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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
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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 ) N5 _' L" D( ~
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    解释的不错' w' O, D" l' ]8 \
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    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。% z" @% }. I9 P% M" x" A6 t
    5 w% Y( g1 `  A
    关键要素
    # S" ]) Z/ F- T* W8 I1. **基线条件(Base Case)**
    . H6 p* i) T3 ]" d   - 递归终止的条件,防止无限循环2 |. e4 |( V+ i4 ]. ?' d% p
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    . V8 ?, W* c) O/ e( w1 s# T" R3 A1 w* s+ Z& |
    2. **递归条件(Recursive Case)**
    4 }4 [' W2 }9 i: N$ n   - 将原问题分解为更小的子问题: u$ R% V+ u) ]1 Z$ R# ]
       - 例如:n! = n × (n-1)!7 O5 ^( z1 f' m+ a3 _, B
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    经典示例:计算阶乘
    . G$ q7 o' a6 Bpython
    % C& _9 k9 h3 U: ?& H: ?def factorial(n):
    ( t# D( M- a) W: B    if n == 0:        # 基线条件* F6 y3 h6 G) I7 q+ w
            return 1; U1 p  _) X; x" y5 J, R
        else:             # 递归条件# q& U% V6 B/ K2 J# ?& t$ R" T0 I
            return n * factorial(n-1)
    5 g/ t0 W7 B9 O" A/ Z' x( W执行过程(以计算 3! 为例):, ^* f! b% L0 P
    factorial(3)
    ; u9 \0 E% [& W1 _3 * factorial(2)
    ' Q& g9 }0 z9 S4 Y6 l0 B# c3 * (2 * factorial(1))
    + D, O+ @( |: q0 \6 Z3 * (2 * (1 * factorial(0)))
    - r: i) x' P8 n  x/ z" P: Y4 @; j3 Y9 P3 * (2 * (1 * 1)) = 6
    * d& d2 U  z6 _% a/ E; o+ w. N
    & H! O- a( g$ ~  e 递归思维要点
    2 }$ [5 u' C1 F: h1. **信任递归**:假设子问题已经解决,专注当前层逻辑. n# b3 @+ I0 Q
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ( {  j4 w1 O; K0 S, I) v3. **递推过程**:不断向下分解问题(递)
    * H  P3 t2 X% J5 u5 ]) h( d# @" g4. **回溯过程**:组合子问题结果返回(归)
    2 L8 n/ [' C4 w" i1 E5 s4 J6 ?2 n, k. d8 v) y
    注意事项. [& V) m2 Q8 y& q# ?! Y6 t* k+ g
    必须要有终止条件
    ) U. |) [4 s2 j9 y' G2 Z递归深度过大可能导致栈溢出(Python默认递归深度约1000层)4 O2 V7 G7 E( F
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
      g7 V, l  l- i/ V尾递归优化可以提升效率(但Python不支持)
    8 _! `4 T2 U# Y* t7 g/ T7 J) u( \' o2 N* P+ z% d. n4 g  ?
    递归 vs 迭代
    1 P2 `' G) ?1 J|          | 递归                          | 迭代               |: S6 o- e% v; A& }2 _
    |----------|-----------------------------|------------------|
    + c" ?0 k+ I$ X0 ]| 实现方式    | 函数自调用                        | 循环结构            |9 w0 _# K3 v) v/ f* a
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    8 O: x+ ?! \% Q* w* V  e| 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         |2 C/ q: @5 X6 T; C3 d. s9 z
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |3 c/ h9 v0 Q5 E! w& T
    & Y0 r, \' q& a  I* r/ T6 Y6 X
    经典递归应用场景" n1 ?. k7 V( |) J3 e6 v
    1. 文件系统遍历(目录树结构)
    * a( t) y* N# ?$ W1 I2. 快速排序/归并排序算法. g$ J# B- W+ C/ n7 n9 {  M
    3. 汉诺塔问题
    + |1 l4 [: Q# X  ^' a: o* _0 m4. 二叉树遍历(前序/中序/后序)
    , B. j2 |  N) f: P5 I9 x5. 生成所有可能的组合(回溯算法)5 U9 _+ |* {+ a, R; b

    6 Y0 \- |( A" B2 c8 |. e/ ]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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    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,
    ' q* l+ u: O: u! ]: E2 T我推理机的核心算法应该是二叉树遍历的变种。, t, y- S- C' q! ]/ T7 w! 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。
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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:
    2 I# l% e' ?2 u. t2 }2 ^; xKey Idea of Recursion; K, z; g4 J$ Y% i7 L! }; D$ i
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    A recursive function solves a problem by:# z  h5 q5 f# h  G3 b/ f: r0 i
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        Breaking the problem into smaller instances of the same problem.
    0 A! P/ D6 n# I6 l# U, G, V, o
    1 r7 s: S( E0 D    Solving the smallest instance directly (base case)./ W5 m, ?+ {2 Z4 D) R& U
    7 y- p, ]* d2 E
        Combining the results of smaller instances to solve the larger problem.; Z! _' A& [7 a* c4 r( v# E$ c/ S
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    Components of a Recursive Function
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        Base Case:
    : B  ~' Y! _; z  D2 Z# a3 M$ H; f$ I6 d" G6 Q2 x9 L
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    8 b; c' N0 A, `4 I- N/ E5 O" Y9 n1 h* P" p/ ]# e& r
            It acts as the stopping condition to prevent infinite recursion.
    ) w# _# c8 p! B' N: ^8 {0 ]- p% y
    4 ?5 x3 k3 W+ S2 d7 R! k, m        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    + A7 W% N# \0 Z2 \3 y0 F! e9 A* K4 w6 ?+ r; `
        Recursive Case:2 e2 B$ T: J  I

    + L7 J* X: b$ Y( ~8 ?  H8 \- A        This is where the function calls itself with a smaller or simpler version of the problem.
      j, W6 [" ?. ?2 f; H( J4 q" U8 J2 ~- `$ U4 X" E- g0 z8 |, y  J$ }0 D
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).! ~6 j9 w6 o- n1 {/ K( H
    " j/ S8 }& U0 O  c: x
    Example: Factorial Calculation) s9 N7 I0 g4 @) Y0 S
    9 Q5 z# o& Q9 u4 J# W
    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:1 d, n6 [9 n. l) q: N
    3 h. i7 Y: |8 A2 Y9 k
        Base case: 0! = 1( W0 v/ v$ N6 _2 `/ F$ \

    , `, I" a4 q" t8 `9 \  d( \- r" d    Recursive case: n! = n * (n-1)!
    + Z; l9 i" r3 ]
    4 \2 s+ \- N, ~: ]: ~1 v' e7 cHere’s how it looks in code (Python):$ `& |; H5 E5 D& n/ ?5 D8 Z
    python
    ; P9 [$ Y; Z5 ]% a6 l5 n& w# O3 f' p+ @- y  z9 K, F
    - D- N% f" t; h' U5 H
    def factorial(n):
    5 d: u. o7 O7 ^  ~2 u& ]    # Base case
    5 C& Z9 T) Y* ]: O& V0 ?$ z- h    if n == 0:' P  Z2 V/ j' U1 l5 T
            return 1
    & D' ?( ?# a& {* Z9 t1 d7 J    # Recursive case
    5 Y. O9 X2 g3 F# R( y: {& v/ j    else:
    5 b3 z' \4 z  h' G! o( n5 K+ G        return n * factorial(n - 1)
    6 x; A% E: O* F& Y- I" S0 W8 t- h" l$ O3 R- v$ o
    # Example usage8 b) Q# A. c$ p6 s8 b0 I8 f+ E, Q
    print(factorial(5))  # Output: 120' v( A" g& c8 c# L) ?/ J& l9 E

    1 P, S) ^; C& kHow Recursion Works$ B4 C, @) u' B4 A

    / [: C9 K1 v5 h6 i5 l0 ~  i! {    The function keeps calling itself with smaller inputs until it reaches the base case.$ X; q  d. O! g4 G
    ( {' z2 b. t3 h4 o
        Once the base case is reached, the function starts returning values back up the call stack.* N, }' {: t% n  H- }
    8 W1 G1 J3 B( V( u
        These returned values are combined to produce the final result.; r# X9 C0 q9 B7 l
    / S5 P4 M* N6 y% h
    For factorial(5):
    , X" o5 R2 {! l9 S  B2 O
    # ^4 ~4 n( _/ J. b1 u0 y
    ! H1 {/ L8 O* _! B7 n5 D6 F& N% M' }factorial(5) = 5 * factorial(4)
    # P3 i2 G, H0 R" s3 K8 Tfactorial(4) = 4 * factorial(3)0 y6 m: ]3 X3 N
    factorial(3) = 3 * factorial(2)
    & l* D* z# u5 t# C8 p4 ofactorial(2) = 2 * factorial(1)! U8 \. v/ e1 M' p( H
    factorial(1) = 1 * factorial(0)5 @$ [/ f  e7 w9 t" x& Y" T
    factorial(0) = 1  # Base case3 Q% S1 X3 t( K1 [
    ) d+ |* }/ M# W& U+ V2 m
    Then, the results are combined:1 Z" v$ I6 q0 \, Y/ c

    , C/ @  K  F) }' H/ F- r- h; N4 f/ j9 Y9 G3 G5 x+ h
    factorial(1) = 1 * 1 = 1
    : {  a$ l' |, n9 Q0 ufactorial(2) = 2 * 1 = 2
    / c# ^8 ^) H8 B) U# pfactorial(3) = 3 * 2 = 6
    & V( Z) _  A& ^- pfactorial(4) = 4 * 6 = 24
    $ ]' I6 u% b* M2 W* s/ b# O* afactorial(5) = 5 * 24 = 120
    & H/ ?1 ^- C5 U2 Q9 W! `" C
    * n7 ^+ E8 }* n" i; v, i9 O6 MAdvantages of Recursion2 R0 @, b  R6 s% t+ X- d
    4 q$ K6 G: q9 S  z5 s4 ]) r' p6 v% g6 Z" p8 e
        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).  Y4 V/ i; T* Q9 v6 Y
    " d+ J( B# j& U
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    3 X) d( k# O9 i. @2 H; V/ j# b1 E  R* ?1 W
    Disadvantages of Recursion
    1 C- R% {! t/ d; B+ H$ d$ l. |+ j+ O2 j  D  a; `
        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.( W/ A; o4 t  C9 M' z  S

    ) G- m0 m7 Q& p' z3 [# D! f& j1 h    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).3 R& h# F& `5 A# i7 [

    3 L% e4 V* C6 W$ i* h9 pWhen to Use Recursion
    2 {$ W4 F  w7 U8 a8 @
    + v5 k, `% W% i" {5 j2 e* ?* N    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).+ p% O( n4 g+ t
    * l7 ^7 |6 [) Q; l! z8 \
        Problems with a clear base case and recursive case.& m0 r& h1 g& G+ q! F+ B
    5 u* k$ h4 l0 _" D, l
    Example: Fibonacci Sequence1 U& t6 h1 c3 f$ e2 w1 ?
    6 Q( p1 V. v8 T
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:; P) v7 @' @9 Q& p' y9 D0 s
    % \3 k" V, B! A% Q
        Base case: fib(0) = 0, fib(1) = 1  i7 z( J/ U& Z9 s! I( p6 R
    ; @: W2 v. [# G3 d6 A2 ~' F
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    ! \: P# Q& i# J- j0 _8 W; ?python
    , z4 L6 R3 @7 b; `; x; n/ V4 B+ T/ C/ m# ?+ ~

    3 r/ G9 e5 K' ]1 m+ Wdef fibonacci(n):
    ( F/ b. A' {4 ^2 B  u: n6 B    # Base cases2 C) F/ o6 N- P. J( t0 @6 i  N
        if n == 0:6 G& A2 O% I( K$ C
            return 0
    1 @# x. X8 Z' P& o% j0 H$ h. Q    elif n == 1:7 Q% e+ L% i  X: x# }; w% m) d8 u& H
            return 1
    ! @* I! k4 W& v" V0 O- a# J5 y" Y    # Recursive case
    ( T! y' R5 f( Z6 ?; g" h    else:+ m$ s& z1 \! _+ T# ^; V3 C
            return fibonacci(n - 1) + fibonacci(n - 2)1 c7 j7 ?9 b. c
    + U8 C$ S; [7 f% F- _8 S  ~2 I
    # Example usage
    % y1 m( ~6 x( b* ]5 P3 a# @print(fibonacci(6))  # Output: 80 Z+ d5 T! W9 E. y
    1 L' E1 s0 S- M! P9 S: a; n
    Tail Recursion+ @9 m& p1 ?$ ]3 t" l, f

    : }; Z( X  p* ETail 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).
    5 ?# w& ^! l- Z! T( `) g! s/ @% C1 R+ N4 v- U
    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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