设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1298|回复: 3
打印 上一主题 下一主题

[科技前沿] 突然想到让deepseek来解释一下递归

[复制链接]
  • TA的每日心情
    开心
    2025-9-8 05:08
  • 签到天数: 3 天

    [LV.2]筑基

    跳转到指定楼层
    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
    7 }6 ^1 M* X. `/ ]% d8 ^0 F& g+ Z: W8 }6 ~' [9 A- x
    解释的不错( O. `0 Y3 D) p7 u* Y# I
    & t' p' k5 U1 [9 E- }4 ^
    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。0 I6 P8 P3 s; @( r4 U8 r
    6 e8 `6 q9 K8 y( H
    关键要素2 U- I2 [4 a0 C- J$ Z0 [% Z  ]/ e
    1. **基线条件(Base Case)**
    * d& `) J9 r* b' H  W   - 递归终止的条件,防止无限循环
    . |  D" o/ m8 j, p- U) L( ~$ N% l   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1# {- J6 g( l+ L( I! ~$ }6 k0 x

    : F) X# F3 d# ~' P  e& B2. **递归条件(Recursive Case)**4 ?' b4 f$ @) p9 P- H9 w1 c
       - 将原问题分解为更小的子问题
    ) U  l. G4 L% n( ?  j   - 例如:n! = n × (n-1)!
    9 @# ^/ V' X2 F& G- b0 Y
    4 K1 I  A( f& ^4 }4 t4 ? 经典示例:计算阶乘
    - N/ A4 [$ P8 _python
    $ W* t0 l% U$ H$ f! h9 L- R  Gdef factorial(n):
    9 X8 ?2 L+ `8 v    if n == 0:        # 基线条件3 h' S, E) o- f) |+ G$ `  c
            return 1& \1 B, V4 ?$ k) j2 ?
        else:             # 递归条件
    ' t# H- t( U! ^: O1 n+ h6 a        return n * factorial(n-1)7 H5 }7 E/ U& T8 q/ o9 e
    执行过程(以计算 3! 为例):* l5 M; q% g9 w7 K6 M- v, D
    factorial(3)
    # s# d* P7 ?0 P' d; [: g& `: e3 * factorial(2)6 S  c+ z" x- k9 \8 k$ y" F
    3 * (2 * factorial(1))* ^, u. D7 I# x& C+ h
    3 * (2 * (1 * factorial(0)))
    . k6 u; J3 _. Y$ ^3 * (2 * (1 * 1)) = 6, s3 y! \. o& }9 r- \9 ^0 ~& {
    & M8 V) @% ?% w' E. Y4 f# A
    递归思维要点
    8 F! P  e  X+ k3 |; D1. **信任递归**:假设子问题已经解决,专注当前层逻辑) i0 }, z0 C1 H9 C
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间): V2 p9 c* ~0 Z! J* n; {# ~
    3. **递推过程**:不断向下分解问题(递)- K8 t, F# o7 O6 J
    4. **回溯过程**:组合子问题结果返回(归)1 m+ i8 E3 X; d- E: B
    3 X1 H' S; z6 |8 [2 G1 Y$ T
    注意事项
    8 D+ |. p' m6 Z必须要有终止条件# Z* J' T, S5 x& [; ^
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层). k+ r2 }2 s# a+ ?
    某些问题用递归更直观(如树遍历),但效率可能不如迭代. l- S% H" [3 L' I9 [) `1 |5 `
    尾递归优化可以提升效率(但Python不支持)3 N1 j9 @3 U/ a) d

    5 M) n# m% _3 g7 E5 H9 N 递归 vs 迭代4 f- a1 U' ?/ C* A( E8 D
    |          | 递归                          | 迭代               |
    # r/ J- j$ U9 L# I/ s- \|----------|-----------------------------|------------------|6 o$ S5 k" d7 u! L. D- C
    | 实现方式    | 函数自调用                        | 循环结构            |
    , d, ^6 K" u" S- Q| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |/ g, K. E! J, g
    | 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         |1 ]7 E* ]2 S7 i4 Y2 q/ y9 I1 N
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    # G3 V& Z$ o2 B. N7 G3 [* P4 p" I( o% h6 a( `7 v1 B
    经典递归应用场景
    & I/ ?; {6 }/ r: Y% K8 D1. 文件系统遍历(目录树结构)
    ; Y* {1 O$ Z  o% |) p! l7 J$ e# e! e2. 快速排序/归并排序算法6 J$ N" k* v# E1 F/ z2 m
    3. 汉诺塔问题7 B9 R" f" f* i+ Y  y
    4. 二叉树遍历(前序/中序/后序); G7 w6 S- a! F2 p
    5. 生成所有可能的组合(回溯算法)
    1 w! t+ O9 ^+ `& b" o- c1 Q, _5 s9 D7 B" U
    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。

    评分

    参与人数 3爱元 +26 收起 理由
    pcb + 4
    老票 + 16 涨姿势
    住在乡下 + 6 给力

    查看全部评分

  • TA的每日心情
    难过
    昨天 07:29
  • 签到天数: 3106 天

    [LV.Master]无

    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,
    + S* d, x! I* k) w我推理机的核心算法应该是二叉树遍历的变种。. t  x9 ]$ H. K
    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。
    回复 支持 反对

    使用道具 举报

    该用户从未签到

    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 s! p) ]- N7 gKey Idea of Recursion  K! M* k' Z* E" H; k' U2 x4 K
    5 a2 A$ y! r5 b) x& T& [  C
    A recursive function solves a problem by:
    5 I& K6 G* Q% y+ \' B; N
    5 j  v1 E, B( A' N    Breaking the problem into smaller instances of the same problem.5 ~- m  y) A; b
    - h8 }1 @1 Z( `
        Solving the smallest instance directly (base case).
    ' M6 m* r2 @. F8 o
    5 f- L# I$ P/ d$ e+ g    Combining the results of smaller instances to solve the larger problem.- I# ?% ^4 Y( U% z& k, |% |# J
    ! f# V7 A% P7 Z
    Components of a Recursive Function8 y' j2 a+ f' N
    . I. X3 \: _5 x2 h! `: z
        Base Case:
    " ^1 U1 c2 A' j! j6 g; F+ W! h6 j4 O
    ! Y) ], l7 n+ g$ P        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.0 |- W4 C/ V% P! y' u
    " x% N/ d, M* D: w: O) v
            It acts as the stopping condition to prevent infinite recursion.! x( b1 F: _7 Z7 \1 k
    2 o7 h  f6 {* h
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1., z; v2 m$ _  T$ Z

    : ]. G% Y# d1 ~1 |    Recursive Case:% M6 c: [# Z4 Z3 C' i' ~) `$ M

    5 C8 W" D+ F( ^' H' r        This is where the function calls itself with a smaller or simpler version of the problem." L) H% j' ^1 e
    ! |7 {" ~) T8 O5 Z9 I! q* t! g
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ' Q6 v& ^& J" l" |; z/ j+ R# h# |# q8 A6 ~
    Example: Factorial Calculation
    & p5 p3 H( u- M% j" r
    $ {- U$ |, D, ]* M' PThe 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:  D' E0 w' C! n& j/ z9 J# H
    0 t, u# u, i9 G8 G" @
        Base case: 0! = 1/ C# k9 V7 ~4 o8 n/ S& O9 f
    - w! H' Q# Z+ G: g) N' G
        Recursive case: n! = n * (n-1)!  [; Z- H7 T" |! Z

    , k( F+ j! V) p* b  eHere’s how it looks in code (Python):3 O% y5 @* d- ~8 y; X$ q' F
    python
    * x' x# |7 G3 Z, P; l8 T% F9 k: A- V  w
    6 N+ A0 l" E; K$ w$ N2 C2 ], g
    def factorial(n):5 E* W1 Z# ?* y: H  F
        # Base case* i6 F1 c3 ^& Q. x
        if n == 0:
    9 H0 [# {' D* }8 X' V        return 1
    * z3 h- C" O& h' N/ S8 ?    # Recursive case$ l- W& F/ z# @/ W
        else:. t% Q. I1 _! w- G0 {5 p9 f
            return n * factorial(n - 1)
      M9 Z+ ~$ J, T: ^4 Q4 a' p0 [' U* d0 b1 p3 ]' M. ^
    # Example usage: G+ Y; }$ F7 X: s% _" Y3 N# f# T
    print(factorial(5))  # Output: 120( Q2 @. s5 E' x! o! s
    # E, s2 p: S2 z2 w8 \3 ^
    How Recursion Works# d4 ?8 l1 E% o, M, R2 l

    3 p. w% g) f& A3 T    The function keeps calling itself with smaller inputs until it reaches the base case.
    6 b2 o0 b5 n+ n* ?& A9 K
    * q9 h9 w3 G# f: b- D    Once the base case is reached, the function starts returning values back up the call stack.$ V$ H' S6 |$ X
    / }# s: l  I8 k. @; `+ Z1 b
        These returned values are combined to produce the final result.+ ~- z1 \8 X: ]; ?% q7 `+ G
    ' \8 o* \- y* o, P
    For factorial(5):
    / y# s; r* o5 L, a  P4 U' G$ A1 e3 Y, T* C
    * \8 W  Q9 @% ^2 m
    factorial(5) = 5 * factorial(4)
    3 _2 Q- W9 c' U! f1 Zfactorial(4) = 4 * factorial(3)) A7 z' i6 C0 n# Q+ D1 Z
    factorial(3) = 3 * factorial(2)
    2 }# T; N, d" N0 W0 Ifactorial(2) = 2 * factorial(1)* Y" B5 g) I  ^6 ^. X
    factorial(1) = 1 * factorial(0)' `- Q9 [) _! l% H
    factorial(0) = 1  # Base case8 q; o& Q+ c% k6 T/ Q
    $ O- [5 ]' v1 S0 \4 z% [
    Then, the results are combined:
    ' g" \( _3 z/ y" c1 E) ]2 `4 m- e+ t! n" y% q/ H

    2 A1 J% V1 p! v& t- w4 ?- J# s/ T7 Sfactorial(1) = 1 * 1 = 1
    ! P0 ^  p4 Y1 S3 k; Gfactorial(2) = 2 * 1 = 2
    2 y' E+ G" i" g; A. _9 vfactorial(3) = 3 * 2 = 6
    . B  ]9 [. [/ l) @8 Q  ]: n/ h' qfactorial(4) = 4 * 6 = 24
    ! B1 {$ S2 U( d. X* ffactorial(5) = 5 * 24 = 120
    ' s# ~( c  U7 w
    5 c/ f) P- l1 cAdvantages of Recursion
    7 u6 J( h/ E  v0 \% X9 z+ U# P# C8 u, |9 k9 _- \# H. u
        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).
    $ Q  I9 v9 B4 i- k: ]5 e) `
    : v5 }1 C. r, `; v- @: O: o: @) z    Readability: Recursive code can be more readable and concise compared to iterative solutions.8 \- Q5 ?  k  M. m& E7 }  Q& o
    6 k& m$ t0 L% C! F0 q$ p- E
    Disadvantages of Recursion/ @6 s3 J. [6 L1 \  T# F

    4 H$ Y' N/ ?& p2 \" A; F+ b9 T    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.8 W, y7 n/ V. F! ?7 ^1 I
    # e7 i( r- W6 A" k7 o0 E! I/ P
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).9 ~1 F( O. S. z  [3 c$ w$ \
    5 `7 t' w% c2 @5 h, e& h  n# a5 i5 t
    When to Use Recursion$ v7 |/ U8 F1 F7 m6 `  P; D

    6 R5 b0 P# H+ x7 O6 D8 v& U" Q) h    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).( X0 [. X/ {- s- L
    # z$ b* `- S3 n  A; D5 n% y
        Problems with a clear base case and recursive case.
    2 r7 [. P$ A' A" S1 m
    & B, m3 F% k, @! v/ E: s% dExample: Fibonacci Sequence& O: F6 Q+ p# y; B/ D& X0 G( R1 y

    % ^. s0 N' @# @  q1 W% jThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    1 j, [$ ]7 k  I: K+ X
    1 L5 [$ I) o  M5 ?+ _( l2 m) w    Base case: fib(0) = 0, fib(1) = 1
    9 {. [. m1 P* q
    % E. r# @2 o- E0 ~    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    $ D5 C, m) y& [4 L' `  E( V
    7 b* K, r" t' r9 X3 O* H3 Dpython
    . N6 j4 F2 O' z! Y8 H6 D4 B9 |: P$ h+ e& g% G

    ( Z, @4 h2 J; {& o6 O4 t" Edef fibonacci(n):
    3 ~2 k2 @: K% }+ k6 ]7 j    # Base cases
    8 V; t% J% k- x    if n == 0:  V& L9 V% f8 s0 I5 r) R! P; O
            return 0
    " T& _0 j* H! j2 m7 t7 y    elif n == 1:; C' |3 K. A& {. n9 m5 z
            return 1+ _: }7 k' l5 m0 L
        # Recursive case
    ; e8 Z1 ^; o+ e8 @: Y9 m" T/ m    else:
    ' J% S: R( l: q9 n6 E        return fibonacci(n - 1) + fibonacci(n - 2)) q% d6 O8 f" `

    , I/ x  Q) [$ Z# Example usage4 o1 ^1 d/ E* y. P- R, Z
    print(fibonacci(6))  # Output: 8
    , S# I  D' @$ w3 L7 T- ]- g# Q2 F! g7 Z' \: {  A( U
    Tail Recursion
    4 f* U2 S1 [$ d% Z4 J
    & q( G: P6 M! E, [) o( n$ fTail 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).* L" R7 S+ a+ Z1 \# W
    4 b1 p" r$ o$ K& [, F: O. g
    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.
    回复 支持 反对

    使用道具 举报

    该用户从未签到

    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。
    回复 支持 反对

    使用道具 举报

    手机版|小黑屋|Archiver|网站错误报告|爱吱声   

    GMT+8, 2025-12-3 06:01 , Processed in 0.042570 second(s), 22 queries , Gzip On.

    Powered by Discuz! X3.2

    © 2001-2013 Comsenz Inc.

    快速回复 返回顶部 返回列表