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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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    2025-9-8 05:08
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    楼主
     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
    ) N6 l4 _. |' E$ {. n" W6 r
    ' U0 T7 |/ B& L& \解释的不错# \7 C2 l. |2 \

    7 }: p9 W$ y' Z  U4 _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。2 g3 t' e4 s/ w! L9 K; K6 H
    / c) Q! ]" N7 j) ^. V
    关键要素5 n7 w: [% a( O2 r7 R0 f0 U
    1. **基线条件(Base Case)**
    4 K# t8 H& h/ Q. K9 w4 z   - 递归终止的条件,防止无限循环& K4 U6 R( T4 J' X' C1 w: P3 B
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1% y, M, H$ R0 A5 F1 D, k4 ]9 U

    & W. O' g! Z* q( O+ V( k2. **递归条件(Recursive Case)**
    4 A  s5 ?6 l4 L% Z5 q! e1 q- \   - 将原问题分解为更小的子问题$ \2 k# ^4 s% [* z! [: U5 d: ^: v
       - 例如:n! = n × (n-1)!
    0 l/ P8 p, E! \; h' d
    # [" O: {% ~& `" G6 U 经典示例:计算阶乘# b7 L. y% t, g. h" G! M
    python9 ]- a+ M2 p& A6 \; c
    def factorial(n):5 }9 ]* g2 R- k- l
        if n == 0:        # 基线条件
    - ]. i& n3 Y& ?  |* H        return 1+ [: w3 Z  p* {1 C
        else:             # 递归条件! H( N# i; y5 h5 }
            return n * factorial(n-1): q7 j0 x5 ]# ^4 A, G, J; [* Q
    执行过程(以计算 3! 为例):
    + i+ \4 `: b$ v( ^6 {factorial(3); S' x' W# U7 d: [% V
    3 * factorial(2)
    ) G, J9 U' w, k! [% Q: _3 * (2 * factorial(1)); `& b: [5 m3 g& Q* g
    3 * (2 * (1 * factorial(0)))
    5 O9 m1 p1 C5 @" [) f3 * (2 * (1 * 1)) = 6
    3 e. B& s, S: r6 ]/ [
      r% p1 k6 b" d0 e/ u, R  { 递归思维要点/ a8 e4 m! i' d* P; `7 g! s8 r' u
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    % P9 o& Y6 ?3 y; {) }2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    1 R2 `' ^) L/ T* |! x+ r9 z6 u# W3. **递推过程**:不断向下分解问题(递)( @" r1 ~1 Z( X; c
    4. **回溯过程**:组合子问题结果返回(归)
    ! l  c: m; a2 f7 _% |" y0 x* `0 B; A
    注意事项
    $ K" A# _+ x+ f& x- ?必须要有终止条件* W& f  t: c* C
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    4 T* M& V) Y. q9 R; ]2 s某些问题用递归更直观(如树遍历),但效率可能不如迭代3 n( y, F/ M: [" @2 L
    尾递归优化可以提升效率(但Python不支持)
    & I8 u' E# G7 g8 F0 j' a( n7 O1 b
      [, a: d1 y2 D. ]* B+ G& a( J1 U 递归 vs 迭代% m# q/ [9 p; E- E8 w
    |          | 递归                          | 迭代               |
    # h; D* w  h% H# \& n8 A, A|----------|-----------------------------|------------------|* t$ ?6 A( @; @4 K! O
    | 实现方式    | 函数自调用                        | 循环结构            |
    ( w- L# G1 x( E) t8 T| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ! m" p% F& y* X| 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         |& C( n) T& i; ^8 x# N; y+ t
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |" P) E' w4 u! y- D* f- c

    ; h) C2 h4 c# t 经典递归应用场景8 D9 ]: _, V) I2 [: X% i
    1. 文件系统遍历(目录树结构)
    $ V$ M2 a7 ?0 ?7 P2. 快速排序/归并排序算法9 l1 y7 x, ?* @3 D/ ^  X0 u
    3. 汉诺塔问题
    ) e9 v: o2 n" G3 K& `! U4. 二叉树遍历(前序/中序/后序)
    $ `' X, N0 x& ~! U* ?9 m' `. I, H5. 生成所有可能的组合(回溯算法), n+ r4 @5 e% C) o

    . y0 W9 L7 d. \: ]  c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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    昨天 09:58
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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,$ e4 i9 X; o1 t1 }7 U* M. N
    我推理机的核心算法应该是二叉树遍历的变种。
    8 i3 J/ j+ b( y. k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:3 q9 w; T8 F5 t# n6 B
    Key Idea of Recursion
    . _0 e1 o' m) H( A1 }/ ^% D9 {+ i( t: }- s8 S/ ~# M# }  h
    A recursive function solves a problem by:
    5 X6 {: d* S" u7 E2 q  r  B# j4 B2 D% ~( O. h2 t7 S% A
        Breaking the problem into smaller instances of the same problem.! F, O& H. @$ F% d0 S

    1 a: N' [0 ]+ E" s; J( t    Solving the smallest instance directly (base case).
    5 X; N) O/ K! O% s- c
    ; u3 {) \" _- H' `    Combining the results of smaller instances to solve the larger problem.
    + H& s& V3 ?* G$ G; x
    ( a* H; p5 ^$ I$ HComponents of a Recursive Function' p) ~( S* h- T, a9 [/ [# ?

    0 X! c8 i6 K$ a8 [) |    Base Case:
    ) t% D" j7 m0 P- ~" R
    ( i1 j9 ~* ~( d/ \        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    & V$ G; o0 v9 G+ B) F9 P% E/ q9 g4 B' V5 i7 O7 [; Y
            It acts as the stopping condition to prevent infinite recursion.' J" w4 a. i9 ?6 \4 T/ ]/ Y

    ) [6 q. Z9 a" c7 s) B" X        Example: In calculating the factorial of a number, the base case is factorial(0) = 1./ |8 I# ?& Q  S- L& y: Y9 x0 Y

    . K) }9 A$ X& r! j4 s1 S    Recursive Case:
    7 ?& Q) x0 a7 h4 _' t# C+ Q  @' H: A$ m, ]. D+ w$ G& {
            This is where the function calls itself with a smaller or simpler version of the problem.# r6 M5 C5 L$ F0 {: w

    ) ?6 }. n! O8 M0 r- U        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    , X. k' I. @% o( n& g
    5 @$ Q& {6 b  _. ]Example: Factorial Calculation+ ?+ B# g4 d1 Z8 @8 u

    7 l0 n  ?6 W4 S* D8 w7 ]+ v9 J1 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:
    5 x8 X' N" F2 h& L1 Y
    & f) Z% P: O' G, y    Base case: 0! = 15 D5 a3 P; o- ?  r* D% m

    * \' L+ r* s6 h$ O8 \    Recursive case: n! = n * (n-1)!
    7 E7 j2 \# G$ U2 g* J+ Z5 _& f
    4 D* ]+ J: d+ R: `4 ?6 N: JHere’s how it looks in code (Python):8 }2 V, ]8 Z: p; z# D
    python
    8 o4 P$ ?" N" h/ U- k' k' k3 X0 r/ V0 }4 \6 B/ W8 O1 Y

    " _' v2 b' E% o1 H( hdef factorial(n):/ @$ w: e& i7 s6 d1 P
        # Base case
    . X# L8 V) |4 ?9 Y7 w1 a  C    if n == 0:
      b& Y0 A4 l8 K" M* \        return 1( M7 R( E* X* b5 J/ l0 {+ |
        # Recursive case
    " ]' k" t1 N7 o8 P* f+ g( H! m    else:
    ) K; y+ i$ x2 ]4 d. T        return n * factorial(n - 1)9 E' C, x. W3 f( X$ q6 j
      r: F5 }5 g' a& c# g! N4 V3 ?
    # Example usage
    2 z( P) m3 e. a' B. Oprint(factorial(5))  # Output: 120
    1 ~* N1 q* S  D0 ?% i  n9 O( q" J" R$ a; Q4 h( w
    How Recursion Works* D% ]6 z! B0 b1 H  ]$ I+ Z) ]
    ( L) ^: E; h5 A1 {/ Y
        The function keeps calling itself with smaller inputs until it reaches the base case.
    $ j* K8 ~( |( c' V2 y* D6 N- L3 C8 f6 n5 o& \' a% S0 z# W: G' i3 [9 I
        Once the base case is reached, the function starts returning values back up the call stack.: p; j) ^- g7 ?" U- i& R

    8 E5 }: F9 l  d, M6 [* F    These returned values are combined to produce the final result.: k' ~, v( Q% \

    4 x2 ]0 I9 r: B0 g# R  z9 `For factorial(5):
    / u% t) h0 o  |& p- B* y! Y' l
    4 X, C2 B0 C3 Y
    & w6 \+ ?- I, s+ w  n: K) sfactorial(5) = 5 * factorial(4)1 ]5 Z; Z% B% c  y  h& i
    factorial(4) = 4 * factorial(3)
    1 W1 A+ D; p0 o( W) d* l/ yfactorial(3) = 3 * factorial(2)
    ! r/ L# o/ B+ n- X, @% ffactorial(2) = 2 * factorial(1)2 }% h# F. b4 E5 M& o
    factorial(1) = 1 * factorial(0)
    8 z3 \. x5 u, {# ^4 Kfactorial(0) = 1  # Base case1 i9 ?6 y8 N7 X- \0 k- M+ b& E

    4 \, Y+ Y" {- T! x+ V' i: u2 tThen, the results are combined:
    5 u: f0 S% T& ?" Q' U6 C. s1 C* }$ L- {

    0 y' u0 S/ _, @- C* B' ^factorial(1) = 1 * 1 = 1
    . R0 D$ p/ D; {+ l. x4 _# v8 Nfactorial(2) = 2 * 1 = 2
    7 B& k/ C8 ]0 s, V  I3 rfactorial(3) = 3 * 2 = 6
    : f2 W$ \# R5 p; k5 b4 @- C' Mfactorial(4) = 4 * 6 = 24. X5 S. r* G6 Z" ^
    factorial(5) = 5 * 24 = 120$ D0 N9 l, R1 q: w2 g! b
    ' J. G' l( C: K' [# R! l( D  Q* a5 ]
    Advantages of Recursion
    : a4 _# k& V% {! Y/ i: S4 ~% J$ U2 z; {8 z
        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).
    6 o& o9 s* n  ~+ b$ ~% \, p: B; u1 Z
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    # z, F5 k# w+ F' g) u' _8 [, w& t" l% M8 z4 w# U' G
    Disadvantages of Recursion1 K% R) e1 N8 o! T" q1 k$ e% t: r
    # ?; f$ Z& m/ ~: J) Y* 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.. y! f/ H: v$ ~6 x" I: i

    ! a# k/ ?# y$ j! g$ z    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    4 R4 s: Z2 |# S# i. W! {  W  H6 ]& x$ [& ^
    When to Use Recursion  i4 Q5 F9 W3 ?2 f( n3 B# k

    # a& `, X( \% ^' U& n) E- S1 S    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).7 ~# A) Y- k3 U# p0 x) x
    ' j0 M- K. m; r0 M# }% t
        Problems with a clear base case and recursive case.$ L5 T9 x; d! x( l# ~9 M7 Y
    5 g+ c0 F' S6 D2 `4 {: q) m+ D
    Example: Fibonacci Sequence
    / x( w" W. U: k8 I2 q0 v( N5 s6 W: V" \
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:7 d" Q* F" X  n

    ! A  S' e1 n, H( p1 Y9 A    Base case: fib(0) = 0, fib(1) = 1$ N: ]$ u9 u: S+ l; ^8 K% _

    ! R+ l9 }1 A: S$ O/ j& i& F& g- n    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    , R. p- x9 ~6 z# n4 \2 G; x5 k
    0 U. s8 p& D% wpython; W+ f( s+ K. o: [, K3 Q  |

    * m- \' C) z1 R% ?) F$ @: R- ^
    - ~9 W9 c. }" }3 Z) G& p8 Gdef fibonacci(n):9 B4 \% ?( R8 {# y
        # Base cases
    2 x; B) [! ]4 h5 U8 K; Z  L, p    if n == 0:
    ( f, ]9 l7 r& l        return 0
    & _# a& V/ v+ j' B    elif n == 1:
    8 H7 D  y: n% Q2 B/ f! ]        return 1
    * |9 T8 `7 n1 W" k# L/ C  t    # Recursive case
    + z/ m/ F6 U+ X' t1 O) W    else:
    ) N" [( _" d5 V' e9 B        return fibonacci(n - 1) + fibonacci(n - 2)0 y0 Q* y4 u4 R* |
    ' h/ z2 ~% E, |4 [1 [
    # Example usage
    ' M& G) V6 F- ^/ \3 W  Xprint(fibonacci(6))  # Output: 85 z0 U5 g( m+ F/ ~/ I
    4 Z, }$ v2 S) z! A: G
    Tail Recursion) y6 s. p" d2 d+ M1 ~

    : G- m" [9 ~- y( O; \5 h" G7 S- uTail 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).
    , g: }' K! i6 P% b& D* C1 z  o! q6 k
    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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