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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 S1 [( i  S# ?% ~3 H, ]
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    解释的不错: e. r: N: M  d+ h: d0 e) n
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    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。0 W2 @) |* l0 b7 g- ~) b7 ^

    : C- a' O' f& t; ~* Z# b" A, c 关键要素
    6 ~$ C/ o1 o2 @7 z: U" O1. **基线条件(Base Case)**! N8 U# {4 U  I6 N9 t" u: C
       - 递归终止的条件,防止无限循环8 T/ p# b, f; W, J8 {7 g
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1: f8 o0 X* G2 l* t9 o& L# Q

    + {0 M  S- ~0 f) P+ U: k$ b2. **递归条件(Recursive Case)**
    9 }  z/ j8 C0 E- a5 q! N( y, Y" P   - 将原问题分解为更小的子问题
    + c, E( O* l: X4 E   - 例如:n! = n × (n-1)!* [" u/ s$ ~: i, ?' G
    ) C5 @- \3 }/ @
    经典示例:计算阶乘
    7 T  a  n3 U2 R4 Y2 jpython
      ~$ _& s+ X! ^8 d% Idef factorial(n):
    $ {9 @4 A/ c0 }# v8 n    if n == 0:        # 基线条件0 ^4 M  \" {9 Z2 N' E' l
            return 1# t! Z% N+ X0 J' \- o0 X! M: a
        else:             # 递归条件+ r6 U/ Z* A8 p  n( O; C
            return n * factorial(n-1)
    * t* S6 G1 N, {" c6 p/ ]执行过程(以计算 3! 为例):/ d: Q2 B% z! i5 e( z/ H2 Z
    factorial(3)+ t; l+ m" Z! F/ ?# \  ?0 I' z6 Z! Y" V
    3 * factorial(2)
    3 Q* b3 K; U% l5 a  K0 C& Q3 * (2 * factorial(1))
    , d& s! r# F- A3 D  m# j3 * (2 * (1 * factorial(0)))5 H9 Y( ^4 d! m/ H, U; ^/ K- Y
    3 * (2 * (1 * 1)) = 6
    & u( J" `6 t3 ]: \1 {  k) V$ ~. g2 f% {- g( x, }5 c1 c. f/ E. y. s0 j1 W
    递归思维要点8 r: Q* C; h- T% M
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    . J3 u5 I! a* r: H( W2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    6 r& @1 s3 \6 M% Z  ]8 Y3. **递推过程**:不断向下分解问题(递)! ^, `$ F# f( V8 y
    4. **回溯过程**:组合子问题结果返回(归)" O- ]8 I, {& Z
    " t& j9 P9 C0 H
    注意事项
    3 l: U7 W- ?6 w( J必须要有终止条件1 T' p* s: ^# s* L, {' B5 X/ m2 R/ a
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ( {/ H* m. i6 i: S某些问题用递归更直观(如树遍历),但效率可能不如迭代/ U% c7 U# a* D) I5 [2 c( ]/ n
    尾递归优化可以提升效率(但Python不支持)) f  G8 q% |  l7 F% W* w* [3 _

    - r1 S' N* h; {6 _' m 递归 vs 迭代
    0 j3 D" q" l0 q6 m|          | 递归                          | 迭代               |
    : k7 w$ p9 `- O3 [" A8 `" W: y|----------|-----------------------------|------------------|$ a9 W5 [5 X  H% h: U) Q( c- A$ a
    | 实现方式    | 函数自调用                        | 循环结构            |
    " N* Z6 g  @" v* s- H| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |4 P8 g8 r$ ?+ d
    | 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         |1 Q# ]' U" Z/ I  b& O- c) k# r
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    7 x3 i. e! D9 b
    * S1 q( R% A$ Q# a' A 经典递归应用场景! f! k7 O3 g7 g7 y% O9 Y
    1. 文件系统遍历(目录树结构)- t$ y0 a+ s8 k: P" ]5 b" V, f) J6 H
    2. 快速排序/归并排序算法
    6 h, o) D  H2 @5 X8 J" ?* C$ G8 O3. 汉诺塔问题+ G. I/ C9 [+ V) D& O1 i
    4. 二叉树遍历(前序/中序/后序)
    1 R& W% \+ ~& R  D5 [& _7 {5. 生成所有可能的组合(回溯算法)* q0 q" Y  s$ f2 {

    6 x5 h$ d! C: l4 l6 x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,
    4 T7 F$ ^2 U$ \3 X" k+ |我推理机的核心算法应该是二叉树遍历的变种。
    ' u7 D! R9 _3 X2 p( p" w& 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:; L, U- W+ U! S6 c2 [/ y9 Z* u
    Key Idea of Recursion: x/ q: a9 t6 f4 d6 S8 `- m

    ' I3 h; p% A/ Z, D# jA recursive function solves a problem by:: `: J+ w8 I* V  E9 E7 Q9 Z1 P7 V4 L

    6 L  C$ c) o# E+ N    Breaking the problem into smaller instances of the same problem.
    " O3 a! s* T2 g+ C; @/ U" `' Q9 {+ ^4 C& y) G) _$ A$ P
        Solving the smallest instance directly (base case).
    5 C) J. s! ]  Y, o. R6 b
    , w# ~/ X" L; t( {/ h    Combining the results of smaller instances to solve the larger problem.
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    5 Y$ G# ^4 B' ?/ BComponents of a Recursive Function8 w" B- P  Q) t7 r- R' E- l
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        Base Case:2 `7 M: o" c9 o% q" c/ S( F
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    1 v) q6 s/ F3 A- A/ t  ^% `, v& ?0 A: V& z& h% r9 q: M
            It acts as the stopping condition to prevent infinite recursion.
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    ! ?: L+ m! Q0 r
    5 _2 `3 h" k- N: g" y0 z    Recursive Case:
    % \% ~: I) R+ Q
    : L6 o; Z) w$ r* q+ u        This is where the function calls itself with a smaller or simpler version of the problem.
    , M* ^5 W' S( P+ v4 Q
    ' s( G) r  ?, y) }( p/ v9 F0 K+ b# _        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)./ f3 w8 N" b1 T/ L& a

    : C3 U$ m1 O$ [2 q8 A# ^Example: Factorial Calculation" f, i& @" q- L

    . I) L3 o( a; x/ x0 mThe 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:0 K3 r/ O& U! ?1 Q6 l- v

    , u- y$ t. S0 g9 h9 o2 H* f: `    Base case: 0! = 13 n* D  ^' ?- P, j

    ! i! S  w) k7 j/ {. ]* ~: _    Recursive case: n! = n * (n-1)!
    . h5 E- A9 v% v3 C  M% ~
    ! C0 |0 |6 ~% u  x: E+ X; g- i/ NHere’s how it looks in code (Python):
    : o# w. w3 L: v: e7 hpython
    : ]; \  C4 H* v! S- _; b, D% p, D3 o0 L: X+ h; q
    / @+ ~+ V' i/ w
    def factorial(n):' q# z5 j( I# y
        # Base case9 v. ?# p0 `8 w* J# u9 M0 z( a4 g
        if n == 0:
    + t3 {: {, j1 S2 ]' G        return 1# {8 O, a7 p# y* }5 y  V6 H
        # Recursive case9 u2 E! R  ~2 T- a
        else:
    " P; i( i) T% l        return n * factorial(n - 1)0 Y) d0 F  U$ K' v
    $ T3 C! _4 _2 ~/ h- f0 v$ N) d+ g
    # Example usage$ F: F! O/ X& Z# V
    print(factorial(5))  # Output: 120% Q0 @# l1 v4 \9 I
    1 Y( \5 m. u( {% A2 g
    How Recursion Works
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    . A4 @$ v& g: l" q  A. b. _( p    The function keeps calling itself with smaller inputs until it reaches the base case.
    + V3 X# v- H& A6 b- V" y/ W5 X: ~1 h( \& v/ C4 Y9 @2 ?9 O% R+ j' t
        Once the base case is reached, the function starts returning values back up the call stack.- ~' A! n8 L7 G) M

    ) {) y; l$ |! O# P: R    These returned values are combined to produce the final result.
    % B- S8 Z7 v6 r  `# r- d" y# ~' C+ X. U! M6 ]4 _- X6 X  i
    For factorial(5):* R4 j, S2 k# @, H& d

    5 ?# v- v; u1 e# x
    5 h4 y  m/ u+ d' h: s8 k$ J6 {factorial(5) = 5 * factorial(4)
    ( u$ S& \0 t( N% f1 m. Y6 ^" L$ `0 zfactorial(4) = 4 * factorial(3)
    " f7 U' A5 j5 \factorial(3) = 3 * factorial(2)( [# t5 `% \* A' p6 p
    factorial(2) = 2 * factorial(1)
    - [6 y  ^; k# L, `8 w: X8 z# w0 ?factorial(1) = 1 * factorial(0)' ^; h& e$ Q3 B* M2 M- D: f. y. S0 Y
    factorial(0) = 1  # Base case4 R6 L5 K1 k4 h

    : D/ |4 b' q, G6 k* TThen, the results are combined:
    $ [0 E& j* K: A7 F5 g6 W/ n* c1 q" z6 X8 e4 F

    + c2 [! N, k1 M8 o% |% s7 h$ lfactorial(1) = 1 * 1 = 1
    ; [7 W5 t7 U! F. }1 c! V% Mfactorial(2) = 2 * 1 = 2
    % g5 p2 D1 }! g2 T4 P1 Sfactorial(3) = 3 * 2 = 6
    0 @  S7 y2 d; {5 q& afactorial(4) = 4 * 6 = 24
      J! X; e. N' I" p0 M/ J" sfactorial(5) = 5 * 24 = 120
    5 e, t: j2 b# O& _/ L  @4 v7 k4 ?# b+ V6 U: G
    Advantages of Recursion
    " W# }9 c$ S5 T6 z5 N
    . s/ M7 P3 m, E0 A    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).
    ( S- w# m9 `6 |6 d) `! b/ a! ]: _3 z# X% y
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    " T# K  o! f5 g
    2 ?5 t1 V. g% P0 n7 w5 NDisadvantages of Recursion! W3 b& Z/ P5 x6 W: K  j
    . K) @# Z8 b5 P# 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.
    8 `. X  p! q/ A2 N% a: U" M
    9 W' M3 S- W/ G    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).% I3 c5 a5 I& ?' F; l# I
    # [, o, h3 B5 U2 s! a
    When to Use Recursion
    5 U9 s! d% J; h, Q# S
    - V# o1 R9 M" M3 t1 k    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    2 e( _+ m& [9 X  q+ q
    ) l6 X& {% q5 X5 M8 O    Problems with a clear base case and recursive case.$ d. {  o4 S5 }: @* O  ]9 h
    6 E8 }0 W- P+ D+ _9 r6 S8 ~
    Example: Fibonacci Sequence
    - z- N( S- e. a3 n+ i8 U- d. P$ g- }2 B
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:+ s- Y) G8 a" F  K9 K" S
    ! C' }: S* w& a$ N3 T
        Base case: fib(0) = 0, fib(1) = 1
    * s' \" c: w( X  }  u1 X9 D; ?, C" u5 a* H/ I; A8 s
        Recursive case: fib(n) = fib(n-1) + fib(n-2)5 q: h9 U! L2 a3 z! ?9 N- K- k
    , G$ R! q% \. U5 p7 P7 a
    python' Q+ v& K$ ~, m( \- c
    . u4 n0 N: Y2 K6 S) n* j8 q5 N

    * i0 O5 r5 ]' o7 m0 s4 `" T9 Ddef fibonacci(n):
    0 J3 I' [) `1 P3 H" y6 Y    # Base cases5 n4 k! z. ]# u% j  m4 v
        if n == 0:, E! h( W" l% `8 R& d% I
            return 0
    : }. I, `% d! ^+ M' t* s    elif n == 1:+ B, i5 ?( |  b/ C8 s0 |/ v0 Z
            return 1
    + {/ f& Q- U- H4 \0 ?+ X# h4 c1 z    # Recursive case- O2 i' n0 c% D5 \" k& _. B9 m) w
        else:
    / q9 o" g8 ?: A9 B3 Q4 E8 @% P        return fibonacci(n - 1) + fibonacci(n - 2)) T& U( `$ U' T/ d: ~

    7 [& m8 `: o# }9 i; [5 ~; ^# Example usage. z% J: X5 j3 Y9 a; E% N8 n
    print(fibonacci(6))  # Output: 8
    ) I" S2 |! m; N. Y0 b6 h0 a  B/ t; T) \3 A" y
    Tail Recursion  V" b( y7 _( b7 v+ ^/ b6 v" C4 t
    + [% Q9 ~& ^1 O2 R9 N
    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).: ~7 {& `2 A9 y/ ?' A" a  S

    4 }6 w# K% m# w6 ]  o# H3 CIn 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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