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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 5 F* y$ b3 e7 C7 m/ i. L

    8 V. |  a9 q( O& \: W8 j解释的不错) M5 }8 S! k( Y  N9 t5 w
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    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。
    0 c, ]  b0 D) X1 A! |2 @0 u8 a0 h$ \& L* C
    关键要素
    / Y* y. v3 J7 y6 F4 G2 j% r1. **基线条件(Base Case)**
    : B5 z( Y9 y/ S! }   - 递归终止的条件,防止无限循环
    . z; _* K$ p9 K$ r. b   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 13 J/ a* @5 {, F# I
    2 m6 V* ~% d! H
    2. **递归条件(Recursive Case)**
    ) z2 Q! E, k: ?   - 将原问题分解为更小的子问题
    9 b) W4 H. X$ ?. A   - 例如:n! = n × (n-1)!, a% ?! Y% d9 b/ Y6 s  f; `  I

    ' d1 m6 r, o" N7 q 经典示例:计算阶乘
    ! Q& h5 Z0 E. @- z% }/ M$ bpython
    % S# W% l2 j, edef factorial(n):
    - A4 m5 i+ _. A# J$ a0 ]) e    if n == 0:        # 基线条件7 w5 T7 v1 ]1 ]* e3 X: }
            return 14 r  Y2 c5 P' J7 \" x  {) R$ W
        else:             # 递归条件
    ! i  J- q: \8 c        return n * factorial(n-1)! h4 n6 ]! f6 ]6 i1 O
    执行过程(以计算 3! 为例):
    5 }8 U( S4 P- u: L) `- E4 ^factorial(3)4 J& p, H% P% f: w
    3 * factorial(2)! K  Q$ q' `8 V2 K4 j- C
    3 * (2 * factorial(1))6 r+ f! N& Y( W
    3 * (2 * (1 * factorial(0)))
    1 H* h- e" S8 I1 h4 N; T3 * (2 * (1 * 1)) = 6
    ) s6 z6 S' f! ?. ^2 G, D
    + I& {; r/ l& M 递归思维要点
    $ F) O  Q5 P3 B  R# @, N% ?, g1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    * w6 f8 i; ?. z; H6 W! D2 C  ?/ d2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ! ?* P1 \6 W5 D1 u& ^7 {* i; n3. **递推过程**:不断向下分解问题(递)
    # [) k2 n6 A- [" b1 K, y( \4. **回溯过程**:组合子问题结果返回(归)
    - q" E/ \, P8 _
    7 E8 b6 x8 ?2 T# z/ C# r7 a2 s5 P 注意事项
    5 @5 t7 o0 t8 n必须要有终止条件
    # {! Z% `9 c' c6 W3 M递归深度过大可能导致栈溢出(Python默认递归深度约1000层)0 s$ D$ `' C1 D. l
    某些问题用递归更直观(如树遍历),但效率可能不如迭代+ |1 v: ^6 |/ k( |8 g
    尾递归优化可以提升效率(但Python不支持)8 l7 R; l4 j* Z# }6 T1 y
    $ r4 J6 `1 ^% O
    递归 vs 迭代
    , W) `7 b% z' R  c|          | 递归                          | 迭代               |1 L. z2 b) f' V3 C& z; d  X
    |----------|-----------------------------|------------------|
    $ s1 m) s' ]+ g1 @& g8 ?| 实现方式    | 函数自调用                        | 循环结构            |
    2 L9 E9 u0 Y' b/ b, j- c1 E% a| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |7 E! @; G2 h, a! 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         |
    ' L& i2 T; R3 }% x| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ( X& l# a5 j# z! [9 A9 L
    1 S, S2 g7 c+ O" j# v 经典递归应用场景7 x+ h: ~0 T1 X; U1 Y. p0 `4 W
    1. 文件系统遍历(目录树结构): L  S( k: [; E# G  E+ P% J
    2. 快速排序/归并排序算法
    & J; b, B& X% g( y; ^3. 汉诺塔问题, U0 |3 l& z3 M8 {& M7 n. u( _; {
    4. 二叉树遍历(前序/中序/后序)
    5 K' }& S. K1 W. T: T5. 生成所有可能的组合(回溯算法)" C# D) G* c! K9 J' v3 j: c' P1 W

    ! y5 R% H4 V! P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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    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,
    ; @9 V- T$ u1 k. N& o( H; {* |我推理机的核心算法应该是二叉树遍历的变种。: y! x/ c( [$ ~- C# i, s8 ~2 v/ P6 d
    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:4 Z* u6 a+ E: ]; `: L
    Key Idea of Recursion: n3 ]7 Z1 f4 S$ a
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    A recursive function solves a problem by:  j6 ~5 M4 n; O0 H

    8 m' l9 s' o1 X8 Z    Breaking the problem into smaller instances of the same problem.
    5 x2 X6 ^1 l- R! j0 r8 M
    4 ~  o8 H9 A7 J* N  _/ R    Solving the smallest instance directly (base case).& u& a5 u- E8 _, p# g
    * V& z* m; s8 h9 o
        Combining the results of smaller instances to solve the larger problem.0 m8 N( k0 |' {3 I

    , r" C' {( I8 zComponents of a Recursive Function
    9 I8 c, R- D/ t: i9 R1 Z  Q) a
    4 a! z, j; b. U! {    Base Case:
    # p3 I) ]& g$ t* W- N* E! w1 w' t6 W9 p% H' Q* Z% q
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.2 p' w8 h9 M- b9 p1 s% R
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            It acts as the stopping condition to prevent infinite recursion.
    - V2 e, _. V. d$ A5 w5 o
    + f% C( F. j/ g% b4 Q# z        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    ! j4 C. Y2 _$ T/ @5 Q- e6 q8 N9 f4 w; H6 G1 Z
        Recursive Case:3 a% W9 Q+ e! b& b
    * @' V" t! M* W2 N
            This is where the function calls itself with a smaller or simpler version of the problem.
    # U4 `$ N/ o, M8 J. Z3 I! z; P0 \2 B- f7 l9 u# E' s
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    ) i. P- q' w2 G; e6 }) B% J; OExample: Factorial Calculation- f: j2 n" @2 T( O. Q+ K, }2 ~
    % k2 J5 M. n8 t3 X6 K2 D, E5 V
    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:$ Z# T/ u9 Q+ i7 Q! s9 H6 D' a6 N4 D
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        Base case: 0! = 1# @& B* v( q- N3 j& c: T
    " n" ?% {4 u# E
        Recursive case: n! = n * (n-1)!
    1 K6 F# F  b- z; M7 r+ g3 z( k6 ]: J4 x& K' a
    Here’s how it looks in code (Python):
    # x- d- I4 {4 N' G% \6 {; ppython& z4 c" Z( k5 f7 U& r7 W

    6 v' }1 |3 K4 w9 y  o- I7 }8 M1 [
    , s% m. {- N& E7 S# g5 N3 }def factorial(n):
    $ V! Y  P9 L7 F/ u0 q+ n    # Base case7 g3 e8 o/ H% |& w6 i
        if n == 0:
    ( [. ~" n: i( b5 K# b, d( O        return 17 k4 n& c; s% A- J
        # Recursive case
    : `! [7 j9 u- r) R7 f, X: q" H    else:# R, d* l6 m/ E) G3 F. \% |2 H
            return n * factorial(n - 1)
    , F4 v! O% t6 U: C# b
    , u, }0 t' p$ L% w# Example usage, D, q! J) |) Y: p# z# U1 ]- y
    print(factorial(5))  # Output: 120
    % E1 R9 l0 z: o: p) A
    0 e; G6 L6 C5 c: ZHow Recursion Works
    8 b6 E5 s# |' [8 w4 ^, T8 G% d" p- V: v% k& \- Q6 F
        The function keeps calling itself with smaller inputs until it reaches the base case.' \5 m, X' u) p. E, L
    ; ^* Y% d9 ~8 x" o
        Once the base case is reached, the function starts returning values back up the call stack.
    0 Z: @. Q4 Y, a( K  w5 t
    5 [: i% h5 A$ ]6 r9 a    These returned values are combined to produce the final result.
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    5 y& a, G. K8 @' CFor factorial(5):
    % v" m6 G! N. C
    + P2 e1 V0 C7 K" S8 q  L
    ) T( j7 N- a! Y2 X3 a# c6 dfactorial(5) = 5 * factorial(4)$ h6 V' ~, g! p8 q7 B0 k) \' o" {8 F
    factorial(4) = 4 * factorial(3)- d) R$ f; G" U, ~* C9 P" Y( f
    factorial(3) = 3 * factorial(2)- ]# M9 }0 }) W& B: |
    factorial(2) = 2 * factorial(1): q, L( S; A4 E2 Q" F# }, G" Z4 a8 t4 S
    factorial(1) = 1 * factorial(0)2 g/ S2 q$ u3 z7 X) H
    factorial(0) = 1  # Base case6 x7 c# e" W8 O; P- z" r
    . o" O  H2 U+ f; q! Y2 J. W
    Then, the results are combined:3 u& f. @  @: a+ k* {# c
    - b$ j8 [9 C$ Y! E/ F, {
    / r) ?# o# C5 `0 G+ d# ?% y# {
    factorial(1) = 1 * 1 = 1
    - f& R& A( k) f6 R4 {$ O; ufactorial(2) = 2 * 1 = 2
    1 Q( T+ o( `! |; m3 {1 d. v* Sfactorial(3) = 3 * 2 = 6: Q4 Q% H# M# u5 p4 _& e
    factorial(4) = 4 * 6 = 24
    ( P) [% L+ I0 yfactorial(5) = 5 * 24 = 1204 b- E# _" W& {* D) \* k( I5 Z& m

    8 v0 X- a& W9 L' p* b. pAdvantages of Recursion! ?& d5 A8 ]" t8 d- o% s( e& t# P
    ) N; d. i: ]- l
        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).
    9 X( A/ P; g$ n6 c  ^$ a% u# N- v  Z& q; @, k8 w: a6 A; o
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ) r8 {5 V; O: X7 H
    ' o0 A* h) w! [( N6 d0 _& z+ h, hDisadvantages of Recursion* n6 L) r  N. K4 ^3 w
    , x5 X7 g. E" ]4 m  s  `2 F
        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.
    6 `8 b6 g8 C, S* D, w& a
    ! U4 v6 _/ J! U. \    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).$ r4 b) I; i9 F' a* a6 D% s
    6 o# i- ]2 C) x5 L, p7 \
    When to Use Recursion
    ; u% z3 `; w. S, o
    - O3 N. C) C, I8 O' Y' Z  k2 z    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ; L& _  b: [4 k+ [
    0 Y* b( y+ i  x    Problems with a clear base case and recursive case.
    ) K# Z0 T5 t  ?- B: d
    5 Q9 F" p. k: r/ v. nExample: Fibonacci Sequence
    $ ~4 ]4 E, Z, M" s# X) I# C
    - ]* V/ N4 q; \The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:7 T$ G( P: z5 u8 n! u
    . x6 c: O7 g" v$ B0 r3 D7 O; X  Q2 V! e5 Y
        Base case: fib(0) = 0, fib(1) = 1$ _* ?. L* @. j' s* V+ I
    1 ^2 T8 j# n/ E. F! i; E$ E/ c
        Recursive case: fib(n) = fib(n-1) + fib(n-2)/ f2 T! T" h2 U( v' I
    : p3 P% _7 y; F% p- `: E0 K
    python* h& R* l, v* K

    ; K7 q" k) O( Z' V
    % t8 b- v: Z) @def fibonacci(n):7 R) F# b. d: \2 a
        # Base cases
    # y: R6 [8 m; S0 n    if n == 0:: z6 a1 ~$ o: B- a1 p
            return 0
    6 _( T; _2 S1 N  X6 f, {0 w# d    elif n == 1:
    " ?4 T% T3 f. E; L        return 1* r* s: E- o* L# H
        # Recursive case. V" D: S$ s0 c, b
        else:* p1 m- ?* @8 R  _
            return fibonacci(n - 1) + fibonacci(n - 2)+ `3 S0 x" t* e0 w) R

    2 H- X* j0 E  ]# Example usage
    * G, H* U+ A& j$ R& uprint(fibonacci(6))  # Output: 8
    9 C" i% U- o" X) [  |# j
    0 h% T  P( @& h6 y8 eTail Recursion
    + Z, u  U% V; i9 o9 K& B5 ^. w7 Q7 K6 }
    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).& Q( j( A; Q$ O8 X3 d$ S7 L' J

      D: C" R3 j) ~: |4 `2 jIn 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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