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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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    2015-11-30 11:11
  • 签到天数: 2 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
    $ O% b. y& d! _! H9 x/ t# C  ?7 _. {( }" r" M: C
    解释的不错
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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。, X  }% i* f% v7 M4 ~

    & K# v' y/ [& ` 关键要素
    ) }6 M/ X7 F7 S1. **基线条件(Base Case)**
    3 W1 k$ A) `* K2 p# i* S2 t' n* g   - 递归终止的条件,防止无限循环
    1 Q5 g: D7 t7 x8 T9 F8 I/ o   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    2 b  y, T4 X( [/ S. n
    & v& V, k8 f6 z2. **递归条件(Recursive Case)**; ~+ p  @% ~. D7 M& Q7 r
       - 将原问题分解为更小的子问题  c6 o4 a. P3 C# t& Z' Z
       - 例如:n! = n × (n-1)!
    4 ]" T- S, ?& \# a6 O/ w# C0 I/ G. i/ D2 a5 Y# ?2 d
    经典示例:计算阶乘
    0 ~% I$ J: [- u* Wpython  F3 w# g" H* r& }" m
    def factorial(n):
    . c9 l7 o$ z; v. \* t    if n == 0:        # 基线条件5 P7 v& W2 I& ~# m9 R, e5 d
            return 1* y2 R: s, u4 O; p- }
        else:             # 递归条件
    ' @6 q9 E& F' e        return n * factorial(n-1)
    4 N! T! E; a6 d0 A3 [8 Q执行过程(以计算 3! 为例):
    0 n2 G" R- U$ j. lfactorial(3)
    : l2 |# G) {/ a$ P3 * factorial(2)
    & t0 g; q4 r. X" |- K3 * (2 * factorial(1))+ N: \3 @- M5 F! |' h' @* S
    3 * (2 * (1 * factorial(0)))* `9 \* s; _% W/ u# B7 c3 }: Y
    3 * (2 * (1 * 1)) = 6! @3 w5 d0 j) @3 R( _  b4 \

    5 e# y* [9 i8 [3 e/ r 递归思维要点/ o- G2 z8 |2 J: ~) T
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    & u* ~9 {7 p! O. s4 o1 f2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    , H; ]: A6 U+ I* I  P) ?3. **递推过程**:不断向下分解问题(递)
    9 D# F) b2 V0 u& @: x6 I" {' G2 r4. **回溯过程**:组合子问题结果返回(归)' t: g- f, G4 g

    " G4 _* C0 s: U' N 注意事项
    $ H# N$ j1 m! W, e* Q5 L必须要有终止条件& L$ P$ S. O8 C" u' l
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层), c% Z9 f  ], R* C
    某些问题用递归更直观(如树遍历),但效率可能不如迭代. ^3 v& c  T- k  C- n1 \; n
    尾递归优化可以提升效率(但Python不支持)
    / {' h7 f- s1 o) S
    " ^# }8 A8 K; B$ j 递归 vs 迭代3 i7 r: |% I; ^$ X8 t0 q$ j
    |          | 递归                          | 迭代               |
    7 c7 G/ Y' B, U4 X|----------|-----------------------------|------------------|; k3 U: Q& C- S# @; D. T: n& A
    | 实现方式    | 函数自调用                        | 循环结构            |4 a. O2 r, \( w% p
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |) v; `6 _1 t( `# _! x4 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         |+ z, W$ T# r- c9 t; M
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |* ^( k# Z) M& j& Q; G
    # ^7 S  Y$ D# P* p# h
    经典递归应用场景
    + H& [% c. v. f' [/ p6 \1. 文件系统遍历(目录树结构)
    * [  r. Z4 G6 Y1 L; X2. 快速排序/归并排序算法. k: c9 _9 P4 b
    3. 汉诺塔问题
    ; ~# A+ {5 E+ o0 a# Q& [; p0 R7 U4. 二叉树遍历(前序/中序/后序)
    5 ?/ D0 F6 P0 `8 m% C5. 生成所有可能的组合(回溯算法)) V4 d  l( f4 A
    1 u" K9 d& y; v
    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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    13 小时前
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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,
    5 Z+ \" P. b6 _我推理机的核心算法应该是二叉树遍历的变种。
    8 Q" i! [( d8 n9 t- [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:
    ! F$ Q" W# T) A1 m  |: YKey Idea of Recursion2 w9 z. u: M1 Y4 y  ?

    % a6 B; M- @* V& b( |. j$ aA recursive function solves a problem by:
    0 E5 K$ ^" Q5 A2 V& s1 |
      Z; z: w/ x* D# M    Breaking the problem into smaller instances of the same problem./ h+ ]% O& y& A5 X' x

    : n0 _& P" i8 E" d0 x  [$ g. v7 Y    Solving the smallest instance directly (base case).4 N0 r. O  q% S  |

    , P. @3 I$ E' h6 w    Combining the results of smaller instances to solve the larger problem.
    7 A( r( M+ T) U+ E* j' V- K7 }* ?- B8 t6 C  [3 k1 l
    Components of a Recursive Function6 \' K5 K; Q2 j* c1 `4 Y) N) P

    ; N5 e& k& ~# O4 ~( D    Base Case:" g8 {' _0 R0 U- t* T

    ! d$ f$ U; H# j& G; W        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    0 l3 \& l& a* c) D. }7 v: B. s  e  w
    + t2 a4 V7 r2 X+ J" ~+ X$ W) U+ y        It acts as the stopping condition to prevent infinite recursion.$ E# F5 u, P5 |
    $ i: {% j0 q) X8 E9 b! F9 O6 ?0 x
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    . B' o' r4 G" E2 {0 c& \2 ]8 {! O; w$ t0 x" `
        Recursive Case:
    & \$ S! n; d  W7 q) ?
    8 l2 f2 V+ x, {/ ^        This is where the function calls itself with a smaller or simpler version of the problem.
    ; h$ K( M0 e7 {( B' R" Y. Q* R8 N* p/ i. i3 H7 c
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).9 ?) @5 @2 ~5 ^; o: x8 E+ c! S# X

    1 c/ D) w( u, pExample: Factorial Calculation
    % F( q" j& I6 P& p( C0 s3 \% y+ r' G$ T9 U- E  Q; _' L
    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:
    * j& J3 a0 G9 L! ]- p- @! J: W& O$ g4 D1 h
        Base case: 0! = 1" ^( W' C6 q9 f6 T0 L" n9 c, m
    4 W! Q; J/ q4 h3 w5 Z; ^5 c9 I( S
        Recursive case: n! = n * (n-1)!
    ( k( V6 J/ i% p5 e; C7 N1 z& @( l! d8 X
    Here’s how it looks in code (Python):
    $ L8 q# i; I6 Epython8 c& g# M9 W( I- W& U' J9 K

    ; \) L/ d2 h; K' ?/ {- p( ?! S9 h. b0 p9 ^& a3 O% W% Q$ y- E
    def factorial(n):) S0 V& I9 k( F
        # Base case+ I: g+ g1 R/ G  Z* j3 c
        if n == 0:7 i7 O2 b* m3 y9 V4 a8 K
            return 1
    # S) O) H* R+ v$ R; e    # Recursive case! _$ b6 {; M7 j
        else:8 o1 |7 G9 v9 P0 k# s
            return n * factorial(n - 1)
    . T1 }# K$ C: R  Z! E6 T8 j* P2 z7 _  M$ }$ \6 _
    # Example usage, Z& o0 r8 @3 L* p4 V2 g5 U* ?1 Y/ i
    print(factorial(5))  # Output: 120
      G3 _7 d+ H) N. P- A) t+ F! e* j, \7 p) g7 U- G. B
    How Recursion Works
    + ~: F3 u: c8 p/ t# J  y+ j1 q3 _7 y2 q5 M& W) B
        The function keeps calling itself with smaller inputs until it reaches the base case.+ F) f. Y9 }4 S* t0 ?

    1 _$ A6 X: j; Z2 w! {' }* I* ]    Once the base case is reached, the function starts returning values back up the call stack.2 m$ X  X/ V5 v& I1 D3 p& Y) d

    + ?9 G& N: d7 U    These returned values are combined to produce the final result." p+ o8 ^2 u9 L& J- Q& \
    , }: I/ P& O" T, G2 e9 e0 [
    For factorial(5):
    9 b% @; v; i2 A" }$ R" W9 O, B# }
    3 d/ Z3 N: S1 Q4 G4 I- }- @  O1 e6 f
    factorial(5) = 5 * factorial(4)
    5 r( M( u2 P& }& Kfactorial(4) = 4 * factorial(3)
    0 Y" I1 w- H, F6 u* Kfactorial(3) = 3 * factorial(2)
    9 S& |2 Y& C* ]: Ifactorial(2) = 2 * factorial(1)9 H& Z, ^$ s& t
    factorial(1) = 1 * factorial(0)
    ( U4 x6 ]' Y- y8 ?  }factorial(0) = 1  # Base case
    5 H% e5 r2 m  S3 J# T8 k' a9 b0 I0 }! `7 f0 A( c- ~8 T2 b  [# P
    Then, the results are combined:
    1 A& u, j/ X" m  P% B' K' A, P& }
    : y" I- L7 e- T' e# q+ f
    7 s! q( S6 H( G# X$ N) ^1 \factorial(1) = 1 * 1 = 1
    , r: i, q# f( k: V* Pfactorial(2) = 2 * 1 = 2+ ^# q6 D; z; z, s5 q) ?
    factorial(3) = 3 * 2 = 6
    $ {0 u- e) ]% k2 R) xfactorial(4) = 4 * 6 = 240 O  ~& c* a* P8 \
    factorial(5) = 5 * 24 = 120$ p' y  n- }! y( Y2 \
    " W7 Q% T" q2 p
    Advantages of Recursion' V- m* X2 q7 Z
    9 L9 h3 b  C5 {- n* [/ c: i
        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).
    " P! D) e9 d3 B3 v: L5 h: y
    ! C1 `: R# y$ G5 l9 b    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    2 Q- v6 g& {0 B; |9 h1 `2 p, `* \# E$ x% y5 H$ a
    Disadvantages of Recursion
    6 M: s8 z6 O: {( [7 d. y, d" S& b& U) D& j" ?% w# d8 z# d* [
        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.% K8 I2 m# o5 g& G9 _' Y' b
    6 g6 {) e  g: m
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).- d0 {) k. C, t9 s1 a5 Q

    - d. J$ c- m7 X9 [5 ~When to Use Recursion& ~$ I) S" o+ y7 y7 h7 \3 a, C

    4 r+ a. r$ E% ~    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    - ~2 J& L( r- f$ X
    & m' T: d: h, t9 J& H    Problems with a clear base case and recursive case.2 E: Y2 _4 y! R7 P+ e

    / Z' W. q' ~6 \; IExample: Fibonacci Sequence9 \4 h* r3 n! t0 ~3 q* }0 g

    ( T/ A8 N- d. g4 N. K$ I- Q6 |The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ! n' i" F# U2 w. A# v# J& Q
    , Z8 Q6 ]. \" u; d    Base case: fib(0) = 0, fib(1) = 1
    ; T0 @! z1 K* V( N1 Q3 M7 |5 ^- ~0 e. M
        Recursive case: fib(n) = fib(n-1) + fib(n-2)9 D+ W/ Y7 N( i  _

    3 J8 t5 Z3 R( p  T" m' w% spython- F' P" G2 G  B- n5 z, a! {7 \% g

    ' `% A/ w+ f! `* H" R! x
    - G  H; @6 K( @: Cdef fibonacci(n):& t! Z  C8 n2 A+ R2 [# M- x2 U/ L
        # Base cases
    ) _9 s$ M- d, V6 t# w) v    if n == 0:. P, Z! U/ Q2 v- ?4 e
            return 0/ p2 Y% {6 |1 k3 s8 N/ Z
        elif n == 1:
    9 @# Y7 \; h5 N- C- c0 K9 J        return 1+ @) J0 G. j$ T; H1 T( P2 E7 `
        # Recursive case! E( Q  D7 p+ w( ~
        else:
    : C/ O5 |" W7 k) h        return fibonacci(n - 1) + fibonacci(n - 2). t* |: S- |- }: l) }$ l9 _
    ; k$ Y: O, e: Z4 M. N# \5 h
    # Example usage! U" H! t" h) R5 t' E
    print(fibonacci(6))  # Output: 8. |8 J- _* e8 U3 B* M4 Y

    9 `: P. o/ S3 \) X# STail Recursion
    ; i4 E4 ~6 R1 ^! }9 `7 d4 B& P/ M  `/ m, p
    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).
    # j/ y" _. M$ z. m3 ?$ o) Z
    0 C( d; Z7 A! G7 F, q" ^& DIn 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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