设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 2567|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 6 q& z. y9 v4 n) J& S+ g
    ( g5 L! \- c  f' m1 G% l( O
    解释的不错/ Q! M% D( D7 A! B( A. j1 _
    / r' g4 j9 g. N# ~7 o# a
    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。: N. P( F0 [& [

    4 l% P) b, u! p; @- z2 U. N6 p 关键要素3 A& L4 `+ F# j6 _2 ^3 C) b
    1. **基线条件(Base Case)**
    $ W+ U# p) e. b1 F   - 递归终止的条件,防止无限循环) P9 C. h* N0 k
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1/ _( ?$ R$ X  n

    7 ^$ E8 g+ v& c3 f2. **递归条件(Recursive Case)**$ K8 @$ I3 q2 G9 V
       - 将原问题分解为更小的子问题0 w4 T0 n3 `# ]5 j! e
       - 例如:n! = n × (n-1)!
    ' [- b6 j4 W  c/ Q. G# B7 p/ F, X1 @7 S. a1 l$ r2 R
    经典示例:计算阶乘, w# ], B* ?( v* N7 y0 P
    python( t: o: G, I2 c+ L" y1 M
    def factorial(n):
    - e! v1 R& d; _0 l( E: E7 R    if n == 0:        # 基线条件
    . N$ B, s. l) j# R2 H/ \        return 1- h/ K1 ]) n: P' z( ]
        else:             # 递归条件' i+ l) M0 k& S' b! v
            return n * factorial(n-1)$ B) |* d5 @' m% |
    执行过程(以计算 3! 为例):( _* b. m9 C* C
    factorial(3): D  y# Q" v8 S9 K7 M8 W
    3 * factorial(2)
    - _- ]" i8 e( E7 O1 \8 S  f3 * (2 * factorial(1))
    " O: u# x+ h/ R. W* ]( H3 w3 * (2 * (1 * factorial(0)))
    % |, e$ J1 g1 y* ]+ n2 y0 b% z3 * (2 * (1 * 1)) = 6; ?$ z( x. s! a& _# X% o0 W

    ! H$ z8 C7 p# |. B* D/ U 递归思维要点
    . S, a8 J( t" x- C" @, B1. **信任递归**:假设子问题已经解决,专注当前层逻辑2 b3 `: e7 k: q  y- U9 L
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间). l& t# i: K' h9 i- F
    3. **递推过程**:不断向下分解问题(递)
    7 y3 D; l# |+ _7 ^5 n4. **回溯过程**:组合子问题结果返回(归)
    ( F9 o5 N: @2 V) o& j" ~1 q: R+ l) q5 u' @$ P9 h5 o  h7 i/ d& X
    注意事项- A& E+ ~! \- E; r9 s& {! d
    必须要有终止条件
    . b# b: l( R; o6 p" \递归深度过大可能导致栈溢出(Python默认递归深度约1000层)/ Y" y2 \4 ]: \& _# H7 @" m
    某些问题用递归更直观(如树遍历),但效率可能不如迭代9 n1 j  U. K2 B- s) S# [9 W) ?
    尾递归优化可以提升效率(但Python不支持)
    4 b' j* k' ~; C% E! ~
    4 M8 l& e" G$ j1 T1 ^$ b- j, [ 递归 vs 迭代
    * c8 l- P8 W: _4 u% ^|          | 递归                          | 迭代               |4 R8 c- k! \2 i4 d# ^& a; j
    |----------|-----------------------------|------------------|5 k, ^7 k1 O' h/ [6 v, x, P5 S- @
    | 实现方式    | 函数自调用                        | 循环结构            |
    2 _# q4 ?- Q3 s: U- p9 f| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |$ A' R+ c  F# Z7 W) |, M9 j
    | 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         |/ x# ?2 S: O  i0 V; x" `' {* g& C
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ! _/ {* \; U! ~8 c1 B" _0 a0 A8 c$ I; x% {/ `; G% Y: e
    经典递归应用场景
    $ \* f* p+ y0 T0 o1. 文件系统遍历(目录树结构)# m" f" c0 v; k. n5 _+ }3 `
    2. 快速排序/归并排序算法, v! i4 o7 p0 N4 w* j3 q
    3. 汉诺塔问题% I; {! o! e7 q& F0 ?5 W- G7 a
    4. 二叉树遍历(前序/中序/后序)8 G8 o* n5 a' r  t) c$ m3 f9 T
    5. 生成所有可能的组合(回溯算法)
    * S& _' @4 w* m& F% V4 m! \
    $ K! h. P* s% r8 n, {7 m8 g2 W3 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

    昨天 07:21
  • 签到天数: 3224 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,0 t+ [5 n& z" j9 j2 F( r& C7 c4 i
    我推理机的核心算法应该是二叉树遍历的变种。( Q+ x# G+ I" q, @$ |% f
    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:
    : f( [, v' S* c: X8 O* rKey Idea of Recursion
    , D  _9 }0 t6 U+ ?" j/ i
    ! g% Y4 M) D$ o( vA recursive function solves a problem by:
    0 ^+ G4 G! z+ y: d& E) v; Z
    : p5 V$ M! e* j: ]* q: Z( X! e, q( r. ~    Breaking the problem into smaller instances of the same problem.
    : L; P+ q, S: O$ _( R, d- O) A2 ^! [# Z- n/ {
        Solving the smallest instance directly (base case).
    + y" n% }1 {$ p( w7 O  q# f% R2 c
    ' J" v' Z* ]2 ~& Q0 l    Combining the results of smaller instances to solve the larger problem.
      I: v8 Z" B) ~5 t" A
    7 v0 x* s, t! g; [6 d* h) N3 GComponents of a Recursive Function0 [. \  n2 F2 {9 J

    0 d' ]8 K. V# i7 E4 ]    Base Case:  t. H6 W' C( L" Z1 P

    1 q7 \+ p* E. N8 K. z* M3 i2 j        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    . S5 Q! f) O$ j2 s% A( R3 n! X+ ^" L  ~  P4 d- k% x
            It acts as the stopping condition to prevent infinite recursion.
    1 w# u# s) g5 N  H6 |& y
    5 [! {' O* x5 b0 i- @; q        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.1 G' F) }8 I( X* {- N

    0 I8 z. n+ x  g% ~- Q    Recursive Case:9 z) b: R$ d* L5 y  B, y

    % I, L) l, `) X) _! {5 o        This is where the function calls itself with a smaller or simpler version of the problem.1 q. [4 Z, ~! {. X  a# z; v
    - O2 ?1 }# {6 N' M! ^8 ?
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    % a) x2 f  A9 P! T/ t6 @, y, E% [9 _- r
    Example: Factorial Calculation
    3 V# p' L, m: F
    " U" H  C5 G2 F# J  jThe 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:
    0 \/ {+ M! V8 o; c
    ) P; u% w% e; g, C    Base case: 0! = 1
    7 D8 O# E6 m& q* I# }
    * V. N. X5 c1 m5 I    Recursive case: n! = n * (n-1)!
    5 b6 J: }; b  g' B! }7 x2 c& d* ~6 _! n  R
    : i& o9 }( s1 Y+ B$ W( F+ f. VHere’s how it looks in code (Python):6 m" A& h. K, `+ ~
    python
    0 G; i( ?; x: X7 E. u) U8 I6 s8 Z. H+ a

    # E: ~; X1 h& W) |/ |  Adef factorial(n):
    3 M1 {- \3 g/ ?2 |6 i( a& B* y! X    # Base case. i, F& o/ ~* s3 {2 [
        if n == 0:8 C" y- V! S2 L) h9 d+ J
            return 1
    % D: n9 s% }" a% S& y7 R    # Recursive case
    & s) e- r7 }. A4 W    else:* ~. b/ V5 E" C& u/ V2 f
            return n * factorial(n - 1). o6 }% e; h, m9 S
    1 \% ~2 K3 O( d$ U" c
    # Example usage
    ! U1 K/ _* A7 T. ]9 k7 Jprint(factorial(5))  # Output: 1205 A  }  Y6 ?* Q
    % \5 N6 ~2 d# s1 d; X; \! ~* N$ C
    How Recursion Works: t6 L( o5 {  e; S1 m8 x

    , g+ H( `# Z2 Y; `    The function keeps calling itself with smaller inputs until it reaches the base case.
    - _# G4 B0 h! t$ \# c- x, r
    5 r* }" `0 Q- O2 i8 U    Once the base case is reached, the function starts returning values back up the call stack.$ B- [! J+ \6 |2 X; Q

    2 l, D  F! v0 t+ A& ^    These returned values are combined to produce the final result., D1 w1 j# m; _% i& Z
    * t+ R' s( J( I
    For factorial(5):
    ; C5 V" L& H  ~- y: z, s& T3 H) A6 d% A4 W- `9 T, _+ C/ s) @

    - e( e" ~' p2 n* w; Yfactorial(5) = 5 * factorial(4)& ]2 p6 j8 c, Q. L  O
    factorial(4) = 4 * factorial(3)$ Q" k- Q' {7 p( A0 h5 o
    factorial(3) = 3 * factorial(2); Y5 V1 Q+ G% n  C* U& ^! w
    factorial(2) = 2 * factorial(1)! Y  l& }+ S  L
    factorial(1) = 1 * factorial(0)
    + E8 \5 L( x4 h& F; H# \6 l9 K6 afactorial(0) = 1  # Base case! P& W/ G& }3 ^
    6 q4 W' a1 ]. T8 U6 d/ y. V: g8 u
    Then, the results are combined:, ]" g6 B. u3 I7 v2 t! W+ p

    4 ]& A' g% G/ V7 [( L5 a9 L2 v; A# h4 \6 ?1 p4 s5 S0 O
    factorial(1) = 1 * 1 = 1$ G9 t0 W3 Y5 Z% X4 z
    factorial(2) = 2 * 1 = 2, S: r7 {' c" G0 C' ?  N, V9 B
    factorial(3) = 3 * 2 = 6
    , m* y% n; a3 p5 X+ Kfactorial(4) = 4 * 6 = 24
    # u9 p9 b* a; T/ B* L- b& Gfactorial(5) = 5 * 24 = 1201 [5 H+ u8 o  u5 E

    & r, S: G! z+ o/ _6 r1 \Advantages of Recursion1 W% W, o3 b1 W- D
    , F1 m, m3 T- r  i7 r1 f
        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).' }$ C: e6 @! h% m& E: S

    ( ?& T2 _0 A& s6 `/ Z    Readability: Recursive code can be more readable and concise compared to iterative solutions.( c. V6 N, Z: z0 K

    # v. e  b1 ]' x# o* V" @- FDisadvantages of Recursion" ]  E. D3 V& F5 ~0 P% r
      P. _" U' j+ K  l
        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.# ?+ l  r* \; r# A; N3 z1 o

    ( Y% y9 @2 _7 h4 n$ v: M    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).8 j- K5 J$ e* h* v9 w

    8 c4 G$ V/ C' V$ ~" Q+ s4 O$ TWhen to Use Recursion& Q% q. L, F+ j# j7 C/ W

    # k: ?. }- U: x$ W- P- s    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).+ g% P4 ]. y& V3 b4 e/ F. Q

    ! v: o. `& D$ |) V4 W    Problems with a clear base case and recursive case.. |! W5 w4 W/ f+ l: ?: x

    $ Q0 W+ F: a( JExample: Fibonacci Sequence5 s/ M  ]* |' g; h
    . |( b7 i% f/ `" G4 y+ i" @
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:$ F; E, `" X# i$ C0 m6 e7 n

    ! W3 U) M1 e- {* Y    Base case: fib(0) = 0, fib(1) = 1
    9 A9 [% Y' l8 ?; G# t! D& Z* z" {
    ) W! \, o) R8 T& Y% D4 I7 M9 U    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    " G$ w, N, s( T2 v! z. N
    6 S0 D( x& U) {python
    % o* k* @+ V1 W8 b: g
    1 b3 s6 y; n5 C5 E) l5 a. o( U; R- l4 t3 c
    def fibonacci(n):
    ; F, j; E2 T1 J  E6 O9 L    # Base cases: I# S# p" K/ i& E  ^% O. L
        if n == 0:
    & X: o) |6 |  T& t3 F        return 0
    9 O% t9 k* M' m7 _6 w; O" l+ t. L    elif n == 1:
    ' D! D$ N) a! U) H$ i        return 19 f6 F+ Y5 x" T! }& l8 L6 M5 l
        # Recursive case- d' }& a3 G& L- g) L! X4 n
        else:! I$ D" Q3 S4 Z5 T7 [' b! t
            return fibonacci(n - 1) + fibonacci(n - 2), u; P/ {4 D; R

    : q* E* O$ Y0 {& S5 N- r# Example usage
    6 J! J  \' u0 z* M, Zprint(fibonacci(6))  # Output: 8
    * F  h4 D5 }% P* ]5 }
    , b" j5 H$ U/ v3 g3 wTail Recursion0 L1 F0 W/ D. W* b$ b

    $ x4 m2 H) c  h: I2 b+ |# N' BTail 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).
    9 C0 b% d* T; c! m! ~/ L4 q3 P% n
    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, 2026-4-26 17:41 , Processed in 0.066944 second(s), 18 queries , Gzip On.

    Powered by Discuz! X3.2

    © 2001-2013 Comsenz Inc.

    快速回复 返回顶部 返回列表