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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| 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
    ; O0 c$ D& g, p8 N9 j# G0 Z$ v
    + m6 o! I' }$ g3 B0 K* ~+ q. n0 K解释的不错7 ]4 \0 j' x, B! }

    6 e$ ?( e' j8 j( C1 P1 c; }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。% s5 ^' L$ B2 F/ {& U# p* n! t3 h+ n
    * @3 y& W2 z: ~! j) N1 @/ M$ z0 G
    关键要素: E# m6 X1 T+ i4 Y( W8 h
    1. **基线条件(Base Case)**. x" U6 \( T# ^7 c
       - 递归终止的条件,防止无限循环/ E. j- F* J2 n; B
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1$ G, S( B) B# T6 ~% X/ d, C! O
      Y2 I. ?8 R5 t( o+ {. W+ l
    2. **递归条件(Recursive Case)**
    , Q& d2 O" ~7 j. f) D   - 将原问题分解为更小的子问题
    - F# N! P; c/ q+ B( e& q! z   - 例如:n! = n × (n-1)!% L, U$ y& s$ W* J% j
    0 m9 _! P9 F2 o3 h/ V, t
    经典示例:计算阶乘
    , D: e% I" Z' Z4 i9 Cpython, a* f: P" y1 n
    def factorial(n):; w. W+ l5 B: d4 a
        if n == 0:        # 基线条件8 M1 E1 o1 x  R+ G% @4 f( B
            return 1
    6 o9 y6 l) }9 H9 {: m    else:             # 递归条件, T! G. ~' d, e1 p+ y& j. z) N: t' \# E
            return n * factorial(n-1); n; b" Q  @! q$ P- T8 V
    执行过程(以计算 3! 为例):
    7 R; C1 n% u% e$ kfactorial(3)
    4 Y; S1 H' \0 W5 _  f- `/ G+ ~3 * factorial(2)- F5 H0 V2 J; D9 [# Q' v: K" O, }
    3 * (2 * factorial(1))6 F- ~% L) X) @
    3 * (2 * (1 * factorial(0)))
    ; D4 G$ i( o% s  ~& E& {" P0 ?2 ~' p3 * (2 * (1 * 1)) = 6: A' ~# a0 Y! [7 O! v# H

    # r$ ?% q8 k4 q* r" E4 v 递归思维要点3 u- o5 F* y; L4 h- g2 a6 A
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ( R) i8 |* N& B2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    5 }6 P& v' O3 v  E3. **递推过程**:不断向下分解问题(递)/ e) B( g( ]% I% p
    4. **回溯过程**:组合子问题结果返回(归)
    ' X9 X5 B# S7 g8 n0 o& s. U; D# B6 C% u% d/ c$ c: }
    注意事项1 o% J# C: A$ h+ G& x
    必须要有终止条件) p/ y# ^) m" n1 \$ ?/ M
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)4 r" @5 B: L8 C% X
    某些问题用递归更直观(如树遍历),但效率可能不如迭代  n2 y6 n7 n3 b
    尾递归优化可以提升效率(但Python不支持)
    % b5 C$ D4 ?: h+ N7 A& e% d4 ~( z/ b- `. j0 H4 S. }# ]
    递归 vs 迭代) ~: N8 t7 D4 J& N9 I
    |          | 递归                          | 迭代               |
    + m/ [: S+ I7 j4 `9 f8 A, j' A|----------|-----------------------------|------------------|
    $ [+ _; h1 `" H# F( m| 实现方式    | 函数自调用                        | 循环结构            |
    3 m; H0 O/ O* r| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |9 R0 @4 e* c5 @
    | 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         |$ H8 [" p/ B% ^
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ! l& S; I- q  s
    0 a+ j& G! _, m& T+ { 经典递归应用场景( K8 t- O6 y& r; g
    1. 文件系统遍历(目录树结构). V" p1 @% t: b
    2. 快速排序/归并排序算法
    2 z3 G1 ^& c- D6 R. P1 Y$ A3. 汉诺塔问题, i- G. q; H* X9 q& }' K
    4. 二叉树遍历(前序/中序/后序)
    # b. c  `4 j8 b, `3 b3 {5. 生成所有可能的组合(回溯算法)
    / M% V5 e7 y/ m7 s. S9 C/ u
    8 C2 e; f6 ?3 O! q9 o  L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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    昨天 07:32
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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,
    ; F7 W/ Q5 |, G0 Q0 q9 K我推理机的核心算法应该是二叉树遍历的变种。6 j4 B. |* J# E) U: z; }2 R! n) L+ 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。
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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:
    . U- h5 K% u/ e# iKey Idea of Recursion" o# I& A& C  g* d8 p. H

    7 p- B: M2 n: @7 P1 h8 TA recursive function solves a problem by:/ ]4 Y  e- B7 D# b' T- A

    9 y' b2 K# _& |! V( }! |- |/ c5 L) Q' R: e    Breaking the problem into smaller instances of the same problem.+ x; T. H: V& ~5 F  w% Y! C0 [" P

    ' L6 U( a0 k- z% _" b+ L  r    Solving the smallest instance directly (base case).5 @/ n3 c- g' l0 t$ y, O
    3 w1 I" R+ Z2 |! E
        Combining the results of smaller instances to solve the larger problem.$ W, e. |( I8 C
    1 ~! M% y# ^7 q9 X0 z# _
    Components of a Recursive Function
    8 K9 Y2 p& _4 R) o4 E# Y, X2 N3 |6 [3 M8 i
        Base Case:
    8 ^9 o/ Q9 Z* ]+ c( s: R3 _5 M7 [1 x8 d3 ]8 v& E/ ?
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion./ S  O7 N5 g  i$ r
    . k' Q6 w! ]8 g* q$ Q
            It acts as the stopping condition to prevent infinite recursion.2 b" z9 @* u% b
    : |# h+ t" Q, u! R# H- k
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    4 X& g( Y; q1 t" @# A* ]
    2 f% x3 q8 S0 D1 ]2 L1 A; q8 D( h! z    Recursive Case:
    ' P# I3 h% O7 R& v( k5 U! [4 ?! p! \* o& \# v" w6 S7 N3 R1 D
            This is where the function calls itself with a smaller or simpler version of the problem.
    * f3 H& w, r, {1 f% r
    , L! H$ g1 e+ O1 S        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    0 f5 S" E- m8 p* ~" }5 L4 u3 s7 }9 U
    Example: Factorial Calculation4 p  V9 x, v+ O
    ' [) u; K* o' Z- _  \" }+ v' k
    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:- w  b9 e2 U1 b. x; s" A
    9 I( k  O0 y3 S) Q' ?: \) J
        Base case: 0! = 11 S' {" z8 }. }

    - k- T8 Y/ \2 n/ N    Recursive case: n! = n * (n-1)!
    0 V6 y3 ^# G9 w, W
    3 S$ ?5 `1 |- \5 x) rHere’s how it looks in code (Python):
    # `. C8 M$ g8 d- k1 _3 kpython
    0 ?$ k4 {! K# w- d& P
    / Y4 E7 b: D0 n' I0 a! G2 o4 h& K2 n/ |7 y/ q; X+ y
    def factorial(n):/ ^7 M  [- [/ ?* \% r8 K
        # Base case
    6 S' F: |  I  j4 K1 w! d) k6 ~" O    if n == 0:7 \1 Z5 C8 Y$ m2 h6 h! X$ A; G
            return 10 ~( ^0 B' t, r2 u* Y
        # Recursive case
    , [( a% z% W7 c8 E2 t) e5 U& X9 I    else:1 {; }% k0 X; q% g( R; i; @" Y( ]
            return n * factorial(n - 1)
      J) j1 Q% p+ a  Q+ q2 D" `" u
    ( p  B6 ^; s- M4 z9 o2 p# Example usage
    + O& Z) ]; ]# {print(factorial(5))  # Output: 120
    4 e3 L* l: ^0 k1 _
    7 J7 r2 V/ k  m' ZHow Recursion Works
    8 O$ H0 K) l0 \- H
    1 p5 [4 i# \7 p- `! m    The function keeps calling itself with smaller inputs until it reaches the base case.5 N2 c- [" N: N/ k* x2 q/ E7 ^
    7 Z* r' }4 @4 N6 v! v
        Once the base case is reached, the function starts returning values back up the call stack.  a$ s9 T- W2 h$ U$ ~6 J" |

    5 Y6 o; y( c+ c0 u+ }2 {. m    These returned values are combined to produce the final result.( u2 i7 R0 ], Y# i
    ) m; Z! C+ K5 b& ~8 x
    For factorial(5):2 o6 d% `. M3 b; a) {5 T' A$ f
    % \% L9 a+ x5 E/ B

    9 f2 w; q) |5 {- r; H4 s# A/ ?factorial(5) = 5 * factorial(4)$ N7 e; A# b8 P8 K" H
    factorial(4) = 4 * factorial(3)
    5 f6 c- W9 c) K( @8 q* D' k) nfactorial(3) = 3 * factorial(2)
    $ a' i  q7 X* B9 o. P' X4 nfactorial(2) = 2 * factorial(1)( M4 x- ^5 I" [, ?
    factorial(1) = 1 * factorial(0)
    6 z. w  b" R- T4 t& z# Hfactorial(0) = 1  # Base case4 @+ q( X  f; v4 C/ f( k6 B' C

    6 q3 Z3 ^8 ~& N, `# R+ D& wThen, the results are combined:3 T. P. X# h: C0 L# o# j, y

    ! B/ b  `1 b; I* w  ~
    : [8 f6 O! a9 u( D, m( wfactorial(1) = 1 * 1 = 1* e: j3 Z+ \! \0 O
    factorial(2) = 2 * 1 = 2" {* A+ K/ v' H+ k! M7 r
    factorial(3) = 3 * 2 = 6
    5 |  ^% G5 E% p9 ^! U/ A" A7 w& _9 ufactorial(4) = 4 * 6 = 24
    ' Y  G+ F: m/ X, Q3 |- v7 Yfactorial(5) = 5 * 24 = 120
    - L* O& Q: M: h( l9 a: V  h* M, {  y  L  v; Z
    Advantages of Recursion) q1 I& l# r( h4 F% L+ Y; T: s
    8 H$ ]/ [+ @9 k: F/ J* v% u- e  G' ~
        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).
    8 \. O- D- w0 T  [& w; O. T
    / y9 C2 ~4 j5 P/ f: Z3 }$ F    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ; d# [  M0 @9 z) r6 V* M* ^: P/ q
    Disadvantages of Recursion
    9 V1 N5 L, a: i3 d) r. V: I6 _: d! S* l) K% r$ c4 }3 N3 Z
        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 ^9 s1 f( J( V/ M% W3 O
    & n4 ~% Q. H4 K3 n/ g9 ^5 q
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).4 D  m' [" e* i/ j* A

    / p) Y5 s, `$ S1 X/ h4 n4 H: k! TWhen to Use Recursion: O( U, v( ^: D. }

    : {# @) T' ]  Z5 D    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).) b3 \8 T/ h3 C" J# e; u* v
    & |3 m1 ?% D5 l' c% g; b
        Problems with a clear base case and recursive case.9 g) n; {' B# l) Z  G: `
    ! {/ h! {5 y4 G9 S% m
    Example: Fibonacci Sequence
    ' C7 _2 h- G/ E/ \5 A% X/ d
      A  x+ t; c5 l1 F6 i( |3 ]9 C# MThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    % c7 e9 B# g% s8 ^# g' P: M3 E2 l" C& F( I, o; l
        Base case: fib(0) = 0, fib(1) = 1( _" Y# S4 S9 m
    + V1 B! H' y6 G
        Recursive case: fib(n) = fib(n-1) + fib(n-2)9 O4 m$ T* g0 i' c) U0 W6 B# c
    9 e1 x4 I+ t3 Y6 z# C7 E; N
    python
    : t  A$ O: d% b6 s+ I2 J3 _( U( [
    + @, W$ K$ G( A0 v- A" n; E$ S; s% u. Y: i- W3 t* T$ f3 Z
    def fibonacci(n):4 j% C4 U( b; W8 `4 q+ k
        # Base cases7 F. W! k% s: i% j3 V
        if n == 0:
    9 P6 T$ H7 a  v9 |  R5 a3 f+ L        return 0
    ! k/ A9 L& Z$ u! h1 c' B% o6 ^    elif n == 1:3 c( v; v0 d5 R% I( b% p, k
            return 1/ ^# X* L: d& e* ^
        # Recursive case
    , r9 D* n3 i( p    else:9 }7 |& R5 q  o& Y4 m$ ?4 `2 p$ _
            return fibonacci(n - 1) + fibonacci(n - 2)# _+ T8 q! c$ Z. Y. [2 k

    - ]1 f" i9 X7 b5 ?: r6 R, P1 O4 @) A# Example usage7 c6 p$ A0 F' k/ y1 ?
    print(fibonacci(6))  # Output: 8
    $ {& p: a7 k! X/ `# i) V0 Y
    9 E+ [9 Y" T  _* V& MTail Recursion
    ( b) W% o# m* c1 t# ]
    / H5 `0 H* a2 l% X5 MTail 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).
    1 f( m9 W% k! M% x% m/ ^! \
    . S' ~. |9 r3 \" G( D5 [' L5 s  aIn 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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