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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
    开心
    2025-9-8 05:08
  • 签到天数: 3 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
    6 T/ }6 X. z" u5 |4 u6 L
    9 A0 R$ ^- G% T  ^. G1 E6 [解释的不错
    8 A4 ?  o! q; [1 T- k! Q& ]* m
    ; s/ e4 o  o9 h+ g- {7 @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。, E7 |) g; Q+ r, B
    2 `6 Y) G9 ~/ z' z0 i
    关键要素8 e$ z# d0 E4 j; T4 b
    1. **基线条件(Base Case)**
    1 L$ P! A7 ^, s8 O   - 递归终止的条件,防止无限循环7 J! j2 a5 l8 J& n1 n; S% M$ f
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    4 y7 E9 }2 {( L" D+ b/ n7 {; q8 J1 `/ |* O7 v0 z2 I3 W
    2. **递归条件(Recursive Case)**
    6 q  ?, h2 w8 r   - 将原问题分解为更小的子问题5 O, F1 }! @! I! z# `- m0 [
       - 例如:n! = n × (n-1)!
    + n, @, Z. P6 [! Q1 p( H  ~* ^7 N+ [8 c
    经典示例:计算阶乘
    0 ]0 O* `6 i5 ?python% }" W% q9 ?! }% C
    def factorial(n):( q: q8 f# U! h- @0 o: e4 l1 r/ t9 g
        if n == 0:        # 基线条件
    : d3 a, C+ [1 o        return 15 G4 P! H& q' S+ q1 v
        else:             # 递归条件
    * g6 o" n$ x/ g% |8 Z        return n * factorial(n-1)1 G& \& l, r% N/ C
    执行过程(以计算 3! 为例):
    - W' p( C5 b  O4 [! J# kfactorial(3)  X2 G. O5 o9 v, Y
    3 * factorial(2)
    0 i/ \, L% I  e, i3 * (2 * factorial(1))
    - `+ ^" e) L2 n! D$ A3 B, Q9 p* o3 * (2 * (1 * factorial(0)))  D4 |) `3 A' f
    3 * (2 * (1 * 1)) = 6: \) J9 z  f% ], q3 T
    % X9 y% R2 S- J2 S
    递归思维要点
    7 J( K4 u% E9 O# }. {6 B1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ! b0 \! n/ s/ Y1 U( P0 p5 p2. **栈结构**:每次调用都会创建新的栈帧(内存空间)8 ~7 O& _6 v( g% [1 m4 U* `' W1 c
    3. **递推过程**:不断向下分解问题(递)' v% L* d) Y" X+ ^
    4. **回溯过程**:组合子问题结果返回(归)$ s  L) Z; O/ X2 K# k; V7 H

    ) W4 i8 h0 [; j2 Y5 X 注意事项
    2 B& C" G! s& s) y必须要有终止条件
    9 g4 b& H# d, f( x* P2 b递归深度过大可能导致栈溢出(Python默认递归深度约1000层)& L4 m2 V/ U4 u! a: b) k
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    4 k; f. t5 x1 J8 L* j' d, J: W' l- c尾递归优化可以提升效率(但Python不支持)
    % K; S: z# i0 c! U7 r$ o( `0 g  y& ]* u3 P: v5 K
    递归 vs 迭代0 R$ i$ Y0 o) Y$ K" d. f/ m. N
    |          | 递归                          | 迭代               |
    1 a; u% K9 X+ S5 U3 e8 q6 H2 F|----------|-----------------------------|------------------|
    - S( A2 B( v6 }! o% z| 实现方式    | 函数自调用                        | 循环结构            |2 D) X1 M8 O$ M
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    4 I- @1 D: R7 W! i* z| 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         |
    4 j: H( _2 S+ j  {( Z8 y5 ?! U! X| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |3 @/ R1 x9 Y. @' o
    2 q  X. v& u7 {( o4 w" }
    经典递归应用场景5 F1 L. _8 M. k/ ^: g3 V; ~4 W# o( ^
    1. 文件系统遍历(目录树结构)
    , V4 n& O9 A7 w$ ^# ]: H, O2. 快速排序/归并排序算法
    " G4 I) U( c& P" f" C$ q3. 汉诺塔问题% G8 t& S; z/ M# p$ g7 J7 T9 D) h
    4. 二叉树遍历(前序/中序/后序)
    , x9 }8 z0 g0 }5. 生成所有可能的组合(回溯算法)
    + ~# u* V) }0 b) }
    : l3 _0 N5 j1 m# N$ ~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
    郁闷
    昨天 12:19
  • 签到天数: 3107 天

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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,
    . Z8 ~) `; R* S! O: F2 h我推理机的核心算法应该是二叉树遍历的变种。! F8 D  a5 I8 q
    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:
    2 k7 ^' H0 H: h$ `0 d8 cKey Idea of Recursion
    / N9 i! p" l2 `9 u7 x# z9 K2 O* }, B
    A recursive function solves a problem by:5 U+ f$ f0 w. O( G
    1 B6 @; N: z2 S. v4 p0 Y: N
        Breaking the problem into smaller instances of the same problem.: U5 D! m! Q0 P( V1 @
    + n$ Y& Y, H. N, q
        Solving the smallest instance directly (base case).
    7 J8 |) B. q3 M) }- g# w# N: X2 P/ s, ?; @
        Combining the results of smaller instances to solve the larger problem.
    ! T5 C. N! H  I
    9 o3 o7 d. m8 {4 IComponents of a Recursive Function0 j* [# e+ s. o3 D

    0 o- Z! i& ?1 x    Base Case:
    ) ^( U, ]9 d9 r; L. P2 Q1 g4 I$ L6 T4 m
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion." d% W7 a3 u8 j! m) \, V
    0 z4 O" m0 D" [  A
            It acts as the stopping condition to prevent infinite recursion.
    3 R4 T' c$ Z4 k
    2 }& X1 P5 y/ g) y2 N( R9 a        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    - G; f: H8 M# z3 s, @' U
    7 h7 Y0 Z7 \; f) C* H    Recursive Case:
    1 V7 U7 \. A" L& b: t' `8 M0 c
    ! Q, L4 g' P7 o+ T        This is where the function calls itself with a smaller or simpler version of the problem./ k( W  ~4 \" {0 T, ^, v  g1 j3 Q

    . R9 H. w  B" ~4 z& u        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    * f& a2 d3 u1 s4 E/ f# U: ^* Q
    " H; D, `) V: g6 L, oExample: Factorial Calculation
    ; J  i8 z5 V) a# W0 r1 c5 W& q6 ~( y! R
    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:; q4 V9 q/ `( Z6 P
    & E- ]% ]) g, D7 K2 A  H. p
        Base case: 0! = 1
    - @- {, |% Q) ^3 A9 S* }
    " t1 @& ^6 B/ h% Y3 o    Recursive case: n! = n * (n-1)!3 S' T8 L. ~0 W  a  L) v( W

      B: c/ \/ n5 [$ o0 U# S) `Here’s how it looks in code (Python):, \; `# M# Z1 b) e0 M, W
    python
    3 k$ U+ [) C+ p0 L- |& I- f, L/ F( x8 S5 j0 Z4 D1 L4 g

    2 M; m$ a2 f9 F0 c5 M# m$ Vdef factorial(n):8 v% u- Q4 W7 @8 E& \5 Y1 M! d7 ~
        # Base case
    7 g# v. J- i- c3 G& [& K    if n == 0:
    # X& d" F" u& ?/ c        return 1/ [- N' L' E/ ~# t4 S) Q
        # Recursive case
    ! Q% h) M+ y4 h3 Q    else:3 h1 C' d1 _# l; @
            return n * factorial(n - 1)
    # i; @1 E7 W2 Q
    ( N, @* C8 Y: N* d+ n6 v& k1 |# Example usage
    7 ^: c( j* J, D, V4 s- r2 iprint(factorial(5))  # Output: 120( X+ _4 I" L5 z; a0 B/ n; x
    : M$ k  B' H/ N) z6 L3 P
    How Recursion Works" a3 d# a& u0 l: S9 x

    - }7 Z6 a5 L5 f- u2 A5 n7 }    The function keeps calling itself with smaller inputs until it reaches the base case.
    , W. _% h, v9 `/ z4 d3 a) R% P9 z& @& B, r: [
        Once the base case is reached, the function starts returning values back up the call stack.  O9 L9 k+ o8 y; X# s6 D

    $ R$ W$ X/ c( f# A/ [0 I    These returned values are combined to produce the final result.
    3 U& B" ]# u" V: a% F
    % ]: V3 f' E* w( zFor factorial(5):4 r; k$ H' v( S4 t

    0 ~; ~. [" p1 d- E- G
    ( z% o% R4 q  O* ~3 z. Gfactorial(5) = 5 * factorial(4)' j" N* A* {7 m! Q0 b. f
    factorial(4) = 4 * factorial(3)) ^* e: a1 o+ f) \9 w) o
    factorial(3) = 3 * factorial(2)$ U, [" q- S  d* T2 f# V- Y
    factorial(2) = 2 * factorial(1)3 `/ J6 l0 q5 [
    factorial(1) = 1 * factorial(0)
    . x" K9 o, q- k# g1 v# z3 |, m* A- _! Zfactorial(0) = 1  # Base case
    6 l+ [' e' `- V- ^0 x. v# C: \
    : f- Z  c1 G6 y6 E* ~* w" YThen, the results are combined:! P4 w5 j3 O$ f! |! n5 _' S' t
    6 ~( F) P" R, i- Z. ^
    * j! k9 Y& V# |2 s! m
    factorial(1) = 1 * 1 = 1( h# V9 ~* j, o; W6 F- q; x9 e8 n
    factorial(2) = 2 * 1 = 2
    $ g3 e! `$ F/ z! W& ofactorial(3) = 3 * 2 = 60 ]: r. g7 k9 B4 Z1 ~3 v) Q
    factorial(4) = 4 * 6 = 24# g# _. |  Y0 @: J/ U' L
    factorial(5) = 5 * 24 = 120
    + m8 b: D. g9 O3 b3 b7 h' ]# F5 B8 o+ c, Q1 ^' N& ]# R( N, m
    Advantages of Recursion
    " d( s# x/ x7 \
      s: C  D9 R, n1 w6 Z    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).
    7 D- E- O+ C, S/ u2 }- y  n
      a4 b6 n8 M& n) [; ]2 {" k    Readability: Recursive code can be more readable and concise compared to iterative solutions.5 O1 i# M) }! r

    ) t, x0 ?# N( \3 n( r$ N0 `Disadvantages of Recursion* y4 r8 K+ E) ?6 C% j! ]( S! Q

    " W' [& _, V* a    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.
    9 _  `' N8 S' g9 x
    2 c/ k" y# G5 X. ?5 a+ A& d7 s    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).$ L2 U& l+ l" p( ?1 _
    ; {9 V3 F$ I5 o% ^) A
    When to Use Recursion
      r6 T; s6 w% Y9 w9 b" U1 ~2 o
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).7 m. F) `" z: \' G4 K. Q

      n, s& e" C2 T- ~" l! l    Problems with a clear base case and recursive case.
    7 ]& Z% S; X$ a1 ?; L4 V) o; L) o1 p$ S. d6 ^+ Y( m
    Example: Fibonacci Sequence" H( {% Y% J3 O( \

    ' ^0 l/ z; U* a0 K, IThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    6 y& q( p) v9 I2 o) Z& j* }0 h) L  P/ \
        Base case: fib(0) = 0, fib(1) = 1
    # Z, k! B* z3 \4 M2 n0 z& b- C5 e3 r  I# B9 t# y
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    , a5 n( u# i7 H" ~3 ~/ Q# j1 T! V( h5 }. c1 ^+ y
    python
    . T+ j% X- ], G6 c+ D
    , G% B4 G6 ?- i8 N5 U4 k  n4 m$ |* k& y2 Q5 L6 ]; `3 n/ ]4 u" C$ \+ n: Y
    def fibonacci(n):9 ]  |& B0 @. W4 b* [
        # Base cases
    , q: u7 e- |# O0 C) D    if n == 0:
    - Q" t& ~) X7 j. k; m" V+ f9 |        return 0
    % d( B  ?3 s+ r* }( ^    elif n == 1:8 O: U* @" J0 T
            return 18 ^, Y. R1 a8 L  A0 {6 b/ ^, E
        # Recursive case7 p9 E. Q8 o3 |
        else:
    0 |1 r. h& k. Q$ S# E5 a! ]: J3 i. {  c  y        return fibonacci(n - 1) + fibonacci(n - 2)# n7 |, u* v/ [9 }
    - ?5 P1 Q4 h4 ~& {' l8 ^& V+ @
    # Example usage
    " [1 S; i$ I- e* J1 j& ~+ V: aprint(fibonacci(6))  # Output: 8
      r1 q2 z( v3 x& p) N$ L1 ~% ]
    4 A8 }0 O( f, L8 Q5 @8 XTail Recursion5 A  `  s9 }+ E( r- I3 Q5 I8 S5 ?

    ( K# j; Y$ r; n& r3 R: j+ J4 X, DTail 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).' f' [2 Z$ b& U! L& @1 j  S% c

    # X( r6 A6 a+ P* A. tIn 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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