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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
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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
    , p" X% Y0 z, T% Y, v# |) B$ S$ y* N+ \+ x' {/ M, K
    解释的不错: W7 L6 I. B- D/ H" `8 H

    9 @  e( `1 U$ c' z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。
    2 t; V5 f, r7 B* h2 V, `: A3 p) @- h
    关键要素
    ( U5 l/ B" m& h0 P  l" P7 \8 W: q' Q1. **基线条件(Base Case)**
    . M- P& m( |9 s7 n3 X2 W; a* q   - 递归终止的条件,防止无限循环" {: _2 q, r/ u! w
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    3 ^2 `+ Q# Q4 D5 l% d/ x- j# V' O7 d9 a: z
    2. **递归条件(Recursive Case)**
    " V3 w, a  ]. E- P  C   - 将原问题分解为更小的子问题/ S# G& ]+ ~  K" s/ B2 G5 T
       - 例如:n! = n × (n-1)!+ t# }6 y/ ]8 B. w

    - c; X& q2 P4 d' } 经典示例:计算阶乘! Z5 j2 v- S% N9 z. ~
    python7 O4 c$ `8 B0 C0 J$ K* U+ b
    def factorial(n):
    ' v2 r% v* K* z- t0 v    if n == 0:        # 基线条件3 L1 ^1 M! K4 O
            return 19 p/ ]; q% Z7 n- V0 x
        else:             # 递归条件
    - t. S8 ^5 x1 q0 z+ |+ f( r        return n * factorial(n-1)
    - h! X/ R/ n# c' W6 Z执行过程(以计算 3! 为例):
    ; e1 P( G# e4 `8 B0 |- o7 O# ufactorial(3)* X  r5 M6 K9 b4 M1 ~' R
    3 * factorial(2)
    , j6 @- `! J3 x# e3 * (2 * factorial(1))
    ' N6 q0 y3 n1 \1 ~2 @* G3 * (2 * (1 * factorial(0)))" m/ ^+ p+ u# g$ ?$ {
    3 * (2 * (1 * 1)) = 6
    " Y' S3 S; M. V; ^& `
    3 y, b6 E2 x1 B1 r/ h$ y) b 递归思维要点
    ) s; Y! A/ R. e9 S1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ! K$ i& x3 R7 C, a3 d( L8 X2. **栈结构**:每次调用都会创建新的栈帧(内存空间)5 d$ i, T( X# `$ m6 l2 E
    3. **递推过程**:不断向下分解问题(递)
    ) ?/ a- k- l6 Y. u4. **回溯过程**:组合子问题结果返回(归)9 U; P  U9 o0 y7 g0 A& u" D  ?0 u4 [+ v

    ( D  }! I. v: m# f* L 注意事项
    3 N, f$ C4 a6 K: q8 \6 ^必须要有终止条件: L7 {" ~) r+ K* l
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    6 D& I) @+ U( \, ]8 l" j某些问题用递归更直观(如树遍历),但效率可能不如迭代' T' g5 J# D  r$ W( j
    尾递归优化可以提升效率(但Python不支持)
    0 E4 B& j; J; j! H6 s8 ]$ n% [3 Q9 y- x9 m' k4 n
    递归 vs 迭代
    ' e& O) Y7 B. A9 M9 d|          | 递归                          | 迭代               |
    ( t4 F! Y/ D" _/ g' z/ n( D7 s|----------|-----------------------------|------------------|
    5 N! B6 v. H+ O: U1 j/ k/ B| 实现方式    | 函数自调用                        | 循环结构            |; [4 D4 K. U7 J# p: y) Q# r5 e5 e3 Q
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |( R8 N. f$ }7 M2 B
    | 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         |3 i% ^# I: I$ @
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
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    / k9 e. Q9 i! k( ]% N; \! \3 R5 G 经典递归应用场景! A1 d( i% P/ U8 Q# }7 N
    1. 文件系统遍历(目录树结构)
    1 D% L& M+ W2 Z2. 快速排序/归并排序算法
    ! ]* l* ^% ~% v+ V1 _3. 汉诺塔问题- V  H4 I! ]) u! M0 T0 t6 c& j2 Q7 Z
    4. 二叉树遍历(前序/中序/后序)
    & ~' @5 T' B! f7 m  ]5 Y$ P  u5. 生成所有可能的组合(回溯算法)+ |, _6 L( Q! j4 v% R) b
      E) r0 M8 F+ j
    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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    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,8 [$ f; C) C. ]
    我推理机的核心算法应该是二叉树遍历的变种。
    " P, A/ x9 C0 @' R' w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:. I2 M, ]2 N! `' j3 s
    Key Idea of Recursion1 c$ o2 E# W/ U$ O
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    A recursive function solves a problem by:4 d3 u. t: q; J% ^& M

    3 R: B3 Q& m/ `  L8 p) Z    Breaking the problem into smaller instances of the same problem.( Z# m! r5 X4 t

    : m1 z: R( K# w3 u' N* M' f" ~2 c    Solving the smallest instance directly (base case).
    ( C2 r/ Q8 P" V1 h  f
    4 H3 l% |0 ?/ E3 N+ f4 ?    Combining the results of smaller instances to solve the larger problem.6 M0 Q) Y' o5 B7 e- L6 E
    % i0 Y/ `4 e3 P0 K/ g. N
    Components of a Recursive Function
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    % X- G0 d& G" b' C  n. {4 [9 x    Base Case:4 p* z- u+ F, t. v

    8 b: ^4 [  x8 p) K7 f- K9 t: }8 H        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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    ) m% ^0 }9 k7 v' @        It acts as the stopping condition to prevent infinite recursion.
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.+ {6 N; s% v3 ^6 t4 O; `- r# d- E

    6 r" ?4 O% P$ C. w  B    Recursive Case:
    ; c; X- ?: K/ }* y' s0 K
    , }  ]1 E( R. X6 h$ G3 A        This is where the function calls itself with a smaller or simpler version of the problem.1 b& j$ h* J2 f; s) r1 k
      T7 C0 H' f2 V5 g
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).; z+ x( ~; M% J; y
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    Example: Factorial Calculation
    1 ]- O% F  N: C! X$ `+ s6 l! S. H
    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:9 r* M5 V: c/ s. {. k$ q
    + g6 m! e0 `8 R- z: w& C
        Base case: 0! = 1
    % ^! F7 w; X# [! ^  {7 i/ E
    " j0 e9 Q6 t& Y& D# a4 l  C9 x    Recursive case: n! = n * (n-1)!
    & N8 s; a, P* I* f
    0 K+ W% Y0 B  c4 H# l+ ZHere’s how it looks in code (Python):
    7 k2 |/ l( l1 l: }5 A8 Opython! X4 Z; @. o' u, o0 O
    0 z$ T: _: F0 r
    ; e7 }- V8 O3 j& L" M
    def factorial(n):
    # O# X8 p# R7 A( j& v    # Base case
    , v/ g' P' }: Q& r, V6 P7 v# D    if n == 0:
    4 `, O3 g0 \2 E0 X        return 1
    / N# D: {8 i2 B% R4 K2 V    # Recursive case
      ^5 q  b/ r1 g+ C. T* T    else:/ _6 P7 b5 p% ^- h% G
            return n * factorial(n - 1)
    ( I! k- s  S' G+ C2 S
    0 W8 T: U0 l. n( J: |! v" `6 o# Example usage! x$ ~8 T, ~0 o5 x! h+ _
    print(factorial(5))  # Output: 120
    3 S( ^4 x7 u. B0 H; c/ V( p7 p& i) T3 r" [& J
    How Recursion Works
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        The function keeps calling itself with smaller inputs until it reaches the base case.
    ! O" j- o7 l: z0 B% d
    ! u- v' e0 r* M& j! O4 |! \0 _# A    Once the base case is reached, the function starts returning values back up the call stack.+ ]/ S$ V& M1 E" l) _0 a
      g$ f% K0 q# @6 F8 i) s5 z
        These returned values are combined to produce the final result.# `2 ]2 i" C( ?. t8 [

    - f: z7 |1 P4 J- _7 nFor factorial(5):1 e5 q: P& d: r3 ?/ H$ x% b
    . q7 U0 \) y+ D7 N3 S+ O
    . j4 J! n( Q5 K& O+ u' I4 r9 p8 c
    factorial(5) = 5 * factorial(4)
    " \: c5 W# E5 s+ w0 ?5 Q0 Ufactorial(4) = 4 * factorial(3)
    : y# G  n& w) I* {! y# Z$ vfactorial(3) = 3 * factorial(2)
    " S3 ~* ~  b0 U" n' u& ]; sfactorial(2) = 2 * factorial(1)4 l; \% @5 u' F* ?
    factorial(1) = 1 * factorial(0)3 @" _) [9 I# j. @
    factorial(0) = 1  # Base case
    $ U  p) l/ {# X. Q6 o! a
    + U8 K4 W3 E1 _% [Then, the results are combined:
    * U  d% o. u9 q9 H9 m) G5 c+ I: G" m# Y9 D9 V' R; _% z: a2 Q$ w2 y
    ( f; R) e3 h" R# ?5 T4 s
    factorial(1) = 1 * 1 = 1
    ( k2 ?) _6 m' h- }' K, \" efactorial(2) = 2 * 1 = 2. ]% `4 F% y0 t: G
    factorial(3) = 3 * 2 = 6
    : v! K+ T0 M' q& J, V3 f3 Ifactorial(4) = 4 * 6 = 248 |3 C, e' I8 r  @# F$ A, k+ ^
    factorial(5) = 5 * 24 = 120
    $ m) w" p! Q8 i3 `4 Z" p) X" G+ z6 Y  n( y2 r
    Advantages of Recursion* M; _. s' L  ^* _

    ; x/ s- e( s' ^- E  {' 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).. b* b# Y9 U( V

    , k, s0 x) |$ V2 ]1 ]( |* g: \    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    0 g: A% }" F% s; c, F" D6 m1 U
    2 A4 k  I. @9 I# c# X- P0 |0 pDisadvantages of Recursion
    # o" G$ w0 g$ c: g/ K
    5 C. P; P8 g; _' ~% x8 F3 V    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.
    / J* w: @, n' v, h! }% q% H& u) D+ H7 e
    : |# D% w; r# Y8 |$ k/ Q  V7 h% G    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    , V6 I* l: n" K8 [( a6 V
    ' c1 W6 l4 c1 uWhen to Use Recursion" {! N" P+ o8 P& f! @
    $ e( S8 r3 d# Q& o- f9 p$ I1 N
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ! V+ V3 Q2 E8 T# G
    5 t  f) D5 s, F( g* w) V    Problems with a clear base case and recursive case.. o. x5 O2 V& a

    1 |6 l6 e: G- A$ ?Example: Fibonacci Sequence
    , X; g) Z0 ~  W$ |+ M) Q! }
    : M+ a$ q; F; pThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    3 `1 `# [# A% C+ H9 n, E; Z: N' g' x0 L" q6 B0 X
        Base case: fib(0) = 0, fib(1) = 1
    + E! d  E: e* W& E) s2 X" D5 u/ z& E5 V9 o: ^$ l
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    ) ^9 z: h. G. R+ E; p2 jpython
    7 J- y# V( o# h
    7 e& E# a5 _' ~5 E; K) X" x; G& R' h3 Q5 \, ^+ u
    def fibonacci(n):7 M& c8 A7 C5 d/ u" b2 c1 O
        # Base cases
    9 x. }& b7 L8 x" W- D/ {' Y' p6 t( |. a    if n == 0:4 }  M! q0 l8 @  s
            return 0# E5 N- M, O/ m' s4 C1 e
        elif n == 1:
    + M/ x$ V6 X, p/ e& q        return 1
    3 F- L1 v; `9 M' Q  V    # Recursive case- H( d7 L7 k1 F$ u% J  V
        else:
    , G) V* v( `; j1 z+ c7 L        return fibonacci(n - 1) + fibonacci(n - 2)
    ) J. N; N; w7 ]/ k  ^( T- p3 ~
    2 r, h4 E9 f' j) `) ~8 S; j1 v3 e# Example usage
    9 Z- s+ d" F: F) Z8 H( Vprint(fibonacci(6))  # Output: 8! Q6 P6 T& g3 @1 K& z. C

    % U% T+ M8 i, D7 x8 D$ B. z6 NTail Recursion$ ~0 b2 t. m. B! R

    # Q# H9 }  O. W; |. JTail 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).+ x0 T) y. p& K7 a$ T0 s6 A
    ; Z( ]8 `8 P6 W
    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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