设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1398|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 ' n. ]( \2 B$ y# a5 k
    : J0 d0 O" m- a+ C  g
    解释的不错
    1 v3 n6 G% b0 k- B. \( }$ }+ H* o# ?, M9 b9 ]% R2 E' r
    递归是一种通过将问题分解为更小的同类子问题来解决问题的方法。它的核心思想是:**函数直接或间接地调用自身**,直到满足终止条件。
    & ]' B4 P7 s5 \) a; O9 \" X) e! N5 j
    关键要素) h+ P2 H  p$ g% y, v3 a' u
    1. **基线条件(Base Case)**3 m4 a4 @0 [  N$ A; t3 b' m
       - 递归终止的条件,防止无限循环* y1 C- m! b5 t$ U* J, F: X  U
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 17 C5 B: j6 a6 @: U

    7 U; C5 D) {7 ~  }4 z3 \. {2. **递归条件(Recursive Case)**
    7 n) \* L' |. R   - 将原问题分解为更小的子问题! y  V4 f& h; U3 C& [: Y; w
       - 例如:n! = n × (n-1)!
    0 B( _9 l" z/ C$ `% v# `
    ! y' q3 Y3 z# W! I7 o- b- E/ ?4 U 经典示例:计算阶乘" [, S9 V" l9 J( r+ `
    python
    ( P& z3 b+ R$ }% G* b7 Zdef factorial(n):( d8 I* K! T* u2 ^3 x2 q& u) R% I; Z
        if n == 0:        # 基线条件
    ; d8 r. ?. m5 z! L% i. z        return 1
    . j) G& _2 I4 c* S( O% r1 {3 @  O    else:             # 递归条件
    3 x! Z8 S- o$ c0 X        return n * factorial(n-1)% C' t" h1 [! V$ r- g
    执行过程(以计算 3! 为例):
    9 v8 ?) w4 K- P0 B+ y2 ]+ F! R! O( ?factorial(3)
    : l! ~; w6 ]+ J$ Q3 * factorial(2)
    ( y9 b" t. ?& u& m9 \7 M3 * (2 * factorial(1))
    ' x4 H% |/ X8 x, N* T+ z" v" J1 |3 * (2 * (1 * factorial(0)))- U1 {/ @/ B$ q
    3 * (2 * (1 * 1)) = 6& n6 V% r7 d' S  K: H

      ^, k& E5 D* B. J4 s 递归思维要点9 F4 ]" J+ M- b% {6 Q
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    7 }3 ?) l$ Z! y3 q2. **栈结构**:每次调用都会创建新的栈帧(内存空间)8 M- f% t8 w4 o, \
    3. **递推过程**:不断向下分解问题(递)
    / Q, W* O* v3 L8 i6 v% D2 T  |4. **回溯过程**:组合子问题结果返回(归)8 j) p: ~0 Q( l  @3 v' Q5 v. L

    4 ~: d0 Z5 D1 P1 m1 B( r  r$ t 注意事项
    % H7 M) Q- h! V8 Z( h必须要有终止条件; v# c& s7 f0 Y' x  j3 c7 b( v& {
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)) g2 D2 g/ X: A  `- Q
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    # n) n1 D, D* t+ O* `" ^5 l# `尾递归优化可以提升效率(但Python不支持)' y5 [: X0 }9 ]

    ; o3 m. q! U: T& T  Q4 w# u 递归 vs 迭代
    2 m) ?- z8 s- Q" j) Q1 s|          | 递归                          | 迭代               |" ?2 j  k' Q) z7 R& `
    |----------|-----------------------------|------------------|
    7 {4 ~9 U9 U* \6 ]| 实现方式    | 函数自调用                        | 循环结构            |' S7 `$ N" _4 B6 z& H, _8 w
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |9 [! C  n' M7 y+ w
    | 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         |
    5 Z; q; Z. j# B6 c| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    * q4 v9 H) E8 n- c
    8 k1 L' P4 ~4 L+ }1 N1 p1 B' z. j 经典递归应用场景8 l9 |& P$ e- p
    1. 文件系统遍历(目录树结构); C  H& ?9 v- q. j
    2. 快速排序/归并排序算法: G6 v% M1 A9 [! b8 ^
    3. 汉诺塔问题. k9 f0 Z3 F) m5 e& l
    4. 二叉树遍历(前序/中序/后序). ^" h6 [9 ]9 o; |) J
    5. 生成所有可能的组合(回溯算法)( K8 Q4 T5 T! ~9 ?6 @) C
    % o! U. c+ G4 y- R! I
    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
    奋斗
    昨天 06:53
  • 签到天数: 3111 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,
    3 Y3 _3 G4 j5 ~# a- @我推理机的核心算法应该是二叉树遍历的变种。
    / h& u0 v. x" w  Y9 T9 W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:) A! G' A" V5 _" C8 l" ^$ x
    Key Idea of Recursion
    / v) P. m0 j1 a, }% _  }* t* H  v* l- w( @
    A recursive function solves a problem by:
    " k2 E+ i4 W# r8 c- k. }, z8 M9 v
        Breaking the problem into smaller instances of the same problem.
    / m# U' T9 q/ Z7 o4 C4 ^. A; z6 J: X6 [& t; v
        Solving the smallest instance directly (base case).
    5 e5 c5 j' O; Y- @( e$ Q4 ]; f  W  K9 o+ d
        Combining the results of smaller instances to solve the larger problem.- Q. [4 q& I5 `

    9 H7 p6 T) z  O& ^Components of a Recursive Function# W( k; ^/ M1 i6 Z7 s: S/ l
    3 m" L; g# Z# [6 g  K8 n
        Base Case:$ T) K& s8 O, n* q

    ' d& n3 `2 M8 Y' S1 t5 i$ G        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    : \% {/ I( S/ [4 u% q2 V1 n4 a) L4 G; K9 [5 l
            It acts as the stopping condition to prevent infinite recursion.
      }0 Z, N7 I* [% ?" v
    ' O, K1 V( t0 u# S& z/ h        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    - o* Y% J+ R0 k: R+ U* U- g3 |, E0 z) t2 X1 E3 d- P3 j* h
        Recursive Case:3 {8 D! {  X& M: ?3 `' |
    " X; M, T+ F3 F& J4 b5 S( k0 x
            This is where the function calls itself with a smaller or simpler version of the problem.
    9 r+ A9 X* i) o4 z0 \* C) \1 r* U# [' Y/ r; K# U$ H2 C8 p
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    8 Y5 P5 K6 }! z4 R" k
    ' j4 ~$ [4 w; k! r) L! CExample: Factorial Calculation
    3 \- v/ @( Y! A+ Z: I3 P' m# b
    2 t4 p7 H9 F/ _' k8 DThe 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:4 Z% O1 a, e  W" ?5 v7 T

    ; J8 }; x. ?0 ^' ?    Base case: 0! = 1- r. k% ]' C* Y; {& z+ Y) f
    - `8 k% P. T) g
        Recursive case: n! = n * (n-1)!! q6 {. X5 B1 G' i' [( p; p4 X2 l

    & ~0 v$ A* Y+ _5 h! ?Here’s how it looks in code (Python):
    ; X1 q* n  w/ @9 c/ f* M7 r& Epython
    % _  T5 ?0 C/ r: f; f+ Y
    % A' I. N' N- F5 w3 b" ~' n
    0 q% c5 L; ^8 A+ c9 y+ Qdef factorial(n):& q, `, \) B) J
        # Base case
    # @6 d* T2 f& _- F) U7 S    if n == 0:. J$ p) T) j, Y+ s
            return 1) U7 N$ ]$ d  W1 X
        # Recursive case4 d' _& s1 x" w0 f7 `
        else:$ k" B0 B3 @, ]; ?
            return n * factorial(n - 1)8 r/ g- Z7 W/ V/ m, f& X! O: i

    ; \8 K! [% E1 w. [1 M# Example usage! B4 x  g, \1 P# ]- B# @/ ?
    print(factorial(5))  # Output: 120% w, o) V$ J# u" J5 h6 T

    * L) s9 p7 j9 ?How Recursion Works
    6 N) K( x: G3 a: q  T. A/ `% ?. b8 x
        The function keeps calling itself with smaller inputs until it reaches the base case.
    1 O  J; s+ s, E) S. Q) _+ O
    . Z5 ~/ j0 E2 t3 d8 q& B    Once the base case is reached, the function starts returning values back up the call stack.
    * P; ], U1 J4 r  G4 }: Z* ?9 Z( x. F3 j, v9 x" c
        These returned values are combined to produce the final result.
    $ ~8 S) d5 b! b0 F/ ]" ~# w
    + N6 p, H6 j# H6 IFor factorial(5):  n8 a6 P1 ?) z* z4 }: x& O* V
    . Z, M' ^( s) ]8 y0 j
    8 F) I% B2 H. C1 E) T2 Y# T
    factorial(5) = 5 * factorial(4)% _8 ?4 F% k% X0 y
    factorial(4) = 4 * factorial(3)( d+ @( d! T* S3 t* S
    factorial(3) = 3 * factorial(2)
    * C: @: _7 ^: _. k. K+ I# I: M4 J  q5 ]factorial(2) = 2 * factorial(1)
    * ~; F  \  B; k* _0 ~$ |" pfactorial(1) = 1 * factorial(0)5 i. _9 p% J7 v! U' S  K' p
    factorial(0) = 1  # Base case- X# o( m2 I% b8 B2 D' v
    ! S; G; A4 D7 ?
    Then, the results are combined:
    & n* D% Z7 g" q* X$ W" O$ G1 C' J5 D

    9 O, C' G1 i) x% O: ^4 C7 Rfactorial(1) = 1 * 1 = 1
    4 C1 f4 U# {5 }, a/ [factorial(2) = 2 * 1 = 2
    0 R1 m4 @1 m" Q* P- N8 w0 L# D6 Ifactorial(3) = 3 * 2 = 6
    2 j/ D) T2 I. g1 t  _, vfactorial(4) = 4 * 6 = 24' V* ^0 v" F4 x# x' B
    factorial(5) = 5 * 24 = 120
    8 ~' C) v" G, ?( q5 G( P2 C/ ?
    / J/ J8 T: O$ qAdvantages of Recursion, j% S' r3 H3 z6 u
    1 T# Q, e  Q2 e$ Q) ^
        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 j& L$ H0 N+ ~" |# A9 r& d7 v9 \; D2 h) C" ^
        Readability: Recursive code can be more readable and concise compared to iterative solutions.* e- L; Q& I* s5 J: a

    & ]' t  O5 S. l' ]6 q6 jDisadvantages of Recursion
      e( s( I7 x2 K6 W5 v1 E6 H0 h3 D
        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.4 e0 O" g) D2 w1 O2 }
    . {8 d8 b: h) y) Z- W9 }8 K) V- n; L6 ]
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    1 B1 d- Q. B/ K- Z5 J
    - {2 s, ]" @( R8 H5 ?5 EWhen to Use Recursion( _, G9 p3 |7 G8 s3 ?) b
    ) U2 x* u6 H$ a' W2 f% T+ }
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).9 ^, Y7 N# _/ k4 c

    - {' l) k2 c* y. ?: m    Problems with a clear base case and recursive case.5 L5 c9 F% A+ f) N6 p7 m
    ; C8 T7 {/ c  m
    Example: Fibonacci Sequence
    ! z/ J7 \: e) a. i5 J! A6 A2 V3 _% l+ i7 O; z" O8 w; n5 y2 M# q
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    2 k1 F6 G& ]5 p" M+ B" O4 j7 f8 P: F. {- d6 s5 d
        Base case: fib(0) = 0, fib(1) = 1' ^) [" l; [9 T. I
      O9 C$ w9 H5 e0 G- r9 O- L
        Recursive case: fib(n) = fib(n-1) + fib(n-2): f6 J* [. c% M

    4 c1 r4 ^- _6 O5 H( W' Ppython
    ( C  n( `0 J$ M0 t
    0 @) N9 Z) T6 l0 N" k& l( O/ h! ?: z3 d  J  k* T/ U, N# d
    def fibonacci(n):
    ( G* ^. w& O0 h    # Base cases
    4 c  [, ^6 N* k: ?    if n == 0:
    , i8 s, s, _& i+ a/ ~2 {- \        return 0
    * w& n9 K- s; Z' L: b1 Y) u    elif n == 1:
    . d/ Q3 @8 v- }$ W1 V! G        return 1
    2 d- o4 b& i1 p8 G! |* P+ P    # Recursive case+ I# `+ f# d. O# \. B
        else:' _0 [, R2 |1 U; ?3 N$ ~  A3 I5 ^
            return fibonacci(n - 1) + fibonacci(n - 2). I7 S& U7 O# e, c( v* W& c3 R9 W
    5 m  v' W1 ?/ l6 i0 k" S
    # Example usage! b/ {9 D1 _& X
    print(fibonacci(6))  # Output: 8' ~1 l  z7 O3 _# n1 O% p

    ( H( D) ?4 |7 Q9 f+ N( Q  o) OTail Recursion
    3 [3 e6 k( e: x. Z
    1 u' s- R- p. i  m" @8 W8 E6 {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).( A" U" d0 X5 Q! G0 D3 s
    , G' O" G5 R9 C% Q! c
    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, 2025-12-8 03:00 , Processed in 0.030197 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