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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 , R* v  S4 C' T9 Y" M
    : _- r2 A+ Y: q' M
    解释的不错) I) j7 J+ c- y. B

    * i& f% {5 \; {" \" t1 \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。+ O! a7 J; @. h  v
    " ^4 t7 ~0 z' s$ v9 N
    关键要素2 ~4 U2 n) T( g( A! Z. v$ f, A: a
    1. **基线条件(Base Case)**3 |) n% J- }) p) K0 v, y
       - 递归终止的条件,防止无限循环
    $ ~# M( A5 v4 \7 {" a# N$ F& h   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1: x$ o8 u4 P% L& T- `7 `
    * Z* e$ L/ ?% Y: D
    2. **递归条件(Recursive Case)**' C9 m6 a/ {' ~% b' C
       - 将原问题分解为更小的子问题- r3 ~2 y+ {# @! e/ K
       - 例如:n! = n × (n-1)!6 O/ e: b+ N8 ~1 i

    5 Y4 a+ r) A' u 经典示例:计算阶乘& m7 y# ?) p  [/ F
    python
    # x2 e6 r* F% mdef factorial(n):( q2 v/ N) q" x+ U- n- E* V/ @
        if n == 0:        # 基线条件* x6 q1 Y6 _7 t; c2 ?8 b5 k
            return 1
    8 d3 k0 t' y/ W    else:             # 递归条件: {# _' O5 B) @% Z6 }$ v! A
            return n * factorial(n-1)
    0 s& J9 P5 F; T& q# S' y3 ?' L执行过程(以计算 3! 为例):
    2 M$ Y. X$ \" u7 Jfactorial(3)# ?+ U2 U6 a' V( q6 X  B# U* ?% }2 z
    3 * factorial(2)
      f4 F- N. g4 o$ x- Z9 J' n3 * (2 * factorial(1))
    + ~6 V9 H+ i, H$ H3 t. N3 * (2 * (1 * factorial(0)))
    ( s2 [& ]1 D) H" T3 * (2 * (1 * 1)) = 6, b4 M# J1 u! x" l5 N

    % w' |6 y1 u' S. o% D" { 递归思维要点. I7 Y9 Y; z8 [$ D3 w$ `
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑$ }( }) p2 k( d6 Y( ^( z
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)' e- O, m" G5 [! j& v
    3. **递推过程**:不断向下分解问题(递)
    ! i" P) Y9 [5 R7 w( k% L( h* t4. **回溯过程**:组合子问题结果返回(归)5 U6 m% l, O7 o3 L1 {

    " H6 J6 x% s9 ^. h; S 注意事项
    1 W& L" f9 I5 V$ u7 x9 [+ @必须要有终止条件
    6 T$ z; s, y  P递归深度过大可能导致栈溢出(Python默认递归深度约1000层): q5 U2 v, |" P5 l
    某些问题用递归更直观(如树遍历),但效率可能不如迭代( I$ D% J' G$ [, [# D1 E
    尾递归优化可以提升效率(但Python不支持)6 l4 l$ o$ f! N, O

    : F, g# U7 \* W; K; a3 ] 递归 vs 迭代
    1 I+ l* v$ t0 x& ~|          | 递归                          | 迭代               |8 I0 H; w: Q* c3 ?
    |----------|-----------------------------|------------------|
      \4 x% _& q# @& F| 实现方式    | 函数自调用                        | 循环结构            |& ^: I3 D/ \$ g0 C! m" x
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |; ~& Z: u/ c# H1 P
    | 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         |* g: M& ]4 n/ o- P
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |! L& a! T# @% c5 }% L1 z+ Z7 K

    ( d9 H  B9 p% D 经典递归应用场景9 c' D- M- H! ?1 K
    1. 文件系统遍历(目录树结构)
    7 c: @6 a' G" p2. 快速排序/归并排序算法
    ! _* x& @" S, `" q$ I& _! C3. 汉诺塔问题# A  A, A4 N6 W: I1 N+ }) A2 R
    4. 二叉树遍历(前序/中序/后序)
    : |% m" Y5 w- S) k+ x5. 生成所有可能的组合(回溯算法)8 y% @, ^  t# P: s. e

    % |! m% ^2 U3 h  d( ~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

    昨天 12:10
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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,; Y! K+ o" R+ P4 @+ [( C
    我推理机的核心算法应该是二叉树遍历的变种。
    $ ~2 r3 C5 C( O* U2 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:2 O3 Y& ~* f& \9 J2 C4 Y  @# F3 x& y
    Key Idea of Recursion
    " A& {* s! J  ~# u! k- b) |/ K, r( u) u
    A recursive function solves a problem by:3 Q$ s& W/ E/ k

    ) C3 R: q' V% N1 n% z1 c4 I, u* j    Breaking the problem into smaller instances of the same problem.
    7 H* a9 N  G+ I) h& @. ]. m
    * G+ [8 G" X0 _' Q5 ^' K    Solving the smallest instance directly (base case).  `; j4 v  [: L4 A+ e8 T6 j8 F' }
    " f$ ~- p4 F( ]$ E) w
        Combining the results of smaller instances to solve the larger problem.
    " E5 P9 r* V2 [  O; s1 G& h6 N2 h0 e( G; N! g3 G0 M
    Components of a Recursive Function* Z* _' \5 U+ ^- X
    4 T3 |/ ^5 Y( Z) U' X
        Base Case:
    6 l* u# b5 k0 c; ~" A
    % o* K, Y3 e; D+ J, z        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    2 l) l5 \2 l( }7 x6 B2 H; N! |2 B  o2 E
            It acts as the stopping condition to prevent infinite recursion.& z$ h& D* @* y! _! i- B
    + ~8 I+ n4 M3 Y7 k* \
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    & b1 z2 }& j, g9 F5 u6 C' l* {5 F6 j' ~; f) A. C) b- A
        Recursive Case:4 w* l7 e) f5 A

    ( O$ M9 U2 l$ `, s        This is where the function calls itself with a smaller or simpler version of the problem.
    ( K5 o# M% ^* N# ]7 B$ ~0 i. S0 |$ h% n7 I; `) i# x8 A+ ]
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)./ ^. i9 e1 D# u  e, P- Z

    4 I, O0 V% A" ~6 OExample: Factorial Calculation
    $ s: z) A/ p* u) D4 M" X# W! N5 O
    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:5 F4 r4 B6 p3 ~' ?3 G+ d# E9 m5 R

    5 n! C$ \: h( Q) a2 b    Base case: 0! = 1
    6 A% R2 f  M7 R& T) d; g5 m3 d# d% C6 m* B: y; U
        Recursive case: n! = n * (n-1)!- Y( Z1 i* w2 j9 `; ?% k+ B4 P& R

      t: d2 e# b  P% h1 EHere’s how it looks in code (Python):
    6 S0 U/ \8 \. [, ]& q/ e; lpython! o( z( ?1 @( w% y0 V9 S  A
      G: f3 R8 Z% j0 e) i; J

    + O# [6 A* r8 I% S) M* o/ Fdef factorial(n):! S6 K5 [' b, n2 o2 B  A+ k; \
        # Base case
    ; \% p$ q0 H) k7 h" n5 G% ~/ R. c    if n == 0:
    4 u4 _1 e7 z& m  e5 ^        return 1: K! q. u, b. X/ C( k" Q# e8 Q# ]
        # Recursive case
    / @/ o- K) }' m$ {+ b: f    else:7 |; I7 ~0 {- |$ W! @( }
            return n * factorial(n - 1)' o3 Q8 G. ?9 E* N! B( _# _) d
    ! W  Q. X3 |! n3 i/ f: ?0 L; G- [
    # Example usage; U( F+ i! n8 |# p6 ~9 m
    print(factorial(5))  # Output: 120' V5 P! C/ {( R$ q$ u& O# r/ F" M

    ( q& h& H, g, x( r5 l, QHow Recursion Works
    $ v# q( A0 k0 \6 W( K% F! S: v3 O
    0 J6 T5 u$ L. E    The function keeps calling itself with smaller inputs until it reaches the base case.! W4 K: K; g$ W7 o6 Q
    ! x1 \3 Z- @: J9 z* k3 a
        Once the base case is reached, the function starts returning values back up the call stack.  _5 i2 d) |# S9 ~

    $ r, C: H6 U8 e& f, }    These returned values are combined to produce the final result.& g# `# O2 F( n) E5 V4 k

    6 }) B" v; `, y( ?/ t/ ~2 CFor factorial(5):
    - R+ z* _7 G6 H- b8 f; i
    ( @8 i( C& D" A/ e2 G. @: ~) W& {8 ?' l4 V& U# p) T9 j
    factorial(5) = 5 * factorial(4)! N. }2 M( E5 ]( q3 _
    factorial(4) = 4 * factorial(3)
    ! a+ H9 ^8 o; e' |2 `) Efactorial(3) = 3 * factorial(2)% B2 {3 R8 L& T6 J
    factorial(2) = 2 * factorial(1)
    $ N) J& y) O8 o; f- h" t9 rfactorial(1) = 1 * factorial(0)! N, u, S/ b# b( h) X. u2 R- |
    factorial(0) = 1  # Base case; A4 ~/ F( g$ Z
    " Y* H* E+ e( H
    Then, the results are combined:; S# P: q/ L  k$ I0 {
    $ G6 z: K8 z. Q5 P; J

    2 M" V6 R. r5 F) A2 }' Q8 afactorial(1) = 1 * 1 = 1" r9 r) X8 v+ z5 O! G
    factorial(2) = 2 * 1 = 2
    8 x9 U% h* I$ a3 ufactorial(3) = 3 * 2 = 6
    , n$ S; B% }1 \- }factorial(4) = 4 * 6 = 24
    ' H- t! t+ @& X" J7 R  m1 z. Cfactorial(5) = 5 * 24 = 120- |/ G4 {/ o; u

    + r: c& T4 H  _% z+ V0 MAdvantages of Recursion' h" s7 P/ {  K0 d6 `
    2 L  ?3 q5 _  V) }
        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).3 w! K7 b: c2 X% Z, ~( x$ X; B% s

    - }* S; v: i1 M+ F9 @. k* z    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    0 A1 X- C/ p8 @9 Z' y. O* D$ z0 l: |% Q! t4 V  x; R. _4 [5 \' R# |! T
    Disadvantages of Recursion
    2 T  y3 N. G: O
    2 o; I8 k/ m* ?' X    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.
    5 e' H, L% ^2 y2 k
    8 u0 T) T! N! H* ~9 a" p- b7 R    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    4 u: F3 }7 }4 i; N4 [  ]' Q4 m- ~
    2 I+ q; g; d8 z, C8 xWhen to Use Recursion: z) g8 O0 D% c  a, \
    $ w, t, ^+ ~, y# ]2 ]; f3 S- H
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    8 i3 Y; p3 {) O3 G) K  {' L2 \! B7 F- C1 L8 _. l( W
        Problems with a clear base case and recursive case.. X  Y4 Z5 `$ b( s" L
    & x) S$ f% s% z" b2 ~
    Example: Fibonacci Sequence6 Q  o  ~# p* Z2 g; e

      \: j1 r0 \9 Y, d$ L' w$ V( k( [The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    1 J  b8 C3 ~+ H) x' a5 v& k* m
    - V; r1 m) g- M8 \+ X    Base case: fib(0) = 0, fib(1) = 1
    $ I! f2 J; b' ]$ n2 p/ Y# u6 W: F# f1 c8 A6 O  k; l6 S
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    6 z- Z% b( E$ F9 a5 u( ^7 d/ f
    0 u% r; F: T, {0 {5 K& H: cpython
    % A  e/ p! e  x4 n
    0 Q9 |% R4 J+ }$ X$ P# Y  g& x$ x6 a# k
    def fibonacci(n):
    ! W% R, r/ m6 @( Y! n) S2 @    # Base cases6 |( o. N  w3 K" i7 q& ?* |
        if n == 0:
    2 [3 ?- m' K* b/ B- T        return 0
    6 A/ K! F  a) V7 @    elif n == 1:' g  G, {( z/ M
            return 10 d1 e& q# P. R8 f( H. V" {: E# M; p
        # Recursive case6 W; s) W8 y2 k: ~
        else:1 W3 }# P1 I; c' k2 y' q' ?
            return fibonacci(n - 1) + fibonacci(n - 2)$ X$ r7 |+ P$ c9 B

    ( Q% P/ g" z5 N$ a. a$ o# Example usage
    9 ?6 n; g5 z" d; h4 g1 }( j8 h( ^print(fibonacci(6))  # Output: 8
    ; P' ?- z, `% J/ S' H, O: p% E6 {: }5 Z- {  d. w  u- }1 h
    Tail Recursion
      L) z/ t, g( A
    1 V- m! a) D4 u, _  u5 |9 STail 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).
    % q, e3 K9 z& g* D; w  r/ U  H
    . ~  _& K3 ~- F+ {0 P4 s8 ?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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