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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 8 W' i, a  d! }$ E: e. Z* n$ M% |
    5 x3 w+ `: T% d2 X1 B
    解释的不错
    9 }' M+ f4 V+ _( J, X
    9 A8 @( y$ T4 o8 f, l9 p: s$ 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。* A3 ]. @  {! q$ w) N1 A: k

    5 E+ E" Y8 ~. E$ P 关键要素9 Q# x! P' J; E5 a% e0 c, X- h
    1. **基线条件(Base Case)**2 L5 |; j9 {" G* E& l5 p* u( j
       - 递归终止的条件,防止无限循环
    ; I5 I, ?/ {' z9 W( \% S8 B$ j7 \   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    . }/ b( R6 I* t+ E( f. Z7 e5 ~% h4 H2 c( V" c, H' c
    2. **递归条件(Recursive Case)**
      R1 G1 P! P8 q% R* s3 L4 @9 \   - 将原问题分解为更小的子问题2 H2 F2 m  m- t% m( k$ z
       - 例如:n! = n × (n-1)!
    % g# c4 S; o$ [, Z7 O, ~3 d
    ' \# z- T: N. P- P& Q 经典示例:计算阶乘& r/ r! k8 I/ a7 A) [
    python
    : N9 _# _: {& p& X' J8 ydef factorial(n):
    ' n' G! k; O; Y0 ]/ ?' e7 ?    if n == 0:        # 基线条件
    4 U% G1 r2 W" Z        return 1: h" P9 `2 o7 q5 L( ?( G
        else:             # 递归条件' z0 L$ G: i% Y! A2 V/ g
            return n * factorial(n-1)' f7 C! P! m0 L0 h% Q; a
    执行过程(以计算 3! 为例):
    5 w* K/ e6 {% q+ `( E& Sfactorial(3)
    * F  T1 U! v; Z3 * factorial(2)
    * I9 ~1 a- n( k# p8 A  z7 K3 * (2 * factorial(1))9 w0 |& `8 \% ~0 C* q) H
    3 * (2 * (1 * factorial(0)))
      ]8 p! s7 u; z: @" l! ?* s! @; W3 * (2 * (1 * 1)) = 64 w0 `& o- w1 n2 |; r6 [
    2 j6 k9 _, s3 K; S4 Y
    递归思维要点
      L) w# S, _4 j1. **信任递归**:假设子问题已经解决,专注当前层逻辑# S1 B! J# m# t6 ~6 d% A* h
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    2 G% o0 |0 L0 N. T2 p" A) j3. **递推过程**:不断向下分解问题(递)- p! X  q! b7 G" Z0 Z
    4. **回溯过程**:组合子问题结果返回(归)( }0 W. h( X; [- i4 |; _8 |# i
    ! a( Z6 c. W; }$ `8 K/ x
    注意事项% d* ^8 j: c. |. i: \
    必须要有终止条件
    . j* u' ~, |; i4 t, W! w递归深度过大可能导致栈溢出(Python默认递归深度约1000层), B# M. \) Z' I
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ' w5 c6 ^9 H7 \! x  `, D0 m尾递归优化可以提升效率(但Python不支持)9 e3 k# F( _& X
    ' q; T6 f0 t' N: P- L
    递归 vs 迭代+ \" O7 Q2 c3 D
    |          | 递归                          | 迭代               |2 S1 X, T# k7 O) c
    |----------|-----------------------------|------------------|+ ~! K* v; x0 u6 I$ b/ k
    | 实现方式    | 函数自调用                        | 循环结构            |
    " U6 |; r& Q, C3 l5 || 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    9 b+ [5 a3 J# i  }- L1 a| 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         |3 S6 t5 \1 Q2 ~0 C! ?" G- G
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |) \: N% F0 Y+ C6 w- r& n# O

    3 V- z! o2 [! ?) {+ E/ h* Q* z 经典递归应用场景& @: L& ?: C4 }- q2 F0 z
    1. 文件系统遍历(目录树结构)
    % [4 u5 ], g  M( S2. 快速排序/归并排序算法( H# o7 r, A( J2 E9 }) g
    3. 汉诺塔问题/ F: E  |' m% d$ J  m8 N
    4. 二叉树遍历(前序/中序/后序)
    0 W: `: Q3 {( y0 F: p5. 生成所有可能的组合(回溯算法)' j' `  M* {7 Z5 ]& x3 g3 r

    7 a$ N3 F5 K5 |! 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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    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 [; J) H. P- a8 @' X+ y( C) i我推理机的核心算法应该是二叉树遍历的变种。
    % H- O7 M' N$ y2 b2 [$ e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:
    9 b) E3 {% Y  S% p+ g, T8 sKey Idea of Recursion3 E% j7 p$ u8 D0 H, q8 `

    : D! D7 K3 e7 w0 cA recursive function solves a problem by:
    ( ?2 C0 L; \' ~$ _& `5 _" b9 {# H! }: \
        Breaking the problem into smaller instances of the same problem.# W' R( _" h6 O8 n( @
    ( f9 _  U9 M# z/ a- W
        Solving the smallest instance directly (base case).
    & W/ ^9 V% ^( }# N
    2 w+ }# Y. z8 l3 A  ~    Combining the results of smaller instances to solve the larger problem.
    3 `1 ]1 O( b; Q
    7 Q* L. q. g' @% Q7 AComponents of a Recursive Function: i% t5 {! x4 K- }! [

    : f1 |! X, m) V. q3 E2 m! I    Base Case:
    0 ?: p$ L, y7 k/ g- l. O: L( n: n$ l) y4 a0 {- x' t: n
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ) }1 A% Q' D* A/ q  z2 S( W( J7 o- o- ^
            It acts as the stopping condition to prevent infinite recursion.! x- h9 D7 G) H! n7 f

    0 C1 E+ `% }$ g4 s: P- N  p2 L        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.' T. {- E$ N5 Q

    + _( M- E5 ]: ]* G, ^& A. _  Y    Recursive Case:& {- F1 b( B. [2 A* F7 {3 [/ j2 x
    ( E2 I8 T. ?# [( p" D1 b# t
            This is where the function calls itself with a smaller or simpler version of the problem.
    3 j' @, p' U2 }6 R' |. l9 q- E9 e* ?$ d
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).9 _' A; S& ~6 C+ @" `* h3 Y3 H
    ( k) P% f- Y& G2 J$ f$ C$ _
    Example: Factorial Calculation1 i- e: m# @7 U4 ?& i
    ; E  y) ?7 L0 }: J7 t' U2 d
    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:/ I' |* n- `: f
    ) w# s" Q: j% q2 {) t6 I
        Base case: 0! = 1
    ) R9 O: P! D: H' ~0 ]. R5 ~( L- a, ~. `2 f4 _
        Recursive case: n! = n * (n-1)!4 d3 V. d8 y/ Z
    " p' g' ]- m. h9 }, _5 ?
    Here’s how it looks in code (Python):
    8 ^0 l) Q% U! \* W3 fpython
    8 L5 Y/ ]+ H/ M2 l- N5 }- P
    2 K( ~' S8 W' u. f$ j; f* `: I
    % X7 ~: P3 U4 a( Q, ?+ l2 Jdef factorial(n):: b  h5 \' N% `) a' g
        # Base case" k* c5 B- }, f% T6 F+ \& ?( L
        if n == 0:# z6 |; R# a! ^( g) [0 _
            return 1) I1 R. z/ z3 c; P, S4 H
        # Recursive case
    6 D& B) S' ~0 G$ I' f6 p" l* c    else:$ e9 q/ l; d7 \3 L0 t% q
            return n * factorial(n - 1)3 p! |0 k+ S' N' I3 Z. V/ B9 a
    # |! G% a; M. ^
    # Example usage
    $ Z% l9 J: x+ [3 `1 O" qprint(factorial(5))  # Output: 1207 i& q8 E/ H5 _
    1 K: o. [- T& G: K
    How Recursion Works
    * O9 I7 O, ~9 [. g, X' r
    6 t' u* c( u! t2 F    The function keeps calling itself with smaller inputs until it reaches the base case.
    0 l9 y/ p# `  z- G$ ?7 Z+ W4 T6 |  n. Z8 B- i
        Once the base case is reached, the function starts returning values back up the call stack.
    . L( c- f+ _9 {, z) X" u$ Z/ X. n  \0 W
        These returned values are combined to produce the final result., Q% h8 V1 _( N  q

    ; L' m, E0 y' l- y. g1 MFor factorial(5):6 m( Q" S# h- r$ _$ c

    . G+ y- b7 y$ A1 _5 @& d5 C* ~' l0 i7 ]
    factorial(5) = 5 * factorial(4)
    0 H; Z1 W& D; y& D9 ffactorial(4) = 4 * factorial(3)# X' V) ]  Z! ~& {! c! T( T
    factorial(3) = 3 * factorial(2)
    % l7 L* n0 e" P8 R) M% ifactorial(2) = 2 * factorial(1)
    2 ~  V! T2 X5 C' r' x+ F$ zfactorial(1) = 1 * factorial(0); E: J' h/ \6 n. Z+ q) S
    factorial(0) = 1  # Base case
    0 {" }8 p+ n# f* ^  i; v
    # Q& h4 G4 B( Z  pThen, the results are combined:
    6 K. M# Z: t, }1 B* y# y2 m! _6 d2 V3 v; s! D9 t
    % J' M& z( q4 L3 g
    factorial(1) = 1 * 1 = 1- t$ ^  f4 s' p. `0 A; h8 k1 p
    factorial(2) = 2 * 1 = 29 l# M* J, s, g( [" g+ ^( O8 `
    factorial(3) = 3 * 2 = 6
    & N7 Q; _% O* B- p! c& o  c/ Ofactorial(4) = 4 * 6 = 24* ~5 P- b/ [$ c  O1 e
    factorial(5) = 5 * 24 = 120
    1 A( R8 C7 C8 ]+ X/ d' `9 G9 Z; r6 i1 l: }8 z1 p# ?  Q
    Advantages of Recursion* n) |( q% P& B# R+ F
    ; J, ^2 R6 L0 ?  {+ G) {+ S5 N# F
        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 Y% Z9 V' [# F9 z! y& E
    , K( M4 @" \: a
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
      z- Q: A: v" C9 m
    - H- S4 C% ?/ m" x) mDisadvantages of Recursion; O$ T0 N6 [. x% `2 b
    2 d1 G+ O; ^+ X4 T4 w5 U* @) k& k
        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.
    0 R7 X( e) Z/ O) w& t, O  {" m# K3 b" P( s
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).# T. x! q5 s* T, s4 H
    " \  y! R2 E8 e4 p
    When to Use Recursion0 G  ~0 G7 f# B5 Q1 N% X; [

    " b0 k9 ?2 T- d6 G; U    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).) v' W+ v! l/ ^1 \
    # @) o4 k+ t% ]- f! A2 t
        Problems with a clear base case and recursive case.
    0 ]% Z0 ^+ o7 [! Q/ f# j$ P$ L/ S( h9 i8 L# E% ~
    Example: Fibonacci Sequence
    6 B& @  B# d" t. [5 E' b) e2 v4 Z4 m- j! M7 W
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ) J. N; a8 o6 G; @% J' |
    8 `0 S: U" _0 {' ^  r- f: |# w/ O    Base case: fib(0) = 0, fib(1) = 1
    2 L( N% j6 X+ O+ g  f+ {# f7 h, \- M6 _# Y
        Recursive case: fib(n) = fib(n-1) + fib(n-2)* |7 T  F# j( r: s8 L- {5 Z

    ! G: j8 K) H" J8 [# z+ ?python
    2 j/ w/ b  v9 T' z- w, w" _" _
    0 f" [+ h3 s6 R. ]$ I' y! `- z* S+ [' C
    def fibonacci(n):# \* O8 ^! h7 H, s7 S
        # Base cases# |; I, L6 G' p+ i
        if n == 0:( \" V$ d* s; U4 Y, x' {
            return 0" z9 j# z; B  l+ ?, k& U) U
        elif n == 1:
    2 t: E/ ]$ r9 `4 Z" `* R% s        return 1# Q( [- f7 R. U1 }0 d) L6 r
        # Recursive case
    % J, f2 B) z3 S    else:. |' R" x. O5 z7 |% \+ ~4 I
            return fibonacci(n - 1) + fibonacci(n - 2)
    . ^5 T, K! _. o
    9 |& C8 }2 S, l* c3 M0 o# Example usage
    5 x# F  |; I% {. j! `7 Z; ?3 y+ ~7 Lprint(fibonacci(6))  # Output: 8
    3 l4 s! Y! R! {% V
    . ~2 E2 R  v1 L: Z4 M: z( UTail Recursion
    * q' |$ s1 j% v& K: J* B
    4 i! c. w6 t3 A1 W: z/ J4 BTail 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).$ Z9 @" U: x, I& p4 y: C7 ^8 ^
    0 {8 q0 q1 u/ ]) u) @# {8 g  \& 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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