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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 3 f+ {4 D/ k5 R
    9 f( L! ^0 J5 O$ [6 G0 A7 F. x
    解释的不错% G/ h. h% b7 w6 r
    + ^+ o: P# y3 }" F
    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。2 F: f8 l2 }  Z* c2 p: U3 [2 B
    / F% S" d' `" X8 B
    关键要素) k  I9 E- j: P, b
    1. **基线条件(Base Case)**! ~% T% N. u# c' k* M
       - 递归终止的条件,防止无限循环
    8 G  ^3 Y- u2 W3 t4 {; W   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ) b% m/ P$ e+ S  d) C/ R7 V" [4 N' V" N
    3 z6 v6 R) v/ H; H/ f# c& k2. **递归条件(Recursive Case)**8 y8 E  z, i5 V/ Z, l3 A
       - 将原问题分解为更小的子问题+ j" u8 \4 m9 m. q& b
       - 例如:n! = n × (n-1)!
    5 R0 ^: w/ B* v- Z9 w' u* p
    * i9 t7 f/ Q2 T% q9 a% _9 g 经典示例:计算阶乘
    % {8 k( R3 Z+ Z1 D7 gpython
    * ]/ U) b8 }$ ?def factorial(n):7 o( \) ]: w/ ?1 _1 Q3 @
        if n == 0:        # 基线条件
    9 K2 k- s& }" K% W/ J/ E        return 1- M/ B- F, {6 t) l. A
        else:             # 递归条件# K, s8 T$ g6 @4 P* @7 ?, o
            return n * factorial(n-1)
    : V9 l& C% Y! F& s7 u; u4 O执行过程(以计算 3! 为例):
    $ b, t1 o( C5 b+ afactorial(3)
    : y+ A/ X4 M& ]0 E' r3 * factorial(2)
    ( c, X2 S1 W. v+ w3 * (2 * factorial(1))
    6 J: }& b+ R/ O% G1 ~3 * (2 * (1 * factorial(0)))
    0 J# X: `# J9 j& e3 * (2 * (1 * 1)) = 69 c' ]$ d7 e, E( z% K4 d9 {

    6 p6 k+ m' [% u; N6 F; R 递归思维要点
    ' V! o/ ?3 Q0 U6 e8 E0 x, d3 ?1 M1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    % I" m: {5 C) w8 G2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    - B- c) C3 w3 o8 g3. **递推过程**:不断向下分解问题(递)
    + ~6 w2 E6 I2 K3 Y) h4. **回溯过程**:组合子问题结果返回(归)' S7 X* e; m6 s5 L+ }# H0 u6 m) O6 {& y

    / O) h& {5 I% D4 H% {( V3 Q 注意事项" l" s7 O; ?' Y+ B$ \- B- J
    必须要有终止条件% Q! o( O% A) A4 Q
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    " S$ y" Q6 w5 ]& g. \  t4 R- M. y* f某些问题用递归更直观(如树遍历),但效率可能不如迭代* s3 w# }9 r4 \& ?
    尾递归优化可以提升效率(但Python不支持)
    ) x5 `3 j  k5 d2 J2 r1 Z$ x2 ~$ i! N: c- v8 |; d1 @
    递归 vs 迭代
    6 {5 y* A& V4 J/ p7 W9 }|          | 递归                          | 迭代               |+ s1 M9 A+ \' `& V. F4 ?) \' M
    |----------|-----------------------------|------------------|7 g7 l' |1 D: E; L
    | 实现方式    | 函数自调用                        | 循环结构            |
      D# L) N* {7 C& g( m# O' J; v| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
      q. W" y% r$ p| 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         |
    : \6 E/ t, q* i3 {# {5 ~. H| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    4 ^5 q# ^" c. A6 S& p2 K3 Q) G! o$ ?; |7 j, y" a$ H6 B
    经典递归应用场景6 i+ u7 _% l5 A/ Q3 O1 g2 r
    1. 文件系统遍历(目录树结构)
    - Y/ c. z. i4 q9 H. z2. 快速排序/归并排序算法& y0 E/ m4 F' w* n1 F
    3. 汉诺塔问题
    * W. ^  M5 C7 U" [% k4. 二叉树遍历(前序/中序/后序)
    ; R/ f' E0 B4 W  `" o* n# G5. 生成所有可能的组合(回溯算法)
    , w4 a. b" {5 w0 ]. f) O' a7 e: p: [2 N/ O5 m& j7 [' G
    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,
    ' [: l/ O0 }0 A% ^/ O我推理机的核心算法应该是二叉树遍历的变种。
    ) I5 K# A8 _) e& F+ Y- }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:
    5 X( S1 B* n1 j; h' mKey Idea of Recursion
    1 _) J* r+ C& ]# G0 g5 O( t
    6 l4 x$ r5 E& C' v$ ?  v, zA recursive function solves a problem by:
    ( @! [& Z2 i0 ]2 w* l6 \) m5 o- \1 J$ I! B6 A0 |& |, N% `. ^7 G
        Breaking the problem into smaller instances of the same problem.
    ' m4 \4 O, W7 Z+ L& N
    " ]7 W) b  {3 C7 l3 X, c    Solving the smallest instance directly (base case).
    1 w) M, Z" k5 S& j+ k5 f
    ' [5 R5 x! o2 w% l( L4 j5 b8 i8 c6 N: }, I    Combining the results of smaller instances to solve the larger problem.) K# w0 D* b" m- |8 X
    # l' q2 G) ]$ j/ H
    Components of a Recursive Function4 K4 b- D+ E9 ^% n

    * q1 a$ C' s) G+ X( P( Z    Base Case:# U) r6 a, }$ V" o! A

    ( y9 ^% V7 G* J        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.& h: R7 F% I/ G" `% N

    2 M, v$ d* K6 P* T        It acts as the stopping condition to prevent infinite recursion.6 N# K- T2 b: ~8 V4 R5 U

    ; O' a2 P' H# m/ z        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    # m5 A$ p( C" T% @, }3 V+ W8 p. l" z" u" J5 I
        Recursive Case:2 R  E4 a% a; }( L+ }4 ~

    & M+ i/ c( p9 M( L        This is where the function calls itself with a smaller or simpler version of the problem.
    & {' [$ V* W) f, L
    4 }. }9 W5 {" P        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)., ]2 D, B( c6 \  O
    9 \& ^$ l4 w' m& G
    Example: Factorial Calculation5 G8 A0 ~" ?$ n# I& V5 q
    ( }1 I" t. i; g7 y$ P4 B# v( j+ j# o
    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:) X* B. r) f' w* e/ d1 M) H

    3 l7 U8 `$ _3 A! F( t    Base case: 0! = 1/ \" V& f+ `, W5 ^
    % H/ Y4 b1 X) l6 a) i" l  q
        Recursive case: n! = n * (n-1)!
    2 W  r% V2 Y! l3 ?
    - F, @$ s" H6 e  e( i4 `; P7 n* o1 m0 yHere’s how it looks in code (Python):- m* ^4 n8 @( c' N4 E
    python
    ' l' j9 A' U, G# v) n% r7 `: A& x6 w" a& I  n

      \8 z# i" D  f3 i2 mdef factorial(n):- b' ^0 ?  {$ v( U# ^! z7 p6 K
        # Base case3 y* S5 p) E# n! \( t% c
        if n == 0:
    5 |% S* i3 G/ _9 q9 I- H        return 1
    ; ?$ I& U6 Z. Q* h, H    # Recursive case& A' m" A/ l- M* \7 M3 v
        else:% a( F& E1 |6 l8 v" B! {
            return n * factorial(n - 1)
    - H# h4 X" C/ |1 y& n+ |- w5 i  k+ S; J
    # Example usage
    5 y  S& Q* o# Q: w- q$ qprint(factorial(5))  # Output: 120
    4 }" F- a" x; N' w8 x6 r1 O1 l! t: }$ W  @& p3 Q6 a7 g4 W* ]' {
    How Recursion Works' L, k* x9 N3 w
    " |  u" ~+ S* N# R0 C- C! T5 ^
        The function keeps calling itself with smaller inputs until it reaches the base case.
    . d. l" y7 a4 k1 f: ~# U
    1 n6 I2 P+ y& N. j1 n2 J0 t    Once the base case is reached, the function starts returning values back up the call stack.# Z, N: }( w, |, n" ~
    : B& [/ V* `& Y& \6 T3 A
        These returned values are combined to produce the final result.. y. _. ^  w' J9 D- B6 |
    2 k1 M0 B, i, U4 J
    For factorial(5):# p1 D* K/ k2 n' d6 u

    9 b7 C: X6 U) V  E6 {
    # m# g0 X, V6 h( N' V. ufactorial(5) = 5 * factorial(4)
    6 v+ w9 g6 X3 t9 j- ^3 ofactorial(4) = 4 * factorial(3)7 a6 [) z5 D5 H( N
    factorial(3) = 3 * factorial(2)
    $ r9 N, i) _5 B( @2 B( }* G$ ^4 [factorial(2) = 2 * factorial(1)
    + \% R  M0 l4 r$ O% nfactorial(1) = 1 * factorial(0)
    4 y7 B- {+ G. ~9 _! Ffactorial(0) = 1  # Base case
    + e7 P# Y4 N! o/ w: W, W
    0 @; e0 l8 C" g6 q! {/ Z( jThen, the results are combined:
    3 H+ j" P" {# X  e1 U$ [; t8 N+ q; _* \9 r

    $ s6 v1 ]7 X% t" F+ cfactorial(1) = 1 * 1 = 1
    6 y" i) e4 f+ T0 Y1 I) Z6 t9 K" hfactorial(2) = 2 * 1 = 2
    * ?' ?& e8 f! V) p+ g9 kfactorial(3) = 3 * 2 = 6+ }4 z: s5 d0 ?9 T5 c$ h# I
    factorial(4) = 4 * 6 = 24
    ( m( |' C/ p4 Q" w5 ^factorial(5) = 5 * 24 = 1207 v0 h" _, [2 o% t9 L' k$ E

    + {' T% W$ U9 @- W' yAdvantages of Recursion
    4 j# v8 b: o/ [/ `+ }% T
    ' z- s1 e9 B4 ]6 x" R: i5 P) o" x    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).
    8 H% q! w, [1 q; R+ t% z, Z# {; I0 J
        Readability: Recursive code can be more readable and concise compared to iterative solutions.8 x, o. {1 l/ \/ F: e- s7 g
    9 Q1 @* {+ P+ ?4 @
    Disadvantages of Recursion
    9 Z, x% K( G1 s$ d& _% Z
    9 i: e* M( B+ c( V8 U    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.3 _; a8 }$ `/ V. m
    ! c. z- X: Z- {3 Q
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    . F; @7 h7 @/ |$ M$ P: W+ f8 H0 R' T% o! K
    When to Use Recursion  c- Y7 F6 }0 @9 x4 A8 W7 V

    . D) J! g9 {- ?& ^! t" S  t1 V# v    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).' S. Y2 `6 `3 Y; v: M
    % s2 k; Z- k7 K# [' v4 }  _
        Problems with a clear base case and recursive case.
    , [2 H# q% A! k
      b2 ~! V0 G; E5 O" O. r) eExample: Fibonacci Sequence
    6 V5 s9 D+ s4 K2 O; G2 }9 D1 i- }* @
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    , u7 t- k1 y+ i
    2 o+ I1 y. x2 A. ]3 t    Base case: fib(0) = 0, fib(1) = 18 ^  e$ ?9 p( g- j% S; |( M
    3 a$ U. s: ^# H! x2 {# S4 h
        Recursive case: fib(n) = fib(n-1) + fib(n-2)8 J1 L) x( v$ M2 s9 H( s+ e

    / e* J" S# u/ ^; w3 jpython
    + q" F' X3 t1 c. U; V; T) t9 }- x2 Z. M$ p; M: o3 {8 T

    8 \4 l8 H. N4 D6 W( l& \def fibonacci(n):
    / C# l. v$ Z0 M5 f1 q+ E    # Base cases
    ; k8 o" C( Q/ O/ u+ V) F# a4 X    if n == 0:. A; [. N! f/ ]# T
            return 0+ j% A! G% k; ~8 l: l/ ~
        elif n == 1:- R! Y! C0 q& W; A1 q
            return 1
    : e# ]5 N3 i3 {    # Recursive case( n0 Z+ u7 Z' |7 I7 O. V8 P
        else:
    # I" E% Y" B2 p" c        return fibonacci(n - 1) + fibonacci(n - 2)3 q$ p3 i6 k0 o& C" q6 R

    ' i' t/ @2 g7 x" |3 I/ Y# v# Example usage1 ]9 \- P+ Y1 _. g$ M0 P
    print(fibonacci(6))  # Output: 8
    ( w. H$ ^; w" {- I' a& y& V# Q! u5 p
    Tail Recursion5 ]4 V/ A) s- b( r& q) K0 G% [
    % G# o& b# a: D
    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).
    & u  ]8 z) ~8 g* A6 p; m+ g% \: b
    ( d% Z- g# N! c7 E( n+ m3 V8 pIn 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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