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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 2 p2 B1 r: `% @6 D

    6 F0 ?; n3 M& d: O* d解释的不错: S# z+ E4 p& v0 R( H# M# }
    3 z7 g( O, l& K
    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! K- Q$ r* X2 [$ R

    , t6 \, R9 t. K 关键要素
    : P8 o9 {" d- J* I1. **基线条件(Base Case)**
    ! l: F) H, C  p7 Y' o   - 递归终止的条件,防止无限循环0 J- V, F' q0 S) [
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    6 K& Z9 s- w% [5 M/ ?, p
    ( j0 \8 e  T) m9 e& o. K2 m2. **递归条件(Recursive Case)**' O# ~$ {* o$ P/ H
       - 将原问题分解为更小的子问题
    9 s6 D! R& f% y. \8 L; T2 Q# m5 D   - 例如:n! = n × (n-1)!" M& ]# B* _+ R1 R% _

    8 I0 E* P% ?$ n 经典示例:计算阶乘
    ; j+ ?# g' \; F7 Qpython7 o: f6 G! F' u, j+ L* Y% y  q  S% ]
    def factorial(n):2 b9 ^( ^) B) h& @
        if n == 0:        # 基线条件
    3 f5 O# F  r0 @! B, ~9 |; |: i4 K        return 1
    6 k: g6 ^8 c, o% t    else:             # 递归条件
    ; H% n8 R" @) Q* n; A8 U8 O        return n * factorial(n-1)# h( c" g0 M" S5 F, f# j
    执行过程(以计算 3! 为例):9 G# T0 c4 \: [, }% M
    factorial(3)! f& w7 ~' ?" D- N) e
    3 * factorial(2)
    / B: }2 n$ o4 n# Z* P5 g3 * (2 * factorial(1))
    : T% ^; j* P" y$ s9 o+ r3 * (2 * (1 * factorial(0)))2 O7 ^) l) ~' m. o4 S8 {/ e6 P
    3 * (2 * (1 * 1)) = 6
    * Y5 a8 u' i( ~7 e9 j) k1 {
    $ i7 d6 ~+ x+ y8 V, C 递归思维要点. Q1 ?: U, A8 D9 s' Z  s$ u
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    7 t6 s# Z$ W" |3 B# e2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    : [- H* V. S7 z3 A$ ?' t& B2 ~3. **递推过程**:不断向下分解问题(递)
    " \" a- P' v0 H) V4. **回溯过程**:组合子问题结果返回(归)0 L# e/ E) D" }& x

    " k0 V5 u# T) F/ }# E 注意事项
    * d' ]4 u. F( K/ K3 ?必须要有终止条件* _6 @. Y1 l: J& k6 e5 w9 d
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    6 p) i% |+ m% ]$ H+ Q: _7 {- R0 `某些问题用递归更直观(如树遍历),但效率可能不如迭代4 S( S8 ^7 R1 X4 {
    尾递归优化可以提升效率(但Python不支持)
    3 z0 _2 F+ C* @$ w4 c' p: k
    ( A- l7 }% J4 K6 o 递归 vs 迭代
    8 t# D$ s; ^# D2 O3 P' F  G: e$ g' e|          | 递归                          | 迭代               |, ]% g* T7 f1 u. Q
    |----------|-----------------------------|------------------|
      }1 b4 ^# F9 P| 实现方式    | 函数自调用                        | 循环结构            |+ p: j) P& f3 ~' H& o8 c
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    0 s6 {6 k; V5 Q. b: i- U| 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         |
    , z% l7 V. c) Y& y8 v& N| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |% D/ p5 t0 a" Z0 r! S1 W# Z4 [
    + h6 R/ y) a+ ^/ V! u% c
    经典递归应用场景/ l4 V  R; v* n1 C: l2 \! k
    1. 文件系统遍历(目录树结构)
    " c, S& U4 d7 z! @5 G5 ?! o7 U2. 快速排序/归并排序算法" u) b" E2 T$ ]4 I+ I% N" e$ [
    3. 汉诺塔问题
    * W! h4 B' ~7 @# u1 a& B4 w7 G4. 二叉树遍历(前序/中序/后序)( B6 W4 F$ e% \9 |
    5. 生成所有可能的组合(回溯算法)+ q, Z+ a9 Q$ b" @

    7 z' G. I+ x$ m6 W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,
    ' ]; S1 r4 A$ v, C  T- c+ p! M! |5 }我推理机的核心算法应该是二叉树遍历的变种。1 w/ Y8 z6 K3 h: }+ \; 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:" X7 Z2 U2 w" @! b8 l+ ~+ s* i
    Key Idea of Recursion
    4 H/ N  z- t5 i* H# x, Y* A
      X" x7 X0 R. S% Q$ }A recursive function solves a problem by:
    / c. H+ ]4 b! C: [4 [# s1 V( o2 r8 [
    $ F- W" O' z* ]. s1 r    Breaking the problem into smaller instances of the same problem.
    ( _5 u- Y9 G6 R  Q- t8 ?1 X0 x
    1 C8 a% I1 _7 t. n$ f    Solving the smallest instance directly (base case).% H0 a" U0 V' X6 a3 @1 e7 Z0 E$ P3 B8 k

    ; k- I# {- P4 n8 d    Combining the results of smaller instances to solve the larger problem.
    7 D5 K8 X0 B* t* v+ D8 p% h* S& }/ F1 j2 ]1 J; P6 T  g( o
    Components of a Recursive Function
    ! e+ j9 J0 ?0 m" Z  c5 P1 E9 N7 [4 Y0 w; k, m
        Base Case:
    : n/ z- L: y% J9 J# {# u( W/ O, G0 ^" ?7 R8 ]' V' y) b
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ' Z# u0 E1 E% y( `# G
    1 J3 O# i- M& {( y2 G        It acts as the stopping condition to prevent infinite recursion.
      R! q& h) F" U! ?
    6 ?6 t! Y5 p6 R7 `5 A+ c- l        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.3 m% g" Y$ h- l. L. D) N* _

    0 D% I& Z3 {* p# B" ~6 ]    Recursive Case:
    % A) U* Y0 u5 f) ?; u) \4 S! ]+ Y# Z& V
            This is where the function calls itself with a smaller or simpler version of the problem., [# X$ c$ {/ c; b
    - Z+ E% ?7 Y* u4 F
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    8 Z7 Y7 L7 _; |# H2 y; S
    ) h3 D3 c6 L) ?5 E' ]Example: Factorial Calculation
    ) @( _: V; u$ x+ w) m" t% Q
    1 E: J5 i' y% I- y$ u7 d+ `8 \8 u2 @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 g) |$ `# u' o2 g6 M4 j& {' K

    + ^6 D% s; ^" _1 N* s7 V9 S/ j    Base case: 0! = 1
    ; ~5 E1 I+ a; ~: W/ s3 t4 o! A5 J  m/ G
        Recursive case: n! = n * (n-1)!
    1 \# f: F  n  f5 U  M/ L$ z# s- P- d. [9 l
    Here’s how it looks in code (Python):8 _2 a- J/ |) g7 d! R: M$ O
    python  }+ z: c  t& P) I2 m
    " X$ Y/ H* }9 Z6 E  C6 x" q
    , z. C. u' q5 }5 x* U6 A; _, J
    def factorial(n):: V6 L3 Q/ E! i! g9 m4 A
        # Base case/ w0 J) x  ]* ?7 {
        if n == 0:/ A. j2 H% T* ?0 b
            return 1
    0 h6 s% T2 V$ @) e- n* U    # Recursive case
    ; o8 `7 B, N1 t0 J    else:. ?. S5 I3 w* }: z
            return n * factorial(n - 1)- q: {: k5 L6 a$ M1 }' T0 L1 ~
    , f1 E) H9 E# [: Y
    # Example usage, _/ J9 t3 P& y3 T% V: a8 A+ e
    print(factorial(5))  # Output: 120
    # _0 `) d, ?: |; z* D% H
    + E4 y. e6 G/ ]. x. DHow Recursion Works0 e+ u# _8 X% {) V! t( P

    # @3 z: Z  `, S- O* e    The function keeps calling itself with smaller inputs until it reaches the base case.
    , e4 K/ G8 M9 X! P5 Q0 _$ d9 O) q4 T& K- I7 T; Z: ?' b3 {
        Once the base case is reached, the function starts returning values back up the call stack.
    ' O+ Z, P* j" g( I
    ) I( Q/ n: [9 _    These returned values are combined to produce the final result.- s4 Q1 i8 p# W( b1 K, J* ]1 T+ m6 N
    $ }% g  `1 x9 {7 W
    For factorial(5):
    : d% _! I4 a) T3 K& A& E5 Y* z+ U& F5 |% d% \( K
    1 _) r, n. R. m0 W$ d( K
    factorial(5) = 5 * factorial(4)  \7 r$ k4 f2 k  a, p& _. Z
    factorial(4) = 4 * factorial(3)3 b% U0 l0 G0 M( u- v7 k
    factorial(3) = 3 * factorial(2)
    - {. b* C' s( \factorial(2) = 2 * factorial(1)
    $ M2 T( {. n5 V4 m1 a: Kfactorial(1) = 1 * factorial(0)4 s* D" ]4 y1 B7 i! l% G9 V
    factorial(0) = 1  # Base case; X/ u( W# K, [# @
    ' o) F, y3 ^: [- K/ t( ]; k
    Then, the results are combined:) D' z' W/ y6 a- L/ Z

      C3 U3 G/ K& U) I5 g0 L! l- R+ A: y* V- s. s! m) Q( q2 W
    factorial(1) = 1 * 1 = 1
    # d; C- l2 u  H+ e6 R# I5 sfactorial(2) = 2 * 1 = 2
    ( ^6 o- Y+ Z3 R  V  ?% ofactorial(3) = 3 * 2 = 6
    ; q; n0 ]0 i5 Q/ Y; W( X5 J: m: Vfactorial(4) = 4 * 6 = 24( ~, o4 R0 i5 M
    factorial(5) = 5 * 24 = 120* m" p6 N) J" h1 o+ A2 r: p

    7 J  A3 A% g4 [8 M( k. O3 u- }6 e3 gAdvantages of Recursion6 L7 l! G5 k$ M) W

    . Q" M- b& o  U* k    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).
    + V. Y. ], W+ T
    1 x9 j+ s; x( M2 \5 G6 V+ u6 ~& l4 L    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    / p  {9 V! W4 Z3 \. L1 t: E0 X
    & l2 _! O# o$ N, HDisadvantages of Recursion
    7 ^) O$ G/ w8 J2 Z7 O
    7 C8 a2 P; ?. o. c: x    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.7 `! E- A' J& W7 y  O6 ]+ H- H
    # ]  E) P$ _* s$ \7 I9 N
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    1 c' [( H& g: {' ~2 \8 `+ y
    + m  t9 H$ m5 l9 b6 ^8 F& }2 a' X0 vWhen to Use Recursion6 G$ x/ ~. Z9 l. c; ^

    9 n# E, E: ~: L: |, R! L! K1 I    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ; \, q2 F6 q! d& `' N; B2 t' ~
    8 v3 E  f2 J: c/ }3 W0 c    Problems with a clear base case and recursive case.
    1 v9 @2 Q' K7 w9 z& q8 E. j, |+ T' M9 A4 e, d3 l$ ~5 @6 Q8 a
    Example: Fibonacci Sequence, b0 ?3 r" ?6 T5 l6 W4 l3 a: `# l
    & B( F( @: a3 o- ^0 n
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ' U$ Z6 z! v# J% v% H8 \. o0 C$ W- D4 j! K0 j) w- I
        Base case: fib(0) = 0, fib(1) = 1
    % n# P% W( k0 j2 Y4 r. u, _) i  z) ]& _7 b# n2 T
        Recursive case: fib(n) = fib(n-1) + fib(n-2)7 w" a+ V( i( W( L+ H# @" e) y% J

    # t. n! H7 k6 I( s2 z1 ~python
    ( p0 b8 z( r% d- {- n9 t/ U4 J: \$ p% {

    % {  J; a" y2 Y. fdef fibonacci(n):; I. }" Y0 o0 e& B/ H
        # Base cases
    2 c$ e( H1 A- W3 W, `9 [    if n == 0:% U* Z1 @5 A  V. e8 p# X
            return 09 ?3 E+ f3 U; m
        elif n == 1:
    * H  G+ G& p5 n. I; P# Q3 d5 \  t        return 1
    1 x. l+ Q1 o. D( w1 w: O9 x    # Recursive case
    ' D6 @& X! W" v+ a- D; c    else:
    % ^! [5 w6 H) N4 @8 u& \( B        return fibonacci(n - 1) + fibonacci(n - 2)2 y; y6 ]" b9 V8 O

    * R+ X+ [* S* c/ y# Example usage# F9 H. b; x- h# v1 ^0 X
    print(fibonacci(6))  # Output: 8
    * O+ ^* M9 a! }, v7 @2 }+ S7 q, ?. X% R% }9 r9 j' {2 V
    Tail Recursion1 Z* v( Q; g. Q- r2 [' u/ i% _4 W

    " {, c- `: c. V0 w: [( OTail 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).* V! h" [: [+ @  b

    # ?7 z" X, ^" S* x' |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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