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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
     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
    + B7 [9 K6 y# q; W5 ?) l4 }  |- }% h# E+ }8 h+ O1 C
    解释的不错5 {8 J. W' j/ ^! j- M, e5 Q3 q' y% T

    9 B' x) N' r; B# @: O; G( v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。
    4 y/ G8 ^2 Q4 b" X8 M4 ]; @
    2 O- x7 K. g, N, p 关键要素
    0 N: ~( ~0 y0 P4 i. x+ d1. **基线条件(Base Case)**
    * r3 u# S9 h) _7 L: o$ @  v   - 递归终止的条件,防止无限循环
    + P$ V5 q. e3 W) h2 d& u6 \   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 16 C+ ]8 v( a4 z) m9 b9 V& L

    % U! U* i1 S6 Y* q0 p0 X4 ]' w2. **递归条件(Recursive Case)**" s$ d( V0 J  [0 Y% Z& I/ B9 z
       - 将原问题分解为更小的子问题
    + X. `/ W( K9 c4 Y   - 例如:n! = n × (n-1)!
    ! D2 l5 [5 k6 ]8 p; j! K( Q; ~% P" Z2 U+ r, t5 k9 t: A
    经典示例:计算阶乘9 A  q( s: h) X$ E5 x' k
    python/ G$ |9 }3 m/ ?% k
    def factorial(n):5 L. l$ Q0 h4 \3 X, N1 M
        if n == 0:        # 基线条件
    . U/ D: c% i- R5 k0 S6 F2 l3 C        return 1- ^, l- G% f* X5 F9 @3 N
        else:             # 递归条件/ Y' k$ j  W! u" b3 y
            return n * factorial(n-1)
    6 w& C) s$ |1 i- ~执行过程(以计算 3! 为例):
    * `. ?3 j+ A1 d$ A0 T& e9 X* rfactorial(3)
    2 R, b9 R8 r* ]3 * factorial(2)( B9 l$ K/ z9 ?9 ?  j6 Y( [
    3 * (2 * factorial(1))4 c/ Q8 i6 w6 U. B
    3 * (2 * (1 * factorial(0)))
      u( j, X7 \8 V. d3 * (2 * (1 * 1)) = 6# G$ |6 g; t% f, g4 u* r

    / V8 J# h7 [( ?# ?" {" t 递归思维要点
    7 K  S, C, f/ o  g3 L. y1. **信任递归**:假设子问题已经解决,专注当前层逻辑8 z1 i" k  \  ^2 r
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    : j5 _# j- \5 \" o3. **递推过程**:不断向下分解问题(递)/ P" V# V' w7 m" \7 g3 T
    4. **回溯过程**:组合子问题结果返回(归)
    9 l* ^" g9 E2 u- ^
    # X6 W$ F' W3 y) i1 L  ]- E( L 注意事项
    6 F) x# U3 I- K2 A; f6 {. p必须要有终止条件  g6 ?: j2 G" `8 l
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    % M6 M3 [2 ]6 |; Y9 G2 R某些问题用递归更直观(如树遍历),但效率可能不如迭代
    " [7 }* A1 D/ e尾递归优化可以提升效率(但Python不支持)
    " c" M- ^# m7 a+ R6 F
    - ~/ `' h4 ?( _# V- j4 b5 N 递归 vs 迭代( V' W3 r8 T8 X( S( d- `( h
    |          | 递归                          | 迭代               |
    / Y5 E) i/ ?. E6 y' O|----------|-----------------------------|------------------|
    ' n* h4 Y$ g3 R( T" J: b| 实现方式    | 函数自调用                        | 循环结构            |, b, `) h: R4 F  @+ F) d
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    7 D6 Y1 m+ k8 V| 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         |$ Y: k2 k7 W: k1 F. F/ m
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |  e" W5 L4 q7 `% W, B
    2 x0 f# y; z7 x7 E
    经典递归应用场景
    : R% w6 ~: e3 \- o6 W% M1. 文件系统遍历(目录树结构)
    & x/ f' J1 R+ o7 e2. 快速排序/归并排序算法
    6 H4 p, m5 Z7 p  X6 E0 }, n3. 汉诺塔问题
    ( |4 f2 @  v# M- B. ]$ x! ^  u0 B4. 二叉树遍历(前序/中序/后序)
    8 z7 {& |" U; O: t5. 生成所有可能的组合(回溯算法)
    1 p6 i2 T9 h; E- C7 T' J* m- g0 d, T) F4 n- P8 W
    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
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    前天 01:20
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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,! J5 ~' j1 }2 @( D* g
    我推理机的核心算法应该是二叉树遍历的变种。: ~: s4 ?% l1 D- @. n" A
    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:
    9 S/ Y3 r5 z3 M2 mKey Idea of Recursion; w) C5 ]! A' y" [

    % m5 S. }% e; j/ \! V  jA recursive function solves a problem by:4 S% l. ^! w3 Q2 T. Q7 a, n

    ! P4 Y" }$ [. t" j& q0 L    Breaking the problem into smaller instances of the same problem./ |$ H7 @, s7 y" \: ]
    + g' ~9 N* _& M( X" g
        Solving the smallest instance directly (base case).0 T* Q# W5 \0 l- Z

    ( E& P; n" j3 Z* F    Combining the results of smaller instances to solve the larger problem.) m0 Q3 x6 ~% k# G5 o, \6 j

    & k/ r# g& W7 L3 [* e0 g6 pComponents of a Recursive Function
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    ) _9 ^" I5 x" F- Y4 v0 C$ b    Base Case:* W: }, Q  U6 w# w) x

    7 I$ j) e" o& W+ G; P/ e6 a        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    * e$ \5 {( ?7 B/ B- n/ p! Y: o) }- N! W: a
            It acts as the stopping condition to prevent infinite recursion.
    ) W8 V0 y/ ^9 [0 B* I. I, u# v4 E  f" Q9 ?5 _
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    $ k9 k# ]; Y( k% m3 \. t5 |9 H+ q+ z' w; O. A. a& d
        Recursive Case:
    : v; E: J; O% w  [, L. o( g, K9 J1 d4 g# S
            This is where the function calls itself with a smaller or simpler version of the problem.3 q* M, t7 {( G, B( |  ~, w5 K

    3 M9 |2 L; O0 I( x$ i' r9 g        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    * e3 r9 {- s. E9 r- }( Y* n3 @) q' @# z; v' U. P& q
    Example: Factorial Calculation! I; f# o( m: z+ Q$ _8 Q
    & x/ [! v0 R& C
    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:
    9 |( ]+ m2 |/ u7 S. o5 R* ~! U  ?; K8 S
    ! F9 f  m9 S& z* V# s/ J6 n    Base case: 0! = 1
    6 Q+ `! }7 W7 a
    1 G$ F$ ^+ R. `% p    Recursive case: n! = n * (n-1)!
    $ v$ [7 Y9 ^! O. d8 ]8 u7 Z# [; I  M* c, ]0 ~  w5 A2 {
    Here’s how it looks in code (Python):5 \- ]* Z+ T" g% N% O! o
    python
    . i3 E& z  [) S8 a3 k0 i9 Z' f9 g( g1 e& Y# H7 y3 T
    9 J8 b4 b, M; V8 ]: m- h
    def factorial(n):9 h( C+ e6 c" @! c
        # Base case3 S" T$ [2 j5 I8 U
        if n == 0:
    2 W! x  e  x; |" g- f        return 1
    " h0 @4 h# e+ E, F. U  n    # Recursive case
    * h" U# N  V  C: b    else:
    2 |9 N# u5 u0 i+ z3 ~        return n * factorial(n - 1)& h( Z* L9 l* o8 E# Y7 ]
    * ^+ n& X3 V) r' {+ {! Z4 c+ {
    # Example usage
    + T5 C+ C( ~3 W! u# Oprint(factorial(5))  # Output: 120" R( I4 `+ e) W  e4 C+ J) J
    / M: N& Z3 X0 H& U+ Q5 ?
    How Recursion Works
    ; N6 N( c- n: ?  ^2 l0 y  X5 v9 Y+ c" c+ l% @5 h
        The function keeps calling itself with smaller inputs until it reaches the base case./ n+ }3 M. M' r2 K! ]3 ^$ u
    / z# r; X: Y' Z  t8 `
        Once the base case is reached, the function starts returning values back up the call stack.
    ( t$ a: R5 j5 y5 C+ v( ]& h/ g+ y/ A7 ]
        These returned values are combined to produce the final result.5 p( T( D( e8 F' T
    8 t0 N9 @) {; _5 g* z# V
    For factorial(5):
    : L, X- w/ g5 O' y7 ]' H: c9 `1 \- B' v6 L( B
    ; U" R& I6 f) A5 u: `# t
    factorial(5) = 5 * factorial(4)" R0 ]* s6 P5 H7 m) f
    factorial(4) = 4 * factorial(3)
    4 V! G4 z7 e: [3 L$ Dfactorial(3) = 3 * factorial(2)/ `) s/ |2 A- m3 w
    factorial(2) = 2 * factorial(1)
    # Q- Z0 z( ^+ X/ M7 }9 g$ M2 q0 bfactorial(1) = 1 * factorial(0)
    8 ^) P9 w3 e& _, D3 |factorial(0) = 1  # Base case
    6 L; T9 d+ a! k1 y1 R% u
    7 F. T. P7 K+ YThen, the results are combined:
    1 r& A4 |5 Z! [8 U: O6 f4 r  \7 l/ f) r4 T
    - d! X, U5 L4 M; [
    factorial(1) = 1 * 1 = 1
    0 ?. M) C# K5 A# u8 D. Cfactorial(2) = 2 * 1 = 2
    4 h% O0 G! z& kfactorial(3) = 3 * 2 = 6
    4 _( m" b! Q2 C8 ~& @( b# L& Qfactorial(4) = 4 * 6 = 24
    $ \4 H) p& O, g$ p. q2 ]factorial(5) = 5 * 24 = 120
    5 T0 R+ }# \* e! N! q$ Y- }1 f) F( @# k/ p7 S+ {
    Advantages of Recursion: _/ u" U& O  u7 T2 w/ a2 q

    9 {0 ^  @0 S* M. k& ]7 R. h; |    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).% {  g% I, Z5 A* K

    ( N3 Z2 M: Z! {; R7 T5 O) F. k    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ( ]9 T8 Y/ v1 m$ ]& A% i6 H& l  \) c# u
    Disadvantages of Recursion
    ; l+ d5 R% i) o
    0 D) w* A, e$ w  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.
    ( o4 I2 q/ t- J) q( A
    , Y" S* |8 |/ u+ h" l    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).! [- r1 u3 [3 u8 I
    * f3 y; }/ Q2 u! {5 @! e. \
    When to Use Recursion
    / ?6 \6 H3 p- Y  o* @' M
    * I4 T) O$ J  H    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).! J' Y& E* K* C) j( ~
    ! B9 O; O0 \$ E2 B  ?1 ]/ r& ?$ k
        Problems with a clear base case and recursive case.: h* w, {4 ?3 F* f0 y  O) V
    & ~' G. B' K. I6 u
    Example: Fibonacci Sequence
    " w7 E% P2 f  e% F9 a3 ?7 @! Y
    1 W7 i( [7 G) q  r; |) Z6 S+ }( ^The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ' r  c, a' n% \  e5 @" O( ?# M0 R6 J+ [0 J# ]0 U+ h# ~
        Base case: fib(0) = 0, fib(1) = 13 A) X/ {* }( O( P1 D1 H

    1 Q, p' y/ z' u+ l9 y5 L. C) M4 I    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    + o" r+ [  B& r/ b% Q4 E+ Q- v5 B
    python
    5 [) @1 U/ O) k% N) N2 ?+ a7 O- v  D5 f. t, T" e) A

    2 j# o* g' A; {7 c6 ?$ Q, {def fibonacci(n):+ G2 c# K: X( `- Q
        # Base cases, B$ q1 @" ~7 Z6 M
        if n == 0:
    8 K8 ~* S6 D6 s( C) D6 m( i        return 0
    & Y$ M9 j& V$ p/ \4 p    elif n == 1:
    5 `% u6 U$ x' z        return 1
    $ ]' p1 h) L" N9 G  K/ T$ s; w    # Recursive case
    6 A# `4 A6 q! @! @    else:
    ! V; I8 ]0 L) i) s5 S/ i        return fibonacci(n - 1) + fibonacci(n - 2)! U! I. E  z' r* p, Q: J

    9 y9 R$ Y3 ]1 @4 N# Example usage
    # Z! t& V2 M( D9 F6 W3 n6 @/ }print(fibonacci(6))  # Output: 8
    2 n  H5 v) W. H$ [" U
    7 q& H( A% k: N" l" i+ |3 nTail Recursion, S1 W+ O. Q0 |2 {+ v' L& P

    4 A: `  F# D3 _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).
    $ p: g0 M! B( `  B8 ?. H$ u- F4 p6 T* r7 l! X5 L/ K4 `! F, e
    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.
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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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