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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
  • 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
    9 z! L$ }  e# [7 d5 z& y4 U5 M( v0 {! q9 n
    解释的不错- y6 P) |3 E  j  {
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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。) R8 {; {, _) I5 `6 Y6 z4 I
    $ V$ p% d: S+ O# s  C7 o
    关键要素
    " a$ v: n6 W- a4 o1. **基线条件(Base Case)**# {! Z: @8 P) I3 X  e( h3 P
       - 递归终止的条件,防止无限循环, c( H- z3 j% N' g- @4 R
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1% u$ X6 a0 E$ J2 v" @; g7 p6 [
    - M) ^: n5 ?! c
    2. **递归条件(Recursive Case)**' k' x. e( a5 I- t0 m6 I
       - 将原问题分解为更小的子问题4 S: D& N/ y/ [. D. }
       - 例如:n! = n × (n-1)!
    * X4 \. m+ e( E4 Y4 l! Z2 ?% |
    6 Q0 o# w  W4 c! l 经典示例:计算阶乘6 p4 o; ^2 z2 S- @5 W1 \: B) ]
    python- y! ^; n! m# A3 P# ^
    def factorial(n):
    7 z- e0 x3 I7 m1 t  B) z  q    if n == 0:        # 基线条件7 u6 z9 g8 i6 B
            return 1
    % K; y' Y( T9 x5 p- q    else:             # 递归条件: S& d0 @, N* u# P9 z; n  P
            return n * factorial(n-1)- J4 v$ m0 N8 _% z$ L+ C; S* l0 Z
    执行过程(以计算 3! 为例):
    ' V' O7 K  P4 B3 F+ t+ O' f7 rfactorial(3)
    3 {- w+ r5 ^% h# P! d3 * factorial(2)- b* y( _. c. ?6 @9 r5 y% w
    3 * (2 * factorial(1))% _3 M; E, ~* G6 K. i
    3 * (2 * (1 * factorial(0)))0 N$ i7 X5 S- Z$ C* {
    3 * (2 * (1 * 1)) = 6
    # v2 ~5 _/ \# i4 P1 ^- ]6 F% e
    , C# v8 h1 \" {* { 递归思维要点
      g" D! V! y) [- ~6 N1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ( B- m5 L. I  x& s. X4 x% ?4 E, `- N2. **栈结构**:每次调用都会创建新的栈帧(内存空间)7 L# r* Y, c* i
    3. **递推过程**:不断向下分解问题(递)" }; Y! p+ `; u3 X6 q1 H
    4. **回溯过程**:组合子问题结果返回(归)
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    : z! @8 A0 ?! ~0 l: ~' S 注意事项
    7 f4 S8 j  l1 q: H必须要有终止条件
    & O9 S4 z: X% o2 ~! e递归深度过大可能导致栈溢出(Python默认递归深度约1000层)+ L, N# D/ _* N* B3 E% }5 |. ?
    某些问题用递归更直观(如树遍历),但效率可能不如迭代) C0 t6 v$ b/ A
    尾递归优化可以提升效率(但Python不支持)/ V7 ]2 d* }3 Z/ {- c6 ?: p

      N2 n/ a; ?2 ^* [ 递归 vs 迭代. c, J2 Y0 ]% Y1 \$ I
    |          | 递归                          | 迭代               |1 p) s" b, I+ _+ W# u
    |----------|-----------------------------|------------------|
      Z& ?) K8 f' f# ^4 f6 E| 实现方式    | 函数自调用                        | 循环结构            |
    $ c1 x( P  m8 l; d% y. Y| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    - `3 j: j% ^1 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         |
    ( N2 D. T# X0 `5 p% ~| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    # E, X2 m) n4 p" p& D' u/ n6 ^2 f% c$ e0 x7 d
    经典递归应用场景
    + u7 O" P6 A. G1. 文件系统遍历(目录树结构)+ `3 Q  R3 g4 O# `8 W9 }
    2. 快速排序/归并排序算法  Y' ^* K4 a- }' ?9 z
    3. 汉诺塔问题
    + ^. W* p; A# B  s. J9 O4. 二叉树遍历(前序/中序/后序)2 u& S1 [. j5 `2 B5 G1 D) H
    5. 生成所有可能的组合(回溯算法)
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    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,; s7 ~+ A/ k. ^6 t
    我推理机的核心算法应该是二叉树遍历的变种。5 q+ g8 f2 t: J! s: j& l+ }
    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 X% Q( K' q3 d$ M  ]Key Idea of Recursion3 C3 E9 z* a) b" D% G- D$ k

    * `! I0 k( ~5 q1 V! z7 n& \' g& `3 ]) SA recursive function solves a problem by:' M- {, h0 {& S# u% W

    " @/ P" N( `) O" I    Breaking the problem into smaller instances of the same problem.. d  U( \, ^  r) |# {) H4 A  K/ m4 A' j

    7 ~: C/ E! X0 M7 x    Solving the smallest instance directly (base case)., [. G' C3 J# x
    9 `# B  m8 B+ F6 j- X
        Combining the results of smaller instances to solve the larger problem.
      V7 ?2 k2 X# e' a" f- y5 z# j2 K, Q2 h" w3 E6 }
    Components of a Recursive Function. Z9 ?/ e# ]) M8 X# f( y6 D9 V$ ^
    / i/ H+ C2 W7 }! b
        Base Case:
      D0 U; o# A4 A3 r/ z, R
    . Y9 C+ O  Z6 A. k. `        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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            It acts as the stopping condition to prevent infinite recursion.; R# R3 ^1 {& P% }
    4 n7 e+ }( U- p8 U0 }
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.# C$ m9 p/ s& V" i+ F" y( h

    3 M8 v9 f% J# W. Y& g+ I* H    Recursive Case:
    8 ~$ t3 i6 v' h/ v" K# ?/ m! N: y6 I
            This is where the function calls itself with a smaller or simpler version of the problem.
    . F0 B' B$ {  d3 Q: o; M) d# e
    , J# Y/ f- Y2 u2 V* v0 Z, V0 b        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).& h7 f( L% [" |. D/ f

    9 s, r4 b4 ]9 a0 hExample: Factorial Calculation, R2 z  o4 k9 a) D

    * f3 d4 D9 z; X* v- U4 a( R6 NThe 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:* \/ K( g+ U0 F: G: c% p7 C

    : _! J2 `& ?+ }% n+ N) z    Base case: 0! = 1
    % s/ h: g# @9 c+ _$ n
    8 Q9 s0 J8 Z2 a- |4 L9 o    Recursive case: n! = n * (n-1)!! c+ V# z2 l! l& O

    3 w5 i2 G" M& l4 M4 KHere’s how it looks in code (Python):: w1 b: O  ?3 N; P, G
    python
    # s" b# Z) W+ R2 U
    6 l# S% w( |+ c3 a" R8 A( B* ?7 k% I, l8 v( l3 S
    def factorial(n):
    - x* T* V, P8 @: j    # Base case
    6 [4 x8 H+ q- t7 S" Y: e    if n == 0:  g2 ]* x% C6 Q
            return 14 K3 l& A! z# \' o
        # Recursive case
    ; a2 [# i7 n3 g! B! W4 G- O    else:
    * ]( s0 ~8 w8 h0 R) B1 X0 r) p' O# _        return n * factorial(n - 1)
    + _0 N- g# `3 x2 @
    , u8 m7 x3 ]# L) j# Example usage
    + V+ ?. a! p+ vprint(factorial(5))  # Output: 120
    . M7 @% S7 f0 x; M# p
    7 W$ k  b' B* p- E4 eHow Recursion Works( x: M1 B4 `/ }! @4 t' }

    7 m0 @6 N; }- i8 `3 {    The function keeps calling itself with smaller inputs until it reaches the base case.
    " f9 B$ J* b6 F/ x  J1 ~$ v- p
    * {1 }- f3 J. o4 w    Once the base case is reached, the function starts returning values back up the call stack.
    * n% @/ Q" w" V1 [7 y4 B- A, J) O9 ^) r
        These returned values are combined to produce the final result.3 v+ ~) W" G0 Y+ @& m8 X! D" {
    : T$ v3 }$ s6 ]% b: P6 w
    For factorial(5):8 [& U: r) S- ^" D. O

    . C6 O- D4 c3 u- C6 L# J# W& g# q; N4 Z( @2 [9 @
    factorial(5) = 5 * factorial(4)
    * ^- C6 h3 d' r; g. X$ B7 A4 U7 dfactorial(4) = 4 * factorial(3)# ?, R) ]- y5 k& }* }  `3 A
    factorial(3) = 3 * factorial(2)
    / p) B& @) h1 V9 o% }1 M& w# v5 jfactorial(2) = 2 * factorial(1)0 S+ k$ W; D* G5 h& Z+ [+ L
    factorial(1) = 1 * factorial(0)
    $ g' t1 ^$ T7 ?9 ofactorial(0) = 1  # Base case
    + w1 H& K0 p1 D& p" C! V* S
    - s' R7 F- Q: P: C+ {* ~, r+ N5 V6 oThen, the results are combined:
    7 b9 u" d) @% ?$ ]+ g8 D& u& J0 a' m' d4 j/ T
    6 w1 @1 n3 F. k6 `
    factorial(1) = 1 * 1 = 1  K. E+ R, G2 x! K: c3 n( f  X
    factorial(2) = 2 * 1 = 2
    1 K- ]  B3 ~" U2 Qfactorial(3) = 3 * 2 = 6
    5 Q2 w5 z. v& u" F% f" Nfactorial(4) = 4 * 6 = 24; I- I) c8 B/ K) m
    factorial(5) = 5 * 24 = 120! u, z8 ]& ~) }/ d
    , G# Z% p0 H0 P! F1 n) O
    Advantages of Recursion
    5 i. T5 m. s7 R/ S% c+ j7 J1 b" @$ i5 i0 ^/ @
        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).
    ' F6 r, a, d2 s  N! I
    & g' ]2 p7 t3 W+ B. ~4 f    Readability: Recursive code can be more readable and concise compared to iterative solutions.. L0 c7 R: |' o
    9 ^; Y" U* N+ e) c7 R
    Disadvantages of Recursion4 b6 Q3 C8 D% K$ `) G, }/ y0 {
    $ x' K$ ~/ e1 u$ T0 n1 {
        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.
      D- U" r! d. s2 F- \6 _, J5 o' r  k7 |- ]
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    + {; F. a, d7 X# z2 n( b+ q4 o
    ! M1 V3 E; t2 ]0 u, {" j* O: a' `) TWhen to Use Recursion4 d- e% M6 f. Q& H' V0 Z4 k

    4 W1 {6 }& Y( J2 m5 {9 R    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
      `6 h" o" P) ?0 K- T3 H2 |1 T0 F' T9 f  T
        Problems with a clear base case and recursive case.; {1 g) f2 {/ u, P

    6 H  |6 o/ y$ G# X9 f2 U# pExample: Fibonacci Sequence
    " J. y: p6 x4 G$ [% o4 @( J4 X) a" U1 \  d% Y$ B" e; D
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:* N( _8 S! K# ]% e

    7 o. W5 W5 \! C; V& V3 q    Base case: fib(0) = 0, fib(1) = 1
      v- B+ a8 q/ N4 b
    0 y( A2 l& f* _+ k( t8 W    Recursive case: fib(n) = fib(n-1) + fib(n-2)+ V1 }! ^" _) U, n
    / I! l0 `1 l2 H  }) g" s
    python
    + c* L2 [7 F; q/ `3 ?" v9 W# H! o- f% O. B2 c) M* P

    8 |; R6 H8 A, bdef fibonacci(n):8 E' M" `; F- f- ^5 B' R
        # Base cases
    . R1 O6 V( Q) v  ?9 \4 D+ U    if n == 0:
    9 I1 T1 k- ~3 ?6 E7 |        return 03 y& S* m! Y6 K  E
        elif n == 1:+ \3 ]" h' `0 `: T
            return 1, a# ~+ e- B9 I9 c, ?
        # Recursive case+ E  C. U7 H$ y0 {1 r' v
        else:
    ' n- m  @. L0 V0 F; Y$ f, a        return fibonacci(n - 1) + fibonacci(n - 2)
    # Z5 [; U, W: O( d
    ) I/ C6 e7 A, @# Example usage
    3 \- q. D& O) Kprint(fibonacci(6))  # Output: 8
    : v% a+ U! o0 s+ c. i- D' ^7 H
      u9 ]; [. K" o1 H- c- ~: XTail Recursion
    2 l7 c6 u2 j5 z( J$ g( O$ l. O+ H, k$ A  {4 K% N
    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).' n3 z+ B5 Z/ H9 x2 T
    - |7 E2 d" u, h7 v4 V6 P, }# b
    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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