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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 * \$ L3 p6 R9 D! s
    5 f6 |( E) D! C
    解释的不错
    9 l& Q9 Z0 @" U( Y$ p+ q* a
    6 }9 H/ @/ C( l6 P# I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。3 Z0 i! B* J  U7 _- M6 M3 C

    2 ?, ~; n; i4 x7 ~. R 关键要素2 b  @% q) j2 P3 Q+ b5 y: \- g$ H
    1. **基线条件(Base Case)**9 _, g$ o5 q: v; Z
       - 递归终止的条件,防止无限循环
    ; H0 l  q" [7 `   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1" h+ ]: Y' {+ I- p
    & x$ T3 ^4 w* a8 _' E! a; m/ |
    2. **递归条件(Recursive Case)**
    ; ?: q# x0 I% E  R" \% x' t   - 将原问题分解为更小的子问题$ [' [5 l: ^4 V# ^$ E; p
       - 例如:n! = n × (n-1)!
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    ; N) R, `, l( N3 A1 t 经典示例:计算阶乘
    * z0 a9 S, }" Y! s/ [0 ~python
    # \8 V  G0 A# Q; L+ idef factorial(n):
    ; I- y* _+ V& e) C8 h& b: t    if n == 0:        # 基线条件: `2 g; f6 T" _9 G
            return 11 f, w. w' Q! Y
        else:             # 递归条件
    0 a  G; n% J& J& T; R! t        return n * factorial(n-1)6 I5 v$ g0 v7 z* g
    执行过程(以计算 3! 为例):. Z/ w2 a( T$ Z# D9 F8 m
    factorial(3)  ]: I/ N% y; p1 O5 F! x! b
    3 * factorial(2)
    9 K8 i) d! @5 @! g9 ^3 * (2 * factorial(1))
    $ K+ o2 X1 L7 i6 P  J+ ]( \6 O3 * (2 * (1 * factorial(0)))
    . ^" j2 s  Z0 W3 * (2 * (1 * 1)) = 6
    . e" a* }4 w3 c! E/ R7 y* s; C/ n6 [; J* q1 y
    递归思维要点# q4 a. D" w2 s& d1 r; t& k
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑! S# V" r3 K  |! N
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)1 w% @8 b) v. h. S) G' Q+ x
    3. **递推过程**:不断向下分解问题(递)# f( W1 V' d* s7 H) a3 U3 V
    4. **回溯过程**:组合子问题结果返回(归)
    8 J* h3 H$ j9 L$ d- w  ?9 _8 i; z) {! e  w
    注意事项
    9 _8 R% z- Z+ |9 b3 a/ H0 G. b7 E, W2 l必须要有终止条件
    ) N& z, T* J0 `& h递归深度过大可能导致栈溢出(Python默认递归深度约1000层)& X7 r( t' ?( Q; l6 Q
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    % R% u' a/ L' p: A" o# N3 A尾递归优化可以提升效率(但Python不支持)+ [2 U& ~9 I7 p3 C

      a$ w# G& M; i( ] 递归 vs 迭代$ p: q  h! j$ V) V: e- ]
    |          | 递归                          | 迭代               |
    % q+ @% A. k! |/ [- I, ?2 S- ~|----------|-----------------------------|------------------|
      P. X$ R! I5 G3 L. {9 _| 实现方式    | 函数自调用                        | 循环结构            |$ v$ I* ]4 V; ~, Z9 Q4 P( B! h& B
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |$ [% t' u% R4 o/ T( Q& A
    | 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 f) v& h$ t: {8 y6 H  B- L5 j7 ]
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    6 e6 m3 Z8 d& a6 z% `  G2 o1 w8 D9 b) k; h
    经典递归应用场景
    6 h  _- }' @! M+ X: }1. 文件系统遍历(目录树结构)
    3 C, f6 d3 P5 G) }. C2. 快速排序/归并排序算法
    4 O& I( E# P8 z3 q8 i3. 汉诺塔问题
    / }- }$ s6 _; X. R$ M* ]8 z0 j4. 二叉树遍历(前序/中序/后序)) L9 d# d; h( e8 @& r/ l3 f
    5. 生成所有可能的组合(回溯算法)# D+ L6 e2 f0 \1 d. ^8 r! k+ _
    6 i. k, T- U  A& a2 J' D
    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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    2 小时前
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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,8 K0 N: Y2 |4 M6 y3 p
    我推理机的核心算法应该是二叉树遍历的变种。
    " B7 M) p6 c1 P- n, y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:
    * y& k* V2 \' k3 J+ qKey Idea of Recursion- t( ]  A, U' O+ ]8 ~! u

    + |4 _4 Y2 n3 U. W3 WA recursive function solves a problem by:
    # t1 J# ?/ I3 Z1 r  N' G1 S- U. M) ^$ D$ \
        Breaking the problem into smaller instances of the same problem.0 b+ @1 P) K# _* @/ _. ?
    ; Q+ `) h( n8 c) J9 m
        Solving the smallest instance directly (base case).
    : b) \2 X$ k5 ]; ]( a9 R4 w) R" @" D5 J! y" t( f
        Combining the results of smaller instances to solve the larger problem.
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    2 C  v8 D# x0 N( A2 b! hComponents of a Recursive Function
    ! y- ~# E+ v5 S
    0 ~2 Z9 }% @3 @! O/ p1 ]    Base Case:
    0 u, w. U( P- J! k4 s  D9 ^6 C8 S- i' t2 C* @
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.* V* h. c6 |) x; y4 c9 d. w3 d9 Z+ |

    : A* I  F; C4 C+ J! Z2 V+ `        It acts as the stopping condition to prevent infinite recursion.
    ' y' Q' x* ]# M# n: o" m: L! Z4 t( y& r+ t) Q2 m
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    : t5 r- Q; H& B. g, j7 z" \- N3 E
        Recursive Case:
    1 F  s* m$ H; [1 f, X7 q
    # U. X4 @- K/ D/ x        This is where the function calls itself with a smaller or simpler version of the problem.% ]: u" _) V2 _2 E/ _
    7 x) \# \* R+ K/ I
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)." n* j$ Q0 x- M

    ' h( r1 j9 w$ O5 _Example: Factorial Calculation
    - Q% y' b7 a1 u7 V% r- z% G+ }) A# O  I8 ?$ v- W4 [
    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$ t# |+ @: T4 y/ S
    3 x9 i) {( a7 D* g! X6 H    Base case: 0! = 1
    + ?5 v2 c( S" j. Z( a& I* x- G4 W! [, c" H+ p6 t7 E
        Recursive case: n! = n * (n-1)!
    ) F4 e% p. t! B8 y2 D
      r# _* l! F4 n* ^6 `/ YHere’s how it looks in code (Python):
    & S8 P  I9 ?7 f" {/ ?3 J3 v; ^) mpython
    0 J. m! S0 `/ M9 I1 O' Y0 y* Y/ n! {  d9 j4 }/ n

    % \" O( l5 Y9 n4 p# c3 \% ldef factorial(n):6 f& m2 w9 B: N% |' Q% J
        # Base case
    ; n+ o# `( U; e$ U/ m6 C    if n == 0:2 e0 j/ d0 l2 j" C3 e
            return 1
    ( [2 e3 `) c; @    # Recursive case$ Q& r/ X, ]+ u/ F; ^
        else:
    : e) g& K1 w% @" _4 S5 W: ?, K        return n * factorial(n - 1)
    0 d/ w( g" ]" Y3 A
    ; N$ t8 y9 m( D% m# Example usage
    . d1 X, U1 z: z4 \) ^print(factorial(5))  # Output: 120
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    1 L) D  b8 P7 k. E$ |How Recursion Works
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    ) r2 p- _+ f3 X4 J9 p    The function keeps calling itself with smaller inputs until it reaches the base case.2 }5 z6 d, R$ g

    5 [0 _* D1 a8 M+ c2 }% j& i' f    Once the base case is reached, the function starts returning values back up the call stack.
    " ]9 }, w7 P& A1 F; N, P% q
    * j3 b% c5 }5 H! G    These returned values are combined to produce the final result.6 u# Y5 c9 J/ Q# e3 k
    5 n6 O7 Z) O5 c' X' I& ~
    For factorial(5):
    7 d+ ?9 y; L/ C8 \. z
    ! C' ], a% `  T1 V+ s1 A  `" e3 O7 K; i
    factorial(5) = 5 * factorial(4). K) v& h2 g# i1 u
    factorial(4) = 4 * factorial(3)
    ) x$ y0 l. R4 W) [factorial(3) = 3 * factorial(2)
    ; A2 U3 l& u0 n+ Q1 dfactorial(2) = 2 * factorial(1)) m* Q. c9 X7 a) A7 ?& o$ V# n6 x! `
    factorial(1) = 1 * factorial(0)
    1 h, h! v" b- V7 u# lfactorial(0) = 1  # Base case
    ' R5 c8 A( m& U& u- n- R1 P: W9 W
    4 M2 Q7 N- w) y) m2 NThen, the results are combined:
      E6 L  R4 \4 C- ~5 r6 {0 ?* N" I: R  g6 O/ V8 a
    8 k- r' }1 ^8 `+ I) h
    factorial(1) = 1 * 1 = 1
      y" R% d7 H$ |" U0 T. N% j6 b7 \; Bfactorial(2) = 2 * 1 = 2! |$ j7 G  Q- H7 Q$ s7 ^) @
    factorial(3) = 3 * 2 = 6
    ' L8 ~8 V$ U9 O2 a/ N- v, jfactorial(4) = 4 * 6 = 240 p% f) P( H6 v: U3 n- V
    factorial(5) = 5 * 24 = 120& {( [6 R; ?$ f1 ^

    # ~4 P) h9 _5 q  `& c! yAdvantages of Recursion
    ' T  }- x+ ~7 Z. {+ S8 v" g
    0 X' {8 P" t# I    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).
    ; ]0 l- v! K! d4 b" _
    . X8 K: E# S0 }    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    5 t6 p" y' R8 b4 A( X/ d2 [- r2 o5 ~# ]" I% z7 c) R1 [
    Disadvantages of Recursion
    , R# R' m" K. b: h6 a9 y/ Z" y
        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.
    * W9 _( Y6 x) b* i( J& v. i8 F
    * C( l/ s7 d# i$ |$ G2 i  h: ]    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ) |7 z6 i1 W3 r7 U/ X& p1 L; a8 [4 E. c/ Q
    When to Use Recursion) i4 D) n/ U/ \# d
    " k( ~+ ^/ t% D  A! \" w
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).* L) }; J1 I' B8 f9 A

    2 Y; N3 j5 A5 q. w0 v* L( V% [+ {    Problems with a clear base case and recursive case./ l! _8 i" |. V! L9 `7 s8 q
    ( V( @; F/ x" n! Q2 D$ Q! F
    Example: Fibonacci Sequence9 R8 J8 k3 D4 D, e4 D4 J5 m

    % O' h0 N" n+ s: E& J: b4 o9 T" NThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    " g3 b$ }/ c# Z) e, i2 e; E' z0 [5 S& j# H+ l1 K9 d6 y$ @: f9 M7 N+ @
        Base case: fib(0) = 0, fib(1) = 1- h: Y% O( Z; f9 A) S, x' @
    / T/ @, G' v+ ~  O
        Recursive case: fib(n) = fib(n-1) + fib(n-2)- I3 m9 i& P* g# a) Y) _
    & |% e/ [% \: h) a% W1 U
    python
    ' d! ?* u/ \2 z- w/ A
    7 T; N9 |9 N# b' x9 `! {2 e
    7 q. p. h/ @/ \9 ]3 Sdef fibonacci(n):7 n. p. G6 K9 \) U3 D  h, e4 X) r
        # Base cases. T/ r/ R! Z. w% h6 g
        if n == 0:
    & ]7 m: r- ~! f4 E6 u+ Q        return 0
    * c" L$ k  }* g# H- k8 b1 [, C    elif n == 1:
    ; W+ D  n) c+ r. }/ S        return 1' f, r, G1 U% B$ \3 V& D
        # Recursive case0 E: m' S% r& W" E
        else:
    / r  U: f( a0 e* A8 C        return fibonacci(n - 1) + fibonacci(n - 2), o! S) k8 A; p) Y0 z" L

    4 i; C% ?) l# n6 Y# O% c( C# Example usage9 l7 I# S" @4 W2 t  y% {
    print(fibonacci(6))  # Output: 8
    + {5 P4 x- c3 v. U* l
    ' c# D5 Y4 x/ N* d+ B$ oTail Recursion0 |5 Z! ?' f" l. c- z

    , I$ w4 i% u6 A  |6 gTail 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).
    ! G# U" D: B. r8 G1 h4 ~. e9 q- X( F3 N2 G
    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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