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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
    # p0 q- Y  R  Y3 l( l/ i+ C4 I$ \2 o" K. p7 ?( @4 E* D0 s
    解释的不错4 O, j; v) ?5 m5 y8 n" X/ o

    - e6 B& V7 A9 ?/ t  l1 E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。7 r# S# _6 b* b% N7 I
    " L( X% u3 ]% S# i
    关键要素+ H) g; S; x/ [1 Q7 |' _7 f
    1. **基线条件(Base Case)**
    ' k; |* H% S3 D/ A% H   - 递归终止的条件,防止无限循环
    8 }# C* Z" w* i   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    - `. x5 G; Y$ H- S  X9 l
    / |' _6 Q4 C% g( Y2. **递归条件(Recursive Case)**
    " N0 z& o% t: G8 j1 g   - 将原问题分解为更小的子问题8 y! z# a5 X  ?
       - 例如:n! = n × (n-1)!
    + ^4 Q1 Q6 _  l4 M. b6 e+ C- p8 F8 j2 \! y  c' R: s. D
    经典示例:计算阶乘- S% M4 d: C' Y+ \" M! K
    python
    % g1 n7 s  n& x, L; O/ ~def factorial(n):
    ; `! i* N/ ?8 b7 h    if n == 0:        # 基线条件% l. A( S& ^+ |8 Y+ x6 F
            return 18 s- [+ c' h+ @# m6 x% T+ l
        else:             # 递归条件
    6 B' D6 X2 {3 T) h        return n * factorial(n-1)
    % ]; G' O4 ^3 D0 m- _执行过程(以计算 3! 为例):" W# J  O6 B! `" G, ?/ Z
    factorial(3)+ G% x$ g" ?, p! x
    3 * factorial(2)
    " t5 O+ M8 n% c( O/ n7 D3 * (2 * factorial(1))
    6 ~6 t- J# B8 l4 {7 S3 * (2 * (1 * factorial(0)))% }$ K; [4 P8 J- j# @4 j
    3 * (2 * (1 * 1)) = 6
    ! Q5 P5 o9 }" X; `0 q" w/ l5 b$ y  W( r8 R
    递归思维要点
    $ W! M  S7 i( _1. **信任递归**:假设子问题已经解决,专注当前层逻辑$ c3 W3 b! E  b# m: w5 ?' i
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)9 @- o5 l6 X6 ^, y) t5 w0 \4 q6 s
    3. **递推过程**:不断向下分解问题(递)
    ; X: r! ?. A2 j" o9 _4. **回溯过程**:组合子问题结果返回(归)
      B7 X# \3 w0 K9 ?' d6 j$ l8 C$ U  o0 I9 f# g
    注意事项9 p; F( M# r6 p# @) y
    必须要有终止条件0 C& T. o; Q+ u4 e3 R. A% O
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    # y* n, d& O1 A3 H- c3 `- [5 ]某些问题用递归更直观(如树遍历),但效率可能不如迭代
    * I) I' s/ G7 W0 Q# F- O$ w6 t尾递归优化可以提升效率(但Python不支持)
    9 D9 \4 ?9 I: q) r# U4 M) D7 Z: A( G& ?6 X9 T
    递归 vs 迭代# c7 B" ^' N$ r2 w2 W' t: d9 N1 ?
    |          | 递归                          | 迭代               |
    6 \& J/ f9 s6 g7 g! X% X1 t3 G|----------|-----------------------------|------------------|: M# u" f& c- u0 I! [
    | 实现方式    | 函数自调用                        | 循环结构            |
    ' t- z1 w/ M: [8 j( I( [| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    5 ]7 d$ n" H3 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         |) W/ y5 W& h0 [8 W
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |5 K! r/ w* Q( w
    . O: w- L4 k: `) _: `
    经典递归应用场景
    ( J; R( a' p2 z% g1. 文件系统遍历(目录树结构)
    3 `1 R6 d+ n. f3 ?2. 快速排序/归并排序算法6 X9 f  a0 c. X& c3 ?
    3. 汉诺塔问题- s* c2 e0 X4 T8 a5 \' V
    4. 二叉树遍历(前序/中序/后序)
    - V6 M! n% B( S3 g  t5. 生成所有可能的组合(回溯算法)) ~2 Z' |( t+ [7 J% R

    * U+ Z- y& E# T! j+ {  r' B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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    郁闷
    昨天 13:27
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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,
    % m, ?4 X* X* O! M我推理机的核心算法应该是二叉树遍历的变种。
    8 |' T# g: ?  v2 d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:
    1 }7 M3 X6 T, G( n& r0 f5 SKey Idea of Recursion+ G6 E- }; j/ S' l# w

    # m; S4 u9 `# C( \, p' I% ^A recursive function solves a problem by:
    4 @3 X5 Q8 |2 |) O$ ?2 k5 I7 \* n% j) D- D
        Breaking the problem into smaller instances of the same problem.
    ! f' C. w. U; j
    ; ^. L, V+ d+ ?0 F5 S    Solving the smallest instance directly (base case).
    6 e8 ]! z- l3 M4 c% m3 w/ E8 N" a+ @1 J3 U. Y& p9 X
        Combining the results of smaller instances to solve the larger problem.6 Y9 Y" W- M. ~0 @9 G9 d- ^# J

    - e, F1 D) b% c9 q) QComponents of a Recursive Function  e+ [3 x: i; a/ |: u, |$ v
    ( i; v/ E& h! |. @' Q- T* Y/ [
        Base Case:
    3 J8 r! ~  s! j/ l. j& m6 Y3 x, ~/ h1 B
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion./ U# v* P$ b; R& }* H  ^# Y2 \4 m
    ' D$ B( K- l- p* p
            It acts as the stopping condition to prevent infinite recursion.' Z1 {7 J% {0 M! ]( W/ M

    3 W3 `' {4 a8 ~+ A        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    * @& ~/ }3 q7 B3 ~* |. a$ K, m9 M9 G8 k$ z- Y  u, ?7 z
        Recursive Case:6 B" X" u, t- {9 e

    , A3 r% b+ v4 y4 S- a        This is where the function calls itself with a smaller or simpler version of the problem.; [, v) S$ ^+ y3 p4 V2 f! M- T; J

    4 h* e* |* A5 M        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    2 F$ i: o4 U. I) ?% l3 ?
    ; b/ P& A% w! qExample: Factorial Calculation9 c/ K; [8 Q1 P4 Q

    : |, V( V$ v" Q, mThe 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:6 I. u0 g/ J/ U  _" s% p
    0 S( B: t! \3 ^5 \
        Base case: 0! = 1( R( _' b# N; J3 g  E$ ]! z

    , ]! [! n6 `  {- ]2 H: `    Recursive case: n! = n * (n-1)!! b- k+ n% {7 q7 K5 U; w
      O& Z4 k' {& G1 n- J, L5 S
    Here’s how it looks in code (Python):( `. d" @) T' ~) A- Z0 [% n9 L. a. k. s
    python% L* L% h2 d1 W- k" j8 {' n
    ( T- K5 l9 V. _$ j) \

    7 U5 I3 A) H! E( edef factorial(n):
    % i) `5 b% W. Z  a4 H    # Base case) p6 P% {3 u0 [2 ^6 X7 F: m0 |
        if n == 0:% N# g  T8 r- k" Z
            return 1
    7 F1 b2 U1 T$ n/ O# P    # Recursive case
    - J( t* d1 A* F0 e5 J5 U8 J    else:" g" `* Q( l$ ^; s5 K
            return n * factorial(n - 1); V6 z9 n  A% i$ a

    , s$ b, ^3 s* N" {& K- X/ o# Example usage! n. o$ j1 J; _) f
    print(factorial(5))  # Output: 1209 y+ u/ M& }$ u% H
    " D7 {' e, o' d* ]
    How Recursion Works+ c2 s/ ^/ O% A1 e2 X- p

    $ O5 l" x+ q7 E" q3 D1 v! J% N    The function keeps calling itself with smaller inputs until it reaches the base case.
    $ ~# t) _% R2 E3 h/ O
    - a7 W5 f" B- {    Once the base case is reached, the function starts returning values back up the call stack./ l' e5 w7 P! ]
    + g' |$ h1 M- i2 g( M2 F
        These returned values are combined to produce the final result.3 S' n- r6 F: c' D

    9 ~* u2 I+ o' s+ NFor factorial(5):
    - ^" h; T, @1 Z' i6 ?, K( X' i0 M3 A6 y$ `2 n7 p2 r  q

      U' O$ {' n5 F  `factorial(5) = 5 * factorial(4); S/ O9 f  f6 z# u0 S" T8 m# }# F
    factorial(4) = 4 * factorial(3)  B- j% z( H3 l# R( y5 Z6 c
    factorial(3) = 3 * factorial(2)
    " e* ?  e5 p" ~% Ufactorial(2) = 2 * factorial(1)
    ( T: K4 m8 _1 l; s; I- Z. k5 t* Pfactorial(1) = 1 * factorial(0)
    - ?6 `; w! o) ^2 P6 F5 p/ Rfactorial(0) = 1  # Base case( O; ~1 V0 [/ W4 }
    2 Q1 o, N3 Y0 p' b0 R
    Then, the results are combined:
    # Z( n. p: a3 L+ N: b: R! L0 w% }9 d) h" W" D+ h

    $ Z# [+ [- {  ]  C5 Rfactorial(1) = 1 * 1 = 1. `1 @' D" H; ^& X& f
    factorial(2) = 2 * 1 = 2
    / l) {, t! ?* U2 m8 Y+ C" g. L/ Lfactorial(3) = 3 * 2 = 6
    ; ]+ Q! X( ]- Gfactorial(4) = 4 * 6 = 248 g' F% E7 e4 Z
    factorial(5) = 5 * 24 = 120
    1 ]$ D7 R+ E9 O+ A3 |  W) p4 l6 @% Q
    , C6 Y6 d% X$ P9 E/ ~# ?: [6 UAdvantages of Recursion: \5 |  c3 D3 t. c: s& m

    - g9 ^  v. m6 m0 H7 Y) }" w3 I    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).4 j3 S/ |* z/ ]( o: C4 H' G" d

    ! ^4 o  G, K' n& f* n1 E9 }    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    . }+ G3 U; E% z9 q! @& W) D$ k  @
    Disadvantages of Recursion& L4 ^3 c8 ^9 Z8 s: n* D8 M& j# b

    ( L0 V- H) t* K4 q/ q    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.7 `: \. m1 `# U! M
    - L5 \0 d: e7 @: h
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).7 q: Z3 V* L0 j  q) {
    ) [0 I4 C5 {: o7 R5 R
    When to Use Recursion$ \# K; a2 ^. J( o3 p2 U
    1 r7 [3 t1 Z7 M% x4 u
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)., j: z4 \4 J: y$ _

    7 b0 G" x4 h: w    Problems with a clear base case and recursive case.
    & A1 }. h2 v; l& w/ g: O0 d7 Z, X5 H2 m/ S% v, T
    Example: Fibonacci Sequence
    6 c1 q* ^9 N1 S5 K* w! b: J# J# I8 R0 A6 Q3 F, h
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ; U  U+ q: }3 T
    + g  Q0 o, P; X% [8 H3 @    Base case: fib(0) = 0, fib(1) = 1
    % q) ~: K, O* ~: j! Y8 Q. b" J9 f3 \/ e9 w  |
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    7 \6 U( l  Y/ f0 c$ w/ h4 O$ ~/ e, {. P0 A
    python; L8 l7 p/ a. W$ H
    $ }/ y) x5 X# W7 G
    ! j& k9 C* F  ]
    def fibonacci(n):+ e6 Z& _) @7 \$ k0 Z% |- {
        # Base cases
    " W& v  D  y. o& r    if n == 0:7 Z) B" j- x- w, Y
            return 0
    ; `$ v& @5 `* g8 x( T    elif n == 1:; K; N! @, u8 C9 V+ U
            return 1
    9 z0 x2 d$ T8 ]1 |. u    # Recursive case" A! Z& j" p/ s: P
        else:
    ! |  R) \2 G# M( `0 P; g        return fibonacci(n - 1) + fibonacci(n - 2)! p; Z9 P9 @; S: o- G3 C

    " M: d; w6 N. g/ b# Example usage
    & z- _, P0 R" G* wprint(fibonacci(6))  # Output: 8
    ' T" [6 I1 h, T& [+ k0 B$ s! Y2 `9 i$ I3 B- [) M$ b( ~
    Tail Recursion
    & t# c: A6 R( I: @* N
    - Y: {4 l3 x0 y# P. A# ?$ _0 uTail 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).
    ! J% m, L' `" d- [4 k; J
    , I/ u1 u! c: @( X  S+ @. tIn 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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