设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1818|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 & F8 E/ D' y* i( X: y

    1 Q# o) C0 h+ ?& O/ r解释的不错# E4 t1 E( T7 k+ L

    " G. A) N. L9 b! \4 Z3 V/ M: J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。! }5 B7 i7 E! k8 N% h: @, C$ ?

    9 Y& U, B! y/ `  c" J/ @4 P$ s$ Z! e& Q# m 关键要素
    8 Q( R% K( D9 c0 |1. **基线条件(Base Case)**9 V7 {; T( y: S# p5 W
       - 递归终止的条件,防止无限循环
    7 T% f  ^- v- f  X1 R   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 13 ~6 i/ m) Q  [- c8 d
    - y' o/ R9 C1 `5 T6 c' |+ B  e
    2. **递归条件(Recursive Case)**
    4 m9 s8 p2 t! Z   - 将原问题分解为更小的子问题
    / ]3 K' X' a0 r- x0 s, A6 i* Q   - 例如:n! = n × (n-1)!  E* n; ?$ C3 N1 [
    3 a5 z& w- ~/ K6 h  @$ `
    经典示例:计算阶乘
    * b6 N) u1 x) ~python8 i6 R* f0 Q3 j$ `
    def factorial(n):
    # ~. h/ S3 x! D8 h    if n == 0:        # 基线条件2 |1 [7 c" k, P2 \$ ?9 b
            return 1
    2 E& @! c$ j$ _* q    else:             # 递归条件4 `  b7 L! B9 C3 K7 P1 O
            return n * factorial(n-1)2 X# M4 u% F" @0 B7 U& O
    执行过程(以计算 3! 为例):* M; y4 U# T) O( B% c1 D
    factorial(3)
    4 {- n5 H) r: _- V5 D3 * factorial(2)
    $ f1 m" R  F8 T5 e3 * (2 * factorial(1)); A, q" l0 J' |/ n4 u5 j, ]
    3 * (2 * (1 * factorial(0))), b  I$ ^7 T% p0 m
    3 * (2 * (1 * 1)) = 6
    / L: D2 V3 F2 q4 D& V9 A/ a, f) V  C" g' C
    递归思维要点
    / y& v5 K, y$ b1. **信任递归**:假设子问题已经解决,专注当前层逻辑& D. I9 w) Y& a' `
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    , d* z6 X' z, z+ R3. **递推过程**:不断向下分解问题(递)! |+ k: L0 u3 X0 z; `' Z- [
    4. **回溯过程**:组合子问题结果返回(归)% u7 G. X8 x* e: ^

    4 D! F& ~9 N2 u$ y2 v. _ 注意事项# N* P) {7 f. I: k+ `: F( N
    必须要有终止条件( U) F) C' z) m7 M! R
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)" a' P% {" N: S8 O9 S8 l, }
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    + e5 P  l4 y8 `尾递归优化可以提升效率(但Python不支持)! w; X3 H% I+ {3 }
    " p1 N% u# g0 i$ a# X
    递归 vs 迭代
    8 n* T4 F) a0 u* {  r: _|          | 递归                          | 迭代               |
    - v( V# f7 \8 R|----------|-----------------------------|------------------|
    0 B! U! c. j+ S| 实现方式    | 函数自调用                        | 循环结构            |  h1 R9 m; }: |4 I7 c% B2 B" A
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |0 G9 O0 G9 }5 Z1 E! k
    | 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         |
    + p% c' e0 P5 [4 w" C| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    - {, F2 p6 k4 ^# n' j( W
    * @+ n2 i6 l5 i 经典递归应用场景/ y  o( }. L: z1 V8 |2 h  r
    1. 文件系统遍历(目录树结构)" l5 }" ~- H( _. L, }$ M
    2. 快速排序/归并排序算法9 z; S& L- Z# K; j8 f) H6 f
    3. 汉诺塔问题" S0 y; y' F5 U, m2 S
    4. 二叉树遍历(前序/中序/后序). B: m8 k7 \' p: G7 p( j/ n
    5. 生成所有可能的组合(回溯算法)
    , R1 j7 n8 m0 n8 e
    % @2 F% O0 x8 \! f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
    郁闷
    昨天 07:20
  • 签到天数: 3145 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,/ c; H7 c9 q+ T& d0 Q, V" B
    我推理机的核心算法应该是二叉树遍历的变种。
    & X2 w: A# X* z+ H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:
    ( o9 ?8 a' ^) l7 J* S! O" F# ?Key Idea of Recursion
    , K- Q  D' Q$ u) a# r2 _
    " a" F- ~7 A) CA recursive function solves a problem by:
    " \4 V" Y, {" [/ I" j
    4 B8 N8 y1 f5 J# @8 y) m9 U    Breaking the problem into smaller instances of the same problem.
    ( |9 |/ w% t2 b3 y2 K% ?
    - G7 M2 Y9 G/ E. F4 y" I    Solving the smallest instance directly (base case).
    ) q" h+ s( ?9 M: y( K% S5 a" g+ D! r% j. l1 _# i* i6 e
        Combining the results of smaller instances to solve the larger problem.1 h3 o" K! X  x. V) Z' [9 |6 Y
    $ @0 y4 o4 ]+ p8 u, m
    Components of a Recursive Function
    & M6 d* ?" k5 I& ], J" N9 u# Y8 g9 `* |
        Base Case:
    & c* w/ W( C$ `( y9 @0 e( ~) @" [' J* i" s5 _3 f7 i
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    , s9 O  @' w: J3 r6 z  S: w9 p
    ' v2 r6 o8 @& w! |        It acts as the stopping condition to prevent infinite recursion.- A3 c% K# i# Z5 |" I: C
    - b5 y& P; W1 U. r
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.: _& C9 L- ?" d+ `1 @2 v  H5 {

    & k8 O7 p/ t( ?6 l) m' H" b# k( `* t    Recursive Case:
    * A# K' N9 C# b8 A# A
    ; d( g( ]; ~$ M7 m' q; F0 D        This is where the function calls itself with a smaller or simpler version of the problem.
    & q0 c, E) c- V4 N. N1 l0 R! D% a5 M% N- y/ n! i/ T+ U
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)., B% H2 W+ E% K" s- R  M
    ; {. I) l/ X- ^) ~5 Y
    Example: Factorial Calculation
      {6 {& A$ [4 H3 Y& K
    : A6 E* S) C& n8 [9 e6 k1 VThe 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:( _* Q+ N! |2 g7 q
    9 f. D. k0 q- ^; @4 @
        Base case: 0! = 1
    $ q. o, a0 H* V+ b1 V+ {* N/ b: T' O" O. W4 W
        Recursive case: n! = n * (n-1)!
    8 Z: l) f5 U7 J
    ; l8 d8 [6 o, U+ ~Here’s how it looks in code (Python):
    ! E( x) V0 ]' O: Hpython
    4 a/ _0 f, F2 F! N# d
    : F( s4 [& K( R# `% b% P1 B$ o1 g/ p, g% p$ H4 Z' \
    def factorial(n):
    ( t5 o0 y9 H8 {    # Base case
    + T3 M; _5 |: b7 Q; P! B    if n == 0:
    % w1 g4 q$ x+ _* V# I% O        return 1
    , S9 v% [" d1 L) E0 }' H, u    # Recursive case- \$ T- @/ q2 ~: B8 u
        else:" I6 ^8 r. p+ }, d  Y
            return n * factorial(n - 1)
    # X% k& B" b8 y9 N' G/ J5 S3 o1 ^3 y% s9 \& E
    # Example usage( g4 l# \& ?. M+ c7 Q% D
    print(factorial(5))  # Output: 120# ~/ y1 n/ C0 |3 h  Y- F
    8 s# x' b! i" H
    How Recursion Works
    : y* \; [# K0 J3 c) b) y. |6 ^/ s% n+ }/ M! y* _# m+ k
        The function keeps calling itself with smaller inputs until it reaches the base case.
    " A! f4 i2 W- i! M; G0 H) Q& {" K# ^, ^# R) {( w
        Once the base case is reached, the function starts returning values back up the call stack.
    8 g* [+ W# O$ j  a$ z
    1 f) w( V1 |, R  c' I    These returned values are combined to produce the final result.
    ; |, n& A+ G% f$ F8 Q" x
    7 r0 |& b( h: k4 E7 ?For factorial(5):0 @" I: i1 M8 d
    , U0 I' R3 Y! g: j% j  P* c
    / Y) R, R* W1 s. s# Q2 n
    factorial(5) = 5 * factorial(4)4 }2 w1 ]5 m- W8 F
    factorial(4) = 4 * factorial(3)0 V5 k- @5 x: @7 @
    factorial(3) = 3 * factorial(2)
    & w, J7 R0 \* J" _( Sfactorial(2) = 2 * factorial(1)- S# W+ y6 [9 n. t3 x! T: r! Y
    factorial(1) = 1 * factorial(0)
    - f7 W  L5 Q4 I  d! U/ xfactorial(0) = 1  # Base case
    # x3 p/ N4 T# Z: _
    $ j) G/ T: E8 |* K+ E1 S) VThen, the results are combined:
    , P2 B. m3 ?2 ^% `* |' u; ^8 d) `/ L0 f
    ' J; y) m3 F: w* \; t: O7 ?, `
    factorial(1) = 1 * 1 = 1
    5 K- I  L7 h3 M- m, g* [  n" P& A" j9 n  Ffactorial(2) = 2 * 1 = 2
    * d: S4 ~  @! Z6 V' e! t2 Wfactorial(3) = 3 * 2 = 6
    - I% G; ?; m7 D) t6 Yfactorial(4) = 4 * 6 = 24
    5 m; z# M9 g7 i4 Q1 C0 g! b, D/ y" mfactorial(5) = 5 * 24 = 120
    ) V) V  V! x# \+ H/ }' ?+ _/ z4 s% K/ j; V! s: ]
    Advantages of Recursion
    ! i! v  v+ \, Z6 V  c# v% f' D: c9 n4 Q9 \# L3 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).5 o8 q6 H+ L' h

    % \' \" G  s7 n: k- T    Readability: Recursive code can be more readable and concise compared to iterative solutions.1 G; O( S' C) _+ O  }( }# c
    7 q0 x; {& u% ?0 C# G5 h( J( U
    Disadvantages of Recursion
    4 L- l( o3 h& R  d; i, g
    3 k/ ]% O, h$ M- A6 H    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.& g6 g' o; G* u" [7 {

    & `! w3 z: n6 C5 s! ?    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).# d0 B4 C5 v4 o# q1 [% S0 Z( m1 q
    ! k: e% E" `3 q9 \; |
    When to Use Recursion/ O0 f/ `8 C8 z( F$ B8 L

    - h# Q% q6 p& K    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ) r& @0 v# z' a" J5 t5 ^$ J# c2 |6 E9 c( M
        Problems with a clear base case and recursive case.% W/ |: F& \, {" N* {( e
    4 q! ^: o) z6 O
    Example: Fibonacci Sequence
    & X3 R# n$ Y' J; A1 R$ {
    ) |! x7 F: F3 }8 d7 W4 m, R" ?( eThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:! w/ b; X5 I4 r2 m0 `
    & ]- h& Y5 R, g: [- c; z" T
        Base case: fib(0) = 0, fib(1) = 1
    4 A; \; {$ E! r4 i
    1 d* {" O; o6 L& a- S- P! q    Recursive case: fib(n) = fib(n-1) + fib(n-2), o2 j0 S& E7 k  S
    0 b7 d1 A+ w$ K6 y. O) M- y
    python
    3 p, u+ G/ z- G$ F! I9 @
    ) r  I- ^$ j' k7 X2 \3 n; ?/ @  o& \, Q3 Q' G1 S2 m0 j
    def fibonacci(n):
    / b" F0 r1 D+ f7 f. I3 `* b    # Base cases' H  n3 G. W" V! D! l0 X
        if n == 0:
    8 d) @2 O7 x* G1 e4 I7 Y        return 0
    # L- u# L* ?5 v    elif n == 1:
    1 }# S. z5 v: B  h        return 10 p: i3 |" b; x- y" t- {2 L
        # Recursive case4 B* c3 x6 c7 P' n" ?
        else:
    2 x3 G1 z. d( D: ?: B' m        return fibonacci(n - 1) + fibonacci(n - 2)3 P( S9 e" L3 z/ A. d
    8 L% l) Y1 R: j3 z  t7 F9 t; I
    # Example usage
    . x, S  m; N9 A8 K  F8 Eprint(fibonacci(6))  # Output: 83 u8 Z4 s/ Q6 T; b9 [9 X$ g, [
    4 I# h$ x" e( l9 B" W, r
    Tail Recursion6 |0 y; A8 M- g& j: C9 i" ]: `

    3 Z) U0 s) @" XTail 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).  K% H( s$ m2 G- ~8 F7 m- P( {1 m

    + U! }: i& [% H& kIn 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, 2026-1-14 01:01 , Processed in 0.029737 second(s), 19 queries , Gzip On.

    Powered by Discuz! X3.2

    © 2001-2013 Comsenz Inc.

    快速回复 返回顶部 返回列表