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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| 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
    : q. R+ a! m2 ?! M1 y( {, I
      B  g+ g; L( |9 H6 ~, F) S' \解释的不错( Z6 n, d* m& v" S- ^

    , d' h4 P& D' _8 E/ M3 n( m8 y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。
    / b& ~) E: U: s- N6 r9 H$ h9 S( W( h+ J' v9 t2 d
    关键要素- I4 ^8 M6 e0 D: D+ Z2 ]8 r
    1. **基线条件(Base Case)**
    . ~& o4 A1 U8 Z: I2 N  w! l   - 递归终止的条件,防止无限循环
    $ \/ u9 b+ M) I* E" d' `   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1, R- t! y3 p% I( e
    : N- c$ {2 A8 J) X5 i. j% R# }
    2. **递归条件(Recursive Case)**; s+ A/ u2 A% C/ x) f% g  Z2 k
       - 将原问题分解为更小的子问题7 G' H$ J0 k# z& I5 K5 ?% E; Q
       - 例如:n! = n × (n-1)!0 [+ m0 v7 l" S& W' R' J) p
    ( @# b; f2 K8 k, ^9 {
    经典示例:计算阶乘
    + L1 r; q' ]/ D$ C' P  H& L- w) Dpython
    $ E% o5 L) L/ l& k0 f; K8 o" i4 Hdef factorial(n):
    ; a* I& x8 T8 C0 E  x9 }    if n == 0:        # 基线条件
    2 F6 L; f1 M9 y4 K: R        return 1
    0 ]6 t3 D/ G  r' E2 X( A    else:             # 递归条件/ Y: w( R; L& D* C; F' K
            return n * factorial(n-1)3 b, s& `; Q( Z" C2 q' y! i
    执行过程(以计算 3! 为例):
    & n; t  v* Q* Q/ afactorial(3)
    " |; ^- Y4 F% Q0 I6 y- v( ?3 * factorial(2)
    $ P2 B9 g1 z# U: _3 * (2 * factorial(1))
    ; r  |8 ?) t5 B/ X" d, E3 * (2 * (1 * factorial(0)))/ r& v3 m- N8 J/ Y! Y& m
    3 * (2 * (1 * 1)) = 6" M/ z; d5 Y7 c2 f1 V( T

    ) l& Y! J( f; b- G) D# D9 o/ _' \ 递归思维要点8 C' V# s2 w$ Q; z% D! l9 H4 w2 x( k* b
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑; J9 i% w$ o+ f; o1 s* B
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
      n- o" `/ L3 F! V/ |9 ~/ O% K3. **递推过程**:不断向下分解问题(递)6 I- p5 g4 I$ a
    4. **回溯过程**:组合子问题结果返回(归)
    3 i# u! V- ~! H
    ; R, C: _! f2 s( \/ W- h, N( N7 @4 v( v 注意事项2 G) r& m% j9 X9 a/ c) \
    必须要有终止条件4 o# w0 c2 D* h1 w0 i7 l8 x+ f8 T
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)9 E# `! W& r1 H) K% i6 C
    某些问题用递归更直观(如树遍历),但效率可能不如迭代0 H  r" r) i2 z7 V
    尾递归优化可以提升效率(但Python不支持)4 H' d1 ?; U3 _/ n8 j4 X

    ( m/ R) y0 o8 {8 F% O6 t' e 递归 vs 迭代/ t/ R, |/ W0 P" [0 q
    |          | 递归                          | 迭代               |
    & z" l9 K  X; Y1 V+ O* Y8 G6 s|----------|-----------------------------|------------------|) Q/ q0 s8 X; }! E
    | 实现方式    | 函数自调用                        | 循环结构            |
    2 E& {( A0 w2 D1 ]$ b| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ! Q& I3 U- d$ a| 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         |
    4 U! i, G8 l- Y# ?| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ( f' l3 e" o+ ~- g3 ]# `$ M; P% s+ I5 P/ N) A, u
    经典递归应用场景
    * `9 G8 Q4 L7 T+ B1 q1. 文件系统遍历(目录树结构)
    # f$ r, {8 Z& |$ s! A/ o2. 快速排序/归并排序算法
    . L7 s8 Z$ R/ p5 E! {1 C3. 汉诺塔问题
    & U6 a$ C3 @* W2 E4. 二叉树遍历(前序/中序/后序)  O9 W+ N4 z/ C7 O% S- o2 X5 C
    5. 生成所有可能的组合(回溯算法)) U$ _; O1 C+ N! @3 a4 R, ]4 b
    5 h; M7 H. n1 ]5 J' H
    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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    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,/ P$ {1 ~" H# o; L0 l9 K
    我推理机的核心算法应该是二叉树遍历的变种。
    1 w) c, `2 G3 n7 M4 d5 s- s" u; z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:( \; J; }, {, h8 {2 W5 b& W- J$ g
    Key Idea of Recursion- z7 G9 R" u5 u; X
    # y3 e3 L2 L' |* @
    A recursive function solves a problem by:
    0 @% `4 F9 M6 u6 n$ ^
    ' L" }& ^9 X- L* {    Breaking the problem into smaller instances of the same problem.% b& e( i' x0 L8 y9 J

    3 i6 ?, |" f+ c0 d$ W    Solving the smallest instance directly (base case).
    7 N- k' L; s$ `$ o+ G
    & W7 B2 {" D; e$ J1 \+ b/ ~    Combining the results of smaller instances to solve the larger problem.4 m% w# I9 ]! c; M/ b2 {

    9 {% N  c6 J, J2 H8 nComponents of a Recursive Function3 G2 l6 u; U6 C8 Q
    ) V7 I  g7 f+ ]  f
        Base Case:* o+ P# a8 r, K0 A9 M# ?/ X

    . g1 B( K7 D0 O2 |9 b- X: W! z# h        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    6 u! \6 Q& P! s3 k+ H$ k. Z1 J
      x( G2 c) [. e9 A+ T" D        It acts as the stopping condition to prevent infinite recursion.$ q7 j' x# T' n  i6 N' s1 h5 |0 y
    ( ?/ A% O8 E$ B; t% _& e9 X
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    + v( ?+ w: Z$ J- M! r0 Q
    7 M" W8 c2 i! M9 U* J: X    Recursive Case:
    9 n; A  g( N% v* l: ?
    ) ?- e6 t7 }" u6 X3 g' [* h1 J1 a        This is where the function calls itself with a smaller or simpler version of the problem.
    . D; h0 n5 r' w- b# O# n% j' r' W, f1 @
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).3 s  e% F, Y& b: Y+ i9 V# ?' A9 J' i

    . |) M; o/ c: M8 ]  ?; qExample: Factorial Calculation
    / C% _1 @) X0 v' z6 u. r% h
    ' Z7 [# d2 C/ l6 I2 lThe 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:" B( }3 A6 Q2 _, p- i# E# M6 G

    2 j% b  u) f8 d: j2 `# p; ]; R    Base case: 0! = 1
    1 b& p: b" D! m$ D7 F- F/ X6 X* x% R/ C" M5 u' D* M) J
        Recursive case: n! = n * (n-1)!
    6 {2 h7 V- [: j1 |% G; J# _: y
    ; S: N9 [4 |# ZHere’s how it looks in code (Python):
    + s6 F& Q2 q/ R5 @  _* l$ mpython  q( F, ^/ w7 w& J
    7 f  U  \, A4 |$ o0 t( l- I
    ( ~  S; X6 }8 X7 C3 j
    def factorial(n):
    7 r  R/ {5 o2 s0 p# c* {    # Base case
    4 ~# n7 v7 L/ _  }9 V" e/ c    if n == 0:
      G6 u' j# U: h' ]; d- j        return 1
    0 l  S$ G! Q1 H& f5 ]9 M( F    # Recursive case0 K5 `7 a* j) H7 W- `+ B
        else:7 Q: M5 N5 z. e' m
            return n * factorial(n - 1)2 v* N9 K' P5 z3 l0 u

    / g7 m2 j7 _2 Q1 L% q. K# Example usage% h0 w$ p) _1 B1 x7 f
    print(factorial(5))  # Output: 1209 Q/ P1 \2 B0 {1 W: m, p

    ; Q% Z, _$ Z* O/ d$ mHow Recursion Works: k+ @+ A# F) o! ^9 o

    . L  F# ]  X0 C' N$ U3 V    The function keeps calling itself with smaller inputs until it reaches the base case.
    2 \: q6 V* \  t, m- O
    ( E/ f$ b1 @; b4 T. u: |3 r6 {    Once the base case is reached, the function starts returning values back up the call stack.3 a: B$ m8 y, W- E5 L; e: e" V
    6 c, p8 I* y0 G% \, v/ g8 t/ l: y
        These returned values are combined to produce the final result., A4 E& {0 v  v

    + d: W/ R( v- W7 m* Q6 o6 FFor factorial(5):! |3 j2 g  M; y
    ! x% Q1 c0 b6 U

    - r/ y; z$ A6 R; C: m1 j/ c9 cfactorial(5) = 5 * factorial(4)
    . S( b" N* d5 H, X/ ofactorial(4) = 4 * factorial(3)
    . H! y2 h- e9 x$ o  R* x8 c0 g- cfactorial(3) = 3 * factorial(2)
    ' b5 g. q# M5 N+ @factorial(2) = 2 * factorial(1)! S# r: X4 M4 v0 g1 o3 o7 u+ _- K
    factorial(1) = 1 * factorial(0)1 s6 t5 p4 a5 t5 p8 l! n
    factorial(0) = 1  # Base case2 W( o% ^5 @4 I2 @4 C2 q2 W

    & ?- v7 K" p7 \Then, the results are combined:
    * h# W' @+ ?0 j3 v9 h
    5 N. P% A( W) \& x4 s
    & P# ~  b/ j) ]& `factorial(1) = 1 * 1 = 10 ^6 N0 J; C  O
    factorial(2) = 2 * 1 = 2
    ! \3 r7 v0 }( a* N& i- d: [4 Ifactorial(3) = 3 * 2 = 6% `+ i5 g8 L- Q
    factorial(4) = 4 * 6 = 248 N( U. F+ a0 l$ y2 K
    factorial(5) = 5 * 24 = 1202 C8 I. }, u! p$ y

    5 e+ F( W1 Y# `2 f! ]0 n; ?& xAdvantages of Recursion9 {, |' c7 y. N4 r  L

    $ K0 W$ W  M+ k  \6 U    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).
      @) c7 q. r$ X5 V* h: Y/ C9 ^7 e- F% p: O+ `: q
        Readability: Recursive code can be more readable and concise compared to iterative solutions.( B. T5 X& r$ I9 }/ U9 w

    $ D; {. e+ @1 j# H) LDisadvantages of Recursion7 A! @  I7 z. Y; E% v+ i2 a- M
    3 h- t$ L8 ~* n9 Z" n
        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./ u3 r9 z0 z5 h/ V) h
    5 h% W/ [' X3 d. M
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ! a6 t- y+ N: G
    % l2 N8 o& [9 `) v# O% [When to Use Recursion" o' R/ X5 v- J6 ?4 _) k4 g1 ^' m

    0 N7 y3 S( p- e% _( ?( K/ N: b: f0 G    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    * R) P5 m4 p, R8 E4 W& t" V8 e2 p& Y6 r* y% b
        Problems with a clear base case and recursive case.
    ; ~( d6 y% r( q6 e* Z& ]% T! K$ ]
    : _$ r( `- Q4 j; z: oExample: Fibonacci Sequence
    ) y  M6 l+ i- \) I3 v- b  u/ ~/ Q2 Z0 t& k8 b0 |
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ( m( T' {2 a/ E. I4 y$ `$ n9 y
        Base case: fib(0) = 0, fib(1) = 15 V, v% {, X& A) [0 x
    0 ^: j/ y$ _2 b& @" w0 z! Z
        Recursive case: fib(n) = fib(n-1) + fib(n-2)+ B& g* \/ w/ `0 [; n/ h9 t
    3 T* W+ G+ x. h, U7 P) l& b
    python
    6 U% l( X: J* C  \% ]5 ^8 q% q7 [! b5 j- B0 v" X- R" D6 O4 S

    / [  d! D# {7 j8 P9 ~- f5 Y( K- Udef fibonacci(n):
    " f+ v) P$ k' b- q% H    # Base cases
    6 }. z& a& B3 d1 W" J. L    if n == 0:
      `, j( E$ D) K) b$ V& m        return 0# w- y1 U5 N" N1 y: e; m
        elif n == 1:
    ! `( x6 U) n3 K! E        return 10 O  s/ `# \1 v2 N  q
        # Recursive case
    , N8 Z( p, S( A( `    else:6 }+ d( g4 C7 G
            return fibonacci(n - 1) + fibonacci(n - 2)
    1 d) |5 m6 v% n5 V: A; r! E
    4 X9 j7 ^, n+ Y( m  z# Example usage2 x% d* E; E; ]& N- h; a8 M
    print(fibonacci(6))  # Output: 8
    ! p. M$ x9 M  _' r3 I3 Z* I& x' p8 i9 ?: G6 b0 ]
    Tail Recursion8 S0 A4 K' z$ g# C1 [
    2 g) g9 `" g# A; C! h4 w" W
    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).; C( l: x. P) y2 J" J" c1 N, |3 m

    ( C$ ?% [9 }: f6 [9 p# V& |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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