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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
  • 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   j2 O7 H6 f' \2 _/ T

    6 g* ^0 C4 U+ o解释的不错: P3 M5 h* |! p2 b, X) o
    $ L# g: }3 j* X" r
    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。: W! f+ H" W. H* ~% x" h8 S
      I" B) R% |8 e/ n
    关键要素
    : c6 f, y2 I# Y4 s3 _7 @  H1. **基线条件(Base Case)**
    ; o' V- ?) E* v" Z2 J- R* G   - 递归终止的条件,防止无限循环
    6 Y! W# x' q* A" z/ k% m   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1$ i5 b5 t& ~/ e1 o  S+ ^: R
    $ j+ W8 S2 I6 _: l0 r
    2. **递归条件(Recursive Case)**  }# N2 f9 z/ @9 Q9 S1 g3 Q
       - 将原问题分解为更小的子问题
    & q1 d# S* t( N+ Z$ k   - 例如:n! = n × (n-1)!1 N5 c  G& A9 {, s9 A* w8 n
    $ ]' b* \; p1 g9 I! |& ], t
    经典示例:计算阶乘4 E# e* k" p* t4 W* x4 D8 V
    python/ l2 H1 j, ^1 M  p. x5 r
    def factorial(n):
    ; }2 N4 b/ }$ R0 e2 i$ ]' k/ L    if n == 0:        # 基线条件
    : k/ I7 L% k/ T& z, n        return 1
    % h  j7 ~' h( u' q' N* \8 F    else:             # 递归条件6 @2 Y% F, w" E- V" L; }; N
            return n * factorial(n-1)
    & |- f5 z1 Q& b4 |. M& w, y执行过程(以计算 3! 为例):0 S( A. r" ~/ q7 i/ g8 F: |. `! a3 p
    factorial(3)
    4 W6 t$ e1 [2 Z3 * factorial(2); ^( n9 V6 ~% O+ i4 [
    3 * (2 * factorial(1))# e0 |) l- [6 K. \. B+ ^5 b
    3 * (2 * (1 * factorial(0)))
    5 i5 q" V+ M. H5 c  Q& z! R( s0 Y3 * (2 * (1 * 1)) = 60 Y0 _' l5 ]) M9 m# z
    8 `* ~" J  T) u) d8 P
    递归思维要点) k# J% U- z; }) o* R5 U* L
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑9 t2 X0 Q7 C% Z$ N
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    9 @( b9 F5 [9 e' @* B3. **递推过程**:不断向下分解问题(递)
    : _' L2 W3 H0 R4. **回溯过程**:组合子问题结果返回(归)
    2 [4 `/ k0 B  J; O9 k7 H3 L: V7 [) L7 U5 n, h, ?8 c; j/ q/ o
    注意事项
    ( ^; n- |/ w( u- X5 j  t必须要有终止条件
    # d, i8 B! l$ U) J  ?; Z递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    . x& l" `4 @3 M$ Q. f某些问题用递归更直观(如树遍历),但效率可能不如迭代$ F4 H! B. Y* x# L, {
    尾递归优化可以提升效率(但Python不支持)  v) S! |/ R' b$ @0 Q: I5 g6 n
    8 X9 K" W. H% k; N6 J
    递归 vs 迭代
    8 k1 g) |# S" Y: R' Z  u|          | 递归                          | 迭代               |2 @( {; {$ K" ?0 L; L; D  H
    |----------|-----------------------------|------------------|  s' \, G7 R. Q! [0 Q7 q
    | 实现方式    | 函数自调用                        | 循环结构            |
    " B% y) _+ ~& n5 B8 \) Y" Q| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |. u# B  {+ d2 E3 \
    | 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         |
    , ]% U; F3 w+ C* X) r| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    + P9 P, V8 i  Q. m; Y' C4 c7 I
    : z+ r0 S0 Q4 ~$ N 经典递归应用场景- Y+ {  w) |/ Z, Q' l9 }+ _
    1. 文件系统遍历(目录树结构)
    , D9 K' Y  p( I9 H9 u! J- d: V2. 快速排序/归并排序算法
    4 M3 a! [+ n/ C* h- V4 |! Z3. 汉诺塔问题4 c+ O! r: F2 i# |6 d$ \. n3 l0 Q
    4. 二叉树遍历(前序/中序/后序): S2 D, v9 i2 Y( j
    5. 生成所有可能的组合(回溯算法)7 J0 t* ~" @2 z6 r$ t; H# @
    ! c2 H) A5 j& o4 \# v1 ]& u
    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,
    + ]1 `/ `& d( r2 Z1 {: }+ M我推理机的核心算法应该是二叉树遍历的变种。. u0 v; V6 W, W6 g6 {! r' C
    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:
    ' T' n5 m4 B4 a6 {" f% SKey Idea of Recursion
    $ F* N) m$ T5 q  C+ L$ D4 W4 T5 [( n8 v2 z9 m
    A recursive function solves a problem by:* v/ F$ k$ k# n. W: ?4 `1 N: \

    % c! w$ f. ~/ n  l: i    Breaking the problem into smaller instances of the same problem.9 ]8 l& t8 Z1 J, e4 r- a

    5 _+ r- F. l. V+ g    Solving the smallest instance directly (base case).; P( X: w# s9 z8 J  A- j
    5 a( n- B) q2 w7 D4 k: W
        Combining the results of smaller instances to solve the larger problem." q; ~7 M9 r* G8 ~. z4 ?
    3 n: L. S5 t" n9 Z) ~
    Components of a Recursive Function6 ~( C. M& z; y0 a6 `3 g% i6 a$ H  ?- u
    $ M0 H( ]$ `/ `% S' w3 U
        Base Case:
    6 ?! i% \+ w  r7 r2 p8 s# D1 G
    0 d% S: N' J% i" j. u" C        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    8 ~9 T8 N# C2 G4 `; E3 s7 o
    % Y' z3 ?# l2 C3 Y        It acts as the stopping condition to prevent infinite recursion.
    % o  B/ ]7 Q: ]& g
    % X  ?" U* r% A- q; f        Example: In calculating the factorial of a number, the base case is factorial(0) = 1./ V' X: a8 N! H: @  ?: X
    , o4 P6 ^8 b7 }( i
        Recursive Case:/ b$ r9 D2 J* Y" \( ^/ q
    , c* u, H- d8 b1 j, m- G) U0 M
            This is where the function calls itself with a smaller or simpler version of the problem.; r; o8 @! B0 w; S7 R% Z# D( p

    8 ^1 ^* H9 n/ r        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).; b, a: z: i0 `
    , q( Y% R7 Z. n- N, ?& v
    Example: Factorial Calculation6 X5 E. x3 O9 b+ |
    $ X: J4 d, ^( I7 R3 O
    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:
    9 [2 ]* F3 I- u2 E  t5 P
    ! V7 y1 n$ e6 v# A7 G& N    Base case: 0! = 1
    ) a$ P$ r" d- S" q6 U
    * L8 J% Q! ~6 o+ C7 `8 G    Recursive case: n! = n * (n-1)!
    8 G' L/ e. W. w* p4 ]) I2 v
    * \0 v' `/ w5 I  ~( eHere’s how it looks in code (Python):, O. N) J5 ]/ a
    python
    8 h# u1 y4 C' [  Y4 J
    6 Q# X7 i1 y- M5 d: {! e- @" l5 q; w$ g2 v7 w: h
    def factorial(n):  Y4 i  W( U" Z7 X1 i  [
        # Base case
    : A! g: K3 t& u$ t% w5 t    if n == 0:+ ^' p0 l% }* }/ d7 H7 i  X" X
            return 16 b' m' E0 N3 t
        # Recursive case$ P* ~" p# S. \2 ?2 V. @$ {
        else:+ F  G5 C, U* m; C6 ]+ A# N2 i
            return n * factorial(n - 1)
    9 X( p& e  G$ D0 P# L% x" P+ i3 q  a9 ~9 @1 }
    # Example usage5 m) e4 ~1 B3 v2 l. P1 \0 {4 Z
    print(factorial(5))  # Output: 120
    ; @- x% |  u- S5 @4 m2 J" E
    ' O* z( ^2 g* Z" M/ a6 O$ eHow Recursion Works, _% r( I* ]9 w

    3 b3 r: |! D% E% V+ c- I    The function keeps calling itself with smaller inputs until it reaches the base case.- r- T  [# v% R+ _
    1 k$ ~6 L+ z0 W6 y( {
        Once the base case is reached, the function starts returning values back up the call stack.
    ( T6 x% X! A) o9 N! V9 s3 a) ]: i3 c9 `7 Z
        These returned values are combined to produce the final result.+ k/ p8 t3 Q2 W/ U" S  @5 z( `. q

    . @" ]9 ^" }* W; x9 V7 qFor factorial(5):" n' @# P# a, `* k1 X6 l

    7 e4 z3 O$ P1 q& L
    ( J* R* C' ^, y* Y: e1 G. r) p3 @9 ~factorial(5) = 5 * factorial(4)1 G( z& ?" a% \: S9 `' ^- m/ ~
    factorial(4) = 4 * factorial(3)
    & A6 H. _. ?* Q! Rfactorial(3) = 3 * factorial(2)
    ( ]! X. b) U- x( y) H' r3 |8 [factorial(2) = 2 * factorial(1)
    ; O3 g# ?3 n* y: ^! @factorial(1) = 1 * factorial(0), w  ^3 u3 O* q: c. K; T( A
    factorial(0) = 1  # Base case9 ]* a; F& {- t. d2 s" ~0 o0 s

    / \- T! ?4 S+ w  `5 h: x$ WThen, the results are combined:1 _+ d( @2 K9 h& T+ b- |
    - {& ?& c6 S  d; t$ `
    0 q9 k( A% ~" E
    factorial(1) = 1 * 1 = 1
    & @$ Y6 R9 |9 [3 Q9 c5 R: B0 n# \factorial(2) = 2 * 1 = 2
    ; H3 y( [1 R# p' Tfactorial(3) = 3 * 2 = 6
    4 c) q! _, T; E0 J2 S4 E4 W7 Wfactorial(4) = 4 * 6 = 245 i1 K8 q0 ~/ m, m
    factorial(5) = 5 * 24 = 120- e- _) k7 x9 G" v
    " n+ Y2 L* X7 a- D
    Advantages of Recursion( W8 p5 p) W# B( }, v. u0 u
    4 T# R0 u! N4 [# s* X
        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).+ r8 P9 P0 |, I" i! e, I
    $ m) l* h4 j8 @$ t/ B& |% ^1 y
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    , n' O" y( m3 c( M* Y9 d
    + p# Z' Z; {$ i7 B4 nDisadvantages of Recursion. j7 z) d" V- w, A* Y

    3 p6 E+ F( b) B8 L, S) D$ G+ H    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.3 H  L1 s& O6 i6 e7 x

    5 c+ o9 f9 i. V  s- Q+ p    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    2 Z! X  R; M5 s$ T6 I' F& K" \! W  ?
    When to Use Recursion# Y* K; O$ Y: V! T
    , g  V+ |4 n, W
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    - C4 @/ l6 f& O% h
    . O+ L7 b5 V8 N, O    Problems with a clear base case and recursive case.
    8 e& U. \& q1 M3 W( f. j& n( ?! _! _1 f
    Example: Fibonacci Sequence
      d. q) J, f+ M; f
    : R0 V2 {2 J' g3 O$ F' @  zThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:, h2 r$ }8 z2 r; X$ r3 Y

    9 }% H; r, G2 \    Base case: fib(0) = 0, fib(1) = 15 c& L8 I1 m1 J! n4 \
    ! f: @- u- ^3 B& u
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    . b/ M% t& `5 X  t: Z
    ; X+ Z: A" }$ ~+ b* E+ w# H& rpython6 x6 ]- c$ L  ^
    1 H6 H) ]2 |7 h/ D+ h# Q
    - \) O0 D1 B# f3 D) N+ O
    def fibonacci(n):* N! t# Q2 p; B- ^
        # Base cases
    : G" M0 |- p/ R# F; E    if n == 0:
    ( z* V+ P2 z6 e  S        return 0
    8 C% P% L3 L5 [- j4 n8 A& b    elif n == 1:" \0 `/ ~- X1 B* Q9 C
            return 1
    ! {0 Z' @+ W$ M+ `: d) n; S    # Recursive case$ t' K4 a" X0 F( t$ T  b% ?
        else:! m: ?& a. c" i* d. a3 y+ H; A
            return fibonacci(n - 1) + fibonacci(n - 2)
    $ d* _) p. i. X) T4 Z
    8 w* n1 B: U+ A5 P$ C# Example usage
    ; {; R# ~8 t+ X9 W1 a& G! Qprint(fibonacci(6))  # Output: 80 m. a/ u. g! Q2 t7 ^# T
    & w7 h$ J, ^6 h" L. {7 u) G+ T! h
    Tail Recursion
    / x- E$ q8 a6 m) U. n$ e
    . M9 [& ?% G% O8 I) vTail 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).2 e# A7 \: s/ G! L$ [, u/ h* r/ a
    9 r/ \; Q& a# u4 W9 F; X" 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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