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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
  • 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
    9 h. k5 f  {- {) N3 z% E/ I$ n2 ~
    , {3 R8 y( G: t+ Y$ _8 s* J解释的不错
    " \0 H7 n5 {$ P) ]7 P; U) P
    6 Q( k6 y0 K! K7 h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。5 K& a$ j+ C) u

    + e( `3 n* }; K9 x 关键要素
    0 A% g1 s5 K6 l' ^1. **基线条件(Base Case)**) e5 r* A! P/ C% \6 U
       - 递归终止的条件,防止无限循环
    ( |. s. F& X7 a) T9 Y   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    7 G* d. T: `  ]6 A/ p; x- m, N
    2. **递归条件(Recursive Case)**
    : c" {3 X; b4 r) f   - 将原问题分解为更小的子问题
    / J9 ?2 X# E7 Y- U   - 例如:n! = n × (n-1)!
    8 {7 R/ {4 l# H. Z5 o) }
    0 {$ M6 D& j# l4 a! u4 [# g* R" ~ 经典示例:计算阶乘0 ?) v+ c9 f% n& C/ ^$ M
    python
    / U5 a8 k( |5 d8 }; W+ i- ?def factorial(n):
    ( I# {+ k) C3 a8 {/ K) f    if n == 0:        # 基线条件
    / Y, l- Y+ ^  }: i7 q        return 1
    6 a3 P3 L4 n. r' K* h8 ]    else:             # 递归条件! Z7 Z& G$ l6 d5 X
            return n * factorial(n-1)% G3 X7 k; r5 p/ d8 A5 E' E/ I
    执行过程(以计算 3! 为例):
    / D6 ~1 J1 w$ N% d$ ^) V0 Xfactorial(3)
    " h% f3 n0 w; e# c3 * factorial(2)
    9 i1 N$ i  H3 B, k& X3 * (2 * factorial(1))
    . N* Y9 Y- j! G; z3 d. w3 * (2 * (1 * factorial(0)))
    ) Y3 u: c. {5 A3 Y9 }7 x0 p3 * (2 * (1 * 1)) = 61 A+ d6 \1 p/ e

    ( l# g) \7 b% N5 Z 递归思维要点  S9 [1 d" T9 O4 w9 p: }
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑% W2 u" l% |- g
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)) I: h; x+ M  R7 u3 E
    3. **递推过程**:不断向下分解问题(递)
    : U2 c, b$ I: ^* `' o4 V4. **回溯过程**:组合子问题结果返回(归)% i5 U& w$ n0 r8 Q; D
    8 U; I) ?' J0 ], G) a( i
    注意事项3 H) J7 G" }: b: w
    必须要有终止条件8 U# O7 |2 b6 W$ X. m! u" C0 N
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)  R% n- x) N6 ^3 d) s
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    : j) J# ~  e  T5 D5 W$ f尾递归优化可以提升效率(但Python不支持)) m) \! w5 ~8 _3 H5 P4 }5 J
    ' P+ Z$ \# W/ z" J/ a, O
    递归 vs 迭代$ e& f, d, m4 ^2 V+ O; U; ^
    |          | 递归                          | 迭代               |' J# g% J5 @) G5 @: D. m+ S8 b
    |----------|-----------------------------|------------------|9 F1 Q/ I4 k0 b: \, k5 K" c$ m
    | 实现方式    | 函数自调用                        | 循环结构            |
    + f' b2 b7 j, r| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |( T- Q7 c& Z9 x2 o! p8 s1 W
    | 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         |/ s9 h; U  t6 i% e6 `! n
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |. x) J: e2 r# Q# G6 M$ O

    8 M  K% V6 {7 g3 f1 c# }' o1 T+ g! Z 经典递归应用场景
    2 B9 a6 R# ?. H* B! l, B1. 文件系统遍历(目录树结构)
      D* H1 c- W8 b0 r$ Z2. 快速排序/归并排序算法; `3 G- @: J0 a" m
    3. 汉诺塔问题, {2 W, |4 ~' P" ~1 ]8 o8 n
    4. 二叉树遍历(前序/中序/后序)9 P& w4 V$ e! o5 B1 A$ \( d5 V' P
    5. 生成所有可能的组合(回溯算法)+ _2 N1 n) e0 T& i  x  r: d
    0 S5 H" X2 R) F  L9 }4 b
    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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    13 小时前
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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,2 C. D$ \' p3 @5 g
    我推理机的核心算法应该是二叉树遍历的变种。
    7 S0 @1 j2 B) Z! a, v* E  h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:
    0 M" q# s; X; u, E; DKey Idea of Recursion
    1 b: `+ O2 W+ a, D% w4 R, M6 f" P7 |3 p% d* |; p
    A recursive function solves a problem by:
    * O5 x! O  a" q0 X. z2 h5 D( {1 B( w8 ^8 b) i
        Breaking the problem into smaller instances of the same problem.$ T6 t5 d9 f' L9 T- \! C
    / w( `5 d. ^# \1 v% I* _
        Solving the smallest instance directly (base case).
    ! q) W+ V4 ~3 h" ?
    1 a* g, `9 P- G) {/ Z    Combining the results of smaller instances to solve the larger problem.$ p% }0 Y7 C) |
    ( P* I, @- ]- s8 M8 v5 E: g
    Components of a Recursive Function+ s- Z- h. r, N/ V3 e+ }

      P9 a  e) u5 k6 C3 Z    Base Case:
    % y6 F; ^/ W7 D
    % v7 Y, P5 U' q0 `  g' x        This is the simplest, smallest instance of the problem that can be solved directly without further recursion., t/ F9 F! p1 S, h
    : l# b5 |3 e9 N; C( Y' ~$ F6 e
            It acts as the stopping condition to prevent infinite recursion.6 X7 S& d6 S0 I9 ?+ S& }

    - b- ~% t& \! o( R* _: T4 ]        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.- B8 f  N, `5 I5 V% S1 W; Y
    8 A! @' M0 O' I- }
        Recursive Case:
    * w7 D4 \5 i" }9 L/ V+ p/ m3 ^3 w3 p. W& A- \( [
            This is where the function calls itself with a smaller or simpler version of the problem." o! I) c( Q! G+ J. Y
    . A9 c; F' F$ t8 j  d' J: j2 E/ a- L1 `
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    7 Y. G1 X* F4 `" P: O3 C
    6 j9 w3 c6 w; UExample: Factorial Calculation
    5 G9 G  ]9 H/ r
    / Q2 v5 b( k7 x# FThe 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:  {6 `4 c7 a7 m6 D  {

    1 W+ j- {# t0 G1 V- y3 l    Base case: 0! = 1" ]2 G0 `1 F! }+ M6 n

    0 \0 a/ C; ~# X: v  }8 ?$ x+ T    Recursive case: n! = n * (n-1)!9 Y  K6 z$ G6 t3 }
    " }0 j6 W1 B& O' I% S, r
    Here’s how it looks in code (Python):+ i# O, D- O5 ]
    python# `2 ]( j9 E3 z, m" p! V5 b
    6 O: l3 I: `# K1 a  B. M. v1 Q

    * v5 w, i8 m9 i$ e: ~- udef factorial(n):3 c) l& J; g* D: g* y/ r
        # Base case: Z& E. K( _( [1 r
        if n == 0:; G2 u; T! J" p
            return 1
    - X7 t) b& v9 s/ B" P    # Recursive case
    3 x3 f5 X) ~7 z) X8 t8 [    else:! T. v5 P1 p/ u' g  @
            return n * factorial(n - 1)/ ^; J8 B+ o* a/ P* _, y8 J- R
    3 i% H* `7 t* m2 A' D7 P
    # Example usage% s( A- g) P* {
    print(factorial(5))  # Output: 120
    3 ]" k# T( f: E# B& T: e& y) \8 @* x& ?
    How Recursion Works3 B; ~; v4 P6 I+ o. k2 G5 ~
    ' S2 i2 T4 V6 I+ Q& M  [3 K
        The function keeps calling itself with smaller inputs until it reaches the base case.
    . m7 K, Z( N7 p2 D0 O3 S
    6 \' H. P% ?9 J) U; G$ z    Once the base case is reached, the function starts returning values back up the call stack.1 v; ~, [  _! Y7 E. i) V
    ( s2 U( A: d8 H! i9 T
        These returned values are combined to produce the final result.. |. l/ T, k& G  L0 ?
    - }6 |+ h0 x1 _2 m, u. l
    For factorial(5):
    0 N! H% ?1 Q0 O; q
    & o, o! j. P) |8 e9 w7 E, k
    8 Q; F" e- X6 Nfactorial(5) = 5 * factorial(4)
    & o7 z0 T, n3 b8 Qfactorial(4) = 4 * factorial(3), M! \" P3 ?  r
    factorial(3) = 3 * factorial(2)
    8 N) V8 C3 H4 y2 T0 a! efactorial(2) = 2 * factorial(1)
    6 \& k/ H/ E2 t1 a* Q: t2 Jfactorial(1) = 1 * factorial(0)
    0 v  k! J$ \! K: u5 yfactorial(0) = 1  # Base case1 X+ ~& \: y) C& B6 t; R' S2 X
    3 o8 B" J7 _3 w* X% u# _
    Then, the results are combined:
    6 P3 U9 ]2 e; i6 _$ ?+ Q
    $ R! \- q* N& t8 N  J, [; z3 s- a2 P3 D' ~3 r
    factorial(1) = 1 * 1 = 1
    7 d6 z0 S' v! p/ P5 _factorial(2) = 2 * 1 = 22 F+ }, V# d) A" x
    factorial(3) = 3 * 2 = 6
    ; c6 O4 N8 _9 }4 gfactorial(4) = 4 * 6 = 24
    ( z5 D3 d4 e' {factorial(5) = 5 * 24 = 120- Q( [7 L& x6 d1 _9 X& |% P9 ~; B

    2 g4 K: K9 r( Y7 }& u! kAdvantages of Recursion
    # k( G9 }( F5 n5 ]
    ; W/ C( Y7 J' y1 j    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).& u" ?/ s1 L0 |) @$ D

    % }4 L1 }( G1 p" b    Readability: Recursive code can be more readable and concise compared to iterative solutions.& G7 h# p, c. U1 x: d1 v; S
      m$ J3 w7 m/ k$ w7 P
    Disadvantages of Recursion3 U# ^, L. C1 M8 a) i! z

    ) O# W/ W1 A. d7 t6 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.
    9 z9 h" z! u# y& c, H4 q( e4 k. ]  X& R! a: y) n: }9 W
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    * F! l. o! \. R5 n0 @# x% ]8 P; P, p7 A$ n2 t9 ?
    When to Use Recursion( _& T& o0 }6 f$ F" d/ b* L
    % S, M, }. }( u. `  X
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    - o! b! L2 V, L$ D) Q
    + T$ I. O2 N) P& L! i3 H    Problems with a clear base case and recursive case.4 Q  Y4 H9 o+ u" d$ Y8 g
      R3 J0 ^5 `# D: b9 i* K' L
    Example: Fibonacci Sequence6 A) J# r! V" ]. Q; Z& E

    - d* m$ W0 e% q0 q& ~$ k* [2 u! vThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ! F% T5 m0 v  T% T4 k& U7 X% l. n& |4 w7 ~5 _& [+ a
        Base case: fib(0) = 0, fib(1) = 1
    9 }) O+ }; f$ D4 y4 y: ?! G2 }& J8 N6 T
        Recursive case: fib(n) = fib(n-1) + fib(n-2)" E2 T  ]/ T2 l' }

    ' e5 m6 Y6 h( s* H* J% ppython
    & z' c. ]' g1 R% n6 f0 A, \# A' E6 k( G1 Y! q6 E  P

    6 N! B8 A: m% ^* }8 ddef fibonacci(n):
    % K' ~7 x2 `* M6 R: L    # Base cases
    2 o  e7 y9 U5 c5 M    if n == 0:
    ' W; ^0 I. z/ g3 S# X        return 0! ]7 J6 {/ }" S- g) b# R
        elif n == 1:5 ?* z9 }( A4 O4 p
            return 1
      g, e$ }  r2 v" K8 _3 Z$ u( s$ |4 m; u0 {    # Recursive case7 }2 B3 H+ \; t6 l, ]
        else:
    ' U4 j. b( K+ B/ o8 u        return fibonacci(n - 1) + fibonacci(n - 2); }0 o( O" s" T2 o) r3 V, ?& s

    / K) h& a4 W: J) |7 W# q* u# Example usage
    1 @6 }, }' y1 w- z% z, Gprint(fibonacci(6))  # Output: 8
    ' ?% m2 z$ K, h/ i
    ' i" h* p5 Y& TTail Recursion
    7 E9 P, ~% A& |7 a# I! f( k0 ?. i0 r% Z  f, j1 s( \
    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).
    1 e8 _- y& q5 u; G* N9 X3 T
    ; u" I& _$ a' j! m3 R! Q3 NIn 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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