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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 6 a; J+ [0 I/ X3 J- j# q  S

    0 U; X8 E2 Z( ^7 s# d解释的不错
    , Y3 L4 x+ l  I5 ^% @( K9 d) w& ~6 d: M' c4 R
    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。
    ! c, d) a5 @* n5 x. D$ C( v7 N, X$ z; M0 _
    关键要素
    0 }3 E! Z' w( m+ H8 ~' N1. **基线条件(Base Case)**- H' C6 k/ D& X# C
       - 递归终止的条件,防止无限循环
    $ `4 d; k$ r" i5 I( w1 P   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ) ], E0 d' _5 t9 Y: b# b# d2 H- h$ s) Y8 z# a- {
    2. **递归条件(Recursive Case)**
    ! r0 i! _* i* z9 k   - 将原问题分解为更小的子问题
    5 M/ D9 B+ I8 B  P$ Z   - 例如:n! = n × (n-1)!
    . G5 o0 W) `. B* q2 i7 [$ ]5 p( N; i* d% P' H
    经典示例:计算阶乘+ L( U9 z; o. q6 O; ^2 U/ v
    python
    # D% |+ e2 A. t$ U( o' n/ Edef factorial(n):
    " A  z1 N2 w' u4 T# U2 H; ~    if n == 0:        # 基线条件+ M4 _, m6 B# J
            return 1
    ( K: y# Z3 M3 m    else:             # 递归条件( E1 }* q3 N( ]4 U7 o: ]
            return n * factorial(n-1)
    5 a' W: T3 n/ A, m执行过程(以计算 3! 为例):7 x6 F0 b- i/ w+ z7 j% O
    factorial(3)
    : E! A& o* b; Q! L3 * factorial(2)$ n+ }" ?" i: S" n0 [! `  @# a
    3 * (2 * factorial(1))' B0 f& F: ~3 |
    3 * (2 * (1 * factorial(0)))& {7 W: j' I. Q  _  q4 G6 }! H
    3 * (2 * (1 * 1)) = 6
    ! }  y. g6 W1 Z# t% R6 \  `4 i* y5 U' C/ H
    递归思维要点' Z9 D8 ]3 y) s0 V5 i
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    8 E# Z, V$ O" c* C" c- v2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    - q- G- K  P6 P3 W3. **递推过程**:不断向下分解问题(递)
    & U7 n: f7 e, f# u6 t; C- v, l4. **回溯过程**:组合子问题结果返回(归)
    # r  B% B$ h% I* S: N1 y
    ) S6 ^; I  W" d/ V 注意事项. G; Z2 ~  g- b
    必须要有终止条件3 }0 H3 t- {! X  ?8 h: s
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)2 N: @5 R" Q# S" [
    某些问题用递归更直观(如树遍历),但效率可能不如迭代8 Z  Q" D! H, N; c$ S
    尾递归优化可以提升效率(但Python不支持)8 Q( s/ n2 t, W; Z6 B" c2 a8 F
    : s& J' X. @$ K: b1 A
    递归 vs 迭代, @$ M4 x/ z  E( q. }
    |          | 递归                          | 迭代               |
    0 D2 z4 F  H; s2 @  @9 W8 [) z|----------|-----------------------------|------------------|) k' @5 _5 @0 W8 }- G0 ?' V. x
    | 实现方式    | 函数自调用                        | 循环结构            |  `) e5 O0 k! y! z3 z0 a
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |" M# m& y% z5 I# \% M4 A2 j. r3 b
    | 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 S' ?6 v5 h/ S9 n7 d% ~/ o2 B
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |1 F2 M1 ?) Z; D1 q& O3 n

    ) B+ d& G6 O" R, l0 H: F 经典递归应用场景
    $ T6 D& {3 D+ d) w( Q$ V/ T1. 文件系统遍历(目录树结构)7 }! ?( ^% ~' l) g2 o0 H
    2. 快速排序/归并排序算法
    * h, N! b/ d8 E3. 汉诺塔问题0 N; B* p9 ?5 J, H, t# q8 i5 ^
    4. 二叉树遍历(前序/中序/后序)
    $ L7 \. P( a( k5. 生成所有可能的组合(回溯算法)
    9 W! n6 [: k( F3 t
    * v# T- j1 L& ]6 z1 S  C* m! Q% M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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    昨天 01:20
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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,' t8 _. t' R6 E, n& a- F
    我推理机的核心算法应该是二叉树遍历的变种。
    7 T' _  _2 j% U, H# g- g0 X% t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 n; L( Y! [. _
    Key Idea of Recursion5 N) T8 P# V2 G7 N

    4 l- x3 {8 Y* Q  ]+ J# PA recursive function solves a problem by:
    : `4 u8 [2 s4 e, {- o0 x9 w; L, n% X3 {) A& y( S, E
        Breaking the problem into smaller instances of the same problem.
    - O% a0 I( B9 a2 q  P
    7 i$ D: V; R$ [- F3 G# x    Solving the smallest instance directly (base case).
    2 t  J) A/ c2 S9 @1 m2 B) k# k4 W
    $ J4 e9 j& m7 o$ T% C    Combining the results of smaller instances to solve the larger problem.
    : J% u! b8 U2 r: Y0 _# x: r5 H2 k7 p- d# C2 |; l' H
    Components of a Recursive Function
    ' V! ^" `0 r& e4 z0 D3 x" A) ^+ m$ q- ]0 B1 d% j
        Base Case:9 i0 P  V. M( b8 v
    7 k& O2 D5 O4 z
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.. B# H, d) v8 Q

    & b) \5 _  P9 \# R: T        It acts as the stopping condition to prevent infinite recursion.; B& t" j4 ~. n: o/ z8 N' e, b# c
      o. m; ^7 l/ a" r6 u
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.+ L/ e. u- m  C& X

    : b" r5 ]2 m- i0 x    Recursive Case:
    6 V( A- X3 Q4 b: l" j* W, ~8 Y9 b; V/ B" ~/ m7 s& I5 K1 U. I0 [2 d
            This is where the function calls itself with a smaller or simpler version of the problem.
    9 \4 i/ N* |( r  {' r3 K* V& J: I4 `2 s1 l
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    / z9 ^8 J% h5 V6 v
    2 T5 [9 @5 E  f  I7 I2 Y! p+ U" QExample: Factorial Calculation
    4 O: m2 J0 r& p! h) Z. w
    9 D$ E1 u+ c* B% }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:- g2 H6 g0 a( e: E" [. ^. i9 C

    8 w) [2 E: J4 K$ Z" b! t    Base case: 0! = 1
    5 F: F# ~1 D' r4 t4 Y  H8 u0 J/ W
        Recursive case: n! = n * (n-1)!
    : |# j, Y( F( Y" j0 m! P- m/ W  @; t1 X9 F% e! H  L1 E
    Here’s how it looks in code (Python):6 X+ E8 \, C( {' D
    python
    4 O; ^7 b2 U: m7 F0 J2 e, T6 `# r4 u% O* o) F8 b- d

    9 v4 z. y* J; S) R: c* L. P* u, n9 rdef factorial(n):/ R* I3 N9 q+ r, r
        # Base case) Y0 ], z# W7 w7 N1 C1 n
        if n == 0:  ]6 U. E. V0 O) ^% `# q
            return 1
    & N: y7 n0 c" r    # Recursive case
    % d8 Z2 X0 X3 R. M0 V) q    else:
    , R4 i! t8 d0 b        return n * factorial(n - 1)
    : H' \1 Q5 \" S/ w' _1 N9 l! z: x3 Q+ P
    # Example usage
    ( o( `5 ^4 f; j" [' |print(factorial(5))  # Output: 120
    1 U% Z/ Y% M; h, X+ M( ~' y' Y& m6 k4 a0 k( l8 j
    How Recursion Works
    3 u, a% @8 c% w. Q# V. ~
    : w, ]/ D2 c5 D' I    The function keeps calling itself with smaller inputs until it reaches the base case.
    6 s! W4 O* c; g  C
    * l2 y& D; ^: l: r    Once the base case is reached, the function starts returning values back up the call stack.0 @+ t" I  Z, j  M: t7 w" t# ^, x

    : {7 D: X( k0 H- X  L$ ^    These returned values are combined to produce the final result.
    3 ~* D9 N* w# s# P1 J0 x* l9 _# T6 ]6 z. o: u: C( E. T/ `) g1 _- s/ j
    For factorial(5):
    % U9 ~  b7 v! m
    2 ], ~' O$ Y8 y/ K* T( d$ p* [
    factorial(5) = 5 * factorial(4)
    8 r! z7 ?  M. d  A# Z# F! F3 ~factorial(4) = 4 * factorial(3)- H- b2 {& s5 H5 s7 K3 k" y
    factorial(3) = 3 * factorial(2)3 U0 @( W8 R, L4 k8 O+ H
    factorial(2) = 2 * factorial(1)
    : H( b# P% S3 n3 L3 Ufactorial(1) = 1 * factorial(0)
    ) m8 X4 c2 Z4 a5 afactorial(0) = 1  # Base case
    : P, p9 f% E3 |7 O/ u- W/ h, V
    9 M/ L$ Z2 J6 w! UThen, the results are combined:
      a( T! n( K+ J
    ; Y. @) v6 {) G
    ) D" @6 s; _4 [5 |, Zfactorial(1) = 1 * 1 = 1, T( w. O' @7 F3 w
    factorial(2) = 2 * 1 = 2
    ! {! {# e- u7 F7 qfactorial(3) = 3 * 2 = 6
    0 d7 J+ x- a3 q' S- `4 ^2 t+ Dfactorial(4) = 4 * 6 = 24* t0 T7 K; W) t9 O% g( S8 \
    factorial(5) = 5 * 24 = 120/ e# i+ B+ f4 i# }

    : A! s) `' g7 _- TAdvantages of Recursion
    ( T3 L6 O1 P1 |- u4 I8 k5 u% P8 N  i. t2 T- `
        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).% k' P9 c) Y) i: f- H% I$ s

      e+ G3 D0 e( t: T: k+ `6 E% f    Readability: Recursive code can be more readable and concise compared to iterative solutions., c( D# X( [4 g2 T3 r$ G" z& M

    8 a) Z0 a( ~, c' q# pDisadvantages of Recursion
    9 n) b/ `+ D* K- f% ]( n0 a# e& W- P: N+ M
        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.
    1 p& x# g& g) \0 Y( y: H  w8 d0 w: j& d$ h2 i6 |. q, h. L
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    $ @3 X& E: [# {* N; j4 M# z# w0 }! L% w, x* J; ?: [: h8 n2 K
    When to Use Recursion4 U5 [; N+ l' g4 E+ o2 j' E
    " Q8 H& _  n8 {, J$ d
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).& h' Q  q8 R( c) S0 X% l4 [6 u
    6 v. ^' M" p7 o! P* P" A
        Problems with a clear base case and recursive case., y5 ^+ {6 S# R" D  V
    ! G  u& [  X8 I' K+ i
    Example: Fibonacci Sequence, i, w* {3 s, d- X) B8 y* ?

    / j/ A- F4 w; i( d7 EThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    - x0 t$ A; e) Q' i
    4 z; ?" B0 f4 Z7 R/ m% C4 [    Base case: fib(0) = 0, fib(1) = 1
    & j8 l3 Z. T, \& _; ?) ?) J1 M1 e' y) N: Y
        Recursive case: fib(n) = fib(n-1) + fib(n-2)- v7 G4 q0 N, R* o
    : P# u0 F$ X% n
    python
    ' u$ D/ E- y! y, f* q' V* i. |$ N2 P& Z/ {: c% p
    # P2 E  |& \9 W& E$ O
    def fibonacci(n):
    " Q6 S# [! v! j& N    # Base cases4 w: K! v! L& M
        if n == 0:
    - K( t1 g0 N5 c; B# _% _4 v" Z/ ~        return 0
    . E, M' m  V: @5 T8 a% J, I" p    elif n == 1:
    % T) F/ F% P1 }1 C+ e! t        return 10 b( \, y6 G# r* Z
        # Recursive case3 Z% x0 e3 Z9 ?  K! z
        else:
    : Y& w9 i- ?( t" \5 t: @        return fibonacci(n - 1) + fibonacci(n - 2)
    6 L% K* }7 p& j( y1 L( B1 T1 O4 |0 F. Y$ q( g- k  ], Y0 K
    # Example usage: }7 f) E; o1 p+ U1 V
    print(fibonacci(6))  # Output: 88 d: d2 V+ R! Y# I* }; x! s7 F

    7 @6 D9 x$ x3 A9 N5 ^8 iTail Recursion
    9 P4 m! q: i9 I
    2 ^0 u  |6 c( y( s6 H0 b+ ^7 jTail 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).
    : U: o" ^' e, A
    7 N' c4 T* }+ q5 F& q, qIn 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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