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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 7 y% _8 b( ]/ E8 A0 G6 E
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    解释的不错
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    4 O. G& }* b2 G3 o& r, ^* D+ }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。
    + U$ }! q- d8 _9 M4 q/ f1 k) |% O" f$ k) W3 @  @
    关键要素
    ' o1 Q$ E) W- T4 s$ o$ B1. **基线条件(Base Case)**
    % @1 M  H' K0 z7 w( ^+ F! K/ e   - 递归终止的条件,防止无限循环
    " Q0 J  r9 ^* S9 i# O   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    / e; q5 Z% H! n: i; m+ r! d' n+ `1 m+ G
    2. **递归条件(Recursive Case)**
    ) J( o" g2 i3 w' f* I   - 将原问题分解为更小的子问题
    % e) {. Q, t, |   - 例如:n! = n × (n-1)!
    - M+ j) R$ d% |4 C& T0 F8 o  z. p! y- P9 }: f- M7 H/ n
    经典示例:计算阶乘! e+ u9 _+ w% S  e5 g5 v
    python% o. M1 `7 f* n/ b8 x7 p  c. O+ T
    def factorial(n):1 Q) I  P5 ]! O9 Y" F
        if n == 0:        # 基线条件* u: Y4 }1 @3 L* s
            return 1
    8 r. |# B; B# j. t4 H9 w    else:             # 递归条件/ v9 w9 c+ Z7 v3 i0 I
            return n * factorial(n-1)& u/ p, E$ ~7 B) E4 x" `) k
    执行过程(以计算 3! 为例):& t8 Y  i$ u+ H3 c/ e4 e
    factorial(3)( n; A! p" ~) a
    3 * factorial(2)& f6 `$ z  `- A# g. P* I
    3 * (2 * factorial(1))
    $ n# g. t+ }8 M3 * (2 * (1 * factorial(0)))8 n% Y( h3 V0 b3 {9 m8 v% @, ~
    3 * (2 * (1 * 1)) = 6' ]7 U! a* ^# C! s4 P
    2 y/ @- K  `: \* n7 c
    递归思维要点
    8 y2 M6 y  Y4 t% o1. **信任递归**:假设子问题已经解决,专注当前层逻辑4 v* F. j! C& y% @0 |
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    : M" j5 ^' B( q# t  @, r. ^& v& X3. **递推过程**:不断向下分解问题(递)' x! R. T# z/ a/ @$ `/ B
    4. **回溯过程**:组合子问题结果返回(归)9 G+ b7 I; t1 _

    # f" o: `  m; M8 R 注意事项% X/ g5 \3 G6 o$ S3 Q1 ~5 E, l
    必须要有终止条件& E$ B0 P( l1 `0 z* p
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    5 b; N0 a; M4 q0 d! c! f8 y6 o某些问题用递归更直观(如树遍历),但效率可能不如迭代6 |  G7 u9 B' o4 x6 p3 X. h
    尾递归优化可以提升效率(但Python不支持)" t! r( k, e. p, a$ N6 s

    7 `  U" Y2 ~2 |, l2 Y% p6 G 递归 vs 迭代7 {7 h3 ?! m/ N' C; J9 _
    |          | 递归                          | 迭代               |; T' N3 z9 {1 U0 N
    |----------|-----------------------------|------------------|
    3 L) [/ s6 m) b5 x# R0 S$ M5 b| 实现方式    | 函数自调用                        | 循环结构            |/ P. X: x. ]+ x9 n
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    3 [3 u9 b& r0 D8 l0 x" f| 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         |! ]8 }* Q' g" w
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ; N; G. M" F- r! o- @
    6 x) ~( v6 F# I4 ^& N7 ~ 经典递归应用场景
    9 t) z5 y* B4 Z! p$ i1. 文件系统遍历(目录树结构)
    % T* |5 p  q; Q6 W" b2. 快速排序/归并排序算法
    % d7 H0 X: D/ K% m  f6 H0 B3. 汉诺塔问题
    $ t1 b2 b  y  e5 Y4. 二叉树遍历(前序/中序/后序)# l9 w  \0 b" w
    5. 生成所有可能的组合(回溯算法)* \$ D9 h/ H/ F4 H
    ( _+ y- z! ?' ^7 K; q
    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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    4 天前
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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,
    : h( M, t# h+ z6 {' ~5 R0 i我推理机的核心算法应该是二叉树遍历的变种。% T9 Q9 f6 |7 |/ Y$ h; 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:/ d7 X2 J& Y9 a5 J% Z& o( o' K
    Key Idea of Recursion
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    A recursive function solves a problem by:- [. t- u# s3 R. ^5 w* k

    - l7 H1 j9 g) {; y    Breaking the problem into smaller instances of the same problem.
    + {; u3 h3 a, P2 {; t6 A/ a3 [3 F4 P) L% F6 w8 S
        Solving the smallest instance directly (base case).  d, P8 S& M0 ^& K' i9 a2 C  E/ M4 Q
    # |" D( _2 e) R; ?' Z
        Combining the results of smaller instances to solve the larger problem.
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    Components of a Recursive Function. I3 o  b7 N$ Q- N" B

    ! }+ U" |6 Z) S; n- [0 S1 i    Base Case:+ f, a) D, W. R, R4 P
    : n. J2 F# F  q9 c$ p* s2 s
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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    + b% e+ \$ B; ~0 g( L- l- B8 ?+ h        It acts as the stopping condition to prevent infinite recursion.
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    ( _. Z5 [) H" i# u. J5 D* ^: i# F        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.  m# {9 }. \: _6 E$ t* i
    % }0 _4 `  h- Y7 [
        Recursive Case:3 Y, V* e2 j# o! O8 h/ n/ p1 Z0 |! \

    ! X. j, H$ Q9 |        This is where the function calls itself with a smaller or simpler version of the problem.) X" h& H+ @9 t

    0 ~& Y9 i! [$ ^2 E2 e        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    . X0 J$ W( Y1 X* c* C2 A1 y& N& {% h) r3 I& L5 N% |; R
    Example: Factorial Calculation
    7 M) M+ C, t% G* v; H! M0 Y
    ) P" D" {( W$ F1 E5 WThe 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:
    3 e+ ^5 l( B' Y( d) D
    " A6 S6 _$ b4 R! A" s    Base case: 0! = 1
    7 f* W( T( k+ L7 K$ y0 m0 ?
    ! N7 B  ]2 @! |+ ^; \- o0 {    Recursive case: n! = n * (n-1)!
    2 j4 `, v# N; e( n+ \5 s! N7 n& ]! Y, g1 V4 j+ L, n
    Here’s how it looks in code (Python):
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    " Y% P, P) B1 f' @7 f. D  i

    5 q( u% q' g6 idef factorial(n):3 ]" p' |" H7 q) {9 P
        # Base case
    % y! O* @, B+ ^# t    if n == 0:
    1 X' t  \1 p; g' J; J8 v! [& s        return 16 q. I& E: d" j( w3 ?5 h" V
        # Recursive case
    . F+ u& c1 y" t6 }, M    else:6 Y* H* V. S* V; j
            return n * factorial(n - 1)
    5 a* J8 d" J5 J( I1 c( W
    . ^9 ]: u7 K3 \: m9 z6 f' z8 ?# Example usage3 c1 D. p( G, E3 M' ]" c0 t
    print(factorial(5))  # Output: 1204 W7 ?1 S& s) e
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    How Recursion Works
    2 _2 F) d/ F6 R: x% b1 n  Z, B' ]" T! O3 F2 k1 k% i) ^  v4 P
        The function keeps calling itself with smaller inputs until it reaches the base case.+ r$ i5 s0 I# ~; o. M7 z$ s
    , }$ G; {5 B! R4 w, c' y) M" T5 ~
        Once the base case is reached, the function starts returning values back up the call stack.# I9 P0 Y5 h8 Q; F+ ]0 i0 a
    * \! N+ O3 X6 X7 O
        These returned values are combined to produce the final result.
    7 _. D8 d$ Z, l. [9 Q0 V8 B
    $ w& @5 K& L$ u) gFor factorial(5):: a/ H/ v2 C! q% T

    % x2 w" s) b2 _9 m
    7 z( U: x3 V7 @factorial(5) = 5 * factorial(4)2 t( A6 z+ T; ?- \/ q0 v
    factorial(4) = 4 * factorial(3)4 O  ]. m) o8 J8 [  h1 c1 A* n$ H
    factorial(3) = 3 * factorial(2)
    , t7 g, f2 w% `5 `0 ?5 z& \& ]% Z" {factorial(2) = 2 * factorial(1)
    % x3 u3 }9 N+ Y9 Ffactorial(1) = 1 * factorial(0)
    * C5 w. h) v7 Nfactorial(0) = 1  # Base case- _( s! e, x$ Z1 _) e0 D

    ) }) x% X, n& @. _$ b6 DThen, the results are combined:
    4 ^' C+ R/ a  i4 B  G: Q, {6 |) J& [2 [6 |& p5 \" s* A

    - w) p/ c& k4 {# w% `' ffactorial(1) = 1 * 1 = 1* A/ x! \. @6 r. K3 L0 k
    factorial(2) = 2 * 1 = 2( u( J8 G: O2 k, w2 p# S/ q6 G* y
    factorial(3) = 3 * 2 = 62 i4 k' T0 g- A) p5 }
    factorial(4) = 4 * 6 = 24+ y( w; e  S3 \; X
    factorial(5) = 5 * 24 = 120
    # W9 R. D+ O7 a. l; m
    / u4 G1 R' t; r! r" _Advantages of Recursion
    1 p* X0 B, n# Y6 h( e
    0 C4 U+ d" `. @3 L+ |6 E8 d    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).  B' U) Z% N- m$ j
    ( r: T7 q* K# U- C, m
        Readability: Recursive code can be more readable and concise compared to iterative solutions.* B( h+ z. S* T3 ~# y+ K; z

    : z; i% B1 K- e# o3 x+ E6 oDisadvantages of Recursion
    $ G; ?7 Q+ z: A2 t" f# u& K$ [: \" `
        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.
    ( b4 C2 w5 v- G5 K( x. Y
    ( d/ F7 d/ s1 q$ R8 w2 a5 b    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).) \: A3 s" _6 m
    4 ?- ~, X9 ]* M; c0 h: X0 Y
    When to Use Recursion6 e: G0 h5 S$ }5 y' T4 G$ y1 t

    / y0 q7 A/ a. f/ z' [  O    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).8 x2 X! A3 d4 ~0 O) R

      }) ]6 f- G/ E/ q! J  g" U. _2 f    Problems with a clear base case and recursive case.
    4 n* j' N- t; f) d5 K4 K* s6 r$ E; B2 m9 j. R
    Example: Fibonacci Sequence
    , s4 E- |5 |( k, F, z9 w: {: {" y8 Q7 P: n
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    " |8 a1 U8 F6 I0 {0 E2 f( e4 M  R+ d9 \0 v4 @6 I5 f
        Base case: fib(0) = 0, fib(1) = 1+ C: O1 a0 h( z5 H% q0 C$ J0 c
    , L6 [9 X6 Q& `3 `1 q
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    python3 c3 z: v1 K1 X) H+ U4 G
    2 ]! a' \( L( I* {  e2 N2 _# `' g

    ( y1 r1 E' H' v( X$ udef fibonacci(n):
    ) w1 y# {, W- L    # Base cases' d/ b, K0 q, t+ y
        if n == 0:- p3 L7 U5 ~3 O5 B2 v4 ]% @( j
            return 0+ W' l) Z5 Y5 M) O) [/ l
        elif n == 1:; R  q( C$ W0 i7 \
            return 1* A8 c7 D+ r' l" S# v
        # Recursive case
    9 `# }4 T! O$ [7 j& f1 p    else:& x* [0 ?  [5 a# r  `& J
            return fibonacci(n - 1) + fibonacci(n - 2)' _. B1 P! G- Z6 T# d( s
    ( G7 R, v# j) `) m
    # Example usage5 Y& R& v+ b5 N$ z
    print(fibonacci(6))  # Output: 8* S+ \2 D. \, q* q& q
    ! R. g7 o; t! \+ x4 x) Q" U
    Tail Recursion$ Y: D/ p0 \  S1 z

    5 H. B/ \. g9 r5 NTail 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).
    6 ?- t2 H9 v8 C) J0 k8 A/ O; Q9 S* Q: H9 X
    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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