设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 2182|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 , Z8 q  n8 G. d6 B5 f* K. |2 I
    # G! o1 l  [8 |3 W
    解释的不错
    2 m0 V2 |8 g" w1 P$ w+ x, O; h6 ?4 ]! ~- Q
    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。
    9 l1 ]( p- N& e: o
    8 {% I( R- z. f* i2 @0 L6 f% ? 关键要素$ n& E. U3 V3 q- |; r$ @
    1. **基线条件(Base Case)**+ Z5 O7 i8 k4 V' v! {# x' H
       - 递归终止的条件,防止无限循环
    ; g* ^/ H( V5 |8 ]   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    7 z6 `9 h- S2 c0 j8 N& q4 c) a: @# O
    2. **递归条件(Recursive Case)**/ P" Q& N, A2 X* n+ @3 z
       - 将原问题分解为更小的子问题1 _: k$ t! J( _# Y) \1 g4 Z
       - 例如:n! = n × (n-1)!+ }7 d- p2 [7 A, U$ M& T

    1 ?) p; \: ?0 F! V: o, u 经典示例:计算阶乘
    4 @1 F* M, F/ F* Xpython
    9 r0 Y. j2 _3 c, h1 I( u9 ]* ]def factorial(n):2 D0 o2 [6 ^! R
        if n == 0:        # 基线条件
    & h! s' p  O) t' f4 L, I4 N& N        return 1
    0 u2 B* ?$ A- z1 E8 H* k% S) t    else:             # 递归条件$ B/ `* X+ f; O' Z0 S; o2 \
            return n * factorial(n-1)
    ! f2 N7 i, z& i- O# T( H执行过程(以计算 3! 为例):
    * J2 X5 {. V, V! r% r0 Vfactorial(3)% f. C- \9 i' m: K$ k* _6 P  R
    3 * factorial(2)8 i: B" g8 Q+ h) H
    3 * (2 * factorial(1))
    7 e, q" D( G" \4 j" M0 t3 * (2 * (1 * factorial(0)))
    7 w; n/ C# L% T( q3 * (2 * (1 * 1)) = 6
    4 |: L) B$ R+ d% I6 c: H. k
    $ U' }) ?# u0 G) J; {3 x8 h 递归思维要点
    8 D" C# S' p1 W% `4 T1. **信任递归**:假设子问题已经解决,专注当前层逻辑  R2 B7 W" h# ^; g% h
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ! q5 I( f! G) e$ M& m; R! Y( l- @3. **递推过程**:不断向下分解问题(递)
    3 g3 h7 F' e( T7 r+ D' g4. **回溯过程**:组合子问题结果返回(归)
    2 ^  Z5 x1 h# `8 o" N6 Z8 d  m2 d% Z' `' |
    注意事项' U4 @/ N8 J% X
    必须要有终止条件% j5 [$ _9 ~% R4 y; Z; {
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)# B3 ]7 R+ K, H9 a3 h
    某些问题用递归更直观(如树遍历),但效率可能不如迭代  F- s; h( X5 Q+ d
    尾递归优化可以提升效率(但Python不支持)
    " e: \' e, E: M% y6 A
    & o6 m  i. a0 E3 [# _) E" G 递归 vs 迭代
    - q* f: u9 ~; Q% N8 D- R  N* x0 G|          | 递归                          | 迭代               |
    # H3 Q3 M/ e6 Y/ \4 t9 S|----------|-----------------------------|------------------|
    1 G7 A' z4 _& ~1 ^| 实现方式    | 函数自调用                        | 循环结构            |) X+ ^- F) ?1 O; m
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    / s' M) C. e) v  a' m| 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         |! J/ d7 r! D/ `( p
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    4 T. I6 }1 f0 R# r6 I- o$ B
    ' t9 u: g& F0 J' q 经典递归应用场景3 L* L' Q# A0 k! y. Z
    1. 文件系统遍历(目录树结构)
    3 s* f; Z3 w( z8 [. a2. 快速排序/归并排序算法
    4 Q8 f6 t: ~( i% `3. 汉诺塔问题
      m) E8 W7 E8 ^$ R8 D4. 二叉树遍历(前序/中序/后序). m' b  @( J' \
    5. 生成所有可能的组合(回溯算法)( a: W, J) }" g( k) {- P1 H% t; ]- J

    # D# g% i0 S$ 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
    开心
    10 小时前
  • 签到天数: 3188 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,
    # f3 x9 E/ T  j; Z我推理机的核心算法应该是二叉树遍历的变种。! H; ?3 i+ W* _; b( N
    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:: |, Q; H) g! f5 o
    Key Idea of Recursion0 h6 ^+ I& V4 c# a5 W

    / b: C  Q5 f: a3 z0 v! TA recursive function solves a problem by:
    9 W- y, W# W7 [( |! N# D7 x# a5 ^
        Breaking the problem into smaller instances of the same problem.
      m  v+ t2 r- U0 P& b1 g1 K* S* k; v  u' _' j0 f. s1 F0 G% k
        Solving the smallest instance directly (base case).' d+ c& }- X2 n! }, \' j9 X7 Z
    ( z. {: r6 u" Z: K
        Combining the results of smaller instances to solve the larger problem.* |( v7 Z, I9 s) i4 w! t6 j

    4 @- q$ ]1 A2 Z; Q6 J+ @4 `: cComponents of a Recursive Function& x  c& k/ y$ \1 R

    " k5 R' `: t/ Z* f6 j    Base Case:
    4 x" N$ \" e0 g
    0 L3 Z* I. ^' N: y9 h/ D6 G3 J$ T        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.9 X* z. J/ F1 f$ e( [0 E' U
    - u- Z; `! h+ S' H: q, ~# j
            It acts as the stopping condition to prevent infinite recursion.* X4 w3 F$ i" {6 A, T

    2 U8 W* O9 r+ a. x$ n  {& s8 H        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.# N5 r: r/ M8 U( s8 H! R5 D: m
    & r) ~$ S- y  y( s/ |5 m2 X; X
        Recursive Case:
    ' h1 o' H5 O$ \! i% |2 {; C' M5 l7 e) S5 |0 _
            This is where the function calls itself with a smaller or simpler version of the problem.
    $ P) o) O1 p2 i$ U) ^8 z
    5 C  K8 G7 x# |2 n8 L! e        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    2 l9 ?; p3 |$ v" d7 `5 w# b; g
    # a# g* r) m3 E9 `Example: Factorial Calculation$ @$ Y) U& Q3 j8 ?; J9 ]  V

    2 T4 ]5 y/ a) L& J% 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:
    1 Q) }1 |3 B! K
    , g; F8 O7 S8 [' O; ^: Z    Base case: 0! = 1- e4 o! O* z9 h: t- V* ^& S7 d

    : u9 N) a8 c7 {. I    Recursive case: n! = n * (n-1)!
    ! N* I) k. |+ d! |" w. }
    % V- _: P; K7 Y8 [- CHere’s how it looks in code (Python):
    , d! _. D2 c, Vpython
    ( T' Y0 M3 V9 p3 `! C' B. T7 u% u& ~- l1 o8 k. `* f

    ' p" D# f3 K) m$ p7 N7 I. E* u+ R7 Kdef factorial(n):* i/ ~% b7 h/ ?" C
        # Base case! z" g: x( Z% N
        if n == 0:5 g- T3 G" G+ H+ d
            return 1- d/ H' R; F/ J9 n6 w
        # Recursive case# ~. D9 r2 g. E( V0 ]
        else:
    ; P4 m) n- A$ `$ Y& N9 f        return n * factorial(n - 1)8 }; N% i+ c) `* q" _' w" O; n$ o, a

    5 \! V0 s0 [9 c$ }2 V' A# Example usage
    ! r6 t* e/ v& Wprint(factorial(5))  # Output: 120
    ' ^6 I2 W# p1 o  x
    + w3 `& P: Y9 e: Y1 X' G0 DHow Recursion Works' ~, d- l$ I1 l- L+ {
    - O; }+ p% ?: ?# C' O6 K; B. `
        The function keeps calling itself with smaller inputs until it reaches the base case.
    / J% n* C: d. ~1 w( @4 p
    7 X) [% F4 [+ L2 k) {5 P    Once the base case is reached, the function starts returning values back up the call stack.) e0 k6 h- g3 `
    3 Y6 h  n+ D/ s( _$ l: ^3 ~
        These returned values are combined to produce the final result.. \# \1 n- s. u' V

    ( j, e- H* f/ G; w3 l/ R+ _' N0 q3 _For factorial(5):2 J, e" Y3 P3 S' D' K: y

    4 k' u" Y0 s/ l/ y
    8 I/ P8 b$ _2 h0 Ufactorial(5) = 5 * factorial(4)
    / }( d+ U* {- o) X' H$ yfactorial(4) = 4 * factorial(3)0 e$ A5 K4 K8 v3 B7 t& f" ~4 X
    factorial(3) = 3 * factorial(2)
    2 c6 t. p  T4 h5 L2 qfactorial(2) = 2 * factorial(1)+ u' L: ]) x$ s$ _8 j0 l  y2 B
    factorial(1) = 1 * factorial(0)
    " K" o( b; g  {0 Q- y5 Zfactorial(0) = 1  # Base case
    . R6 Q9 S7 s4 q! a- Q0 x0 C6 y3 ~6 r( g' g
    Then, the results are combined:
    * e5 Y' X! ^6 n6 p/ w* ]" i! A9 O3 B/ F7 k6 d

    " j% `1 u; I' n" j( S/ `9 sfactorial(1) = 1 * 1 = 1. i2 z6 ]# l4 a7 a: o) Y
    factorial(2) = 2 * 1 = 2
    ; ]! s' `& F% Z, a1 Wfactorial(3) = 3 * 2 = 65 _9 x, r  B- B  u+ F# S4 l
    factorial(4) = 4 * 6 = 240 I* ]0 u. M5 {3 A1 n* A
    factorial(5) = 5 * 24 = 120
    4 g. Q& f; x8 e0 L# B: b# ?1 p  w( C/ r# W  T5 i# \( O9 n7 L/ d: _
    Advantages of Recursion
    ! x4 z. @* M" V9 q' e( b( Y
    + {( t( W4 Z1 t5 B    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).
    " ]% I# y- b* z, a( L; ?9 k3 N1 R/ d* A- x- `
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    # X. r% n' L! G4 R$ T# d: Q
    # E; c9 M& i/ X) ~0 }$ U/ y5 N6 Q+ @' @Disadvantages of Recursion1 \, X- }# p5 f: Y, G# K6 d

    5 C2 a) y+ ~, T; P7 v; E" p    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.3 O2 L6 [* G- B7 H9 J/ X$ {& @
    * o0 w2 Y) H- c- c4 W0 M
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    % {2 g1 V1 m9 e/ ?- S$ g! `" N
    6 v: |3 C% G' B- d% {( f+ YWhen to Use Recursion- z% T0 o( Q7 Z& u  k9 A/ z7 Q

    ) T% T: d3 y; G' Z* t& V0 x5 O2 s    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).! W3 U" E: }6 y4 [1 s4 y9 E, }
    / E0 W: T  a; `; u: v- B
        Problems with a clear base case and recursive case.
    : t2 B6 @  m8 z9 s3 a6 {
    + ^$ ?; A5 U, Y* J+ ZExample: Fibonacci Sequence
    1 f1 u. \8 Z. d1 P  V4 O$ @5 i, }- `
    # s( {" {7 {( d$ f) w9 oThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:( A1 K- l9 K$ C% [6 w4 B

    & U/ g) L4 e$ _4 f- u9 ?$ h/ n    Base case: fib(0) = 0, fib(1) = 1
    , V9 e; Y- p) M) U
    # T, l/ N9 f7 C' [! o    Recursive case: fib(n) = fib(n-1) + fib(n-2)6 U# Z- c/ V7 z% q! A2 b4 b3 z6 C

    2 _6 t$ \% k- Y( e" q* Q/ dpython
    - c% e) E8 U0 ~- g5 [; [7 Z9 z# i/ Z# w2 V  _
    # E7 y0 m' J* v6 x* G
    def fibonacci(n):. n* U$ J! \; [$ h$ z% F" M
        # Base cases2 W4 V" [$ ?! k
        if n == 0:
      |7 @, M2 J1 o1 y; B- X        return 0
    6 j% ^  R7 V. w  v2 n8 a. d    elif n == 1:: P7 ]9 E; \$ }
            return 1: `3 r  U% C" {
        # Recursive case, _' p" _# k! W8 N! w
        else:
    7 Z* o' f7 @" [) L. w4 G        return fibonacci(n - 1) + fibonacci(n - 2)! R/ N% O: ^# I# D# G

    3 L- W# }' R3 x# Example usage
    ' Y# M5 @; x4 S' J- q% a# z9 Cprint(fibonacci(6))  # Output: 82 O* Z7 y8 z- s! d4 `. g
    3 D% r. ~0 N! k( C$ ^6 u2 I' Z
    Tail Recursion
    0 X' e8 t! H7 B1 W, D5 G/ M% G9 S! O0 o1 ?- x" F  h; }5 ^) r' S
    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)." R5 E, V( M& b/ x

    / J7 i4 |5 G; }5 G. BIn 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, 2026-3-1 10:14 , Processed in 0.070031 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