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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
     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
    ! k; r$ V( f% E4 L+ |4 `* k1 d3 j# d% A! k5 {# g
    解释的不错& N* q6 L1 V- o: W# m5 f

    9 [$ s* v7 S- y" ?4 l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。& m! P% i) {& l9 L7 b$ v6 s

    ) `0 t" C& k' J* P4 E 关键要素
    9 H$ ]- a5 {4 q1. **基线条件(Base Case)**
    % g$ x+ u6 e( L4 {   - 递归终止的条件,防止无限循环/ m+ X% n0 y# k2 a0 F
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ( H. a7 J1 U# A" [, e; q4 V) c+ D7 Y$ _
    2. **递归条件(Recursive Case)**
    # \( e# M8 Y1 T   - 将原问题分解为更小的子问题, _& j: }2 {2 W/ Z7 Q: I8 w% @: u) V- J
       - 例如:n! = n × (n-1)!
    / m" \1 d7 [$ i
    % s1 G: S) Y' T  m' @ 经典示例:计算阶乘' w7 s; v. F2 Q4 i- B; H: h% P: \9 u
    python% t* Z5 ]5 {# y  }1 B
    def factorial(n):
    2 I" U: B- m1 T% E5 Z; V    if n == 0:        # 基线条件
    : E3 q6 m$ t* a  X3 a" [" [        return 1
    ! K9 j! z6 O% v- L+ ?/ Y0 \" ^: }    else:             # 递归条件( N. N/ B" d* s; \% S
            return n * factorial(n-1)) X$ j  V: B0 {  T5 K/ o+ d3 ^; S( w
    执行过程(以计算 3! 为例):
    ( o1 L* c" x: V( Y+ Jfactorial(3)9 \. M! a0 v7 S( }2 V
    3 * factorial(2)5 R* l0 `- C  }4 L, S0 j
    3 * (2 * factorial(1))% M# ]/ J, u, Q( |9 V5 j1 l
    3 * (2 * (1 * factorial(0)))
    . b3 [5 ^( T0 d1 K3 * (2 * (1 * 1)) = 6/ l1 X: V+ q+ q4 H& H

    ' ~: D8 ?2 c# D9 `, v 递归思维要点: I  w  j) a: }- ^. B4 ]
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑8 @1 z3 U" j7 S9 Y# L; s
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)  g) J7 s; U( f# `2 ~1 ?
    3. **递推过程**:不断向下分解问题(递)
    8 p$ H: n+ ]" p9 b6 z4. **回溯过程**:组合子问题结果返回(归)
      r' z- \" _3 G! q
    ) z4 |3 J; m, E# Z2 k 注意事项7 b6 }: v5 N) k4 d* u
    必须要有终止条件
    3 w% T8 @2 ]% Y递归深度过大可能导致栈溢出(Python默认递归深度约1000层), E2 m0 J1 K7 o
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    4 B$ z1 g+ g( @4 F8 W7 }* ~5 [尾递归优化可以提升效率(但Python不支持)
    3 o3 T& `6 q- ^/ g. ?, }7 P& H/ ]# d3 `' H0 u' \4 C+ o
    递归 vs 迭代
      v/ m" S- Z+ u3 i% X2 l& M|          | 递归                          | 迭代               |
    6 D, i/ G, k$ K7 W|----------|-----------------------------|------------------|
    : O" ^5 T8 S& c: W. L' a2 ~' C| 实现方式    | 函数自调用                        | 循环结构            |
    7 x. `% b3 s  [# C| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    4 A3 @- _( H  y" r/ G# Q; f8 x| 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         |9 n8 c' k* c- N) x. t& ]
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    , {8 j2 K' s, b, Q3 @( b$ H* D4 E5 E# W: U2 w' `0 t* j
    经典递归应用场景
    / |2 u5 Q( D- w4 {5 n1. 文件系统遍历(目录树结构)
    ) |4 e( T. n6 E2. 快速排序/归并排序算法+ E3 w& z) a3 m: `
    3. 汉诺塔问题" k8 O7 C3 \6 t9 S( p: D
    4. 二叉树遍历(前序/中序/后序)/ m! E, E+ z3 q: t" ]. {
    5. 生成所有可能的组合(回溯算法), v  d0 R1 G  x/ F( C

    # S) h  z& p$ S! ]% l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 08:51
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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,
    & L2 V6 e1 _$ @2 @3 T  b/ t; A7 O我推理机的核心算法应该是二叉树遍历的变种。$ P$ \, |* h; E% L# z
    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:& ?) D* G% Q' `; ^9 G
    Key Idea of Recursion
    # N% O0 j7 e- s7 Q2 O" E8 P8 |
    % S0 o  O; M8 o/ V) V, jA recursive function solves a problem by:
    4 `. {& j7 X8 K: @7 c% s" C* Y, Y
    6 H8 h; B% I0 w/ U    Breaking the problem into smaller instances of the same problem.) J  \. P+ t- l: h& ?. ~9 k7 {
      J2 B6 E7 V! X
        Solving the smallest instance directly (base case).
    , n" n; [4 w' }: W, I) I* ]. F% H+ t  F5 Y. `9 K6 V
        Combining the results of smaller instances to solve the larger problem.& A$ S2 w# ~( `% n4 q7 c* A0 C

    + }# X6 n6 w& M3 \' O( UComponents of a Recursive Function
    5 v& Y8 Q- g" f" _' f0 N  h  d6 ^4 h
        Base Case:+ X! m! @* D( }# g3 W7 a$ {* \6 [
    ! `1 ~& W0 u  {
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    9 X3 S* r  B4 y: ~- S2 V/ q
    1 g: q4 u' V5 o: F, z# q' \: C        It acts as the stopping condition to prevent infinite recursion.( m7 X* Q0 k: S; |; B6 U; O% B
    % B* \) F) N1 O
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.6 ~# J) _' s8 n6 }! x- t) o
    % L& }" p+ B" r0 o" N& o9 C$ K& z
        Recursive Case:
    % }8 h# {+ |- Y. m/ }+ x' t0 b, Q, ~2 {! |
            This is where the function calls itself with a smaller or simpler version of the problem.6 h) r2 X* o; o
    * _) T* y9 M, P* D* u% N9 r
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).  u; w3 q- U0 e$ ]7 ^9 W. K/ h: c

    - j1 g8 K4 N4 i8 y7 w) R) v3 OExample: Factorial Calculation4 a. D+ ]& Q$ v

    7 y2 N/ Y* P; p% NThe 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:
    2 q4 c5 L2 `4 O  W- j0 v3 m: e4 k$ H$ `- V  @
        Base case: 0! = 1
    & ~5 ]$ y! N% p3 m; l* w% B( {2 _9 ^6 C- W+ M" G, W7 W8 m: K
        Recursive case: n! = n * (n-1)!
    / Z1 \  p4 p1 W) Q* Q; R, p' f" U1 R
    * n7 _4 B! f3 I/ D; Y3 F( S' t8 GHere’s how it looks in code (Python):) z' A& Y) c1 ?6 A. o  U, W
    python( \& D" z9 \5 Y5 v( u0 w2 Q5 A

    # A4 |3 o+ D# {/ S. j
    1 Y3 P+ b1 _% g* {/ wdef factorial(n):
    * a2 V! Q1 P4 L! w    # Base case
    : ^# s" h0 `+ Z9 z1 M2 ?    if n == 0:) e+ A5 d5 U% r% ~1 M
            return 1
    - M" V, h' c, k+ q0 _5 y    # Recursive case
    $ ?6 I8 q" \. ^# J  x4 j    else:1 d0 ]8 j& b# M" j& F
            return n * factorial(n - 1)
    3 s& H0 L  M* ]) B/ d2 x0 }# R6 ^8 v5 Y' e5 M
    # Example usage
    ! e% d9 i2 `  \& |/ Pprint(factorial(5))  # Output: 120/ P* p8 C$ n2 H2 ~8 _% ?7 [9 q
    4 }+ M* v/ L  P1 \% z) u
    How Recursion Works3 U) U# l# d1 ^- W" l$ U
    + x& r; J- y+ a) r3 T: g( o- F  Q
        The function keeps calling itself with smaller inputs until it reaches the base case.
    ' D* r& r: {  v. E" a
    ; m, K9 x: K1 o. F    Once the base case is reached, the function starts returning values back up the call stack.2 V. H* `6 v3 d7 {3 R; ?; U7 `
    3 L* \9 f3 O7 ]# P0 F( E# s
        These returned values are combined to produce the final result.
    & ^& z# W; W; n% Y1 @
    2 }& a4 v) ]9 [5 G7 TFor factorial(5):' j4 @; V- c- [% ^& Z# T

    ) c; j- T5 g9 s9 b1 q5 \0 N9 V
    3 X7 j' G7 u! nfactorial(5) = 5 * factorial(4)
    . ~' U8 H  c$ @( h, \9 h! F6 e/ Efactorial(4) = 4 * factorial(3)
    + s9 w1 N  f3 l* [2 F0 w+ @factorial(3) = 3 * factorial(2)* w, s: u, k3 [4 x& ~: Q5 U! e
    factorial(2) = 2 * factorial(1)  r6 g% }7 `9 Z; O2 U
    factorial(1) = 1 * factorial(0)3 J# m+ A0 Q  }
    factorial(0) = 1  # Base case3 R+ T8 l+ ]9 ^! s

    . c1 \) `; C) M+ a7 xThen, the results are combined:% z: @7 I  F$ s4 T8 o

    ; l3 h3 k+ N& f, a
    ( F3 k5 N8 s# {) H) |5 i+ ]$ w2 x/ ffactorial(1) = 1 * 1 = 1
    . _0 j! t# q: pfactorial(2) = 2 * 1 = 2
    ( e$ t# `. j6 O+ _factorial(3) = 3 * 2 = 6- z; ^1 M& L1 h5 _* y: l
    factorial(4) = 4 * 6 = 248 j9 U3 F$ U' f8 @$ |4 j
    factorial(5) = 5 * 24 = 120, j1 a# U: t- W  @8 z8 v4 Y

    ' m; w9 E% Y- I" ~. m% `4 ~Advantages of Recursion
    - [) \+ t. i7 i( W# D- k4 e
    6 Q% u# A% k2 m1 V$ X4 c0 \    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).
    ' N( D- [6 y0 Z6 ^$ V& e: ^* ?9 B4 j- {% }$ j
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    8 _: F2 ]+ g1 }
    ( H) d( }0 D! X- I; C* x$ GDisadvantages of Recursion
    # z; A% D6 b5 l6 C6 ^* ?
    5 K" ^/ m! H, d' p: x* f8 F    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 N3 x* y* p2 f

    0 j! L9 g8 a0 O3 b3 p    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    0 [9 S6 V8 z+ @+ d$ C9 Q
    / x+ s4 X- g1 u; I9 w! f& dWhen to Use Recursion) J# V; d/ \& B3 j( Z+ T
    % K( ]9 l7 {2 |
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).& |8 y! s8 R$ u# e( r1 \/ Q
    ! z5 l1 G: X0 q$ d$ r
        Problems with a clear base case and recursive case.
    0 @5 a7 R3 ]' D3 L4 i# l7 a9 Y/ ?/ P. q( \, o5 d+ x
    Example: Fibonacci Sequence9 @7 j2 N+ P( K, Y" J" L' `# A
    , b* Q- F0 N8 a8 K2 R- J
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    . M2 b) I1 A$ O( }) I; A* [$ M7 t
      d4 P: Z. L5 G# P$ E/ ~1 c, n    Base case: fib(0) = 0, fib(1) = 1/ I) P! R7 W2 ]
    $ K, o4 L2 C. z
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ' S2 @1 J" K3 Z3 B1 T2 T9 x! ]6 C% I
    python
    ) g+ o& R& e, v( S* t! Q' ^9 x" f) z  ~" v( {  ~3 Q, _0 _

    5 h* X; C6 V7 _0 A8 V/ z/ Ndef fibonacci(n):
    6 B6 U! M# _* `    # Base cases% h( u* B% Y5 _% r' u7 W) U
        if n == 0:# X; o! M& E* t8 M# b# ]/ _& W
            return 0
      A: f& p; a% M0 h    elif n == 1:
    ( P, y: t' R) j; s% O2 H        return 1
    4 O& e8 I$ `6 F* O) h' a, t# j  U: }    # Recursive case
    ; U) f3 X+ q+ Y9 d, O0 I$ Y! y    else:  r3 A1 G: R7 b: p3 E
            return fibonacci(n - 1) + fibonacci(n - 2); y/ E& A7 M* U/ g; ]' X
    9 [  F9 R% A" s( q
    # Example usage
    8 c2 k& d8 r' Zprint(fibonacci(6))  # Output: 8
    6 a: ?: R# g" h2 T8 r0 u) P
    , z: \# o. |( I; p/ DTail Recursion) `. J$ G" [" X- ?% d& f4 V

    ' A1 V3 G8 i7 q, ?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).& \" K2 J" l' O1 n  d. X
    : Y! Y+ ^) L5 t7 h/ b9 h3 {
    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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