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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 7 Q$ K$ D, m  ^

    3 r; w: K- n) w" l( L- b% J7 q* E解释的不错
    9 X  _  t6 p7 h8 z
    , [5 m# Y: \* P* a+ D! 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。
    ! r2 e4 c% P* u$ `1 |6 R8 V6 q! Z" ~3 W% Q  B4 ]* f2 w
    关键要素
    , ^) h" F* ?) o. J2 Y1. **基线条件(Base Case)**2 @' @4 l$ B$ R& U) ?$ W. W
       - 递归终止的条件,防止无限循环7 l0 ~7 R" m% e% }
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    : G3 b* J4 U; [* @/ d8 L+ G! `
      P7 N1 Y5 I7 i2. **递归条件(Recursive Case)**
    + m% r, j' |7 w' \% Y8 y( T   - 将原问题分解为更小的子问题
    8 n2 |" f4 L5 ]9 V. P   - 例如:n! = n × (n-1)!  S4 k6 P, o* Y2 q/ e9 z
    ' ~: g# N9 H& x* H+ P  ^
    经典示例:计算阶乘$ j2 y1 H  S$ e5 ?7 n* h
    python
    & y' \3 A8 [5 F, A' G: vdef factorial(n):) z2 I2 ^* O8 }2 C5 K8 m: ~
        if n == 0:        # 基线条件# Q& r: D7 G/ ]4 t6 G7 x3 U
            return 1; A3 m: c+ G+ e7 `4 ?" `6 ]: `/ _( v
        else:             # 递归条件
    0 J# b% G8 m& q8 a7 U/ v& }. s        return n * factorial(n-1)4 P% }1 i4 [7 ]; d& k
    执行过程(以计算 3! 为例):
    # T1 ?8 c% `' S8 x' rfactorial(3)* y3 t+ y  p3 F! n
    3 * factorial(2)
    ( y  E2 O+ ]" v3 * (2 * factorial(1))1 j" r3 [2 b% y4 s
    3 * (2 * (1 * factorial(0)))6 n. C% l; E7 h* {5 S& p7 q# ~
    3 * (2 * (1 * 1)) = 6! g% l8 i' R6 @6 C' R$ ~

    , q6 Z2 L7 n+ i 递归思维要点+ {* W1 Z' }0 r$ X. S9 O8 H" ?, ?; u
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑! u% R& o" x% {
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)8 `( {: C( l/ f: O
    3. **递推过程**:不断向下分解问题(递)) Y6 N+ h$ a: @; a( ~' s' Q
    4. **回溯过程**:组合子问题结果返回(归)7 X6 Q* G, C( b  @  `9 F

    , k- D5 y  ]2 V7 i, L% o 注意事项
    # ]) e9 U2 P( [: P1 _; F必须要有终止条件5 G5 |- T2 q, o# a
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层): v% A7 [1 d. U7 ]: Y* ~' D7 r# _* v
    某些问题用递归更直观(如树遍历),但效率可能不如迭代0 T) H* @, X/ G" Y7 h5 Z' H) P
    尾递归优化可以提升效率(但Python不支持)
    3 a0 p/ \# G) U- f
    4 ?5 r) {, c4 R& w2 ~ 递归 vs 迭代: ?! k8 j+ E0 |2 n2 m6 E3 d! e
    |          | 递归                          | 迭代               |) I3 s; ~2 `' C; z0 i" \
    |----------|-----------------------------|------------------|4 w3 X9 G( O9 a; [4 R  g2 r
    | 实现方式    | 函数自调用                        | 循环结构            |' E2 h, I" L& J1 i3 Q! g9 k9 A
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |' t- {$ t2 D2 N
    | 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         |
    & Q  X& R& a% ?: s) n( N+ M| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |1 X% i3 u& V) v

    7 [* m. P* d7 t* {/ O3 o& o 经典递归应用场景
    ' Z( C! E0 _  c0 v* W; i0 `1. 文件系统遍历(目录树结构)! E3 v  m( ]3 {
    2. 快速排序/归并排序算法9 m- g: S2 b' F3 }. H# T$ E5 l; d
    3. 汉诺塔问题4 z: P/ G9 G( F& v0 Y0 G! y% [
    4. 二叉树遍历(前序/中序/后序)
    , f" i6 c. i; B3 C3 N) q  R5. 生成所有可能的组合(回溯算法)( _3 l: A) D; H8 A
    ; k1 n- x' ]- P  W: 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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  • TA的每日心情
    难过
    昨天 07:51
  • 签到天数: 3161 天

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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,
    6 c0 j3 i# u. b# e我推理机的核心算法应该是二叉树遍历的变种。# y/ ^9 \4 v6 X
    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:
      q' t) |  n( i+ y6 Y0 JKey Idea of Recursion; Z) Q- R! |. @! Q0 C

    ! C* D% i1 V% |8 v# L$ N# kA recursive function solves a problem by:# ^/ p/ ]/ z$ \# z* r7 s& ]

    9 r" u  ~7 W1 n& z$ W) y+ o    Breaking the problem into smaller instances of the same problem.% P: r6 y( {9 k( \' p
    ) Z* o' A$ @$ e( H
        Solving the smallest instance directly (base case).
    7 A3 d  [! v! Z0 ]7 f3 ^1 }- z; B" `2 P0 `
        Combining the results of smaller instances to solve the larger problem.- b8 K: d6 Q. L: q8 k- ]

    2 |& _  q# {# {# u9 Q- DComponents of a Recursive Function7 O4 V) X8 h2 Z) i9 q' Z+ @
    - }. k0 X5 j# N5 j. g# G
        Base Case:4 r  @/ H: G! B
    3 o3 f  B: c! q
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.4 f5 s/ Q8 {- e6 e* a

    8 ^4 v% L5 M  x/ Q) c" c% k        It acts as the stopping condition to prevent infinite recursion.
      `' x/ K6 {- f! w4 o8 p/ b9 m% i2 z( T/ H( k: I& z
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.$ V3 n- B& x" C8 U7 m, g) F+ L
    # N. V0 Z4 ^2 Y3 E  C
        Recursive Case:) x! C; @/ j7 M- M1 I# ~

    0 Z9 G9 s0 \1 A* c( f        This is where the function calls itself with a smaller or simpler version of the problem.
    . a: X) n" N/ ]
    4 c6 f4 m; ^4 K        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).3 ^  J' `- s" V

    : X1 S, o6 @4 Z; c" }. m8 y% qExample: Factorial Calculation5 e; j" J' A9 C# b4 j, ]
      Y9 D9 D' q: C* `$ u
    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:
    7 m# ]+ S' Q) U$ e$ Z5 D% ~
    2 Y" _5 K, Y2 z    Base case: 0! = 16 `5 A( a: S+ O* A: `! Y; I

    * K: I. F5 {, _# O    Recursive case: n! = n * (n-1)!' i( v, A# P; f$ V2 T, X
    * X; D1 X5 ^' ?( i; L
    Here’s how it looks in code (Python):
    ( N; Y, T; n; m2 H5 |4 Mpython
    / o2 A/ j% D7 l2 b9 s& w, R
    4 a. _; X6 @4 r0 ~6 f8 X1 e' _0 f) }6 ?- u
    def factorial(n):9 h: N- T5 b% m# `
        # Base case+ e) s, E% e6 `# X; w9 C' |
        if n == 0:
    # C/ r" j7 e  A        return 13 a2 l) ?/ j6 h
        # Recursive case
    3 \' X  P2 x7 Y1 P# Q# S" P! g    else:  I+ I: d' Z- I4 @' }
            return n * factorial(n - 1)2 k- t! [( q! ^  U3 e6 g

    ' ]3 W% x  j# {/ r/ c# Example usage! x9 }3 Q0 i; F+ `% @
    print(factorial(5))  # Output: 120; N$ E1 Q- C+ O

    ; `; P& l8 i9 J4 v  h7 i* ^How Recursion Works- }* h; i" S* e7 n7 }

    # |- I( Q6 G2 w0 K* s% L    The function keeps calling itself with smaller inputs until it reaches the base case.
    ' Q. i$ Y. q* d+ H! ]7 n/ Z
    , e; R4 w; k9 F& A' ^) ?    Once the base case is reached, the function starts returning values back up the call stack.
    ( N- B8 |+ y# K$ V) `7 Y0 H, E9 w0 y9 G; l
        These returned values are combined to produce the final result." B* ]2 k; a. h3 Z1 _

    & z% W! ]2 v* B  q2 uFor factorial(5):9 X4 [- |$ X7 n1 A

    ' o; m) @- k& T7 X& E
    + Q- h! {+ H) ~$ Tfactorial(5) = 5 * factorial(4)7 ~* l4 p2 `  y8 o
    factorial(4) = 4 * factorial(3)0 L# }" Z+ T/ i8 \8 O+ W4 q
    factorial(3) = 3 * factorial(2)
    1 H9 P- @; d/ l7 O$ afactorial(2) = 2 * factorial(1)6 ?! }& V$ O. a' b/ V$ }3 r4 G
    factorial(1) = 1 * factorial(0)
    6 G" j1 m( e9 M: x. F7 Pfactorial(0) = 1  # Base case
    / V& U$ }. d+ [% p0 |# X& t
    1 {3 L7 {/ V; L) y6 C0 Y( `) @9 M2 iThen, the results are combined:
    4 U) A' M4 o0 b9 P9 a; X+ h
    : q0 I) j- j4 P) o. }
    + U: K6 N- H5 R- E( ?) l5 hfactorial(1) = 1 * 1 = 16 i' i: C0 E8 S, n
    factorial(2) = 2 * 1 = 2
    ) Z6 Z8 o1 h3 ufactorial(3) = 3 * 2 = 6
    $ B: E; L0 G0 v  @+ t- M9 f' I: Tfactorial(4) = 4 * 6 = 24
      {# Z! |$ l' ^% _factorial(5) = 5 * 24 = 120
    % Z  d1 G9 t" R
    5 [) v8 U+ r, v- d* u4 s' TAdvantages of Recursion
    : T! X2 l( {/ A0 N2 x1 E2 Q+ c0 b& v: K7 H
        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).+ H8 w6 M+ f% n! @) D% Y

    3 T2 W5 L5 @2 X    Readability: Recursive code can be more readable and concise compared to iterative solutions.% M$ N3 i# F' G$ a2 c! Z. W% {

    / l( |+ j: I# n% l' z, zDisadvantages of Recursion) X" y- C% X- R$ ^

    4 h: E/ t9 n# O/ R* f& z0 w    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.# l, T! A8 @; e, e/ w
    2 X2 |& `# ^6 t/ ?5 s' h" H# \, Z
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    , M$ F+ a2 w( }6 {- {) i: ]4 r7 W! r! C+ N
    When to Use Recursion2 X, X, q; Y; W( d9 z* B
    8 l- r* J7 Z/ r
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    . q& t3 O3 l$ i; Q, k5 _+ d* K1 M& p; i( T( Y6 c
        Problems with a clear base case and recursive case.
    + g1 Q0 T; y6 J2 J* d# G6 i* e  f" @* C/ [) O4 B- |, c5 z; }+ s
    Example: Fibonacci Sequence. B+ O6 g% J# a4 U
    5 e% R0 g  {) ?* X5 G$ E- c
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    . @, ~- x+ o1 V6 n% I8 L7 @7 A+ v4 ]; A0 J2 e& X
        Base case: fib(0) = 0, fib(1) = 1" |; I) u5 u1 F

    - k8 i% {( B& C8 H& O' n    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    4 R" h% s0 |2 J  a" O& N: q% @, u- z  Q* g* z* F) X- R! w* v8 O
    python
    5 y7 _3 g* z* H3 h: \& _9 @% E8 r- z1 u7 `1 w5 n  ]* ^/ a/ P, r9 j5 e

    ) r3 m  q& Z8 ?7 {  `& P1 m# Pdef fibonacci(n):- \/ r8 Y6 ]4 r  z
        # Base cases
    # a" v6 \1 u0 [/ ?* V; o    if n == 0:
    * U8 P# a5 b& w! C, N        return 0: ~- p) L; ^% B3 [8 r0 z  H: K" d
        elif n == 1:# a! V* N: y4 Y! a% a3 b
            return 1& v% ]: o" @% l+ o
        # Recursive case
    * X% z2 T, |, U    else:2 w* w, `! {( r, v' @2 ]0 R
            return fibonacci(n - 1) + fibonacci(n - 2)$ }/ e# a5 o$ o& ^0 f5 B7 k
    ' D. L* R0 G1 J; d0 l5 ^8 v
    # Example usage& M# G2 G" a9 ^) S: I0 J  w( I- g
    print(fibonacci(6))  # Output: 8
    * v" k1 k; U7 k5 ^& q8 u' O+ W# T+ {
    Tail Recursion
    1 b# o/ e5 v3 R  _4 {) m* F# ~' R% V) u/ A3 f; ]8 C% j, H
    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).
    5 [7 A2 F$ l. t% l( {( g; _" T/ j3 n+ l* |8 U
    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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