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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
    " Y( R# F* ?2 x) w$ R' e5 ~) K- M- D5 y: Z2 G, I- C  U
    解释的不错& D1 ?( ]' ~! R  \4 Z  X& E( j9 Z

    " Q/ e! R/ Z( ?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。' ?# e% r* f( u* x9 a
    7 e( h& l8 z5 ~. q/ |$ _. }
    关键要素# d7 n0 c8 c, m; r* G
    1. **基线条件(Base Case)**
    0 }( I; A. L; Y5 O7 I; r   - 递归终止的条件,防止无限循环- H: J, w0 \0 U% p. I7 l
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 13 T6 ^1 H1 [: W& o1 I
    / N! T( `; {0 P6 e! P- \+ X
    2. **递归条件(Recursive Case)**
    + x; q- ]# t6 l0 [/ s6 o! C9 j   - 将原问题分解为更小的子问题& o+ Y5 ~4 U2 i4 ~& H
       - 例如:n! = n × (n-1)!+ {) e' {2 d7 @% G; J& q# C* S# Q# i8 `

    4 V+ _& Z4 M) X) N% d) F( ?- k 经典示例:计算阶乘( n* @3 y9 U3 l$ A9 E
    python
    ' u% [6 e* r! E) ?* H0 ndef factorial(n):
    . Z* w+ A9 |3 [6 x' h    if n == 0:        # 基线条件
    ' v: R  M/ V4 S5 j        return 1
    . @) h+ A0 v2 K5 c7 Q8 Q7 P    else:             # 递归条件
    . B& `( G# F0 ]# j3 ]        return n * factorial(n-1)/ F9 c* A1 G4 Z* l) T
    执行过程(以计算 3! 为例):7 {" {2 s( P9 y7 B
    factorial(3)8 K- |: _( p$ m7 D
    3 * factorial(2)
    - Q/ C# N" y: h; a3 * (2 * factorial(1))/ e& a, ?6 _6 j6 K
    3 * (2 * (1 * factorial(0)))
    2 f: |) p$ B1 y$ }! u+ n3 q3 * (2 * (1 * 1)) = 6: w) f. b& \4 [3 U( c1 T
    1 B9 c2 K% l' k$ i& @
    递归思维要点
    ; ]7 z/ X* @. `- |1. **信任递归**:假设子问题已经解决,专注当前层逻辑9 N9 b+ K) l9 V! X& b1 V
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)4 k) C; i5 s$ B1 _' H' e, w9 e
    3. **递推过程**:不断向下分解问题(递)
    " S6 ~% L7 j. C0 i6 n4. **回溯过程**:组合子问题结果返回(归)
    4 U0 u# \  r4 j# s/ c: M1 T5 ~0 ?7 @2 a) E
    注意事项
    . Z) A, T1 n( L  ]; K$ L% P) k必须要有终止条件
    4 r( a. F7 n8 t) \; p; O. b2 ~- `递归深度过大可能导致栈溢出(Python默认递归深度约1000层)" L% g& F4 C! b
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    2 u. |8 o6 o9 T8 z尾递归优化可以提升效率(但Python不支持)9 a# O7 \: P! V  G( E( Q2 p
    / W% y6 n  k% t' d
    递归 vs 迭代
    4 f: M  e$ I4 A# y|          | 递归                          | 迭代               |
    & F- F1 Z* ?: K2 y3 a% U' D, p|----------|-----------------------------|------------------|' |  @! B& D5 P( m3 ^! ?& W
    | 实现方式    | 函数自调用                        | 循环结构            |
    8 Y( k5 `5 o: f2 F! x! c| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
      F% L) J, J8 g! j1 ]/ X& {& b0 i| 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         |
    - r0 a7 u% a& L: H, _' l2 {( f$ B| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |% e& [9 U$ q# V; V( B8 I! T$ v

    $ {1 t& d6 b3 C- l) B/ k- j% _$ { 经典递归应用场景
    9 o/ Y- x2 h  @8 P5 n3 T1. 文件系统遍历(目录树结构)
    9 V4 `/ F4 W% Q( M  K* O2. 快速排序/归并排序算法
    - T" t( H( Y  o' t. X& a3. 汉诺塔问题
    , R+ O9 f9 P; G' T5 Z& X4. 二叉树遍历(前序/中序/后序)
    1 j& M+ E$ \+ G$ j4 Z9 q$ q/ O# M  d5. 生成所有可能的组合(回溯算法)% P$ L2 A6 D2 ^& w7 W+ E

    6 x# r6 f7 g9 X3 h: M9 o5 I, j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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    前天 06:54
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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 \5 g. e3 u) U4 Y5 w( G1 @我推理机的核心算法应该是二叉树遍历的变种。
    4 A) S" u: ?8 N9 R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:
    ) z. Q$ L2 w* S$ t: ~1 _Key Idea of Recursion
    ' `3 E' f0 u. a4 k$ a* d% }* B/ W' T8 f' q$ f) F$ v
    A recursive function solves a problem by:
    # f! @6 V/ U3 w- S! G. d' }3 r2 L) f5 `
        Breaking the problem into smaller instances of the same problem.
    ' @3 f' l/ i6 |* n8 o8 P+ ]1 W$ M: C) j' s
        Solving the smallest instance directly (base case).7 L" u* K- {. _6 g& ?8 ^  `
    4 o8 o1 q/ q4 f0 z, A! F" ?- F' x
        Combining the results of smaller instances to solve the larger problem.6 t, L! ]- Q6 h& h( p5 |
    5 A/ L  w) J# i2 _& U/ y% j4 l+ J8 Q
    Components of a Recursive Function
    3 J) x3 ]4 B4 j: `
    & q" |- G8 E  \2 J    Base Case:
    ; p( j/ w  U2 h2 p- _& O! I$ K3 ]. r- W0 r
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    % q" q5 t' u# |( @3 X2 J/ j7 K5 _' H: Q: M2 W
            It acts as the stopping condition to prevent infinite recursion.
    6 a8 h' ?- N0 ?$ j) |
    ! l( Y% S. [( g) V0 G        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.$ I( j7 {2 j( [3 {8 i( v# c6 I0 l

    ; I2 }+ [7 |- l, \    Recursive Case:
    6 l! g, G5 `1 N5 O5 u6 ^8 o6 R5 N' E$ `7 I- \* r& x, b0 n
            This is where the function calls itself with a smaller or simpler version of the problem.
    % y8 j) e2 t4 C0 d  [# S" ^+ b- w
    6 R1 H( ^0 m! g3 h6 A' x1 l        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ; \2 v/ p) l9 G) R$ _) U( R# e9 r! S; T& c* S
    Example: Factorial Calculation
    1 F8 r2 W" S! L1 ^$ F! G. _; n
    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:
    2 N2 c3 I* R8 i1 a1 G( |( F$ n6 ^# {
        Base case: 0! = 1! b7 P0 t. x; u4 e2 ^0 L% D5 {

    ! Q5 _; P6 U" v9 p+ u    Recursive case: n! = n * (n-1)!
    9 l( c6 K& j6 J3 o9 G2 Q
    ' a0 e' t0 \+ @5 V. QHere’s how it looks in code (Python):
    4 [' |: y% j& P+ q; o8 s9 mpython: H1 \, c9 Q2 n; b- D
    - ]  Z; c' Z4 z/ F- H- o

    ; Q) A  ^+ E: p- wdef factorial(n):
    # _8 v9 {( V2 H8 Z( P: M' y    # Base case
    % h- Q  x- }% Z2 r    if n == 0:6 y, K8 p" J! J9 G6 ^- G0 @+ {
            return 1
    * p* a; U$ k  ]8 _9 G' R; W' ~    # Recursive case
    / u( X& I% M! i7 p    else:" n) O/ V6 C. Q$ i6 x
            return n * factorial(n - 1)3 B( E- X: J) Q
    3 X! h* L! Z1 m/ I  o$ y# p
    # Example usage
    % A- R% [" b4 O/ ?5 r9 }  z5 Zprint(factorial(5))  # Output: 120
    : q1 l% x* `% ~. |" W6 H/ N( }  S. W) O4 p& H* e$ o7 X  H2 q, O! T4 r
    How Recursion Works% V/ M# g" j8 ^" t$ \: j* J( d

    9 H5 l' e- Q6 s  n    The function keeps calling itself with smaller inputs until it reaches the base case.2 j$ S' S7 a* G3 N0 V& p

      @/ z- c# F: e! N+ L$ H5 \/ M    Once the base case is reached, the function starts returning values back up the call stack.% g1 t  u& J* B1 m8 j4 c) u
    8 V' e& t4 X% C8 {: t/ x
        These returned values are combined to produce the final result.( d6 f6 L" [1 _2 ?
    # d6 A. b, }8 E( P) n; h
    For factorial(5):' F2 ~+ P+ [9 {3 N* m; x. p1 F8 ?

    - @/ h+ c: S, M' C  s! K2 ]) I% I* p9 q
    factorial(5) = 5 * factorial(4)3 r7 g5 L# \& n7 J, K
    factorial(4) = 4 * factorial(3)* N5 A, j* R% B+ a' Q) a" ], Q
    factorial(3) = 3 * factorial(2)3 |! a. u9 t4 x/ E: D, S$ \
    factorial(2) = 2 * factorial(1)
    $ b$ S4 m9 d  p- Pfactorial(1) = 1 * factorial(0)
    : ?' A- P2 Q6 G# p7 R! kfactorial(0) = 1  # Base case
    : J0 r" _- D6 R+ d
    $ U3 V$ g& W7 B  N+ r9 a) K% KThen, the results are combined:
    , m1 [1 W9 h0 X) l& L) F% J% n" N/ l! e* L  Q

    $ b  F- u2 ^  C( h4 j( p/ kfactorial(1) = 1 * 1 = 1$ y1 c1 ?2 h& z3 Z- }& l: L7 t
    factorial(2) = 2 * 1 = 2
    , D4 C1 }- W# i0 e8 q* o% gfactorial(3) = 3 * 2 = 6( c' n; A( f& ]% m. j0 {# o
    factorial(4) = 4 * 6 = 24
    , I+ y3 L4 d) G+ w* qfactorial(5) = 5 * 24 = 120& }3 h2 n; g, \% o3 B; J' q( D0 D
    5 @8 {6 Y3 H2 V" _+ q( ^
    Advantages of Recursion' Z- S3 ^) {& ^8 X: w
    ( R1 H6 P; F3 q6 k
        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).
    7 w' a  ^' I' U% c4 r
    % f9 }+ L& Q$ R/ \7 I    Readability: Recursive code can be more readable and concise compared to iterative solutions.  H1 s/ m: O8 O4 P" H0 P

    + a5 @$ n8 O- Q4 Q4 TDisadvantages of Recursion+ B" v4 ^' a! T- o- n
    / F8 F) @- A' |6 o4 {
        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.
    : W/ z! ?% A- X( {2 ]
    * M3 p' @2 {! D2 t$ ^& D    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    6 x- y' f# m3 k$ K' A& G+ n5 @
    * b0 ^* Y  ^1 s! x' r$ _" cWhen to Use Recursion
    & D7 {* ^9 ?( C! I! F3 k7 G1 v- O+ z' g' A& f8 z# L
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    $ w9 W0 P; C8 Y% w
    3 D9 Y' n, {; ^' E' g2 ^0 d2 B    Problems with a clear base case and recursive case.
    " J+ B  V6 B5 `9 X' E$ J, U% x2 Y( V% F/ p. }' K5 ~
    Example: Fibonacci Sequence, ?: E& v& h3 r: C, e  O

    2 k; b1 Q! J/ F4 J7 Y" J5 ]The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    3 ^! s  _  e% j, G
    " d3 \. H) ?* C6 H: ^9 S    Base case: fib(0) = 0, fib(1) = 1
    ' e) R0 o+ |7 l& N7 d! T& W* {9 J% `& L5 a9 q2 Q1 ~
        Recursive case: fib(n) = fib(n-1) + fib(n-2)  D" g. A- E$ V4 \! ]$ w/ n' w( e; Z
    + G* M* Z+ P* A! [1 W
    python. P9 G' P, [# s7 ?9 x# s  N, z

    ( K/ L* ^' ]4 _7 |  {
    9 A$ a5 s# X0 q. u5 Q' m/ A! P( cdef fibonacci(n):6 j* D8 i% m( L# N! t. {. c
        # Base cases" K( J5 K8 m" \& E2 r
        if n == 0:
    ! r5 U0 i2 g8 \        return 0
    ' \& m7 e9 [7 ~2 Q" {% }    elif n == 1:6 j' L; y) h/ _$ G6 N: P8 p" @
            return 1
    ) m% m: {4 h# Y- [    # Recursive case% @$ r* q% c' b
        else:# v2 x. N6 H( k/ b+ v; @! @# b! ^; S
            return fibonacci(n - 1) + fibonacci(n - 2)& _* y2 h. f3 U" H2 n1 v( ~
    ! Q0 ?& g  B9 z6 t2 |
    # Example usage
    " T4 g% d' g& }print(fibonacci(6))  # Output: 8
    ' a* u4 R1 z7 w& R% N2 [) R. K
    ; h) }- z  B; j$ }1 JTail Recursion. t$ a5 n8 O$ e) B# ]% Q4 L: U

    0 h1 F6 V% i' }" n1 VTail 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).
    ) p: b3 @/ N& P4 {9 d) Z5 q2 w7 p! A3 X1 b0 y7 x4 _! T
    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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