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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
    ( n" i# @0 R$ t7 d- g1 e: |! o" U+ A5 h  b% o! n
    解释的不错
    . @' }/ F3 A, c# p0 D0 ~7 V+ l6 ~) M% G5 P. 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。* n" p7 r) h7 W# p  ^2 Y6 \
    - r  ~) F% s1 q8 j( D( C* u. j% M
    关键要素
    7 M" E- D0 w% v6 h. g2 _  U1. **基线条件(Base Case)**( W% B+ j0 ^4 N, A* E: w
       - 递归终止的条件,防止无限循环  A- G7 L2 M! O6 ?, x4 t* N
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    9 {' _* r2 L; G0 D9 \  _0 _, W0 J5 n+ A9 M# P) b. l- E
    2. **递归条件(Recursive Case)**" n  J1 W7 g2 i  p. ]; C3 V
       - 将原问题分解为更小的子问题" I* m, K8 M, `" P9 q
       - 例如:n! = n × (n-1)!$ \- p4 U- S/ s# C

      Z3 z- j0 h0 M* i. Z 经典示例:计算阶乘
    ; T% `) O1 p: N) H' t4 ?" c/ Hpython
    . j: v5 }% ?1 S1 H; r7 Z. d, Kdef factorial(n):' d2 V" `  K- p6 E* w3 [6 q1 M
        if n == 0:        # 基线条件; D+ Q# a6 r& Y  M
            return 1) a+ Z& `7 p# B+ Z  r
        else:             # 递归条件
    * C) z& p" Z* ]7 V# F5 `. {        return n * factorial(n-1)
    / B0 b+ k% m7 A% Q5 j6 [执行过程(以计算 3! 为例):
    - z: E5 c/ {) B# d* ?factorial(3)) j/ I. J; D# K* k
    3 * factorial(2)1 S1 g& L$ ~' Q4 @' H! n& A6 x
    3 * (2 * factorial(1))- s% d. j' V6 T! }
    3 * (2 * (1 * factorial(0)))" C, E5 G( `6 h: u* m
    3 * (2 * (1 * 1)) = 6- |# F9 ~; n! d& f. d
    + I' f; o& y" ~, q' S, a
    递归思维要点- M; ~0 B5 y( q: s7 \+ a
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑0 y& C' W6 C2 ?5 A* N5 r) [9 f
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)7 d' Q" |9 t1 q+ ~/ M
    3. **递推过程**:不断向下分解问题(递)
      U, z8 s# v9 H5 e' Z% |2 R4. **回溯过程**:组合子问题结果返回(归)+ k. y. k% x. D1 E+ z5 r

    0 S( j  Z6 g* f7 G9 v) D 注意事项* l* Z1 r4 |1 m. P7 X* D
    必须要有终止条件
    * j& y4 l( i6 e: \% T0 X递归深度过大可能导致栈溢出(Python默认递归深度约1000层)' {, z2 M* p. E) B$ b% q3 D
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    6 ~9 R7 k! T9 l, E8 @& z1 ~1 v尾递归优化可以提升效率(但Python不支持)7 w$ O9 w' g' N* Z, }

    ! d! O4 O- d$ o- g. u0 a( R 递归 vs 迭代1 G5 t6 Y/ I! h' d, i
    |          | 递归                          | 迭代               |
    $ P' y4 u9 ]/ R1 I|----------|-----------------------------|------------------|7 I% ]5 }7 z! Y- n
    | 实现方式    | 函数自调用                        | 循环结构            |
    - j# U: p# Z4 t8 o| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |3 E4 q. C" D1 ?, h" Y+ L& d
    | 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         |
    9 l+ v, Y; T/ j  i% k& h$ t) r| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    * q# m2 {  q) e, E6 T0 ?; X6 V; \6 G% d) U# e: C& Z5 o
    经典递归应用场景
    6 R2 u" ?$ K+ L' P$ {2 x1. 文件系统遍历(目录树结构). {. w/ p3 ^  @0 _7 J& K5 u
    2. 快速排序/归并排序算法
    * s, s) g7 v5 S) @& R5 t3. 汉诺塔问题
    0 [+ E/ o2 d, H1 X- _1 s4. 二叉树遍历(前序/中序/后序)8 H( f$ }  y* Y' k0 h; o9 c1 z
    5. 生成所有可能的组合(回溯算法)
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    + H: M6 K3 ?$ c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
    慵懒
    昨天 09:58
  • 签到天数: 3148 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,! T  Y; W2 V7 {: {8 }9 u
    我推理机的核心算法应该是二叉树遍历的变种。
    : t& l; A  j2 t) {( w. _$ d# L- ?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:
    7 Q8 f' }, E# W- [Key Idea of Recursion
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    . s$ U4 d9 ^. o6 P$ p# QA recursive function solves a problem by:8 H6 i# c0 l( V5 ~
    ) r$ w! N* {, v6 x
        Breaking the problem into smaller instances of the same problem.5 f# Y, n8 H+ S7 B( ~1 V

    ) `4 Z6 g) o7 Y) C2 H    Solving the smallest instance directly (base case).
    ' v( Y- }: s! d. j* S
    # W6 q4 A# B6 w8 f% L# E    Combining the results of smaller instances to solve the larger problem.
    $ o( j" m; O2 ?5 ?6 ?7 x% I2 F( w3 V4 u9 l
    Components of a Recursive Function
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        Base Case:1 H, s8 g9 P7 Q& u/ }( C

    2 B3 Y6 m; G* F        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    8 F* Q5 a' U: o& h5 ]6 X4 L/ {3 [3 L
    5 G( }& p+ h. d" Q  t1 w        It acts as the stopping condition to prevent infinite recursion.. ~& }. P! p+ P2 Y7 f0 V

    7 c- I5 g; \- _6 d3 Q        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.; c4 w  D3 S$ r3 p
    2 W, Q( r& G+ d1 k9 h
        Recursive Case:
    * M% S( p3 ~* z) X0 w$ a) u2 h2 @; Q" q5 {( {
            This is where the function calls itself with a smaller or simpler version of the problem.
    8 F# z7 p- l6 Q5 N- N2 B0 V) V- [" H
    % k$ |/ {- k6 o4 h6 K! K. D/ d        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).6 Y0 J) _/ [' w7 v

    5 |; c$ |* @/ r: }( |: U5 Z3 uExample: Factorial Calculation
    * M/ [, u$ }. I; l' _0 n  F; U: E
    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:( _; b8 \5 V4 @
    / \$ x" [/ f* ]
        Base case: 0! = 1' k7 T0 [0 H. d

    7 Z5 `" U- L9 K: b    Recursive case: n! = n * (n-1)!
    ! W4 w" \- e; C2 a# b8 c, }% O: l6 q
    Here’s how it looks in code (Python):/ Z: P0 v4 J1 d( P4 q6 \
    python
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    1 e4 u5 S! ?/ x. W- T7 Hdef factorial(n):
    7 v$ C8 @8 M; j    # Base case3 M- `3 v3 A5 L& q9 _) M
        if n == 0:3 c1 [  p( \, f. w6 j' A9 ^5 d
            return 1( i; y  H# y7 H+ `
        # Recursive case
    ' U* w4 g1 q1 v4 t7 m    else:
    ; e! h6 M' I1 w8 L4 e, a6 `% R        return n * factorial(n - 1)
    & k. ]: H) }; K$ o7 B1 ]$ r# G1 c# u; M
    # Example usage
    0 b$ X6 c4 U( r0 `print(factorial(5))  # Output: 1207 W  |% F% S8 A  ^/ @/ j, W
    , b, a1 u4 D0 R& R$ c8 k5 e
    How Recursion Works, ^0 h+ F" x0 a3 x# \# O
    ; ]1 v1 V/ T$ z! x# S5 A" S* v
        The function keeps calling itself with smaller inputs until it reaches the base case.' I9 V) }: h7 [/ E' ^" q4 R
    $ n/ ~- Z) N2 V( @( g  c
        Once the base case is reached, the function starts returning values back up the call stack.
    ! x2 B$ d7 K8 D* U, I1 X; g/ P! O: B
        These returned values are combined to produce the final result.4 j  h( o  w. S, r% B
    7 l3 V+ b; X& P5 N6 N1 W
    For factorial(5):
    # m& H* H0 q, J9 z' a8 r
    , }5 h1 [4 d% ?' c3 I  q
    & O: t& Y, a. kfactorial(5) = 5 * factorial(4)/ a' a/ i4 V/ l+ n! c. C+ X1 O
    factorial(4) = 4 * factorial(3)
    & j" n& x7 n7 D8 ?) Qfactorial(3) = 3 * factorial(2)3 i: \8 G  t: _- U
    factorial(2) = 2 * factorial(1)
    8 H  i, D" [7 f, H+ L" N/ \& h' K# Sfactorial(1) = 1 * factorial(0)1 @- z+ Q6 L' P/ z2 g3 B
    factorial(0) = 1  # Base case4 ^" P1 f# R3 j; W% L. I
    9 W' `( r. ?/ _$ {
    Then, the results are combined:
    , t9 a7 Y; Z4 {0 r6 A/ N! }4 W; k" N/ u5 l

      s1 C# ?: i1 N8 ^1 P3 q3 b0 t3 H* Ofactorial(1) = 1 * 1 = 1( ^6 r  D8 n6 d( \  V
    factorial(2) = 2 * 1 = 2
    / A, d. e5 [2 z% K5 D1 p+ Q1 zfactorial(3) = 3 * 2 = 6& b! Q8 k" Y$ P3 |2 S; O# z
    factorial(4) = 4 * 6 = 24$ ~: @; ^1 j: Y7 \" d6 x
    factorial(5) = 5 * 24 = 120% c. K7 O9 Y$ m

    * B7 q* v2 s9 s3 \1 G( FAdvantages of Recursion
    : d; G/ L, k8 m& a9 Q" T1 m( b. _  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)./ r; D# u" z: A4 P

    2 ^* C  O( K( N' o5 z3 {    Readability: Recursive code can be more readable and concise compared to iterative solutions." g: {1 H6 T8 D+ r
    ) N' a+ g4 s# b4 D  b6 U1 ^
    Disadvantages of Recursion
    7 K% s8 s& k8 `2 V; i; N2 }. t
        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.2 R6 n% X4 m3 u1 R* R
    $ |! C1 |9 ~$ W
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ! b  Z$ x! Z% G- D3 Y; N7 `3 j' W& |8 r
    When to Use Recursion) H+ M9 Q; y7 \9 J# Z
    & p9 ^& S2 W/ \1 k
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    - {& H1 p8 O. U, ~. r+ T& o5 y/ z0 ~/ _8 j) C7 Z" l
        Problems with a clear base case and recursive case.4 X  D' h8 a3 y  S( @
    ' }0 `, F: N1 v$ Y  d. {& E  [: e. ?
    Example: Fibonacci Sequence% y1 q5 m* e6 o

    " G2 d9 e! R4 V5 N9 ?0 }; zThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    5 B8 d' ^6 b5 r: g3 ]9 E2 B# F% X9 J# B3 s; V$ R: _
        Base case: fib(0) = 0, fib(1) = 16 x/ H) o6 O4 `: u$ C
    ' O) m% {$ n2 B
        Recursive case: fib(n) = fib(n-1) + fib(n-2)! |8 l( s# F: L1 \2 i7 j5 _( }- R
    9 u5 a& h1 ~2 p5 A% q  }
    python
    $ p$ c" `3 Z' b9 n# H
    * ]1 _4 U  N4 i
    # M5 Y& u- C: ndef fibonacci(n):# s% Q0 u  g8 i+ {( z) N
        # Base cases, J( Z" Q1 S: @% B) j" U
        if n == 0:
    ' q# n& t" ]7 ~, K0 O        return 0
    ( T6 _8 f" d6 B    elif n == 1:. a, b! p2 r- s$ L- X/ @' x1 O
            return 1
    ; b+ Q$ x" O5 B9 s1 o( d4 O    # Recursive case
      }6 p3 l, p; J) q* p  v5 _    else:
    / k2 D* S; \9 t8 y        return fibonacci(n - 1) + fibonacci(n - 2), ]) D) ~! {* i4 O9 U/ w

    & Z6 J0 f9 v  J# Example usage  O1 o9 I7 E! @# I  e: b0 @
    print(fibonacci(6))  # Output: 80 K3 t! h$ B5 ^+ z% t$ {4 Q
    & z) \; ^( V3 \0 M
    Tail Recursion
    ( i- C. W) ^1 [2 k: p/ j, p" s, W' Q3 |. T" r* |
    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).
    # D, m# T* g; Z: Z3 j, \; S5 R- a: J) `5 ^7 a
    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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