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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 - q# O8 z0 w5 S1 B* L

    " V9 p; Z4 X% f* S# t6 |解释的不错1 W1 l1 G7 W* ]

    ! s/ |! l% R' A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。* ]/ e9 v: d' ]$ @" x/ Y* V

    1 c. J! x- ~: t# h 关键要素
    # _* X; h! }1 _8 ~2 J9 u8 w( z1. **基线条件(Base Case)**- Y1 Q% N0 c- K5 G0 ^
       - 递归终止的条件,防止无限循环
    7 O# x, l8 B( a8 T   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    $ f, I3 Z' Y' P' \1 v2 K$ K5 p+ w  Q4 p" X
    2. **递归条件(Recursive Case)**
    + G' E* P& b7 ]& T; b  Y! W9 K   - 将原问题分解为更小的子问题
    + ^9 H& ]  B5 c8 D+ d( x   - 例如:n! = n × (n-1)!  e3 I! W- o6 h/ C& Y: o
    # {+ T5 l6 R7 Y3 J! C) y
    经典示例:计算阶乘
    6 Q8 J+ m) M3 W; H2 L5 V3 z9 Y" qpython
    ; n" v& T! U; |: r; C% T' y7 _* S7 gdef factorial(n):+ O  o) Y2 S# @
        if n == 0:        # 基线条件9 w0 f( w; b4 f+ q& H+ \
            return 1
    + K  a  Y& S% H* C9 R/ s' A    else:             # 递归条件
    , y) r- z) ]7 q4 D9 P  j# a9 z        return n * factorial(n-1)
    6 R6 v) c8 d9 L1 g; w执行过程(以计算 3! 为例):; M% U0 U5 d/ A; u) g
    factorial(3)5 S' K6 L, y- K. ^3 `) h
    3 * factorial(2)
    9 x* b8 D1 ~6 {, s, p- x9 n2 {: x3 * (2 * factorial(1))
    . ~* N8 p! ], X4 o3 * (2 * (1 * factorial(0)))
    % n0 q7 ^# e8 C/ Y8 l3 * (2 * (1 * 1)) = 6
    - k4 T8 Z( l8 a4 Q- H
    ( ~/ Q; z8 W! M7 ]2 r" l- A 递归思维要点" u" `, V  ]7 ]0 |  B
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ! V7 U# F" n# C# S* I1 N4 g( k1 ]& j/ b2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    7 f+ m$ R) L# s1 W9 H3. **递推过程**:不断向下分解问题(递)6 k7 I2 ?3 x) ?! ]  y' ?$ u& ~* F
    4. **回溯过程**:组合子问题结果返回(归)
      m. \) p- J" p6 c! B: A/ s2 t
    0 `# ~! W$ c: V* x% ?4 V1 g0 q 注意事项( t+ m" M: b% t, g
    必须要有终止条件, p' A. q1 s" g& j# N
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)/ r6 y, j* x. d- t( x4 {
    某些问题用递归更直观(如树遍历),但效率可能不如迭代% ^2 i% F* t2 n/ D, U3 f
    尾递归优化可以提升效率(但Python不支持)
    % U& C" B3 k( U5 \5 U" d+ ~) C, Q; ^; c) g8 W- Q8 p' g
    递归 vs 迭代+ B' W" U+ h1 |8 v
    |          | 递归                          | 迭代               |; u" [* y/ {2 [% C! N& Q* u
    |----------|-----------------------------|------------------|. T& r6 a' o! s  Y
    | 实现方式    | 函数自调用                        | 循环结构            |: i" ?+ F* w+ T
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |6 M; G/ z" e2 t8 i$ f
    | 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         |
    & u# l# A! @! n3 n| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    7 I! P6 V6 l1 Y( Y# L" P8 ^
    - y5 ^6 f$ g6 ?7 H: F 经典递归应用场景) N& U# I1 i' T6 ^+ Q) `
    1. 文件系统遍历(目录树结构)
    % m' _. J8 b' K2 Y4 A# ^& n6 B2. 快速排序/归并排序算法
    , j" M6 B) ^* P# q3 O7 i2 I3. 汉诺塔问题, b5 N9 `' K7 c& r& W
    4. 二叉树遍历(前序/中序/后序)5 [3 w& B" C$ c
    5. 生成所有可能的组合(回溯算法)
    " T# j, Z1 s: g) T8 h+ {& O5 h: }5 J* O/ U; 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,
      i. ~+ o; q' ]6 q7 ~我推理机的核心算法应该是二叉树遍历的变种。# J/ H0 d0 Y1 I% D
    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:
    . D) x6 [. Q6 W" }& u! a- d& L5 q1 ]Key Idea of Recursion
    / U0 {8 y5 j. {* x( j+ L1 C% [9 M0 h8 q" ~9 y& o$ |8 p
    A recursive function solves a problem by:, @& U. t! t( k6 j: |/ `/ S1 Q

    . m& h; D7 ^. ]) Y- A    Breaking the problem into smaller instances of the same problem.9 m  p3 W! N# x$ W, }

    # f) F- p. c3 \5 o1 k5 [    Solving the smallest instance directly (base case).5 {4 g3 C5 x/ B, x) e7 z

    ! Z8 D* I) K3 o+ m' s) x$ s2 Z4 g- g    Combining the results of smaller instances to solve the larger problem.3 i7 @( L# u$ h# i4 Z2 Y, X
    5 b% V* i- n+ {. d0 B
    Components of a Recursive Function  r% `  y4 R# r8 X7 [4 s$ R

    # k& S! g3 `  e) R    Base Case:4 {! y5 b8 u4 {

    9 c7 ~" D9 \( B0 Q9 y9 v$ R        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.; N. l5 g- ?$ @2 x" B
    . b  Q# V1 e8 T  A
            It acts as the stopping condition to prevent infinite recursion.
    ( H7 `/ t5 U" x. Y0 M: U0 T
    7 i' {5 s! S' F        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    4 {! F1 o% |: \+ c2 T
    : H# z( j; s0 |6 @# a" B0 I$ p# {    Recursive Case:
    ; ^- b, j( Z) y8 z+ g2 S/ L, o$ v0 H/ W9 O9 D
            This is where the function calls itself with a smaller or simpler version of the problem.
    * F2 A: V$ \1 [' K% F( ?" ~: U# t
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    3 g9 n1 B/ |9 U4 ]7 I
      k/ c, p1 L- C5 VExample: Factorial Calculation" T! v: y. W  M& ^

    1 z; F7 e, F# s, Q8 ^, c1 M( DThe 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:0 l# B- ]8 z: g" l5 w! S: O. F

    2 P5 Z$ I# b& D9 F    Base case: 0! = 1
    % Q  f# T7 P# d& k& z" R1 p, V7 \
    ) b) K9 ]8 u6 d3 ~$ u! T    Recursive case: n! = n * (n-1)!
    ! |7 P' q' `! ~/ y( k: v" |5 Z' G' K& R$ J- Q, }; _* f
    Here’s how it looks in code (Python):
    / \7 u- ?2 w( U, J# Ipython7 m3 v* D+ Q5 W7 K

    & w, Z: P6 F9 g3 t  |$ g$ @# f; v
    & B" J7 m7 n; j8 e  {$ e; o. l  Cdef factorial(n):" J+ T  B( R: N* p3 t  P9 K3 f
        # Base case
    0 B6 W7 w/ t/ y1 T    if n == 0:
    : e0 A& P& m2 S) H" [        return 1
    ! B, a( b. U6 t* x( o    # Recursive case5 Z, [5 I3 l- @) a9 ~5 g2 C
        else:' |5 a2 n  \/ m# I  ]+ V# b% Y
            return n * factorial(n - 1)3 f5 g& \3 B" {9 F

    / b( ^. s* n- Y3 P# x( U" i# Example usage
    $ V' Q( o; H1 U+ eprint(factorial(5))  # Output: 120
    : L( Y" }( |1 Q7 h3 B; l" T7 |; j. W# |
    How Recursion Works# a" ^3 K; |$ q# y7 c

    9 G8 ?5 `% L7 L) y8 y) K    The function keeps calling itself with smaller inputs until it reaches the base case.
    2 k$ W5 }" T6 K  D& e! A. V
    ! K7 k5 a/ s. f% Z    Once the base case is reached, the function starts returning values back up the call stack.
    / v* f0 Q3 C- R0 p& @- n: R# `$ q" _! ^3 l  k3 m( D- m
        These returned values are combined to produce the final result.
    + X8 `! {( P; F: K/ U2 S( k5 \% \
    & v5 ?" y) _: M) N& o8 p4 ~For factorial(5):/ y5 o& K! A" K. A: s3 v+ K* Q

    / E! ^6 P9 P5 x4 n, A+ G
    ) |  I( t/ n; ?, lfactorial(5) = 5 * factorial(4)
    ; y6 I; n: F0 d  n# O0 r0 qfactorial(4) = 4 * factorial(3)( {+ q* e  y* V. S7 W) \
    factorial(3) = 3 * factorial(2)0 b+ [  D( ^5 K1 B
    factorial(2) = 2 * factorial(1)
    " m  R, a! T: V+ A6 p  W: Ufactorial(1) = 1 * factorial(0)6 U9 U" {: `. Z3 S+ W( p' e4 H$ B
    factorial(0) = 1  # Base case
    % z# j9 {& h4 f4 x0 k  b+ G( X6 e; f# z: X- ?0 v
    Then, the results are combined:6 j3 A* S3 \$ c9 x! i! ^

    ' Z' z$ v7 c" x' K$ i1 n0 g
    , U8 n- k8 O  U/ R7 mfactorial(1) = 1 * 1 = 1
    + h% T' c! S$ efactorial(2) = 2 * 1 = 27 r. B3 x% H2 ^! q% g7 q
    factorial(3) = 3 * 2 = 6
    2 I3 v6 u9 h, C4 E: U$ ffactorial(4) = 4 * 6 = 24
    $ v4 ?) N) e9 ~3 }$ ?; j, cfactorial(5) = 5 * 24 = 1207 {  B  G8 L/ r! E# E' I
    , v6 ?/ P, _! v4 V9 S* R3 Y; P* \: G
    Advantages of Recursion. T: ?( X6 B4 ~2 s( W7 o

    / }9 g5 ^0 _! v( B' L: O% q* O    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).
    , y3 @6 R6 R4 x! y+ i8 n; r. Y& h( R- c
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    $ ?8 J; E( g  ~: g& F* _
    + D" o' X  u9 H/ e5 HDisadvantages of Recursion
    : r. }* T8 F6 p4 u+ M3 L8 |8 @: S0 Z  [" j; G4 ]. G
        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 U; y  l1 U1 M, D) X- N: |0 {+ u1 }$ w8 S0 H( E; x* B
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).9 H' z8 d/ B" ^3 a
    ( l# \5 y+ X* _: n' l, t2 p0 S! q4 U- r
    When to Use Recursion+ [/ k2 i5 j% }2 @! j
    7 o  ^8 t/ y* W  z
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).2 i- o' \6 h/ N/ [6 w. g5 J. {
    4 u4 c* \4 c2 }: R9 b
        Problems with a clear base case and recursive case.* ?9 D/ V4 X6 c5 _9 q' T
    4 N4 A2 ]8 K1 d$ i& Q; r8 l
    Example: Fibonacci Sequence! p1 ]( P& U3 W
    , d; `5 O8 B! F  H# x! h
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    7 ]7 i1 u9 F7 `1 \9 U3 p' i4 h! l/ Q# C' C6 t
        Base case: fib(0) = 0, fib(1) = 1! v  R( ~: c* B2 z& ^' Y  H' W
    9 n8 O' [/ _7 d: h/ r' E) r
        Recursive case: fib(n) = fib(n-1) + fib(n-2)1 ]0 {* w7 N# X4 m3 n
    ) N8 n+ A; T- E* @: ?
    python
    . B8 p0 z* Y! [' z+ a7 F/ O
    ! f4 z! g0 J. P
    # T0 l+ b* N) Y  Wdef fibonacci(n):
    ! T$ Z; i7 c$ E% n! }3 F    # Base cases: Z1 x2 ?6 }7 j% l5 m' z# J) a* A
        if n == 0:2 n& @' l5 D" k/ J% W2 ~
            return 07 ?6 p; w' B% x
        elif n == 1:- ?6 ]. f. U. v1 T
            return 1' t3 X6 d# N' F0 L8 e  m7 J
        # Recursive case
    ' ~# T4 C. V  ^    else:
    " P3 p( K: |  M, |7 Q        return fibonacci(n - 1) + fibonacci(n - 2)
    & k6 A  h: j4 N9 R" ]
    9 ?# r1 @1 A$ F& a# Example usage
    % K3 J. [6 L* W! ~/ g( D, x0 Xprint(fibonacci(6))  # Output: 8
    * K; a# y5 l/ S2 g5 D5 p0 E
    % c8 F8 }& t, @& h. K- P2 a' N; ]Tail Recursion
    ' O5 E+ x  k6 W' h% j" ?9 g& ]
      m4 @+ p% |% j6 S  YTail 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).
    ! C" k1 w. S! j* n  ?
    + s$ J; W( {) h$ jIn 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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