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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
    + t2 j6 S* [" w' {1 v$ c2 E; N$ \# u4 P  P% J& h
    解释的不错
    ) V5 r- ]6 X* k7 F8 O
    ; o/ Y3 b' z3 ?, A. G6 v; Y6 w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。
    # I8 z/ |5 s4 \. ^9 k; p$ }! {8 E* [0 F0 D
    关键要素
    : M* }" A5 U. v# z! l4 l  ~1. **基线条件(Base Case)**
    . `/ U+ X0 _+ K; b8 J' y- D; M$ [) G   - 递归终止的条件,防止无限循环
    / i5 Z/ z, m- i/ b   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    % S+ m* @7 B/ f' m1 L& ]" X7 v: N
    - U" E6 D# i: `3 x: y2. **递归条件(Recursive Case)**
    ; p6 B% s3 y) g0 ?5 O# ]" t   - 将原问题分解为更小的子问题
    " B# {1 K6 E# N3 h! n, l/ ^/ k   - 例如:n! = n × (n-1)!
    ) v5 z) V) E! u" Z7 {8 d6 g( x
    0 p3 L: r( c# A( s, y 经典示例:计算阶乘
    7 b6 R+ W+ L, [  w$ D, Qpython
    - G( F" l% V5 v. d  Sdef factorial(n):
    # F$ K4 O- B  ]4 I4 }; u( X8 A5 s    if n == 0:        # 基线条件
    2 ~+ A+ U- J* H, g( Z* M( p        return 1
    8 S6 M5 b/ r4 @' k% b    else:             # 递归条件
    5 u" Q8 c. R0 O7 Q# @        return n * factorial(n-1)
    - F9 `6 }& \9 \3 m" a6 q5 y执行过程(以计算 3! 为例):
    / z/ W, W4 L* H$ d0 D7 h0 R1 x" kfactorial(3)
    ' V$ k% u' g# q! d( n+ l3 * factorial(2)( m3 J* g0 h1 B9 `, \/ E5 b- R
    3 * (2 * factorial(1))  U# g+ I* V* i3 W. G
    3 * (2 * (1 * factorial(0))), y$ }( t6 i$ {  j# c9 A& k9 P: k
    3 * (2 * (1 * 1)) = 6
    ! l) }* |2 g" v% H( T( b" U2 x6 O+ D
    递归思维要点
      v7 ~: T" A7 c+ t% M0 A. K1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    1 ^5 L$ @7 t7 e/ y7 U2 ~! H; @2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    - e( q1 E+ p. J  K$ V  T4 `3. **递推过程**:不断向下分解问题(递)
    % p4 H( h$ Z% r5 y8 Z1 \. g3 a4. **回溯过程**:组合子问题结果返回(归)
    2 @6 S* B9 T0 J  S& I
    - y. L6 ^! R6 v, `& N8 i4 w 注意事项9 ]' X; S# T3 g1 V) V# }6 x
    必须要有终止条件/ v$ _/ P8 `& _; E/ k0 w3 `
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)( J, n  l# w' G- N. \
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    4 _( K" d' s5 x  D8 z4 j! o3 K7 p9 c尾递归优化可以提升效率(但Python不支持)
    % N6 }' H# {! h8 M/ p
    ' d& E7 y0 ]8 b/ y$ B 递归 vs 迭代
    & p$ W& A) Z0 N8 q* X|          | 递归                          | 迭代               |
    2 V0 ?8 {& h3 T. G# V2 K7 m|----------|-----------------------------|------------------|
    9 p' G1 a% w! o+ `( m( P| 实现方式    | 函数自调用                        | 循环结构            |9 {' X3 Q# Q' a9 f9 L7 s/ t, ]
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |3 Q* _$ E3 y* ?' F
    | 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 E( w0 j: m6 m$ Z| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |& v) m; U/ {6 z; R5 k) f0 M: a
    ( s# i3 w0 }1 \( t
    经典递归应用场景
    1 Z' _# r5 M2 h: Y8 n+ e1. 文件系统遍历(目录树结构)
    ! V: x( j$ x% h: y1 S) O2. 快速排序/归并排序算法7 [' e4 M0 e# C& `# q
    3. 汉诺塔问题. r! W  x6 p' V
    4. 二叉树遍历(前序/中序/后序)* V: A3 p3 t! y. Z) C; p3 k+ P
    5. 生成所有可能的组合(回溯算法), Z$ J3 C3 N7 `, C

    . a$ N( K+ V7 j8 C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 06:52
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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,
    7 D8 [/ R& H7 D我推理机的核心算法应该是二叉树遍历的变种。
    ( ~  ?2 B" a0 B. M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:
    * r7 o5 o  T& {0 B) yKey Idea of Recursion
    / f' Q$ y# b7 A* ^0 z6 A2 D5 C% L( x: a# J* [
    A recursive function solves a problem by:
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        Breaking the problem into smaller instances of the same problem." k3 b: E' [9 G6 g$ v* U6 h  E

    9 S6 [; Z+ Y# w1 b! C" K    Solving the smallest instance directly (base case).* a6 d. \0 d& k, S3 C8 v
    6 _+ T& ]7 H7 y( I' m1 @
        Combining the results of smaller instances to solve the larger problem.- l# \$ J) T0 J5 d' G" z

    $ ~) I+ n' t  U) a3 c' M( ^Components of a Recursive Function
    . u: y2 f* m* O1 D6 P: P' O* G  T% L) B8 y* ^
        Base Case:
    / |( ^& Z3 _* R3 t# l8 i8 M5 [: q; K* p) C$ S) a! O) \; A- v
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ' k1 D0 [2 l. Q& b
    * D/ _7 V4 R' ^5 Y+ f& z        It acts as the stopping condition to prevent infinite recursion.+ M% E: t8 h9 @; `
    ! Y4 u7 O8 z; E. m, U9 u1 l
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.8 u8 z1 r8 b. ~2 N7 d

    2 U# m5 d* ~7 C  Y. w- |    Recursive Case:7 ?3 Q0 ?& q: J
    8 Q( O: t8 V6 e0 ]2 A
            This is where the function calls itself with a smaller or simpler version of the problem.  e% Q  M8 A6 \& h

    # |0 k# K6 A' Z6 h' D# e6 z/ o        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).2 o) L$ P+ f# A) {2 h
    # ]6 o4 N* Y4 e+ v% G' I" A8 w
    Example: Factorial Calculation% \$ Q; h6 `- G8 T0 X% E5 x5 o0 E
    ) B; k* G% Q; D: j1 W
    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:
    8 d; [. V$ B0 M; ~
    % m, ?$ r& r1 @6 j    Base case: 0! = 1
    5 }. j# x7 w  ~) C' H( v$ g& O3 P! d* d
    0 q! m2 a% P. f! [6 H6 L    Recursive case: n! = n * (n-1)!
    # Q0 @6 M' `+ h6 ?# Y  T, [; F! A# G" b  ]
    Here’s how it looks in code (Python):
    , J% @( z8 l2 H0 K2 K0 W# vpython) O# d/ Z  D& C4 Y8 \
    " [* x- \8 `" X: j7 |4 j
    4 V6 ^8 W$ R+ @" s+ a. R& h
    def factorial(n):8 l+ S8 c$ g# U! K! t& _4 e6 O/ n
        # Base case
    8 I& _5 i# n7 A- x! W9 |    if n == 0:5 D: n7 q9 Z7 c# I
            return 1
    7 ^* B( _$ F4 a$ {& K( S- E    # Recursive case- ^. H3 M0 q) G% k8 n
        else:
      ?4 W* O# A& ^3 l6 v, F        return n * factorial(n - 1)
    ( c: \, e* i* p8 _7 Y$ }' O/ ]! J  X# M  [
    # Example usage
    - i  W% E2 y" ^+ h' ]print(factorial(5))  # Output: 120
    " n- W. x8 V5 l% m1 P6 n: l0 h& M/ N
    How Recursion Works
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    2 y1 Z: m6 ~/ f& d/ W    The function keeps calling itself with smaller inputs until it reaches the base case.6 v/ B2 D& m5 r" R- C& s4 }
    8 W; ]) U& ?5 A* [: s% v! O) y
        Once the base case is reached, the function starts returning values back up the call stack.# z$ a+ S$ u2 o1 x# c

    - w' S6 I' V  d) ?3 n& p    These returned values are combined to produce the final result.
    $ Q8 ^7 y1 x, w8 i3 n
    0 x% B+ p- c* X0 l, qFor factorial(5):; e  H& G" ^- E, u8 B9 W- g/ T

    6 \' @3 a0 J! Q" ^$ l: w& J1 ]  O% e+ {) ]9 V& P" b, W
    factorial(5) = 5 * factorial(4)5 O+ r; ~$ _5 l$ b" R/ ?
    factorial(4) = 4 * factorial(3)& q6 k$ W9 \  H4 q( V% N* n% R' n
    factorial(3) = 3 * factorial(2)0 R' ]' O5 d* A% K* l% Q
    factorial(2) = 2 * factorial(1)' Y0 \( [1 H" R; [; a
    factorial(1) = 1 * factorial(0)6 X  x* C& _' H& x/ r
    factorial(0) = 1  # Base case' T& \% Y4 n: g* P" X6 G. S. M
    ( L, W; E8 t7 L* e; @3 n
    Then, the results are combined:
    7 V6 L2 k! I  [5 A& Y( N/ W; [& u- P, E& f$ |
    4 x) C( d2 S2 W; \6 g. S0 }
    factorial(1) = 1 * 1 = 14 @. D3 h: s$ R' S
    factorial(2) = 2 * 1 = 2
    # m$ q8 ]8 ]' L8 e; j) d) Hfactorial(3) = 3 * 2 = 6
    / l. A- v, `" e) ]factorial(4) = 4 * 6 = 24' X, Z" ^3 q! t( W, e: t# V
    factorial(5) = 5 * 24 = 120
    * Z+ G6 \/ n) G: K: o  c9 G% ^" L1 c/ ^* c
    Advantages of Recursion
    5 n# c; A: u9 R3 D2 P( f+ w7 _) d. t4 A; M- M
        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).& l. R2 v; H: l" l
    $ N6 q4 ~2 T3 k/ q2 p" j5 U
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
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    Disadvantages of Recursion; p0 H$ h+ H2 d; e0 o1 j
    5 k4 b) q! }# ^6 ?, d
        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.
    " ?9 I  g; P' ^+ O* n  T* Z
    4 R  j2 U3 a/ s2 G    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).4 x' V4 m/ A& P( H
    ! r1 K0 e, ?& j/ N
    When to Use Recursion
    " }% z; t  ]4 p; w
    # C' o# I: U' ], G5 U    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).' u4 i- v* J8 q) X
    : ]+ f9 J; h* t7 ]4 X
        Problems with a clear base case and recursive case.
      a( ]( U/ t+ c# O5 @
    - c) u% Y, }5 @0 k9 i" `5 fExample: Fibonacci Sequence
    / L* d. L+ m* d  K6 |! v
    , H; n. [5 b1 s& CThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:9 p7 z: ?1 a/ f' y: d/ l" I9 |. I
    ; ?5 I1 ~1 Q  b
        Base case: fib(0) = 0, fib(1) = 1: |/ j( A( o/ H" b5 `
    9 E0 f3 K* @) ?1 z" p7 y
        Recursive case: fib(n) = fib(n-1) + fib(n-2)' c# P% S+ p0 C# Z
    $ `9 ?7 f% C- p5 Q+ b
    python1 r2 I4 A" H0 |# A# _; _
      E& y( S9 f. V' T% _$ M" z0 `
    " c2 c+ h( k( G1 O& q2 K
    def fibonacci(n):
    ; b3 ~& l3 s% ?% y8 g! H& `    # Base cases
    4 j  e7 Z: l% T/ N0 w8 G    if n == 0:
    ' i" o5 ^0 E: D5 L        return 0  V6 T& c  f* u* n0 M9 y
        elif n == 1:
    ; V9 b5 J0 F3 e4 C' C! a        return 1
    - f! t% X1 ]. g$ F    # Recursive case
    3 l: n! c3 {4 D. k/ ]    else:
    . n' E/ I3 |, D' ]        return fibonacci(n - 1) + fibonacci(n - 2)/ T, s  p: M8 ^0 x9 S5 B! Z2 o) E
    - g( F" S" p0 g7 m5 [1 Y; W7 ]
    # Example usage% u8 l0 B' M7 ]3 o8 S3 D. t
    print(fibonacci(6))  # Output: 8
    / ~6 ~# n+ B. j* Q4 F. W3 Y7 ?) J) U2 U7 U% E; I: y' A) I
    Tail Recursion
    , w" {6 _- I$ I, @# I+ @" O: x2 A; h& x  F: z; ]
    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).
    - N' x$ q4 w) ^
    . }) D2 h5 x2 H5 eIn 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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