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[科技前沿] 突然想到让deepseek来解释一下递归

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  • TA的每日心情
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    2025-9-8 05:08
  • 签到天数: 3 天

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     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 ( l- S& t- a1 H
    - p- M  J& w; @2 n- x# c: W
    解释的不错
    3 c/ F2 C, J, t+ p1 P1 a+ ]7 b/ u$ y3 K5 [6 T6 A/ D$ K
    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。) y9 ]: z' c! Z- w$ S- I

    . \( C' k7 c* W0 n8 e! x; R/ i 关键要素
    ) O& D5 x( u* M9 B  |. `1. **基线条件(Base Case)**: f" ~) k! g) ]( B% f7 N2 J
       - 递归终止的条件,防止无限循环# y0 A- L9 m9 h$ E: q
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1* e9 _. j2 _8 p* n4 U" P+ U) C( U
    + x, D6 \, Q! y* H
    2. **递归条件(Recursive Case)**
    3 l8 }, d' k9 W! x$ }, s, S   - 将原问题分解为更小的子问题# h( k, m  C& i4 w; U# e. r" j8 x: a
       - 例如:n! = n × (n-1)!
    / M; }/ h3 B& D4 P1 O
    + t; s6 r, \9 Z 经典示例:计算阶乘
    5 P! M2 z$ |  D/ {" dpython5 F2 B* l% }: ~
    def factorial(n):1 l$ ~! I% g- L7 r5 t
        if n == 0:        # 基线条件
    6 q' S7 T2 d7 ^" z        return 1
    / ]5 A$ G1 ?) }* j) M* i& M    else:             # 递归条件5 \2 {6 L& n$ d
            return n * factorial(n-1)( p) w1 X% }) h8 \
    执行过程(以计算 3! 为例):  `8 W; c0 r5 L4 c8 ~( O$ N
    factorial(3)
    2 J: e6 v9 s- y3 * factorial(2)" @0 r3 V5 P8 T8 x- n+ G& A  Z
    3 * (2 * factorial(1))3 e! @8 W( O: j$ H9 m2 [
    3 * (2 * (1 * factorial(0)))) [( b% e+ f# h& j7 q& k
    3 * (2 * (1 * 1)) = 64 I8 @& f& O& F+ s: x, Y% \
      R( x  {  k& x) l; G. ]( I" C1 F
    递归思维要点
    & p! b' T& F6 J2 z5 ^# P1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    . F& b! l5 l8 g5 O, ^: f2. **栈结构**:每次调用都会创建新的栈帧(内存空间), M, Z+ T2 f- y( |& H' q
    3. **递推过程**:不断向下分解问题(递)' d2 |5 z7 _! b6 c/ }
    4. **回溯过程**:组合子问题结果返回(归)
    8 n! e5 i* J2 G6 ?" F3 A: a" A  G
    注意事项
    ! K! z& H1 I9 t; \8 T- J' z2 A% G必须要有终止条件
    $ b: f. Y8 B- {0 @递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
      ^" `* ?9 n  {# j某些问题用递归更直观(如树遍历),但效率可能不如迭代- u! ^; M' ]1 Q
    尾递归优化可以提升效率(但Python不支持)
    ; }+ Y6 {; W+ T2 a8 o, E4 B1 {% J$ d+ k" r
    递归 vs 迭代: g; t9 v6 l% E; c) Q9 f) k+ H
    |          | 递归                          | 迭代               |
    0 }1 ~2 P+ c- o; e|----------|-----------------------------|------------------|
    ) M7 i. d3 _" Z( o) U9 Z" z1 `| 实现方式    | 函数自调用                        | 循环结构            |' e; P6 |4 B* {, O1 _( [! r# b
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |$ O) r' }; K/ ]4 ?. k" h
    | 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         |
    6 v8 P3 q/ v) K' \! J/ k| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |* q# o( [# Z4 b9 [3 u- P5 G

    " K$ X4 L4 L* g9 p' ?# u2 m4 ? 经典递归应用场景( A) F% K, q6 Z! `9 B& ^: U7 \
    1. 文件系统遍历(目录树结构)' R3 p  f. c# x. J( P8 z
    2. 快速排序/归并排序算法" Y' S+ e! i$ ?0 E" L3 E
    3. 汉诺塔问题
    ; V/ g' y: N( L0 [+ w; r6 ]4. 二叉树遍历(前序/中序/后序)- T( I* r; F" r% e" V* t
    5. 生成所有可能的组合(回溯算法)
    4 ^3 l+ {: K2 Z' z: |4 F
    ( Z1 s  M# f0 N/ B; \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。

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  • TA的每日心情

    昨天 06:44
  • 签到天数: 3104 天

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    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,
    ! `5 g, K- p. I4 |' |' v- V我推理机的核心算法应该是二叉树遍历的变种。
    3 u9 t- D. W% o, i/ b% d" o) g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。
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    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:8 J3 B1 _- g5 C- x
    Key Idea of Recursion0 O8 [( r2 m# A- h

    1 C( S* a! K+ y* FA recursive function solves a problem by:, ~0 H9 ]# E$ @( t

    % X, k) x" S% {: H2 J$ R    Breaking the problem into smaller instances of the same problem.
    & }# H; I. @4 P# v, J3 N1 l1 O9 z2 y
        Solving the smallest instance directly (base case).
    + f% O0 |0 L9 v$ O; y
    0 u! z( k9 d- R: P- `6 g    Combining the results of smaller instances to solve the larger problem.5 z/ S; R) t7 V

    7 V# ]; O8 p3 d, h, w* B4 fComponents of a Recursive Function
    / O' U( X  b) A% I& q3 p
    - w* T9 k' U' I    Base Case:  S: u: s7 [& K! g$ h( {) S( D: T

    - |0 e" V- R; q5 y# g6 M! j        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.% x+ d4 ]# `$ H3 R: u$ _0 e$ `
    * f, A; z+ X. |! e
            It acts as the stopping condition to prevent infinite recursion.
      {) C6 |) p" C6 |) |1 c
    7 Z: h& C. R; n* G( b% P        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.2 A- d" `, b+ M, {8 O8 h& r
    & [# ?/ D5 x9 I" c7 s: y
        Recursive Case:: N7 s: n- H1 \" f6 M+ Z8 K4 l
    " q" ~; _; i0 E$ ?" ~
            This is where the function calls itself with a smaller or simpler version of the problem.( R1 G+ v6 X7 a$ W0 i

    8 e" Q* B3 y4 [0 ^        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).: r/ j. F5 A9 `7 l) h! {

    ( X+ S; Z8 e/ s; `" A3 t2 AExample: Factorial Calculation1 O7 z4 K& X7 B) c5 g2 y$ ^
    9 G0 i0 |9 J# P/ w5 @
    The 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 f( h$ Y# I0 X) W- T/ n) f* x

    6 {; X: Y; W% E9 M' s! W0 O& [    Base case: 0! = 1" }3 F. M+ @; N  d! i

    - R3 q( J' @- {% P' A. W, w; Q3 Z    Recursive case: n! = n * (n-1)!4 j. l% L" A: z8 W

    " j, y; \+ D8 Z7 ?* U" yHere’s how it looks in code (Python):
    7 e3 m; M) `( e' A0 h3 c# ~3 ypython0 Z3 J" o8 I2 ]' o! {7 Y! v

    * d+ m- P% y7 j! `& ]$ ~4 ?, M4 M0 }  I; C
    def factorial(n):) c5 R& d4 I. z  C
        # Base case' \  B+ f! \9 {" {0 H, w
        if n == 0:
    $ w8 _7 Y; f% T        return 1, Y2 k9 q8 \: r* j
        # Recursive case2 u' N) u( z; W  R/ V
        else:; N5 f; s  k: P
            return n * factorial(n - 1)# E! r7 @3 r1 O5 l: f) q

    : }1 I- `# k- T& v9 L1 E# Example usage# `5 p1 z* N, c% H
    print(factorial(5))  # Output: 120
    & O9 {# ]* T& q* K& `) s: K/ u
    . K0 g+ @+ k6 R5 NHow Recursion Works
    # K4 y8 q8 @+ V; j* l8 K  u; D# z1 r0 j! H
        The function keeps calling itself with smaller inputs until it reaches the base case.5 N) V& `5 ]& N' v! e9 h/ _3 F' {

    4 Q) o' R! m8 ^3 D& R6 a    Once the base case is reached, the function starts returning values back up the call stack.: c7 O* _, {  _1 n8 @

    9 E/ A. M! Y% O! p8 Q* M    These returned values are combined to produce the final result.
    0 w+ R' X4 R/ I$ a: y: B# G7 D! S9 `& `/ B* R/ O' L# y, m! o
    For factorial(5):. q, T( z/ p# x' O$ s; Z
    . o4 i* q: q( m9 M2 A9 x

    ) u& z5 O7 B$ ^* c8 gfactorial(5) = 5 * factorial(4)' D8 j2 }+ S4 P% Z
    factorial(4) = 4 * factorial(3)
    3 f$ i7 F% f& J. _! b1 R$ T6 Wfactorial(3) = 3 * factorial(2)9 y; k5 Z6 N! ~$ o
    factorial(2) = 2 * factorial(1)) h% X0 y. z; }+ l
    factorial(1) = 1 * factorial(0)
    3 R4 I: m5 t: W8 t; x+ S6 |. cfactorial(0) = 1  # Base case
    & |% O% Q/ Q. e1 r/ O
    7 Q- i6 B' e5 k# u2 HThen, the results are combined:+ q3 w: v/ e2 D- G* R( \

    / }# U, g9 m) C; j8 k; m5 `$ V" ?+ |+ _# a3 I' y4 [8 {
    factorial(1) = 1 * 1 = 1: j6 \% m. _) O6 Z
    factorial(2) = 2 * 1 = 2; n" ~5 q% D) o  M$ [
    factorial(3) = 3 * 2 = 6& a" M0 r$ v# r  l5 y
    factorial(4) = 4 * 6 = 242 K0 Y( u: ~- v/ \
    factorial(5) = 5 * 24 = 120
    * K& l7 E/ {$ `/ S0 L2 c, [* y. k7 s* Y4 y3 ~4 t! D' O0 A1 O6 A
    Advantages of Recursion
    4 Q- L+ T5 X2 h9 O1 k6 M: d% f3 F, {5 M9 }: a  w+ V/ R3 O
        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).$ p) G& D7 ^  c
    0 I) B) j& H8 b& S
        Readability: Recursive code can be more readable and concise compared to iterative solutions.8 ]+ V4 w: V9 d

    2 X1 T9 H$ s0 a7 |( CDisadvantages of Recursion
    8 f' p5 O9 m8 Q$ W; q0 t2 w' x1 Z1 R. ^3 k5 ?
        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.; Q# z# \7 D0 w8 c% E$ M
    7 _# Y7 {2 v7 t% I$ g. a, P
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    3 t9 s# n# T2 g5 Q. f
    , }) [" F) s2 y; c$ F3 u7 wWhen to Use Recursion& b8 U  e, P5 O* l$ M
    0 \1 X, F% O' q2 ~: y1 l+ u
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).* y5 u, Z& @% h* Y* C. K
    ' `' ~9 J6 ^! v- z2 N3 j
        Problems with a clear base case and recursive case.
    : D6 {0 s* `5 y! I3 ~* b; {  n1 {- O$ S* |7 }* r7 t0 w3 t- x
    Example: Fibonacci Sequence
    " R) _0 H% l- M% V% t0 L! @
    * Z2 @& a4 w: t3 k3 PThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    7 k/ t& a$ k. x3 R/ g, u' D* }: j6 x
        Base case: fib(0) = 0, fib(1) = 1! y* ]$ q& {4 C( r% ~- H8 }) r
    ( t( L$ c1 m; d+ @: }; ~' u
        Recursive case: fib(n) = fib(n-1) + fib(n-2); x% g' U8 S! E: E2 Q

    * ~& u. z7 B* H3 u$ F" ?% vpython
    8 ]/ i# J+ {9 o0 O5 w; E- W: Y) M1 `7 d9 r2 s5 L, S

    ! ]2 I+ k* W# v3 S- o# Zdef fibonacci(n):# S, l) p6 Q. v1 c! @$ p- l
        # Base cases
    9 @' ]( ?4 ~* Y% _    if n == 0:/ b' [( z" F" c
            return 0
    , L$ {/ n! C2 [* }) T    elif n == 1:
    - I! E# |! _2 v( e' @) L        return 1" _2 N! g6 T, V8 J
        # Recursive case
    " I( E. U( ?7 }' A* t6 K    else:* M0 Z  s. \$ U; W4 t2 W; S- g
            return fibonacci(n - 1) + fibonacci(n - 2)
    ( U3 s# z& C! M- E5 x/ e0 Y% d
    / ~% \6 [& l! M7 ^/ b" ~# Example usage
      K$ y' U# {7 a( C) g: q7 eprint(fibonacci(6))  # Output: 8
    5 `2 z: t% ?& Q1 u. s4 L" s0 _' b( [: h" p) B0 c) c+ q
    Tail Recursion' u+ @9 y2 p  I' [$ c
    . ^$ \$ K: J4 l
    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).
    0 [- d1 N8 o0 E4 V* j! ~
      Y- ~& ~" M; Z0 `. }. X1 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.
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    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。
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