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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
    & |$ K' |" ^/ i: x3 H" E$ u0 S5 A6 e) o. E9 ]9 m/ Z
    解释的不错5 l5 `1 l, V* x- w1 |
    8 G! m; r8 O" I& d) p* R( R
    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。9 g5 }: m% A9 z, _. H: s
    + z& |0 ?1 D. n4 n) H; G) e
    关键要素
    9 B( `0 ]& a) n6 _1. **基线条件(Base Case)**
    & W+ O' b+ Z* A0 R0 ]   - 递归终止的条件,防止无限循环
    0 b+ f0 T. Y" U/ l( o0 s   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1( ~* j) a0 }4 o% v9 I! i; c  L

    7 u( ^" _0 |5 p2. **递归条件(Recursive Case)**
    % h5 f) C+ Z) v   - 将原问题分解为更小的子问题
    & O- m; ~; |. S   - 例如:n! = n × (n-1)!$ X* P. {# u# |3 Z3 E' b, A
    1 o( U% K5 n8 B' V
    经典示例:计算阶乘7 [3 D. `7 z7 p' n
    python
    / Q) b0 d% e+ A6 A5 B( ~) F/ Qdef factorial(n):
    " `# _3 `1 z* |' @- @    if n == 0:        # 基线条件' q$ A/ ~' D" ?. J$ ?
            return 15 Z+ ^; r! r8 E# Y
        else:             # 递归条件
    5 _1 p' f  P3 p" B        return n * factorial(n-1)/ u, ~; l; i4 E5 m
    执行过程(以计算 3! 为例):* ^' a, p/ ?; n( W
    factorial(3)
    7 Z" f% n. q% f3 * factorial(2)
    , t  i; c0 h/ y9 q3 * (2 * factorial(1))8 E* Z- f% c, h: \& ?$ k
    3 * (2 * (1 * factorial(0)))! r" e- P9 \# Q/ T) O/ {, B$ \
    3 * (2 * (1 * 1)) = 6) M2 _3 m1 D3 u

    4 J& x& S& _; V9 Y* J! F 递归思维要点  Z& E( b8 E8 N2 F* X1 E
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    7 m# Y: V! Q" W% A+ t# G; Y2. **栈结构**:每次调用都会创建新的栈帧(内存空间)4 D7 b3 d& d# f/ f
    3. **递推过程**:不断向下分解问题(递)5 A6 M" W$ {4 B" Q- r
    4. **回溯过程**:组合子问题结果返回(归)
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    4 X3 s# n' Y4 @ 注意事项
    - ^' e. D8 B$ h& |0 h; d必须要有终止条件
    7 a% F( p7 t1 x) R$ T" R. n; s递归深度过大可能导致栈溢出(Python默认递归深度约1000层)& v; Q. c0 U+ ^& E) g
    某些问题用递归更直观(如树遍历),但效率可能不如迭代& s/ o1 A1 \1 v5 ?) _, {  |' f
    尾递归优化可以提升效率(但Python不支持)
    ) b. t& a1 |3 `% n/ c' q$ t) F8 b% ~& J/ J. Y/ \& Y
    递归 vs 迭代2 c4 c9 E7 ]( u4 W9 C, `
    |          | 递归                          | 迭代               |
    7 l$ W0 b7 X0 {( U# B# X) b( [|----------|-----------------------------|------------------|+ P5 O; h/ |. P8 a* P7 F
    | 实现方式    | 函数自调用                        | 循环结构            |. [* f3 `4 `+ M) |% D  _& {
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    3 E- P1 G8 h0 H8 G8 a0 z7 ~: X. i& N| 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         |+ b6 y3 ~) R; t  X0 D* E! R
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |& d- @: D9 Y" n$ m( G

    9 X8 n' B/ E! W$ Q* c& d/ ?( S# m$ [. H 经典递归应用场景
    $ ^9 L8 s3 h# U& N* I8 X" D8 e. `- K1. 文件系统遍历(目录树结构)
    3 q: L# y* m) B% @' j9 a- s2. 快速排序/归并排序算法, A" l! h" m. H+ H) ?
    3. 汉诺塔问题5 o% G* Z3 h8 R9 y; g/ n! |
    4. 二叉树遍历(前序/中序/后序)
    4 C+ X4 L1 a/ h" V' }& O* g5. 生成所有可能的组合(回溯算法). B) R, P6 i4 k" W& G5 k
    $ c  L5 ?9 Y& r) d8 @( ^6 L
    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,* N8 t9 F# ~, ~) w  |# k! Z6 R
    我推理机的核心算法应该是二叉树遍历的变种。
    ( v- z+ o& m0 l% N8 \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:
    ( D9 ^! l0 [; f7 ^' u0 _+ C* vKey Idea of Recursion0 v% N- g  i; Y# {2 O! P, X1 O- h
    / i3 `) b# X( ^% H6 L  l( ?
    A recursive function solves a problem by:* {' e& g$ d( E' r/ W9 {2 D3 l

    , U0 I# L8 V& Y, w- q$ b    Breaking the problem into smaller instances of the same problem.
    $ r1 k, r4 ]7 R, F; H- Q2 R6 o& R3 v
    * J, w" b+ A. q3 }2 b    Solving the smallest instance directly (base case).0 I' Q1 @/ O: L  R" u* k0 k

    ) H3 Q" {1 M  `1 [1 c) Q2 t    Combining the results of smaller instances to solve the larger problem.
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    % \6 O" C4 v( M5 Z) \Components of a Recursive Function! u+ P; U1 I9 K+ [1 p

    ' U1 B& o* N& |- M. N, T0 G    Base Case:
    ; Q( L: t; z5 |" w! p# L# U  v8 r6 T5 n+ J3 u* F' u5 r8 ]2 B
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    $ I/ f5 E' z9 x- d5 B1 }5 k! {/ E$ Q
            It acts as the stopping condition to prevent infinite recursion.
    " Y, ?  A9 I8 {$ U% ?, u1 K& [
    ' J4 s, g: k' E3 M8 I5 s. ?        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.( r. ?6 f. Q5 [: I, e
    # J8 i0 L1 B0 C4 @2 d2 O
        Recursive Case:
    9 }; [* @* G: P" Y
    - s0 M- T5 n1 ~! ^4 h3 {$ t        This is where the function calls itself with a smaller or simpler version of the problem.( A2 h' |, |' V7 w

    1 _4 ~- e; i. w1 s# v8 F        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    1 x2 |: r6 r$ Q& |# r, ?, Z- J+ z0 p1 h) m3 C; u, x8 ?
    Example: Factorial Calculation
    6 ?- r5 N! a1 I  G+ p6 S' [# ~8 i# L* E, R. }& ~8 `1 Q. ~
    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:
    8 O0 l: F: C, k5 N4 I* P) Y5 r7 b% b
        Base case: 0! = 1
    ; R4 I0 d9 ~, l# u; N6 I2 \+ Q' V; L, Z' G
        Recursive case: n! = n * (n-1)!( R& z4 L' y" Z2 S/ A" M. A9 G! O
    ( ?0 P0 ?( {4 |% P$ ~* R
    Here’s how it looks in code (Python):
    : z2 {' g# D$ H. R6 q& dpython4 Q, d- B) _3 C$ L4 T, n0 W& y
    / @3 m0 k1 F( S
    : F5 I3 `$ H: x% i
    def factorial(n):
    / Q; E1 P0 v8 U% s/ ?3 _; l, e    # Base case
    ! {" B. H: ^5 b    if n == 0:
    5 N5 G+ A! Y6 ~4 V9 d6 v        return 1
    + n. s9 ?0 m1 O$ e# q& Y: V9 ^+ Q, h    # Recursive case
    - Q( K+ v: e" e& f4 J# e! R    else:, a# g8 t# I1 P) W
            return n * factorial(n - 1)
    . ^) O. [& _* v2 y8 R: t
    - h0 {+ u5 H4 r! m- M2 s' ~# Example usage
    . R" m& R# d- d) v! [5 lprint(factorial(5))  # Output: 120
    / x2 Z# Z! x9 B$ l
    - a/ o0 d/ B/ H1 M& u" `; XHow Recursion Works# p" R7 W8 x5 ~) M2 Z9 H# E
    ' X$ d+ [8 p9 y' `$ H
        The function keeps calling itself with smaller inputs until it reaches the base case.* n. }7 N5 P2 z6 v8 W! k, v+ s
    1 c9 m) F/ [* M/ `% B; h
        Once the base case is reached, the function starts returning values back up the call stack.
    / T+ ^5 J7 v3 @6 l$ B, c) e
    / K$ Y2 b5 A2 i0 a    These returned values are combined to produce the final result.6 ?/ D* k! C1 b* k
    2 B% T' l5 h, {  \3 o* d( r/ m
    For factorial(5):5 l  @3 k  Y8 S0 Z

    / \* I+ D9 r( u+ a3 \; z3 K+ W
    factorial(5) = 5 * factorial(4)
    , p9 |; O- a' C- I& |5 ]factorial(4) = 4 * factorial(3); A+ _: V0 s* ?  ^2 W
    factorial(3) = 3 * factorial(2)
    # y$ s  a  U/ zfactorial(2) = 2 * factorial(1)
    ( Z, D: h: h( Xfactorial(1) = 1 * factorial(0)& w* x# n8 g2 d2 r+ u
    factorial(0) = 1  # Base case9 e& E$ S5 G0 {
    ; R9 \/ U- x; Q
    Then, the results are combined:
    + ]4 c: ], j0 Q7 Y& h
    2 R/ w) x9 H+ F" F) J$ ^- a  B; S/ ?6 Z; A( j" H9 V
    factorial(1) = 1 * 1 = 1* F% g9 Y' A' _+ b+ Y$ {
    factorial(2) = 2 * 1 = 2/ J% ]5 A5 A* a! X/ |% ]
    factorial(3) = 3 * 2 = 68 O7 c8 d! s% T0 Q* g
    factorial(4) = 4 * 6 = 240 \) J1 U% s( C. p
    factorial(5) = 5 * 24 = 120
    & T+ A5 y" S& e; q. V8 J" l: k! C! C7 ^, M! M
    Advantages of Recursion
    2 d$ ?9 a' [, c; {0 j5 t- t6 h8 ]0 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).( A. `& H$ R! G3 {/ Y, P

    # S# @$ j$ x8 D  s8 i2 n    Readability: Recursive code can be more readable and concise compared to iterative solutions." o9 X  `' w9 n' y. s( j4 O

      P$ B, A2 n$ s. G) HDisadvantages of Recursion
    - ^5 y) T, i. U5 c; r& f7 M
    ! L% i  \+ d& \8 ?) e) I( }    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.
    0 e- m! ~4 Q; M9 \7 l# W
    $ N3 @! O- g2 Z, h  k" J% N    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).3 a! n; }# `1 j/ y9 d
      S+ G+ }5 L5 o8 H' ]" p
    When to Use Recursion1 t+ ]! c- j2 R+ q8 U! ^+ p

    8 T' A+ R' @0 y" u' `" \; b    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    6 l  V* \3 w0 _2 @9 D" ?( f% x8 f( f  t- R
        Problems with a clear base case and recursive case.& V5 m* m! y5 p# S) m9 Z3 O! w# X

    4 w; V7 [: ]% _( ZExample: Fibonacci Sequence
    ) Y! k; A" Y/ R* e
    5 q* {+ O$ R, N$ P0 k6 n, hThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    * h( R7 z8 q0 T/ `4 ]& L+ C0 e* v& B/ h  B6 z
        Base case: fib(0) = 0, fib(1) = 1& e# v1 G! G+ h% b3 u% l1 M

    - J" l8 L/ Z5 w$ A% {' v    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    python
    * t( ]. e3 Q" z! D
    8 ]6 \  O* |" v" g+ _
    8 K- t, {2 @& [% Kdef fibonacci(n):
    , `/ v% r/ f; T    # Base cases
    2 C8 G; B3 l6 H2 k4 @    if n == 0:
    9 c0 `- e/ L% @$ L        return 0  B  Y: \/ E" J% ~6 g" [
        elif n == 1:/ }6 F& W. ~2 K9 X) z2 r
            return 10 C/ C* ~2 A8 r0 g* H* X9 h
        # Recursive case
    & ]! T9 F( w, \# U9 i# V) e    else:
    9 E! L* p2 v7 ~. }. h- Z        return fibonacci(n - 1) + fibonacci(n - 2)
    $ T. o- ~3 G8 i7 _7 i
    3 F5 m) u2 `0 r; O6 C- @$ T" h+ ~# Example usage; d. G- y% ?' F4 S" A
    print(fibonacci(6))  # Output: 8
    2 ~1 r' A0 ?$ `6 Q6 U- _
      C$ g4 w! L. m) n9 I6 s! iTail Recursion0 `5 [% U; d, D, M. A

    : F9 Z# l4 B1 `5 {" y$ RTail 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).
    9 r' d5 U( \* W+ P) f2 \0 A
    7 g- E4 X4 b" q% ~0 N% r4 ^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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