [' z: a$ r    # Base case
    $ v. l% u( p( p. t# m. X" @    if n == 0:% q9 U7 T+ t! d) C" J" N
            return 1
    0 p) K6 j! Z; v. q. Y4 p    # Recursive case
    ( O, D, p4 Z' j$ P& X4 J    else:; T" V( u% Y, q0 p, ]$ q% \1 Z" J3 L
            return n * factorial(n - 1)
    / w' x9 Z& j6 ?  l6 ]
    - n( Q$ E# o- E0 P( z3 L# Example usage  n) q  k" s9 }: I0 [2 }4 ?/ ~* r
    print(factorial(5))  # Output: 1203 G& m5 u* K$ H) l
    6 D; |- U4 v$ V# ?
    How Recursion Works
    " ?2 \9 J/ E* K, m& N4 t) r* I3 W: I; a# P) Y! T0 J
        The function keeps calling itself with smaller inputs until it reaches the base case., q7 E) ~# Y( W- j' N
    + Z& u( ~* {: o. ~: ]
        Once the base case is reached, the function starts returning values back up the call stack.
    # c+ }& F1 u2 P1 o
    / H7 r: h& a6 v. m" \# T    These returned values are combined to produce the final result.
    + H: t. s% g# S) ?( b# T2 a3 D8 U5 A' c6 v, y5 o6 S6 i
    For factorial(5):1 `/ [& l6 ?% ]- @  @# a. ?' P

    & l8 U: e) ~/ b! `; f1 ^* \; b8 d. p" V) x  \
    factorial(5) = 5 * factorial(4)
    9 J' }7 m- L' S4 }7 T2 K, u" x/ bfactorial(4) = 4 * factorial(3)
    8 f1 ]& P3 i9 T# ?factorial(3) = 3 * factorial(2). ^+ g3 U; e( {9 z! w
    factorial(2) = 2 * factorial(1)8 `4 V4 m8 ]9 u; J3 y" s3 V7 R8 f
    factorial(1) = 1 * factorial(0)
    ) ^6 L$ s9 r8 a. d' Q- o0 Ofactorial(0) = 1  # Base case3 \' y. {0 n) u6 M- k; e8 L

      c# o6 h0 Z: W! FThen, the results are combined:
    ' i6 P# d# a2 f$ W7 q/ u5 H, j2 U7 W) Q2 @  I
    0 l$ Z3 E0 u8 J3 m
    factorial(1) = 1 * 1 = 1
    ( j5 G3 e- o  o0 ^/ \' L1 Hfactorial(2) = 2 * 1 = 20 H# E6 [0 r2 b, W# x& b
    factorial(3) = 3 * 2 = 6
    6 L0 m% ~/ E9 ~factorial(4) = 4 * 6 = 24
    0 k" o+ v$ S* y* B5 z! X+ p, l, x5 Lfactorial(5) = 5 * 24 = 120
    ' H* l3 }! ~7 ^. P( r$ @) m3 c9 e) T) K5 G1 X  F" b
    Advantages of Recursion
    & h7 u+ X& p$ q3 j& q" W
    % e9 G' ^$ S( T; i    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).
    9 A" j, F8 D; A+ m$ T8 A2 g* C9 j) b+ W: l
        Readability: Recursive code can be more readable and concise compared to iterative solutions.5 Y' {8 I8 O' c, |; G  G% K: ~

    - ?, I4 F1 D  n( r+ DDisadvantages of Recursion4 r6 {4 b7 @- A; \/ ]4 t
    - C0 U- ^. G4 G) S  X5 N0 ]" d
        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.- e! Z  r: {4 |+ @* J

    7 d( i1 h/ ^- I) ~1 _+ X6 p    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    # Q$ \; A1 Z/ Z
    # \- |' O, u0 X( u# M3 B7 h9 jWhen to Use Recursion
    . f: O! T' p: W2 L  T' _
    8 c/ m5 e/ `2 @    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).6 v4 q. C8 q: B9 Y% V6 k

    4 o' S# V! {" |* \    Problems with a clear base case and recursive case.
    ) D- a' C* b5 h7 q1 c8 I
      c- J) z; y0 C8 |Example: Fibonacci Sequence
    ) A3 m) u/ }$ G& d! ~' g5 ^
    2 j9 {- m9 R% m# h7 ]The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    / U) ]1 z0 `$ d2 C: U2 ?
    " Z* B& B# G- R# U3 O  k    Base case: fib(0) = 0, fib(1) = 1" K8 d6 h* d- b8 \0 A$ g2 ?3 g& ?

    6 z  G6 l+ a+ K    Recursive case: fib(n) = fib(n-1) + fib(n-2)+ ^- T, X# d  b6 j' x

    * y  M% Z( g" ^( Ppython
    # G/ O# Y6 R* P# [, T
    $ D' {8 m1 G% d, x. f% R$ w  K* f
    def fibonacci(n):
    8 s! ?9 D# X% m    # Base cases) {' L" Y3 K# _5 D) U. z" s
        if n == 0:
    & Z, z5 B% n" t* T        return 06 K2 w0 B" D* D# s- v4 E3 |- N
        elif n == 1:' q. v. O% x% W, Z) E
            return 1  e3 ^" H& w9 q' F6 ~
        # Recursive case- @8 o5 l$ R9 x
        else:2 A6 P" x% X* C; n' g. R8 u  u
            return fibonacci(n - 1) + fibonacci(n - 2)
      a* u( }: b# J+ y2 S
    ' A1 {2 d# C+ X2 {$ A5 Y# K% O# Example usage; [, _: X* K( b& G" G6 ]" E( n
    print(fibonacci(6))  # Output: 8
    + ?$ R8 r% m( [2 d, z3 w0 m6 E  A( c: X: S' H
    Tail Recursion# b$ m: M7 a# m+ X0 K' I6 @
    6 i5 V$ b  \* X1 w, ?6 X5 D* H
    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).7 a$ v3 h6 m5 Z
    3 K0 ?# e( C; l1 y4 f, \8 L4 H
    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.
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    地板
    发表于 2025-2-2 00:47:27 | 只看该作者
    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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