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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 & j& M9 u( L- `. q. P2 s6 S

    $ b1 o& x9 s# g2 e解释的不错! M) E2 B& Z/ X: \6 J9 A. v- _
    8 \' u$ B5 n0 B+ u2 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。
    ( J& _! t3 P' y4 x
    3 {& Y  q* N( O: H6 M 关键要素
    - W2 X+ n/ Y/ S6 ?, w1. **基线条件(Base Case)**0 @; l3 q; |- m
       - 递归终止的条件,防止无限循环3 E& F( N4 M9 n* ?; v8 [! x/ l0 }4 [3 G
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    0 A# G- k; @8 l4 M
    # {% h; ?& C7 M% r7 ~2. **递归条件(Recursive Case)**: c! u0 X3 }+ z3 x' z8 S
       - 将原问题分解为更小的子问题& K$ i# u: D* h1 P# h" h
       - 例如:n! = n × (n-1)!% i$ ]# O8 |' b- V" }1 a; T8 x
    ! \2 k! |& j0 T' ]: i
    经典示例:计算阶乘
    * p7 A: F" n$ Z' X+ T4 \8 l+ Spython
    ) B7 V9 e. D8 Xdef factorial(n):6 K7 Y% m) F. ]: ^" r6 u
        if n == 0:        # 基线条件
    " y" V0 @5 ?  a        return 1
    0 ]- T1 {$ E9 U# f. Z; e" l( s  K    else:             # 递归条件
    2 A$ {( [' v; R2 h& C        return n * factorial(n-1)' D) S) F' U/ j3 e  o
    执行过程(以计算 3! 为例):
    / i; m* s. n2 M' L% G) E! k. z# p* R0 i$ dfactorial(3)
    9 _5 E& n+ z" I' h3 * factorial(2)
    " W7 k/ T$ }& [/ `: n9 |1 j3 * (2 * factorial(1))* G% K) e" b0 a9 G3 P
    3 * (2 * (1 * factorial(0)))
    0 d' W6 P1 Y- d3 * (2 * (1 * 1)) = 61 o/ a; B5 h/ U3 U4 m  o
    4 [6 Q& j- D# }! o
    递归思维要点
    4 Y5 n; p, {9 J* _3 K1. **信任递归**:假设子问题已经解决,专注当前层逻辑1 h: _4 m" N/ y
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    8 O5 L* s' `- V7 B& z) E  [3 F0 q3. **递推过程**:不断向下分解问题(递). |$ F  C$ \2 J- C" k  L
    4. **回溯过程**:组合子问题结果返回(归)
    ; f7 E8 W8 b* t( ~/ W+ n; r- m1 n- O& o/ q- d3 _0 Z+ J" l) Z
    注意事项
    3 r) K4 ~  l2 m必须要有终止条件
    8 L+ r' o/ Y9 i8 |& h( U/ g( P递归深度过大可能导致栈溢出(Python默认递归深度约1000层)% k  X* m& v, }( `1 V
    某些问题用递归更直观(如树遍历),但效率可能不如迭代% \, ?8 m  m6 ~3 w+ Q
    尾递归优化可以提升效率(但Python不支持)
    9 S7 j- A. q( j7 l2 V1 p3 s+ w! }- D4 v" _3 ?2 m5 l; J6 g: m1 o
    递归 vs 迭代, m' a* M) t- D/ X+ x
    |          | 递归                          | 迭代               |1 P$ H2 H2 I& t% k3 W( d) h
    |----------|-----------------------------|------------------|; t8 n; W" }! w9 Y1 N
    | 实现方式    | 函数自调用                        | 循环结构            |
    3 ]2 I# q. I7 g7 n7 C: |6 S| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    & h8 y4 F; C( ~6 S| 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         |
    + p1 @( |! z! @5 C| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |. P, y) u! V+ A# g% y* I
    # N0 {7 M. {4 P0 O1 c/ d; R
    经典递归应用场景5 N  H$ E1 |/ Q7 }3 K5 o3 V+ V' {
    1. 文件系统遍历(目录树结构)
    ! m$ d: K0 x: L" W2. 快速排序/归并排序算法5 \8 }8 V) o% z8 W2 }. C
    3. 汉诺塔问题, }5 f$ L" `6 h. N: p4 C3 R) D- W
    4. 二叉树遍历(前序/中序/后序)
    # Q, s. K0 e8 @# ]( S3 r" V5. 生成所有可能的组合(回溯算法)
    # }4 W/ T6 |5 h$ M" O. l
    0 q+ {3 D5 W# ?% H- }/ y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

    4 天前
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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,+ p) T5 A, b% s( N/ X4 m
    我推理机的核心算法应该是二叉树遍历的变种。
    3 M% i" V' B9 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:! _4 G( y% P" |! q9 ]5 n& M
    Key Idea of Recursion& N, e4 [" C3 X/ r: |  E9 N
    ; o6 U  u$ Q0 J
    A recursive function solves a problem by:3 r  P! z2 L* t, v9 E# k: \/ H3 @
    2 Q- H/ a6 R2 X! l1 V" G6 M
        Breaking the problem into smaller instances of the same problem.6 K4 w( g& R4 w# a+ M
    ) ^1 q' ^: E; w
        Solving the smallest instance directly (base case).
    - Y4 l7 ?3 @6 L8 P/ r- |! K/ ~7 B% \: O$ H5 k$ ~: D+ t
        Combining the results of smaller instances to solve the larger problem.2 c; ?5 t; `" @" R# x; p! K
    ) C% S( Z% X7 D7 r/ j; p
    Components of a Recursive Function
    " I& o9 Z4 ]' g
    ( F; J% D# M5 s5 U& \- ?* N    Base Case:
    ' V: i" F* [* M* I$ c' g- |
    5 n5 X7 M2 M7 m% z& ^: @        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ( x  ?. x0 t! a8 F- ]2 i  u4 }1 ^" F9 c( s+ f0 r7 T& }: H
            It acts as the stopping condition to prevent infinite recursion.
    : @9 S; V$ f. v: b; A0 F* ?+ S: j' a( q0 e- i/ J3 ~
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    5 Z% A$ l9 ~4 x5 l+ d+ A( X+ b  N* U3 l# [5 f8 Q/ Q
        Recursive Case:5 j1 \; d7 c2 U* O, H9 q- G

    $ r1 }, ?% @2 [0 A( P, d- i        This is where the function calls itself with a smaller or simpler version of the problem.
    2 A+ n9 S. G- a$ Z' I- g8 n8 ]
    & `1 l7 L' I, e6 u* l0 o" V        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).* J6 }% F/ M5 U+ Q
    ( U' E- x6 z) Q+ j) g2 \* V7 v
    Example: Factorial Calculation: k: P+ w$ \( P6 X- @7 z2 d

    2 d4 p- g6 S5 M5 c6 t! H; UThe 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:
    , b8 G2 L$ }! Z  P- \3 D4 d5 N$ d) b0 |9 N
        Base case: 0! = 1! n) ]+ ~* n, Q$ m+ j& V
    ; _: Y+ p- j6 z) M
        Recursive case: n! = n * (n-1)!
    ' K9 M* f7 X( F$ f9 B3 {, s) h5 Z1 L. m  {0 O" |0 u$ j0 [
    Here’s how it looks in code (Python):, ~8 W, N8 T& L( a
    python
    ) j, F/ U2 p  G5 M
    6 {7 E* M& [9 l/ T# Z2 O
    9 s$ V2 ^9 D+ b- V* fdef factorial(n):
    : F* T8 S3 z2 j' P* |) d    # Base case
      w8 }; M5 D# P$ l) n, v) D' h+ E    if n == 0:6 |4 ?3 `  P5 S) m" Q: @# C
            return 1' }" l! D  H; F1 |* O; A- @) }
        # Recursive case
    % Y3 O; l  M) f# Q0 R8 {) b; }5 s! X8 R    else:
    4 N7 {+ r1 \, K        return n * factorial(n - 1)9 B: J( j4 U7 K4 p# A+ ~
    4 J) ^6 M5 _" E8 J* _5 B
    # Example usage
    4 F3 u) O% ?4 m) }& X. |- g, pprint(factorial(5))  # Output: 120& A$ W8 C  `9 H* |
    : c" B# S# k; T+ D; z* [
    How Recursion Works, Q3 k5 {0 h5 V3 U4 s

    $ s% d+ j# U" R- U    The function keeps calling itself with smaller inputs until it reaches the base case.
    % j) g/ L) u8 O% D
    % ^0 j$ \* _3 G! n0 `6 V    Once the base case is reached, the function starts returning values back up the call stack.* |+ o) L- c" k6 e/ b" l( ~
    7 |8 V& ?8 n9 Q3 N9 A
        These returned values are combined to produce the final result.- `$ d& b" b- ~; V

    0 g' s" k: `9 L2 H9 FFor factorial(5):# j6 I: k1 u" w, D2 m' E; [

    ( f# j9 e' d$ f' K: u: y9 P* C. q7 G- N$ z+ y7 X
    factorial(5) = 5 * factorial(4)6 _# T' W2 E7 x9 R. m: i, s2 R
    factorial(4) = 4 * factorial(3)
    % N! Q7 E6 x; [% K5 rfactorial(3) = 3 * factorial(2)
    2 M5 a4 F$ Q/ I# Q2 }factorial(2) = 2 * factorial(1)
    6 Q$ W3 p) U8 Y% o' I! Ufactorial(1) = 1 * factorial(0)
    9 |+ v4 d! o7 y6 @factorial(0) = 1  # Base case
    " C! W- r4 k8 K" x( u) n& ]/ `+ P& t+ \7 x( j+ r
    Then, the results are combined:3 q3 K7 S+ ^+ `/ b8 F

    . z( a$ \: A3 K1 O' u5 j
    4 c  y6 n* N7 @& _0 E& g# q% ^factorial(1) = 1 * 1 = 1
    ; R, r* F  z3 B/ I  v1 M/ tfactorial(2) = 2 * 1 = 2( Y) a1 T" i: q- a4 W
    factorial(3) = 3 * 2 = 6- b* f# N  U& y+ G, l( R
    factorial(4) = 4 * 6 = 24
    ; p: ~  m. D8 _% z6 b  lfactorial(5) = 5 * 24 = 120$ Z8 V- d) G9 g8 ]
    1 K5 r, O( \5 q
    Advantages of Recursion: R" |( `  S* M% g0 n

    ( d# a+ D8 q2 J" W. 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).7 a; B& ]5 `# w

    ' R$ S2 w: S( `2 I8 N9 V    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    & ^" N/ Y. k+ t$ i  }, x0 @
    / \& P/ ~& p7 t1 o7 r" gDisadvantages of Recursion
    8 W  m, v% K' c5 B) ^/ i) r' u4 d! T7 |) _& 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.
    - V3 B$ J3 o' `2 v& `3 }* L- f& [
    $ ^$ i( n; a: u, b. A# f% j    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    4 ^. t; y( c! q. B' h- e& Q( W/ K2 _( b; T! W4 `
    When to Use Recursion/ x8 M7 K; s; {! v6 Z' c: J

    ! e( ?7 X& L. p" p    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).& a" [8 R1 _4 {  f7 N. [& _

    $ o. \5 ^5 l: |# j* [    Problems with a clear base case and recursive case.
    # m$ k/ S) _, n
    " \6 W1 F# {5 w$ x7 \Example: Fibonacci Sequence
    % F) z4 Q3 ]5 R- Y4 }+ T
    . T0 J! a, Z3 P$ A5 J& DThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    , p1 y/ W, g  y& n# U% X) I# h  f+ D* g: q& E
        Base case: fib(0) = 0, fib(1) = 1$ s' r! }0 |2 o

    4 k( g/ k& W0 X$ V) V! |1 |- N    Recursive case: fib(n) = fib(n-1) + fib(n-2)- v* k4 I+ W: k
    " J, N' r2 @; y- t& Z, P
    python
    0 [8 N# v$ n% S  }: `# g& h
    7 o% v5 ^5 M9 e0 I2 U# z% q, z% a1 ]! D2 r2 n* v0 D
    def fibonacci(n):7 c2 I( i7 Y2 T" P
        # Base cases
    8 R4 P4 ]5 I6 t4 P9 f. [6 }    if n == 0:, O$ j% @, X( S7 N
            return 00 ~0 Q  u% u7 K: j
        elif n == 1:
      R8 @3 q; c' [0 u* z+ Z        return 10 f- q% P& m1 F+ X* V- Z
        # Recursive case
    $ }4 G2 N, z2 @, c  ]- }    else:, F0 P- U  P! [  E( E0 G
            return fibonacci(n - 1) + fibonacci(n - 2)
    ( i( B% y4 m. J- m* T4 M, o; o1 C' J9 _* G3 e
    # Example usage8 f4 N) \! f, l; O1 W: t6 U
    print(fibonacci(6))  # Output: 8
    5 r, f3 j# T" N" i) ^! b' T# [# P( W
    Tail Recursion
    ; d, J' Z' S4 ]
    2 L; }- @7 i& N, z9 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).
    " B! V) R  d+ }& ?  l, t) e  C5 ^9 s. n8 N+ j9 T
    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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