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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
    # R3 o( }- {$ z$ I$ ]* Y6 [9 l+ z5 J. e2 `5 `( i* Q
    解释的不错
    " b+ s, e" R* T) ?6 @" l2 s
    # F$ g/ a' Q  b. m- ?; O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。7 x8 d, o/ A9 @4 o

    6 ?1 X$ ?; a) C' G$ R 关键要素
    # X+ q. g; m8 X  h2 G/ z9 K1. **基线条件(Base Case)**: N( i9 ~3 W, n. t1 G$ a1 \5 J" s
       - 递归终止的条件,防止无限循环
    5 ?5 ~3 D: E! c+ ?/ p6 O- Y4 X   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 19 R7 m( y7 p+ S- t: q

    0 k5 a# C: _9 A  W. L) c$ Y& H2. **递归条件(Recursive Case)**2 ?: e# M5 X2 P
       - 将原问题分解为更小的子问题; f& i8 ]% z) v' v0 [
       - 例如:n! = n × (n-1)!
    " h" }; }+ X) N/ q( ]3 n
    9 \% O: C0 N5 Y( C& M8 j 经典示例:计算阶乘, `5 P5 }& f0 L0 y5 ?+ H
    python" ~+ W/ t7 x6 ?- h4 z  n
    def factorial(n):, L9 a$ u* Y# x  r1 z' r2 v
        if n == 0:        # 基线条件
    / M3 T4 G  ^& G. }" M        return 16 n& h6 M' M- H" x5 w# i
        else:             # 递归条件% X/ o' r: _, f/ \' V9 S( r
            return n * factorial(n-1)
    ! j) }) |# \- s+ A9 f% n9 Q5 x) D: O执行过程(以计算 3! 为例):
    " f4 A+ ^; p$ i  ?3 wfactorial(3)
    - T% ~1 }+ F- |. |) k5 x3 * factorial(2)
    9 q$ S. K$ T- K! Z# i* m4 ?3 * (2 * factorial(1))1 C1 Y( S. g' p1 X0 N9 i
    3 * (2 * (1 * factorial(0)))6 ~6 ?# ?) i4 A! F3 i
    3 * (2 * (1 * 1)) = 64 m" s' b/ q$ ^) L, l

    1 V: @" M6 K6 L5 A, d8 T, e, ` 递归思维要点
    9 E$ W) {" R' F# T: Y- H1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    : j1 ]& \, B' k+ ^" \2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    / T- n3 n- F  ^! [% ^& [3. **递推过程**:不断向下分解问题(递)5 Q! c" Q! E8 s
    4. **回溯过程**:组合子问题结果返回(归)
    0 Z  Q; p0 l, n/ r" [; I# z8 g  G6 [3 x' l6 m0 y4 V
    注意事项
    5 P" |, V8 x8 u- B8 S% Z& I必须要有终止条件, i6 x; S4 R' `7 _. z
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)( p: b/ a; {. R. ]6 v, k
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    6 z/ N/ O- q1 Y* P' @0 R2 O尾递归优化可以提升效率(但Python不支持)
    " v& S* c" B; O% F1 z7 ~( B1 K
    7 g. ^0 w9 ^/ f" q 递归 vs 迭代9 v) E, t6 b( g5 o) x8 Y) h
    |          | 递归                          | 迭代               |) M$ p( z$ j4 q0 h
    |----------|-----------------------------|------------------|
    9 m  C4 L. q6 ?* \+ {2 L1 {| 实现方式    | 函数自调用                        | 循环结构            |' K- q1 J% {+ |9 {$ e3 r
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |% G: s% Z8 p1 ?/ O
    | 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         |$ N2 s; I/ M/ H' z
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |! _" y+ S7 f. g, D- |

    % A: d1 L9 `, {; z 经典递归应用场景
    % \/ L$ j" t$ _1. 文件系统遍历(目录树结构): a1 m9 j5 _. X2 j! _
    2. 快速排序/归并排序算法& `% ?9 A: X& i
    3. 汉诺塔问题0 v/ ^: G* b, P# T  U5 Z) s
    4. 二叉树遍历(前序/中序/后序)6 T; A& L; s, [, s
    5. 生成所有可能的组合(回溯算法)$ N, X5 |' P& W8 b# h8 M
    6 {* x! w7 \5 I+ v% M% 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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    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,6 \4 V0 c3 I) b: R  `
    我推理机的核心算法应该是二叉树遍历的变种。
    : Z% s: N$ }6 i  {% @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:6 n0 L$ r5 Y5 @) {
    Key Idea of Recursion* H/ l* A  N5 I% c/ O
    ! x  ^& Y7 j8 _+ E
    A recursive function solves a problem by:4 l: a  A" J# P! o1 N

    4 d9 g$ `; N7 I2 p4 p! m    Breaking the problem into smaller instances of the same problem.  ?* a  J3 i+ k( l1 q+ a

    : P. r6 G0 c# x    Solving the smallest instance directly (base case).
      E: c3 }. O% u+ A) X) v. G  N* u4 M& i% m8 J. k8 X  D
        Combining the results of smaller instances to solve the larger problem.9 d, k" {; F0 t( Y1 j8 L

    ; _  ~) W" L7 x! q8 BComponents of a Recursive Function
    3 Q8 e' C5 I1 \& W7 I
    $ H: a* ~% s. _" Z+ j% m! |    Base Case:
    * f* ~1 Q! O0 d9 p3 m
    0 Z3 d( B& V" o- ?5 }- j        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.3 a; W; d: H* l5 g1 B- v; l

    0 }" J# n' ?- T. H        It acts as the stopping condition to prevent infinite recursion.
    & R* T" }* H. x7 e% E9 w6 l
    7 s5 c" \2 h0 {1 p        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    - R$ U9 u  U# x/ }( U3 @. W5 K- N* W* D* Y
        Recursive Case:! l- [/ X8 ]9 h+ ]% k

    ; H6 v( s, k: `        This is where the function calls itself with a smaller or simpler version of the problem.3 [: ]* M. ^$ ]2 Z. F1 x
    5 d) m) E" r; z* `3 t# |, _! p
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).8 J8 z1 k" L3 q

    ! K' `) V: N. U, J. QExample: Factorial Calculation
    0 V" H1 W! g3 M% {8 [8 P, o# p# U& I( ^& W
    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:# g* B) D) ^& v; k! M  L% S' T1 r8 d
    . Q# @) d5 R  Y! m8 Q$ `  F3 u1 y
        Base case: 0! = 1
    # B' x/ ]' W! b& M; d/ z) C% d5 ]8 T9 n' G1 G1 L, V; R* r% S
        Recursive case: n! = n * (n-1)!
    ) Q; w7 Q$ o4 M# q$ ?. R& H3 G5 x9 o
    Here’s how it looks in code (Python):. ^4 a% o( X' D4 Z: T4 G/ B# W
    python
    - t' ^) ~6 E$ e6 X# ?3 c! A' K4 j* r

    ; Q+ _( U/ z7 y0 [- _9 i, hdef factorial(n):. ~9 D+ R6 ?) R* \6 I0 P
        # Base case1 \' l& n* g* @2 o! |* m- B1 b
        if n == 0:
    # ?3 [0 B) N* y8 Z3 v        return 14 j0 [& c. m. E. |- [: o
        # Recursive case
    3 M2 O0 E! _7 r0 P2 [    else:3 }9 E. r$ i1 y( }5 Q
            return n * factorial(n - 1)
    + V# b/ m2 x% w/ v5 {; V
      L5 S9 U- H# P* o7 \# Example usage  p1 b, {( T1 I
    print(factorial(5))  # Output: 120
    ) N/ T, M; i+ D: e9 N2 [" v" {5 i
    : X& }6 u5 V# E; H: A6 {7 Z" dHow Recursion Works
    0 _7 n+ o& \, u0 Y4 K8 Q% g1 y1 [3 D! h$ A1 f) Q! U$ E* W
        The function keeps calling itself with smaller inputs until it reaches the base case.) O: M! L; V5 R; c

    % t9 U' |8 n. j    Once the base case is reached, the function starts returning values back up the call stack.
    2 ^: S- ?% ^5 `
    : l. n( o- c9 `. t% Q) [    These returned values are combined to produce the final result.
    ( t  r9 S* O) T3 t" K
    5 M4 k! C; [. o' }9 a5 {For factorial(5):
    7 Q3 B1 d  k, |2 Z* e6 F7 _' Q5 d& }- U% e4 ~6 M
    # _4 l7 @! }% k: U
    factorial(5) = 5 * factorial(4)
    / S( G# k, w  ~9 ^. d! afactorial(4) = 4 * factorial(3)3 G7 _& h7 N0 P& H" H
    factorial(3) = 3 * factorial(2)- H5 x; o" k# p4 H% @
    factorial(2) = 2 * factorial(1)
    0 L8 J+ I5 g- ]( O) B6 L  Z% Q+ Ofactorial(1) = 1 * factorial(0)+ J# v1 |% b, S- L7 w
    factorial(0) = 1  # Base case
    # R- v+ L' R' L9 {9 E) P$ `8 ~6 t* }" [; T7 q
    Then, the results are combined:
    8 M% q& L# \3 Y: Y1 P4 v8 ]
    0 f( \/ k$ ~' N/ N- \! Z
    3 V& w  I  t% r# e0 ?" dfactorial(1) = 1 * 1 = 1
    9 E( J/ x9 R% u1 kfactorial(2) = 2 * 1 = 2
    - E: P  E; |8 V! q" d/ F! ~  P# Rfactorial(3) = 3 * 2 = 6
    1 i) U! Z1 ]! c: [  Jfactorial(4) = 4 * 6 = 24& |- k' s* R4 K0 M- t5 d
    factorial(5) = 5 * 24 = 120
    0 V$ U; ^5 F5 S$ l& B
    ( q% W7 m8 u& U' }1 Y% n% {' E# o+ QAdvantages of Recursion' e8 b' Y( I2 A% H8 K6 J6 f* O

    , m' X% o, K+ Y( p, }! s, `' p" _    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).( ?6 r5 m1 ?% A7 l
    # ^$ _0 y* {9 U& U
        Readability: Recursive code can be more readable and concise compared to iterative solutions.+ d2 M4 w! R7 N$ K1 }& Q

    # X3 x" i2 S0 o) B- ]Disadvantages of Recursion
    & \+ ]. N: z5 f5 M: t8 `
      e' i' N0 g1 u3 ^* N    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.
    $ Q6 J8 i3 t' D0 p# X0 z2 p, O% s: [- ?: F9 o, F( m( f: m- J
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).9 F9 d0 z# z& M: j
    ! C3 W1 H: G- v
    When to Use Recursion# p: r& i: t9 x7 O
    : [5 ~# f  h6 L! ]/ G
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).# w' n! \# A1 s. n% L. @- p

    . O  p6 H0 `% ?5 }: }, k( z    Problems with a clear base case and recursive case.  `8 g5 a. k3 \, ^" \. O8 i
    8 |# p& p  g2 z! R1 O/ ]! d
    Example: Fibonacci Sequence# u' }* r8 v& v& U2 I4 r) _7 v
    ' ]2 u4 W1 [: }2 y+ ~% t5 [
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:0 q3 i# a0 {9 N9 ~* n
    " u' r( M4 r  s8 x" q  Y9 E
        Base case: fib(0) = 0, fib(1) = 1
    . \: M. g: {2 G8 f9 q' S2 x( `9 c1 a$ O2 m. z) t+ Y' |
        Recursive case: fib(n) = fib(n-1) + fib(n-2)* @. J9 D( m7 z$ Z5 Y( _
    . ]1 Z! |; E) y" d! j8 O- h
    python0 M( J- V+ l6 o7 M* y
    % K, R8 `& J% Z8 ^7 m; l4 \
    ' D6 E: d3 ?% k4 |2 F8 r1 c" T7 t
    def fibonacci(n):
    ' K( l  A, O9 [    # Base cases
    * `. h+ b) ~) e* C    if n == 0:" ?8 U, a# c" H
            return 0: J1 f& I( y; Y% i
        elif n == 1:
    : s* m, z& `. p* O( T5 ^/ _        return 1
    % J% W( R) [" _9 w" d    # Recursive case# c$ V, a8 X3 o, O5 I3 J
        else:8 m% z) E5 y" |! C; |
            return fibonacci(n - 1) + fibonacci(n - 2)
    & j9 _/ |) j0 B# J5 c* G
    % d- r6 n# ^9 F. f1 \$ b# Example usage
    , P3 i2 }" K" j- |print(fibonacci(6))  # Output: 8! _8 y6 i) O8 e+ ^& ^+ [7 v- @7 _9 g8 K, I

    % j' \. R/ N1 i' X+ ~6 KTail Recursion6 M6 ]2 ?; Z4 U( `. k

    / D; b6 C# N: P# s3 h. cTail 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).
    $ |- b( F0 p2 p) v! e# @0 N" H) v  K
    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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