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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 $ J1 N% B) q2 Z* H$ v" ~$ _

    9 P3 C+ l  A3 ]& V" C解释的不错) B* E  h9 Q. c( j4 @6 M0 Q

      v4 P6 [) V8 {! J5 z$ q7 L/ C: _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。
    % f6 W0 v! d% W5 D
    % y  V, N0 G2 g1 } 关键要素' z0 ?" q& b& K. F" m4 V5 |/ X9 s% G
    1. **基线条件(Base Case)**
    ! K1 o4 ]. Z1 z# t7 \7 ^* j4 f   - 递归终止的条件,防止无限循环3 D% Q. _) L, A
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
      y- F, A- R+ T& ?. O5 u7 \+ p( Z( Z
    / Y1 d% X( I, h! r: j2. **递归条件(Recursive Case)**1 z- w8 s. G- C+ [5 k6 @
       - 将原问题分解为更小的子问题
    : S6 T2 C2 }  X1 Q8 F! i0 \5 v: s. ]   - 例如:n! = n × (n-1)!
    - A, I0 N5 q+ Q' I4 M
    7 R  U0 q  a' q  t& h 经典示例:计算阶乘
    0 n4 G- F+ U- Z9 N1 F7 H0 z# |" @: Opython
    - W4 s$ g/ B  H8 _  [# udef factorial(n):: T5 x) A: w6 z5 _
        if n == 0:        # 基线条件
    3 o+ k0 G6 _6 @& E1 \2 D        return 1
    $ O$ }8 z; {( r8 L    else:             # 递归条件) x+ y; [2 R; |3 J, n- G
            return n * factorial(n-1)+ x  i/ U# s5 s
    执行过程(以计算 3! 为例):
    ( J! k* o) i# r2 @factorial(3)6 w4 }& \  z% \9 n7 W4 C
    3 * factorial(2)% i* b' l& |9 p* J8 _1 n) z8 U
    3 * (2 * factorial(1))
    9 x" z/ e2 S2 ^( z3 * (2 * (1 * factorial(0)))7 M" e( Z9 e, l1 C' B& G
    3 * (2 * (1 * 1)) = 6
    , X+ t9 c+ f* C+ R5 I' _" [# V- ]$ [  o$ \5 D" b- G; |
    递归思维要点* f1 F; R8 u# p
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ) `$ }* e8 G+ V1 a% ^$ U$ n2. **栈结构**:每次调用都会创建新的栈帧(内存空间)4 {$ ~& j0 g& h: U# F, m
    3. **递推过程**:不断向下分解问题(递)
    ( u5 b$ E9 |& m0 M2 L4 g3 j4. **回溯过程**:组合子问题结果返回(归)+ K9 g9 L6 Y, k5 k

    - g2 z+ l( ~3 H# ~' r7 l 注意事项& C8 e7 H0 R& g$ c) ~* T& o8 p, d
    必须要有终止条件$ R8 ]& Y0 b& x# G
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ! `( d1 y, t/ ^) }# M某些问题用递归更直观(如树遍历),但效率可能不如迭代5 f1 U2 h& G! m& f( w, ?3 Z
    尾递归优化可以提升效率(但Python不支持)
    + P; d" K3 T3 a6 Y) L. y  h' Z. o' T6 l6 R0 `
    递归 vs 迭代% |9 e( T" z: g% o+ u) O' G4 e$ l
    |          | 递归                          | 迭代               |# _4 G" k8 K8 B: b5 ?8 q0 X
    |----------|-----------------------------|------------------|. w2 a8 n$ S! f
    | 实现方式    | 函数自调用                        | 循环结构            |
    + x% S$ K( N$ j* I# j) N. ~/ M7 _| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |6 ~- F- u4 T" i& 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         |
    0 z* f- a/ V' y7 x$ \4 ?| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    $ X& F7 Y$ K- K: I& j7 W% Z2 T" }1 t5 q+ m& D) @3 z
    经典递归应用场景
    ; i- L5 Z% O- w' V/ o7 y  q" c; Z1. 文件系统遍历(目录树结构)2 J  o3 z' Y2 n$ R, X
    2. 快速排序/归并排序算法& T  w! u$ C2 L  P
    3. 汉诺塔问题8 W" l! T/ ~- U) s) u, t
    4. 二叉树遍历(前序/中序/后序)% Z. b# _- R$ G: g7 w' p
    5. 生成所有可能的组合(回溯算法)7 B8 P3 P4 B' u" b  a
    ) d; |+ G8 E  u% L' E2 Z
    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 09:41
  • 签到天数: 3168 天

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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 C4 w" {4 P+ t8 L! }5 T  }+ s我推理机的核心算法应该是二叉树遍历的变种。
    , C( ]% ?0 X4 M9 F9 z3 ]9 ]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:
    & c1 v# D2 m- E. k5 D" E4 kKey Idea of Recursion
    2 d8 r" B- B2 z9 c. U& w) {' c& F# x( Z- Y" O' @
    A recursive function solves a problem by:- a& A  y3 F# r

    / u* u* W# `. |. d4 @    Breaking the problem into smaller instances of the same problem.
    / H. ]1 B6 Q# A9 ^5 A1 ]" s# @6 Y; `
        Solving the smallest instance directly (base case).0 _0 h4 a8 j6 a5 V
    4 J7 }+ @7 [* \$ J
        Combining the results of smaller instances to solve the larger problem.# J, a" c9 S, v1 \5 d: B

    8 a2 T- a/ ^- a. b7 b' oComponents of a Recursive Function
    * x( V" O, n. F+ b3 n5 p# C. y8 S# N  T5 V) G8 ^! }( I( g6 ^
        Base Case:) _5 d0 M* B  [$ N7 K

      w- z$ W/ o, I2 t        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.* b8 }0 p. f  W( a
    " E; M+ s: z) ~( k9 y0 ]
            It acts as the stopping condition to prevent infinite recursion.* J$ H" p, y. D; F0 n: c

    ' e  D( K0 @4 A, q/ r; l+ e3 O        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    0 U2 s; \' ]0 G9 e
    ' L$ B; C2 z, ~    Recursive Case:! f' I5 O4 \* z
    $ j) k3 k, B' @$ k; P" I" R/ e: ]
            This is where the function calls itself with a smaller or simpler version of the problem.+ S- }' H9 z: ~( S8 z. v
    ' ^+ o1 A- Z6 p; M
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)." b# e/ K/ F, q5 j; o
    # x: n3 }- W  R& S/ r* R( {
    Example: Factorial Calculation% A& S+ W) \4 l
    ( v: a& X$ K# U9 K
    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:" {7 M% d; Z' C" ?. C/ D' [+ D

    + t: y% r: R4 E    Base case: 0! = 1
    ( c* A3 b7 }1 b: h5 W
    * [9 d+ {" T+ A# G' t0 k3 v& b    Recursive case: n! = n * (n-1)!
    * F- b) N5 w8 V# E% c
    # C( G& }) e% G5 f3 y0 w' ~+ L" JHere’s how it looks in code (Python):
    6 q% V# z( l- k2 B4 Vpython, ]* A( [/ u8 g9 E
      J+ b: e' R  x: i! [% m* ~
    # M; P5 n( d$ U3 g( p9 P
    def factorial(n):
    : _2 \# p+ m1 [    # Base case$ p% h6 @1 S- H8 q3 J
        if n == 0:" T- x" h! Y3 x9 T3 P. v" w
            return 1
    ; c6 O) u7 m6 M$ m, X2 f% o    # Recursive case
    - c! F, @5 m) R/ V# R    else:: N+ b, B" J3 b# Y7 g5 O: O5 Y3 S# x1 _4 l
            return n * factorial(n - 1)% J2 ?2 S- I5 _7 x

    2 D$ m" q8 O  x- ^0 J% @3 x0 b# Example usage
    6 l2 W/ V# {' h" G3 P3 v1 bprint(factorial(5))  # Output: 120: W$ @; e, q0 X/ y

    + d7 Z: i' z9 l5 HHow Recursion Works" w9 @* e9 Y( }2 C: }$ v' ^; V$ R
    ) o7 K9 L* N8 }- R4 D
        The function keeps calling itself with smaller inputs until it reaches the base case.. X% W0 m, V& c1 F8 C

    4 v  O  {' D0 r, @1 i( F    Once the base case is reached, the function starts returning values back up the call stack.
    3 H( H, p/ v/ ]  T& W& K$ N" n  W* z! y- Q- Q- r+ H
        These returned values are combined to produce the final result.
    0 h1 m$ Z) z) X7 x. L3 M
    " A6 o. i! \# d! y" ^  ZFor factorial(5):' h5 ^2 I* p" S/ J. u2 E3 ]
    $ D9 l" E0 A/ }8 o

    2 R$ N( b3 b9 G2 v& d: }6 V. gfactorial(5) = 5 * factorial(4)) [6 k+ B' K4 M" n( b! U+ E/ I$ j
    factorial(4) = 4 * factorial(3)- J  i/ a# b  v. N' J6 D1 r( J8 o
    factorial(3) = 3 * factorial(2)
    7 b' j; k8 N4 [+ Z# Mfactorial(2) = 2 * factorial(1): C7 G& ~) W  G9 A7 f) q1 }
    factorial(1) = 1 * factorial(0)" D2 \5 E7 X0 M: b/ U& T" x) V
    factorial(0) = 1  # Base case" y( f, x& h6 a+ O. J  \

    9 I1 |; \& n& C: @1 uThen, the results are combined:' Q( X- J, z/ q. Q8 z2 q: r
    7 ?0 C; h: b. e* c) k. Y
      d. ?7 J9 ~; F- p1 |3 \
    factorial(1) = 1 * 1 = 1) D# h8 q1 z+ Z  B5 X6 q3 E
    factorial(2) = 2 * 1 = 2% T6 y3 k$ E) M* P" W8 U. E
    factorial(3) = 3 * 2 = 6
    8 G; q* x6 M( ~$ v9 C0 ?factorial(4) = 4 * 6 = 24. _' D+ Z; G" c6 Q  b
    factorial(5) = 5 * 24 = 1209 @( [* L2 L" R9 @- D* _3 {- P
    + c, S% u7 f5 C2 g9 ^9 f
    Advantages of Recursion
    5 D! y/ Y! K( R' a( m& F
    & [  Z: `. `4 Z    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).0 x; P) y5 Q$ `8 l2 ~" N1 Q/ I

    - C& U0 K* r' Q# l7 R    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ! [+ j, h6 o8 p; f, h% ]( r
    ) i: R2 H* B7 O; V& |Disadvantages of Recursion
    " t8 \6 `! K, }! G! N
    1 q  B2 o8 |- p    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.
      |' B/ ]1 Z+ h9 }9 o# N! Y& y7 j( d; Y
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    + y" c* \; n1 _
    ) {6 Z- S( W( n9 JWhen to Use Recursion0 s' n9 g' V+ U' c& B/ {2 A; H) @

    ( C  t  q1 w% Q5 T7 h6 z    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ! W( g8 D* c( q" S1 V" |3 U( q5 ]- y9 V/ W# z
        Problems with a clear base case and recursive case.6 `9 i0 V6 k8 s) u8 R: F
    0 N  ?, X3 ]2 p% K; A7 w
    Example: Fibonacci Sequence( }' d) m3 |( S+ M  R
    4 u+ ^3 s. ^% s9 {
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ; g8 {0 D, C7 L: ]% {2 m! ?6 W& J( g
        Base case: fib(0) = 0, fib(1) = 1- N7 l8 O. ?. D0 }  `. I: I

    + P& Q5 O# o# f; y    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    & g  R/ {: ~* s
    ( d" t' h, m1 T( D* z) K0 R9 mpython: r) ^. Q) ]6 ^# k

    + A6 q; z8 E7 ~) ^
    3 B9 \; Z7 t, ~5 Mdef fibonacci(n):
    / D9 O. A* ^: O& q$ D    # Base cases
    4 P0 |* K1 J5 V- v, T0 i2 [: g    if n == 0:( T. ], t% F; x  w5 H
            return 0( l) s) B( z& F! M2 [
        elif n == 1:
    $ M, _/ [2 I9 B) s$ g! I$ t7 p        return 1
    " J. k2 ]' C3 p- a) r# g; c    # Recursive case
      a, v  K* @6 H6 S    else:
    ' Z' G5 ~5 B! _/ k$ I        return fibonacci(n - 1) + fibonacci(n - 2)6 \7 o/ u% l1 m* f6 w

    ' {% I9 o& c/ d# `# h6 }% D# Example usage
    8 U  Z' @% G6 q# m& gprint(fibonacci(6))  # Output: 8) O/ u3 @1 e, d; @  C9 N
    $ P9 ~$ v0 y7 {' X/ {1 x
    Tail Recursion
    ' q" ^; \+ W2 Q- J: s  E' k; Q6 o8 \2 g# H
    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).
    + R2 Y$ K* I4 r6 n% r  b% w4 ]- q0 |8 v6 {" 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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