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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
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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
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    3 s# J+ K) {" w) {  K解释的不错% K2 T; m6 g) w0 Q2 Y/ z$ K2 X

    # q$ d# I9 `5 o4 f% J3 ^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。6 w5 I# q* F+ b# T/ t6 s9 G5 o2 Y

    ) f. f2 P: }2 X& M* K# C 关键要素
    % j2 @8 h$ A/ K, G2 k# M3 b& |1. **基线条件(Base Case)**
    / @* Q6 o/ V! n* C/ ~   - 递归终止的条件,防止无限循环$ r4 i0 v: X: x. K
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1) \. i7 R$ j+ V* b

    # h) ~6 b9 y& a' J/ H( \2. **递归条件(Recursive Case)**; O' P3 v, C2 T& S
       - 将原问题分解为更小的子问题
    9 _. n$ \7 h4 m! U% Q% f4 f) [- r/ {   - 例如:n! = n × (n-1)!3 z6 }8 j( X: s5 P
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    经典示例:计算阶乘& M, v" F5 v2 k9 D
    python: M8 s8 q- \" P! X
    def factorial(n):9 w; A( Z1 ?3 a  E4 p, s
        if n == 0:        # 基线条件$ L3 _1 i: o) _6 T5 X# @4 r6 [
            return 1( C% p( a4 u$ J! B$ U" }
        else:             # 递归条件
    0 q+ j! D% ]+ c1 y$ j        return n * factorial(n-1)4 D8 s8 U: n0 P- ^: q& t
    执行过程(以计算 3! 为例):( B( q) p3 u* D( }
    factorial(3)( V7 S. Y3 _4 F; H5 g
    3 * factorial(2): u3 s3 S  d# l/ ]
    3 * (2 * factorial(1))* p  M  a8 v" e: s, d2 O& ]
    3 * (2 * (1 * factorial(0)))
    ( D% ?/ r2 K5 y& f3 * (2 * (1 * 1)) = 6
    # \( H" ^9 h8 h7 b1 L1 k# Z7 n6 \( y
    递归思维要点: N- J6 e- N, _. X0 y1 M5 |
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑# E$ R1 Z, q, N7 B  y6 g& v0 h; X
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间): k& F. k1 R( Q. U, Q5 i
    3. **递推过程**:不断向下分解问题(递)& a, u: V9 w$ r- U( V. l) R
    4. **回溯过程**:组合子问题结果返回(归); v; B5 b- n# u  b  b
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    注意事项/ y$ P% `! P. U% \6 E
    必须要有终止条件% {) z) ]1 J  R' ^1 |
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)+ f0 Y* e! ]" X1 g; u
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    3 @' s  |% ~& B6 p# L% i/ s: Z尾递归优化可以提升效率(但Python不支持)
    , x! v- P/ u$ Z' |, [+ Y
    9 B& @& h/ r; Q7 U 递归 vs 迭代
    2 F1 W9 L0 ^  v; X9 o|          | 递归                          | 迭代               |6 G6 [4 A$ Z6 A, s3 f
    |----------|-----------------------------|------------------|9 E' \4 h; }; A8 Z. E2 s' D. h; ~$ y
    | 实现方式    | 函数自调用                        | 循环结构            |4 G" s7 c  ?1 p
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ( C7 N5 B3 S3 N# Q4 j) {  e- |; Y| 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         |  s  o. G' k7 {- N* X5 j
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |' t. b( {9 O5 b3 h

    1 ^; B6 f8 J) j0 p 经典递归应用场景1 [9 x$ N5 |2 d
    1. 文件系统遍历(目录树结构)0 W: W4 D" y* i
    2. 快速排序/归并排序算法% |  y1 z* A5 @4 `. M6 X. B) ], P
    3. 汉诺塔问题
    7 Q/ w! `  ~. B4. 二叉树遍历(前序/中序/后序)
    5 z& P! c; A5 ~+ e; \6 S" {% |5. 生成所有可能的组合(回溯算法): W8 p. g  M7 X, k
    + N+ H; _9 z* p
    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,
    4 i4 {  i2 d0 E! `0 b( q我推理机的核心算法应该是二叉树遍历的变种。% p% r# l! O. E) I3 f4 j. g3 G( i* 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:$ x+ C) y# J0 j( S" c( x$ c8 f  e
    Key Idea of Recursion
    + `0 e' t2 V. X+ i9 j3 c( J5 _0 J  }! @( Y" j2 B4 w
    A recursive function solves a problem by:
      ~. D1 H! d& X$ t9 d
    2 C, y/ O5 S- c8 I+ f8 f) u    Breaking the problem into smaller instances of the same problem.) W. [5 A) u: L1 I8 a

    & f8 Z' P, m: ]2 Y; m9 Y    Solving the smallest instance directly (base case).
      h* w: |& a$ q, ]" `: {8 S' b8 j
    5 Q, c) ?' a) s8 |; m$ I1 l    Combining the results of smaller instances to solve the larger problem." o' G" N0 M5 c1 T; N8 I

    0 |$ Q0 ?( Q+ u/ W8 J8 q7 DComponents of a Recursive Function# j0 n2 j) Q; s/ D" \9 @. ~! J! `
    0 H3 K& s- r6 V5 T' u- d
        Base Case:
    - }3 q+ [8 `  F9 V$ |* v9 p' v" d( p/ w( _7 E9 W
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.- z5 M; B  g3 X4 W
    " n) l8 }1 G: e* R
            It acts as the stopping condition to prevent infinite recursion.( O7 j/ O# _* H( m/ @: u

    / m6 W6 C2 {2 w5 N8 L: n9 j        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.* @$ @# F! I$ v) R9 x

    9 C$ \: d8 k8 i& G$ ^+ a* j    Recursive Case:  N' Q0 C) D! P. _- f5 j6 D

    5 F6 Y5 Y. M, p3 _0 p/ k3 ^& f4 a        This is where the function calls itself with a smaller or simpler version of the problem.
    # h3 l, t/ `: g$ O# i: e9 u) C9 U; s9 L5 d
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).' K7 C/ S" f" m$ ^
    . {% q' x4 o/ F! f5 F/ @
    Example: Factorial Calculation  E9 }. o: l; i7 J
    3 ]# l- m1 ]2 u! o/ R
    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:  W1 t5 w! D  D

    : W( A$ @$ ~+ W8 h: {    Base case: 0! = 1: i5 |% J  u# k* G! W
    9 `6 |/ R1 a. b; x8 k
        Recursive case: n! = n * (n-1)!& d# `4 M9 t' z* N( B

    9 I9 P: p8 B" E; |% G" k3 sHere’s how it looks in code (Python):* b+ N9 B) K2 v  J
    python6 P, ~+ ~+ G" `5 O9 ?  o

    1 N% }$ x" r( S, g
    9 c* u- s6 u& E4 Vdef factorial(n):4 [4 c3 Z& m( x& K
        # Base case
    + `8 z! E- `1 m8 \# L0 M; C    if n == 0:* H, o! P8 H: D# M
            return 1: r+ V9 h8 |4 q. S5 R0 V, S$ y' K4 e
        # Recursive case* m5 J8 b7 s! G; o: @' Q8 N
        else:
    % R2 A3 K) S4 A5 W( e* L        return n * factorial(n - 1)
    6 ~3 Y6 a  M% c# q$ Z1 l* o: k" f7 N' s% u% ?7 @
    # Example usage7 }/ K& R2 W+ j5 t
    print(factorial(5))  # Output: 1202 g& K9 w# {1 ]$ c4 N& O

    2 w7 t9 G7 I+ N# S$ C7 eHow Recursion Works$ V$ o7 S! `' ~& F8 \/ p4 W, j
    & }6 c& d( J( B7 y- f8 W
        The function keeps calling itself with smaller inputs until it reaches the base case.
    & i) t4 U+ J5 i2 D0 ]/ a2 x& R  P5 S! h9 H
        Once the base case is reached, the function starts returning values back up the call stack.$ _0 S! _4 ~. S! t1 i; N* B

    & P* Z/ P- t) `, z5 y* f3 R    These returned values are combined to produce the final result.
    6 w6 w8 @- y9 ^  h5 d
    % l2 m0 f6 N8 [( e& tFor factorial(5):) T6 d9 `, G" |, d

    . R/ g: z( x7 Y& I0 h
      u# P2 Q1 I4 N* tfactorial(5) = 5 * factorial(4)
    ) T! H7 D* H, A/ s" M2 H: n3 i: N/ Afactorial(4) = 4 * factorial(3); _8 o- m1 r( I% v( W% `
    factorial(3) = 3 * factorial(2)
    0 f' S* q( W" W$ D9 Cfactorial(2) = 2 * factorial(1)
    # ~, }# S8 |% s- S9 \factorial(1) = 1 * factorial(0)
    6 U6 Q' H$ d; I" i7 |! z) Zfactorial(0) = 1  # Base case1 E+ U7 z& a( D

    ) r" f6 ?4 T2 aThen, the results are combined:
    . A0 `" C; B4 R: z0 w% A: D# a* A) V6 L& A, l' u
    4 E- X  m0 ^$ m3 A
    factorial(1) = 1 * 1 = 18 |9 j$ A0 U" c5 ~4 C  z
    factorial(2) = 2 * 1 = 2$ F( p' }; I: l! R
    factorial(3) = 3 * 2 = 6) D* w3 J" q9 ^2 V
    factorial(4) = 4 * 6 = 24
    7 f8 ^7 F  J1 z1 y. q. bfactorial(5) = 5 * 24 = 120. }# o$ r& |$ c% u" f, C
    ' |& s% n$ k: S4 g' @
    Advantages of Recursion2 T- [1 g& _7 W5 [/ I

      A: j, }+ z0 N* i( C    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).
    : M( R* Z0 g# x. D1 C- Y) t0 H; w7 `; r
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    + Z- _( e- p- c, B/ f4 s5 `& E; f. x1 S: A& {- q5 c8 d* Y
    Disadvantages of Recursion
    5 Z4 H" q; j) n0 s5 R, L6 B. b
    " n' g. g) Q7 t( Q9 e; Y- ?+ v    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.0 i/ z6 ~3 y1 ~

    * F& y5 g, l& Q; y    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).* P, m! V4 y  ~5 k1 j
    ; c8 P8 e8 R; n0 D# S
    When to Use Recursion1 k' m  p4 a8 P0 v
    / R& u4 H; S; H  f6 r- D
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    * O4 s$ e3 h% m$ {1 v8 B
    1 |0 f7 i& _5 t    Problems with a clear base case and recursive case.
    & @* X8 X) [# u4 E" D; o; i; ~) M
    / j/ X2 P, O# [1 V: H6 R8 c* _Example: Fibonacci Sequence
    8 \5 w' O! H% ]  G6 }8 w5 S
      i% F$ |2 w8 j1 lThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    $ q* q( q1 Q; p7 S
    4 T# e+ U$ h8 `3 `+ r4 E! u6 B    Base case: fib(0) = 0, fib(1) = 1
    ! B0 @6 X7 n0 l8 T2 T7 j  B
    + Z" n! ~( V! D    Recursive case: fib(n) = fib(n-1) + fib(n-2)6 j: E" c# d/ t! F
    6 n9 d# T. P6 z  d( P
    python6 A' s) O$ C. C- a+ X  o" f/ t4 P$ v3 O5 Y

    . i. _  j0 ?7 f& a* @+ b6 N/ o) l5 p  I" s) z( [3 D
    def fibonacci(n):
    6 W, j, V1 U6 M* Y    # Base cases2 y- Y2 x& _8 ?3 {9 V% y- w
        if n == 0:
    3 G7 I* g7 b" W        return 08 K# Z7 E1 k6 r# j
        elif n == 1:
    7 C5 V; f. x5 M! f7 u        return 1
    ( Z7 s1 ^5 t! m2 Y) G7 K/ |    # Recursive case2 ^8 ~; Z. g. u. ]
        else:# x- X7 n# S) C5 A5 |, K! R) }
            return fibonacci(n - 1) + fibonacci(n - 2)
    2 E- C5 ~8 x( x$ m- h
    6 X. _; E( P3 e+ z. O9 b# Example usage8 ]8 f0 b- B2 I: e2 h
    print(fibonacci(6))  # Output: 8) B  k5 E- y# i% l- {

    4 M* ]6 [$ H4 TTail Recursion
    / V, Y: X: h6 S' q  k( j9 I0 o/ ^* r4 a5 ~! O' m2 Y" }; I
    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).% k# F7 B7 N4 b" L$ _) r
    % P, P/ `. G$ \2 D# ^
    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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