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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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    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 0 {/ V; M9 y& K: y: g& j: M4 w5 f

    % {+ ^7 y( @7 t5 P2 p解释的不错( W; V8 V& k8 a$ O: v! H5 d7 j  {3 w

    4 ~4 [9 z) k5 r. q; @5 d4 s  s* t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。! S3 s: q3 s% {# G( K' j; t

    , I9 ~5 c4 G5 z6 u: C 关键要素8 _$ i+ ^+ ~8 q- B% S, m
    1. **基线条件(Base Case)**
    9 v2 K% c' C* A! B   - 递归终止的条件,防止无限循环6 e0 d* E& U( E* D6 B
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1- [' N& s% F% n

    0 k3 |1 j% R0 }9 L7 \; t, e# e) n2. **递归条件(Recursive Case)**
      R3 E) l* D3 Q0 w4 Z   - 将原问题分解为更小的子问题
    % f* L1 Z# c4 S8 x   - 例如:n! = n × (n-1)!( U0 b& Z" E0 d" x+ u

    % U, [$ O1 [3 w0 q( L 经典示例:计算阶乘
      F+ {* t% E* x; @python
    0 c! H, u3 _/ S9 ]! qdef factorial(n):
    0 Q) ]) M6 a$ C( ^  x2 U: W3 ~    if n == 0:        # 基线条件  r2 M0 |, [' O1 \! ]8 T8 W6 u& W
            return 1
    7 Q/ ?0 S4 [4 H/ z% V" p! y/ O2 ^    else:             # 递归条件
    & w( a" ~0 F$ h8 a        return n * factorial(n-1)
    ( h* d; V, t- w! \, y! b. K执行过程(以计算 3! 为例):
    2 V( |) B; e+ {" D% m' T* L- t! Tfactorial(3). @- O! O6 F/ ]. N3 G
    3 * factorial(2)4 i' `* s, J! N/ D' H
    3 * (2 * factorial(1))/ y# T6 t9 v% i% Q8 ~8 c/ y3 V% X8 e
    3 * (2 * (1 * factorial(0)))1 b5 y; [/ r) L
    3 * (2 * (1 * 1)) = 6* P8 ^4 r$ z# ^# o

    - X$ J- m; S6 b" V9 M. l& K- _ 递归思维要点; P8 \8 C6 [6 m. H, o1 d& {
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑( i4 p* S" e' K9 t2 y3 u
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    : A8 B  c- s6 }( Q3. **递推过程**:不断向下分解问题(递)% w) J: I. W* O+ N
    4. **回溯过程**:组合子问题结果返回(归)
    1 M0 o) h" v& l& q1 X
    2 N- y2 i0 y0 w3 r6 R  f 注意事项% b# j, ~! c9 Z9 X
    必须要有终止条件
    , E9 v" D- E. c1 t- i9 b5 `递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    7 q/ B0 u( S0 F8 s1 p' |* p1 j某些问题用递归更直观(如树遍历),但效率可能不如迭代
    # B8 ?/ h; c9 }0 E5 x/ F" T& i/ V尾递归优化可以提升效率(但Python不支持)( v8 f1 h8 k* [; S7 d2 d

    3 _1 V/ w' a  L7 z% `, F: W9 X 递归 vs 迭代, g8 c- t& H9 R% d
    |          | 递归                          | 迭代               |7 v/ j4 ?1 d4 S
    |----------|-----------------------------|------------------|2 o. U- f4 f4 }2 h; w# c3 A
    | 实现方式    | 函数自调用                        | 循环结构            |
    $ W/ s' O3 I3 H5 w+ o  g7 X| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |& L! p  ]' J, K7 j
    | 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         |
    1 j- i# u. L9 M' L# |( n* U| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |& S# a, ]1 |! e( f
      a  y& R4 t1 B4 h/ N& {
    经典递归应用场景% A# D) r# B' x  o- Z) [) K! ]" @0 e
    1. 文件系统遍历(目录树结构)% U% P& ~" A; z! x+ v2 N9 L
    2. 快速排序/归并排序算法; P. f7 U' c* \! ?: ]2 R4 v1 z& U
    3. 汉诺塔问题0 F! E# ?! \* z- i% \, b1 u% b
    4. 二叉树遍历(前序/中序/后序)
    & c5 c: d6 C# ?# W, A5. 生成所有可能的组合(回溯算法)+ ?- U% E2 i: B) I  S
    6 N( q( B! f8 s0 o* J2 }
    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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    昨天 06:41
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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! F' U! I, S8 ?# u$ p- q我推理机的核心算法应该是二叉树遍历的变种。
    # _8 M! s  F1 @, U: Q( y, F1 r+ G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:
    + m9 q  I. `8 a) w4 R6 iKey Idea of Recursion) z7 X. D2 ]  L9 m& Y+ G$ A

    4 g3 M/ _" l7 J7 kA recursive function solves a problem by:  L* {- i: n9 T. ~0 Z2 J$ q! w
    1 f8 \- v2 ~+ S( C! s2 \* f. W
        Breaking the problem into smaller instances of the same problem.
    ) S  R. V8 k% M, k' w0 Q
    9 e  H# g: }! Q; \- ~; E! I    Solving the smallest instance directly (base case).
    ) B! h3 r& i2 s4 J/ o5 q4 L
    3 d9 B; Z6 u! z0 {    Combining the results of smaller instances to solve the larger problem.& U6 l8 Y3 N# V+ U2 X6 L! z

    5 ~( m9 O$ k+ _5 w' lComponents of a Recursive Function" h: u: F6 C0 F% P
    1 ^1 n9 x, e6 U1 `/ Y) ]0 C9 W
        Base Case:' X* V( E' N8 C- t- T
    5 y2 d' D) @  n4 K: l
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion./ ?# Z3 |, T6 w" u( H. @& N

    ' b5 l' v6 d8 g* K$ {7 [" N* |        It acts as the stopping condition to prevent infinite recursion.8 z  U$ g5 @) _/ i, n5 L
    " B/ z3 n' q1 |3 r2 E
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.* T( X8 F. z* u9 r: g3 D

    ! ~/ R; \4 T7 k: q    Recursive Case:+ u5 h% ^+ h8 C# o$ P

    1 d. c" f" V. V. z% M+ d        This is where the function calls itself with a smaller or simpler version of the problem.3 c% F# G2 T% c8 g
    8 x, v, k2 w+ p' n; ^1 n9 P
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    0 L( I# M' Y7 z; V% U# X4 x8 {' l/ v7 [' d1 R+ b) j1 Y
    Example: Factorial Calculation9 j% B5 L) t; D/ f, W/ f: A
    & k' O0 K  ~5 l
    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:- E! p; `  X; \. ~; Z" S9 K1 q

    ; T: h* |' S9 y1 s  P/ n    Base case: 0! = 1/ u: ^) _$ ?- J5 L! q

    8 L9 s4 T& B. d    Recursive case: n! = n * (n-1)!% q8 j  u( \# G+ y/ ]
    6 g( _9 A6 A2 G% O' x
    Here’s how it looks in code (Python):
    ' L! C9 _$ E! apython
    $ {. x3 t8 N6 d8 M9 t
    / R1 K# T% f7 k$ G
    ; P2 D5 W6 ]' W2 d) ~* T- kdef factorial(n):9 _( d+ R* [0 n% p2 K
        # Base case* Q0 L, }' R" J0 D% L) t
        if n == 0:
    2 T* M( k4 F2 f9 ~( u        return 1
    7 X4 U$ W4 c2 [* D    # Recursive case% N# F& A! B- }- W  t1 {
        else:
    " {# M0 z+ X/ v7 E% k3 K! Q        return n * factorial(n - 1)
    & P5 D4 \: \4 W# L/ b, E2 Z/ `! O
    2 l' C5 q$ ]$ g4 h! A# w# Example usage* E$ @: J6 A$ A. c3 c" P% `, \
    print(factorial(5))  # Output: 1204 v# D# `4 v8 p" k; R0 a$ F$ h
    ( u2 Y4 u) D4 Y
    How Recursion Works
    . d' ]( S3 p' R* ~
    0 T$ j3 Y, N- A1 L  }7 ~! F/ M/ s    The function keeps calling itself with smaller inputs until it reaches the base case.8 @. t; u; [1 V# S

    5 [) y" M; Q5 A    Once the base case is reached, the function starts returning values back up the call stack.9 w1 [# C- p) s9 G: q
    " r" k3 V5 X; y; J) C) i+ G0 p
        These returned values are combined to produce the final result.
    ' l: |* [" C  [  f: f& {5 _& ?3 }' O( u* y2 t8 v* [
    For factorial(5):$ W) k4 B  ?  W; F5 t: M
    - T) C# c9 ~; M

    4 `( S) g5 F- J" l5 P; p; n% Ufactorial(5) = 5 * factorial(4)* @! h* H5 W4 H$ c: F. x. M
    factorial(4) = 4 * factorial(3)  r1 t/ P( [( P" ?; c; @: R  I5 a
    factorial(3) = 3 * factorial(2)/ m( u# _7 G; d2 x' U( S
    factorial(2) = 2 * factorial(1)
    3 ^- W9 [, T4 Y8 ?4 u+ J# V0 ufactorial(1) = 1 * factorial(0)5 w5 I3 P. M+ l6 y  ]3 O8 T; u. K
    factorial(0) = 1  # Base case
    9 z3 ~9 J- o5 _& O, }  y2 x9 }/ T$ h1 r. o, F! a
    Then, the results are combined:- O8 W; c) H( ^% D& V

    ! `, f* T1 o0 K, i
    + r- e2 m6 i4 U! [2 p1 d4 Z( Ufactorial(1) = 1 * 1 = 1
      @! @2 \0 L) v: Ifactorial(2) = 2 * 1 = 2# Q& \0 g4 ~! I  ^# B
    factorial(3) = 3 * 2 = 6
    * ^4 n0 r; g" _factorial(4) = 4 * 6 = 24% C6 P# L# U. V, j! F) F1 X. L
    factorial(5) = 5 * 24 = 120
    ) F- J/ o1 C2 n: L; R! y  n5 \
    + N- H# X( h& DAdvantages of Recursion0 U7 d( ]; x# v8 t* o5 W1 z

    ; ?6 x& Z& |) U* t4 e2 C    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).' x" ?; x; M0 J0 x; }& n! m, B
    , C$ O" G  R+ U. s# N7 A  {, Z( ]. ?
        Readability: Recursive code can be more readable and concise compared to iterative solutions./ F  Y8 ]% }' f" R' b0 P

    - J' s: q0 }6 @Disadvantages of Recursion1 Y/ D& m$ F6 r0 b/ A

    & Z; E, E! z, t/ 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.& D1 Y9 I& t# R& ~
    $ p- V+ ^" H" s* }- X
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    7 d6 s2 Y: s/ l9 ~. u/ @4 m' O) ?0 V
    When to Use Recursion
    6 ]9 t. a* r0 Q; \( f, o
    1 k% A9 m" p$ \. {    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    1 A5 X; I  h2 |* x8 D
    ( N; W* \1 G7 v' i4 ~1 X    Problems with a clear base case and recursive case.
    , ]/ X8 J) x; q
    $ G/ a4 c+ u" T* D; O/ T& TExample: Fibonacci Sequence, g+ n; g7 H7 p9 S& }! {1 B
    8 X. S+ x) z  s; \3 C6 l
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    * Q0 v! N6 G( O5 Q) J( m! |) t4 q$ h8 s" u
        Base case: fib(0) = 0, fib(1) = 17 ?1 Y7 Q+ p" l) y' _

    * j9 s! O8 D6 V& e: s    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ! j# \4 f/ s+ o% f6 G
    % V, ^& _3 E/ z" zpython
    8 a; }! }, U. [0 f: |! _0 S8 s8 S; J8 U& W

    4 G$ V" i4 D/ o3 U& q- ]def fibonacci(n):1 q  n0 ^. B: I0 B% y9 v' ?
        # Base cases
    7 S: Z& `/ |% O+ G    if n == 0:
    0 l2 t7 \/ g' t$ e7 I2 j4 u" k5 l/ m        return 0) z7 S# a) Y6 A7 l+ ^: v2 p% {
        elif n == 1:9 B: g4 Z( p* h( i: d
            return 19 q6 t$ |8 X) \
        # Recursive case6 g; }  u9 B9 U( [4 f
        else:
    0 d) o* h; |- }8 F2 y9 H        return fibonacci(n - 1) + fibonacci(n - 2)
    + \! K; I: _6 l- [' U' f" E& T, T
    # Example usage7 m. v% M1 Z7 S4 e0 W1 z
    print(fibonacci(6))  # Output: 8# N* D, C" p. B0 x8 |
    . g6 l+ i0 r/ n! _7 D
    Tail Recursion1 T( }, N% d# j8 S

    $ ^4 A8 t. F: o1 f& _+ M1 t' r" 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).
    $ v- \1 V! O: v4 E1 Q: K
    7 ^! c' h- |: t" V# ZIn 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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