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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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    2015-11-30 11:11
  • 签到天数: 2 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 - e  _& B- A8 W0 F3 g' x1 y
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    解释的不错
    2 o! d. B6 q0 P3 ^9 o) y; b$ |
    % r5 Q$ V/ n7 x" a' q' _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。
    1 i3 u& c8 v5 U6 l; \7 c9 [  A# s; T: S
    关键要素
    # N6 x+ O1 k$ j0 a5 F, K; Z8 Z1. **基线条件(Base Case)**: S; r9 D; f* t' Y  M1 S0 O/ s) T# P
       - 递归终止的条件,防止无限循环1 Z8 P$ A, f6 ~: y+ m+ t) v" C
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1+ w* G: \4 H2 W4 }
    - Z; H; i( X2 u" w$ |9 _
    2. **递归条件(Recursive Case)**& x( S3 j$ b  R7 {+ F. L4 V0 g5 Y
       - 将原问题分解为更小的子问题
    ) M7 c9 v0 t2 m5 g/ {* v# f   - 例如:n! = n × (n-1)!
    % J) K6 b: F3 y4 m$ {/ F' ?3 C( l9 U9 [, @
    经典示例:计算阶乘, z& a$ M5 ~. F2 \. h* q# B
    python6 l! z7 w9 P  p
    def factorial(n):
    9 p$ R9 ?0 K; M2 ~0 T4 a    if n == 0:        # 基线条件& R3 }& M; D& u( S$ O  i
            return 1) b' W* u) T4 F: H3 [( b2 b
        else:             # 递归条件* u4 T( c6 U$ y' d) ^) B
            return n * factorial(n-1)# p* ^! }4 \; R$ M" T$ U  [) r
    执行过程(以计算 3! 为例):! f! v$ v$ C; {/ e, R
    factorial(3)4 e3 N# V& A8 ?& }. o, q
    3 * factorial(2)
    & J- l3 p3 y6 v$ ~8 \3 * (2 * factorial(1))
    8 _* V% _% |& t/ }3 * (2 * (1 * factorial(0)))  @1 C% _3 W. Y/ s, k: c2 ]
    3 * (2 * (1 * 1)) = 6& F( }( _  x& i0 V) x

    - A5 P( }2 n0 n/ O0 K 递归思维要点% q2 I( I( u, R- E3 l( R1 E
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑7 O0 b8 u" u5 E9 |
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    % Q1 {' @) W' M3. **递推过程**:不断向下分解问题(递)* p/ D: E; k8 h! y+ t" P  V9 O
    4. **回溯过程**:组合子问题结果返回(归)6 S! L% s3 M, b' H
    7 i( w$ s7 Z/ U$ v. v4 A
    注意事项% x; @" y/ C. h* |5 O7 h( M/ I
    必须要有终止条件, B/ _1 D5 D0 t  P3 _3 p( G, E
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ) n; u! U2 d  g某些问题用递归更直观(如树遍历),但效率可能不如迭代
    : d+ `! V; u5 d! F5 |/ v1 ~' @尾递归优化可以提升效率(但Python不支持)+ b( v# O0 s' ]/ `

    7 u5 [) E8 i; y: G. g 递归 vs 迭代/ R0 z* G7 m' l. H1 e* B
    |          | 递归                          | 迭代               |
      i9 o& V' `6 c, [|----------|-----------------------------|------------------|" k( g: U7 }, ]( D9 `9 [  p
    | 实现方式    | 函数自调用                        | 循环结构            |
    2 t# l& d+ N2 X9 c2 t$ m" s9 U| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ; a9 H+ o; I8 F. 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         |
    ; T" Z+ v/ s" W" m: l| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    1 K; w* k+ N( y# W! K9 @0 X- Q. h" ?/ Y# c, Y' B$ T
    经典递归应用场景
    9 b  H( |8 I) h- P; _3 i, L+ J1. 文件系统遍历(目录树结构)
    ( h: _) O+ P$ A2. 快速排序/归并排序算法. d: _$ D4 n0 p! V9 O7 a! p$ P: P
    3. 汉诺塔问题
    . ^* i* v9 W) c& p* Z5 F4. 二叉树遍历(前序/中序/后序)( A( m5 x! M+ ~% s; _- W
    5. 生成所有可能的组合(回溯算法)
    $ U0 \) a' _0 d6 T
    ' t+ Z9 N* W3 O6 \# O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
    慵懒
    昨天 08:22
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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,
    % b1 ~: s# L6 T; V3 n1 R我推理机的核心算法应该是二叉树遍历的变种。3 s5 w1 a# \! Q) b$ I) t- S
    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:
    ; v. F% P" {5 B1 x  zKey Idea of Recursion# _/ |2 i& h. V9 p1 _5 Y+ r. [
    3 H; r+ r/ k3 |
    A recursive function solves a problem by:
    0 E5 I$ e2 w5 n, f! p4 A7 q  B2 c  V) z; }" G) d2 q4 }) u
        Breaking the problem into smaller instances of the same problem.7 }8 q1 u4 B# Y) q
    5 n5 z% \, W) o) d0 D) e. Z
        Solving the smallest instance directly (base case).
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        Combining the results of smaller instances to solve the larger problem.
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    . q* p* |$ J- g5 c  N9 N( u- ]Components of a Recursive Function4 v# a5 [0 I4 L
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        Base Case:& t& v% G( J! R+ f9 O/ T
    ) P2 |2 \' g) \& E$ `! F5 }
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.* q: a0 z' E* h, O% W9 }! D

    7 g+ |" S0 N  d; E        It acts as the stopping condition to prevent infinite recursion.& o# d/ j  d5 J5 L, \4 A
    : ]  r& P) [5 R" A& [$ n& u# Z
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.  @& c3 v9 _+ J- D% `
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        Recursive Case:1 {' T" C8 O: H# X

    7 R) e; E9 q6 v0 \2 J4 b) E  }& h        This is where the function calls itself with a smaller or simpler version of the problem.
    & p& i4 u4 S# z) |+ b/ x/ D" e/ D  K+ m. U& E; P7 y
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).6 I& v* Q# n* ^+ s
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    Example: Factorial Calculation/ A, T& d# d* v) r% k

    # L" a# r; q7 kThe 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:
    . @  j4 U7 B$ Q! v  ]! R+ F! P( {7 O( m( \9 g
        Base case: 0! = 1& W9 r, K' S! f
    ) t6 r, L; L* G; ^' ^9 t  N
        Recursive case: n! = n * (n-1)!
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    3 ~% Z, ]: w6 x/ D  ~: JHere’s how it looks in code (Python):' J4 M3 M5 m1 I) v0 E
    python; ~- U, \9 Y1 |9 Y8 B( Y1 B
    , m+ `6 x; f1 }; r

    * b: v6 g6 Q4 l$ r: Y  U. ?3 Fdef factorial(n):4 e" S. C& y2 F$ `* t0 m
        # Base case! R# d) I1 H2 S! [' a
        if n == 0:
      m, I' ?! E' S$ k5 r& x' k% |3 \        return 17 c% e- O* Q! u& {% C; B: X
        # Recursive case( K; O4 A- B) g' r/ j$ R
        else:- ~8 G, W9 F8 b# V* G: \
            return n * factorial(n - 1)
    $ Q. ]/ H  u1 p- R/ ]' g+ d2 d' I* R; N# t" E# A
    # Example usage
    3 ?: B, Y( J# d! D1 G" gprint(factorial(5))  # Output: 120
    # i) `; P$ K9 G8 Y
    1 U3 u+ c* m8 l& O3 xHow Recursion Works  ]7 {; J5 R8 @5 h1 V' N

    9 I0 C# {; Y' w    The function keeps calling itself with smaller inputs until it reaches the base case.
    7 ?6 z* n7 C8 m. p" x$ H& t+ h1 q+ S  z. k/ l
        Once the base case is reached, the function starts returning values back up the call stack.
    5 A% h! a. |* ^: S! s
    3 l, t9 r+ n3 y: E% A( E0 C" U    These returned values are combined to produce the final result.( Q0 Q, T9 E* a+ m( Y2 ~2 {' V

    , k% m# v- Z- Q6 Z: {( e" \8 [% JFor factorial(5):5 _3 p) C: k+ P# o' J
    " r2 B0 t' Z( F4 X# Z7 P* G% K" K
    $ L9 s) a; L0 |- z& M
    factorial(5) = 5 * factorial(4)9 D* W/ A4 o* [+ F& d0 n
    factorial(4) = 4 * factorial(3)
    : B( ^4 c+ t1 {3 u# e) L6 A$ o  Yfactorial(3) = 3 * factorial(2)& ?3 p8 Y' B% I8 @7 V6 b
    factorial(2) = 2 * factorial(1)- p0 y: u9 t# F
    factorial(1) = 1 * factorial(0)% r/ J- H6 |$ |5 p
    factorial(0) = 1  # Base case+ Y# \4 r5 f1 k; S$ T6 l( E% B

    ( N3 ~# X5 s. v, u2 ?' @Then, the results are combined:* }; g. g+ Q4 [' `& Z
      \5 d- ]7 l3 ]# [$ h8 b  q% O
    . _; e$ n5 U; c1 c- x+ v5 u
    factorial(1) = 1 * 1 = 15 ]* \  j* H( E0 M
    factorial(2) = 2 * 1 = 28 [, d  L# g8 N+ ^
    factorial(3) = 3 * 2 = 6) n/ U# j# ?, t" @& [+ L
    factorial(4) = 4 * 6 = 24% W. D: A2 L. e; [2 ~. v
    factorial(5) = 5 * 24 = 120
    : B6 r, r% I4 ]# S( |# W7 K! o# }4 o, A; m
    Advantages of Recursion
    / `" q1 Y- v, m1 ]; p" `: R
    9 c! v3 l# x0 g& r    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).
    - O4 P+ N5 p7 t  K; a& H' b9 j
    - N  R1 ~3 Y' r" o. I% w    Readability: Recursive code can be more readable and concise compared to iterative solutions.4 H) A! [0 \. l: }2 G
    : w: W' s  M% i2 v
    Disadvantages of Recursion; m& x" j: f9 l% \* g& T
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        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.
    & k' k- }. f) A3 _/ X+ Z3 |) V" Y4 s6 t) {' B- k* g
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).2 |6 Z4 T4 p$ q- V& D

    3 T$ C2 c! G- C) x( a) QWhen to Use Recursion7 H+ F3 {' X$ Y& y
    - ^7 L: ^' @: S8 R
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).% n. T( A) W7 ?# v1 r& M

    / r4 q  H) x. u6 o% t* K    Problems with a clear base case and recursive case.
      W$ `& k" @8 q1 D' {- [5 v7 a0 V/ [
    Example: Fibonacci Sequence$ C# o2 F$ f  W# z3 K9 ]8 q

    & ?& X& O9 h! VThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    1 H* r4 _/ v% x6 z5 v6 \* G! V( ^8 b% M
        Base case: fib(0) = 0, fib(1) = 1
    8 ?2 o2 a; \- e4 b# k  ]1 k) G  W; n+ l7 Y# ]
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ! F  m% m6 j. G# v+ |3 r0 o  d! d. ]& k% W6 p) a
    python9 F, h3 M. n0 `9 G) H

    . u; T/ [. |. ^1 }$ s- I, D2 I( ^; G% ?7 H  n/ B
    def fibonacci(n):
    . K, G; @+ Y! p4 W7 y    # Base cases4 ]6 ?. l2 p1 @4 ~' K2 h5 d
        if n == 0:
    1 h- [# K% ^3 Z        return 0  R+ L+ L- Y8 _+ D
        elif n == 1:/ @: A# `* G! [
            return 11 `1 c+ b; B: r& p$ e
        # Recursive case
    ' \( P, M3 B( [2 h% n1 A# Z% j    else:
    7 M' A' F3 ]) E1 P+ q+ g8 p        return fibonacci(n - 1) + fibonacci(n - 2)
    ! x+ _6 @- W$ y) W# {' ?% _- f5 J+ H: ^0 w0 _. s6 d; A
    # Example usage$ T& q8 l+ F4 H) L
    print(fibonacci(6))  # Output: 8
    ' t: n* q7 X1 D. b6 K/ h3 A! [4 z7 {- D3 Y& T! `
    Tail Recursion
    * o# v  t1 u; l1 Z. u& E8 l
    + |1 {, X3 J, i. DTail 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).
    * g4 i* j. D2 U5 Z4 ?' R% P4 K) C9 c) ]4 h- P4 O: d
    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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