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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 * \% I7 q/ ?; q+ d2 ~  f
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    解释的不错
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    ; E9 E# W; K" G" H$ h, E$ \2 s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。4 g+ q' O6 d# |! X* `  u
    ; n% p3 h5 `/ W2 r9 t3 N
    关键要素0 j5 l) K$ @4 P! T  ]+ c5 A
    1. **基线条件(Base Case)**4 K# M" A1 p. f! n
       - 递归终止的条件,防止无限循环2 G1 @' ?( X2 m* S1 m5 ^" D
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1& e5 v* \2 y& Q7 y& E7 \6 f& F

    1 Z* H' M( G- v" ^, p: i" y2. **递归条件(Recursive Case)**
    + W* [% [6 R  \3 Z   - 将原问题分解为更小的子问题
    % Q/ p% f  S% H9 L' V# w   - 例如:n! = n × (n-1)!  Q$ C! p5 Q% v, L; F7 a
    8 P) a0 n6 j: A: L4 W2 j% O
    经典示例:计算阶乘
    2 n2 M# C- s1 r0 S# J# D( `python
    4 w# K2 L8 q' M4 r8 Y. [def factorial(n):
    + Q3 w- T8 t; w    if n == 0:        # 基线条件
    0 G4 e! m/ J" ?  i$ L" o) j! f        return 1
    5 G" h) O1 J9 Q; I9 l4 r    else:             # 递归条件
    4 ]3 n% M: ~+ A        return n * factorial(n-1)9 `7 \7 P) ^. Q4 G3 h1 F: O
    执行过程(以计算 3! 为例):
    ' X; T8 f0 M! b- ifactorial(3)
    * Q: @% g7 F+ S" Y9 @3 * factorial(2)( v7 N$ S) l* b; a% J
    3 * (2 * factorial(1))
    . q  o) k- h9 T5 \# m% @3 * (2 * (1 * factorial(0)))) g+ f- d# H/ x0 ^2 V- v
    3 * (2 * (1 * 1)) = 62 @" R; [- T+ n( q- z

    * d4 M7 a7 f& b- B# \2 v/ ^ 递归思维要点
    $ R/ U+ Q1 N, a6 g1. **信任递归**:假设子问题已经解决,专注当前层逻辑5 U0 N; h) t. J% H4 I. d0 B* a# `) V
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)6 s" ?# o* P7 i( M8 M. W, s" M
    3. **递推过程**:不断向下分解问题(递)
    8 z/ l  _5 A4 w& F6 R1 H) p: B9 N4 F4. **回溯过程**:组合子问题结果返回(归). W( R! K1 ~0 G
    + u+ n0 u; n$ z3 r
    注意事项
    * s2 p. F2 s. m, w4 d" r) H, {必须要有终止条件
    . ]: G3 J" V6 \递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    $ @5 s2 t$ L1 f+ h# o# L某些问题用递归更直观(如树遍历),但效率可能不如迭代
    6 p9 e! q) ?( F尾递归优化可以提升效率(但Python不支持)3 u6 z5 u: `+ B. d: [6 }3 ?' `1 a/ q
    % l! g( c* @" |" O( A6 s
    递归 vs 迭代
    3 L5 _7 P6 D+ h. u  V% q* i0 X% Q|          | 递归                          | 迭代               |
    + Z( Z1 G/ k3 l" p& A; H7 \|----------|-----------------------------|------------------|
      S" L- W0 g8 i6 P0 e- T# Q9 g| 实现方式    | 函数自调用                        | 循环结构            |; ]7 v8 ~8 W# R0 s6 `: A" M
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |9 o' E4 K- f9 }" J! }6 o
    | 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         |0 q' a. v- \- U8 [
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ( O; \. ]" h, Q+ ?6 v0 }+ Y' |$ Z5 k$ }
    , f7 H) [/ M* X8 @( D3 Y0 V( I 经典递归应用场景# B8 `# U7 J! z  n0 c
    1. 文件系统遍历(目录树结构)
    2 P* p7 Y" c9 w% G2. 快速排序/归并排序算法  k+ N, l) q4 V2 Z& t& ^3 \
    3. 汉诺塔问题% ]1 A0 w' C4 h1 l% j; L3 ]
    4. 二叉树遍历(前序/中序/后序)1 l6 w% ~3 c3 f& _3 A
    5. 生成所有可能的组合(回溯算法)
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    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,! v# G/ W5 O3 i" c  z0 F# a7 H
    我推理机的核心算法应该是二叉树遍历的变种。; M7 x' z5 `' U
    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:* D7 R- g* i7 z, }
    Key Idea of Recursion
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    , a% a6 O) V& S5 R+ M6 uA recursive function solves a problem by:
    * M. v% m( M: G  c  y6 A
    . F' y! B( z* R5 A: b  c4 W7 P    Breaking the problem into smaller instances of the same problem.) d5 Z. H$ K  q% m) o0 ^

    1 N; \7 N! h3 U- j* R6 {. g    Solving the smallest instance directly (base case)./ X; r: P% I+ U9 o
    . R8 j, x: f6 z! {$ g4 R
        Combining the results of smaller instances to solve the larger problem.: P) L  i. A) p' B

    + |* B% _6 f) E4 X7 y: @4 YComponents of a Recursive Function0 |* Y3 [( b  N' r( h, n
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        Base Case:
    ) r6 K  j$ R, p+ D
    0 ?8 X& P$ ]: B& q) L        This is the simplest, smallest instance of the problem that can be solved directly without further recursion." u( h0 \+ V! `, ]/ r) X

    ) i  [6 q: D* h$ A7 s# @        It acts as the stopping condition to prevent infinite recursion.% z  B* [9 |8 P1 z. T% o! E
    9 ?3 r6 N( f) H7 K& ~
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.4 y6 U# _, P  s; G% K2 B, |

    $ a4 ~3 ~6 l0 q/ t    Recursive Case:
    , T5 t& H2 p! D' r. {0 w9 _5 u! K, }# `6 L  ]. d$ F9 p1 E) |
            This is where the function calls itself with a smaller or simpler version of the problem.
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    . @# R: d0 E1 W% Y9 t( S; T        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).6 m( [# X: z. t
    ) C) M4 h  D3 y* F% S; p
    Example: Factorial Calculation
    + |2 w" v( C( y" p4 `4 \) J0 z2 l  {! I+ `8 I& _' w
    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:+ e" z; y* A! j% S5 w% V! i
    . E9 V1 z' G9 D& `) V6 ~
        Base case: 0! = 1$ B& u. p4 t# i# c3 q
    & B) u! O1 b4 ~3 ]4 k* B
        Recursive case: n! = n * (n-1)!; }% ^5 p/ J8 w; A; M% T8 v6 s5 \
    ! q2 p4 F: p  p! |
    Here’s how it looks in code (Python):
    5 N( N/ \& z7 n; ?& \3 q" ^5 ypython0 T. i& }7 d& j- h3 i2 i2 q, B0 k
    ! A. y3 S5 v  j3 ~
    ' M0 p: i: p' k  [% T; G
    def factorial(n):3 {" G1 B" N* \* Z7 \6 w
        # Base case
    5 X2 b5 ^( X7 }4 k6 A6 i    if n == 0:
    6 \2 ~$ A* S; D- c8 A        return 1
    7 g. f6 P+ }1 y4 v- W! v    # Recursive case) i) l; Z1 d, s" s( \1 |
        else:, a2 Q) Z9 O$ P
            return n * factorial(n - 1)
    8 A: T$ D( E" @- _/ u3 K, n4 {* @& Q0 n  i( N7 E
    # Example usage
    0 x3 I+ p% L/ _5 ~9 w2 L; vprint(factorial(5))  # Output: 120, C7 M; z. s' @' ^0 m

    2 P! q% k; \6 O! a  q( N: LHow Recursion Works
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    ' k# Z- a  ]% P" x* T& W    The function keeps calling itself with smaller inputs until it reaches the base case., \/ T2 d, B+ M4 ?0 {9 @# ^
    3 ]! d7 @6 ~( n4 [+ z' ]
        Once the base case is reached, the function starts returning values back up the call stack.
    + L, n  ^1 X  ?: z0 ~: |2 K& j% S, V" ^" t0 c9 o* Y
        These returned values are combined to produce the final result.
    % j* Z* ?5 \0 w9 f9 O4 D
    - {  f* `; O/ F0 j$ x0 t( X/ n* C$ TFor factorial(5):
    - u0 \7 g; i2 L: }' ^# V7 C
    ! Y" ~9 l# l! }2 R2 \: U$ C* c6 O$ I4 c3 B/ V" @: k
    factorial(5) = 5 * factorial(4)
    3 E7 W6 R" A5 s2 ~factorial(4) = 4 * factorial(3)
      k* x% _/ S( }) g6 r$ j; l; Yfactorial(3) = 3 * factorial(2)
    7 J) Z% l& b2 Afactorial(2) = 2 * factorial(1)
    ( y9 b  k- d% lfactorial(1) = 1 * factorial(0)- M2 `9 D; ?( f% E5 c( B3 c! j# Y
    factorial(0) = 1  # Base case
    ' T8 S0 s! L3 x5 I) |' u
    ( i4 S2 h- V8 X/ aThen, the results are combined:
    8 [. T) c- ^4 j3 N' p! r- N/ {4 n$ q! F

    9 T+ q4 m% E7 N. y7 \factorial(1) = 1 * 1 = 1
    5 d: M! ]8 r& G; D* ufactorial(2) = 2 * 1 = 22 ]8 w- n! i# y; l! U
    factorial(3) = 3 * 2 = 6
      c, }( s& Q3 t+ B( J4 ~factorial(4) = 4 * 6 = 24
      ]6 o, j8 I  J" a1 c# a$ f4 u# Yfactorial(5) = 5 * 24 = 120% a; W, m- V' m7 m
      o+ F9 P2 H" a4 T& M
    Advantages of Recursion7 k& ~+ a' J8 h6 p3 ?) Q; ?

    5 A. m$ A+ j0 s  `    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).
    6 U2 d' h1 }2 p& I6 |$ ]9 E4 _/ w7 D" n8 z2 E7 v% D+ n
        Readability: Recursive code can be more readable and concise compared to iterative solutions.$ V- v2 q2 w0 [7 E& F6 g& F& p( B; K' S
      K/ a5 o9 k! S. L
    Disadvantages of Recursion
    . [* o( `- i4 i3 A  C, Y; u
    0 ?! E0 D  \4 t, M, Q! `    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.( e( [! ?/ x4 J% J9 X/ K' D0 K2 f' Y6 W
    4 [+ k' Z3 i+ q! n% m4 a
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    4 j# Q3 Z8 v4 z+ h7 a" \5 q5 n8 d, a
    0 i  ?2 V3 h; nWhen to Use Recursion& R2 }. a8 |. S% \) ^9 F1 i
    6 U; _) Q& V/ D
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    " l6 C% v8 |, K5 w3 r. ~7 K0 }4 G$ z2 R
        Problems with a clear base case and recursive case.; V  t# r4 }. _0 @# F

    : n" c4 O9 i2 h* bExample: Fibonacci Sequence
    4 M0 L& O& W) A9 H9 f6 J1 ]( Q- i9 c  e
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    2 m5 p3 {" j2 h/ x; c" P% w) G/ m! m# F  c, `6 L6 Y9 A8 s
        Base case: fib(0) = 0, fib(1) = 1
    8 R* V$ F7 }3 Z9 x6 J3 Y
      |2 e+ H1 j. j( s' {7 F/ k: ]- Y+ ^    Recursive case: fib(n) = fib(n-1) + fib(n-2): |$ x( A! r$ f
    ( b0 Q/ ~9 h. S4 d5 ^  E, \
    python
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    : d( D& N1 x9 R7 H9 Y3 C: t' \% G0 [
    def fibonacci(n):, g$ X( u' }, A% v8 d! b
        # Base cases* O$ `6 z/ s9 m9 S0 ?
        if n == 0:
    ; {  m, C% Y+ s- N9 X+ d' [# t  n        return 0
    - f5 _# Y8 Q$ }& m2 `    elif n == 1:
    / r1 v- r  g& X$ z$ @8 U, H        return 1
    4 [/ L0 P% i$ q. J, M    # Recursive case
    , J" X  V. X' ~2 K4 X- c    else:& o: m/ J' R# s
            return fibonacci(n - 1) + fibonacci(n - 2)
    9 D2 Y- Z; v0 ~6 h$ u
    # ~- [$ A  u0 H8 b+ p3 _% |( g# Example usage
    " U( T% D# W; Y7 a5 q9 e) J- X2 v4 D4 Pprint(fibonacci(6))  # Output: 8
    ) f: h- K. _% y" E9 H4 s2 P
    . u2 i7 ^. w7 W+ a  zTail Recursion
    * t8 _# s( o5 h0 N
    + ~) t8 ~( C8 WTail 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).# x7 c; C0 C6 w
    % {" V0 n* x1 \2 @. q; i3 e
    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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