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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
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    解释的不错" b8 \- Q$ s! I8 l) J0 C+ J# V+ I+ d
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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。- v* W* f% H" f' K. V3 m4 z# g

    " C# {! C( b9 h1 D' q$ h4 D+ \. i 关键要素
    6 L, c" R  {; W% _& P0 Y1. **基线条件(Base Case)**
    0 I5 o: O8 S1 W% d. l. M- t   - 递归终止的条件,防止无限循环3 I; D  C9 G: K- g# w
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 18 g1 {5 o4 T' H

    2 p8 w) F% ^, }2. **递归条件(Recursive Case)**/ @/ l  t9 \2 y( J" t* ?( o0 r
       - 将原问题分解为更小的子问题/ J$ m+ _8 K( P8 G
       - 例如:n! = n × (n-1)!# B3 U3 _/ U, ?7 G9 S
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    经典示例:计算阶乘7 Y' i* `5 I0 {! ^) u
    python
    ! E. Q' T- l$ n0 D% e" L, f& `+ pdef factorial(n):* ~0 }" l# S) e' a! }
        if n == 0:        # 基线条件, w4 J; m+ Y0 M/ z4 Y2 r
            return 1* u! s" x/ E4 ~& p: Z* q/ `; o
        else:             # 递归条件6 y; u& O! c6 D7 h" P3 }$ H9 Y
            return n * factorial(n-1)
    % t! U4 ^3 L+ l" H6 ?9 h2 K执行过程(以计算 3! 为例):
    9 s9 O7 c8 }, B% q. t& tfactorial(3)" v' ]- `! p' b! C, ~
    3 * factorial(2)9 ~; r  E( S4 T. l0 i
    3 * (2 * factorial(1))
    : L: I3 `; ~2 Y3 * (2 * (1 * factorial(0)))7 q4 {9 i7 e) C% P: ]4 I2 w7 @
    3 * (2 * (1 * 1)) = 6
    - G7 I0 g3 j9 W; I4 Q# V/ M
    4 `& @! S( |8 N/ n. A" P- o4 @ 递归思维要点7 B  o4 R: |" l2 f4 T! s
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑: h' T- x$ i& Z: N5 N6 n- P( ~
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    . \0 G* T7 E0 f( G8 v- q3. **递推过程**:不断向下分解问题(递)
    + S  M5 \( d$ s, i4. **回溯过程**:组合子问题结果返回(归)- @& P2 c* _% ^8 W; ~

    $ k! p1 R# b) y, l) a, T2 j. E 注意事项
    : k* G7 O7 m2 f必须要有终止条件
    9 v/ B- z3 R) ]* X" Z递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    . D+ u0 V/ H! d/ W某些问题用递归更直观(如树遍历),但效率可能不如迭代6 U) T8 q; B( l& O
    尾递归优化可以提升效率(但Python不支持)
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    递归 vs 迭代
    # i- V0 d* @& L; W|          | 递归                          | 迭代               |  r0 U  n1 f- s1 x# h9 Z. h: s: d
    |----------|-----------------------------|------------------|% F( U2 }1 }- a- F. g5 Z
    | 实现方式    | 函数自调用                        | 循环结构            |
    / v+ _' h0 y- E& F| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |9 R' u- j, ?6 S/ m2 G
    | 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         |
    0 E$ f: W) q! N- O% U( r# J" t| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |. r* m$ r# X+ ], G8 m
    & h0 G& G. S7 s
    经典递归应用场景
    * p5 D  y$ e% j1. 文件系统遍历(目录树结构)" L& `3 t2 z) z% E9 _5 R4 l! \$ D! f: X
    2. 快速排序/归并排序算法  n! L) ?2 Q& f
    3. 汉诺塔问题
    5 Y2 M5 V# s2 ~* |) n- N, G" z4. 二叉树遍历(前序/中序/后序)+ V9 f. g$ K- |- P0 Q, F
    5. 生成所有可能的组合(回溯算法)* |6 e# ~3 T# S2 E( ]0 ^
    9 c5 n* h" {7 R* r9 N7 j$ c
    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,
    1 j0 P! T/ }+ v) m! C我推理机的核心算法应该是二叉树遍历的变种。
    + f, b3 I1 m0 c; B+ U6 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:! Y' J. @  M: P8 k" q5 H. B! ~: m2 T
    Key Idea of Recursion
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    , q8 c% v- h0 Y% d3 i, RA recursive function solves a problem by:: A2 n+ t5 V7 I* }6 o. O
    / T8 ~' ], C* S1 I8 ]+ `
        Breaking the problem into smaller instances of the same problem.
    5 e  @. u$ l9 G& q+ G
    # u- N/ A7 a, W& G8 x) Z1 y  f& z    Solving the smallest instance directly (base case).
    2 B: g! Y3 b- R2 P& S0 k
    # C7 J4 D( \/ q( u1 O- A5 g, p    Combining the results of smaller instances to solve the larger problem.: S( r( c) |, z( Y, q; s& |, q% k  H5 d7 f

    - {, l1 K: O( u" W4 c+ pComponents of a Recursive Function
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        Base Case:+ v+ w; c  g. }5 l
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    : @; B, E: ^/ h7 W7 Y' N: e3 @& Z8 y: X; u: y& t2 \1 I
            It acts as the stopping condition to prevent infinite recursion.
    % V, o! L( o; T3 P& ~: ?
      M, a; t2 j; H5 I* G  ^' L- r        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    % Y0 V5 P# O" N" Q9 v2 C+ n6 c! u3 }2 \* M4 ^+ `
        Recursive Case:
    ; E" q# O- x% z3 `2 q! ^1 ^! B8 a$ J8 G- G: r( c
            This is where the function calls itself with a smaller or simpler version of the problem.2 B0 j7 q. a% w% j9 b) m
    7 O, E3 @4 R& v) C7 B6 N8 M
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).$ F% C6 a3 V! ^; U

    6 f. y. X* q# n6 D# q* wExample: Factorial Calculation
    0 w* k0 E. l5 O% H0 i: z. \2 o) X6 h8 a! p7 F) U* X
    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:
    ) t* b8 h" d* ^& P, J* T. e- j) h+ h$ W9 V& K9 S6 E
        Base case: 0! = 19 b& d( ~; _8 y( e

    4 m5 N( f0 o- V; r1 J5 Y1 O    Recursive case: n! = n * (n-1)!
    % J+ }; Y( t: M$ B) n0 R% f% j; f9 p# }/ m/ M% v# W2 ~
    Here’s how it looks in code (Python):  x( m; U" b3 h( {' p
    python" j9 T- D5 H/ \6 T% {
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    8 I; I, q8 o- y: l. `8 P
    def factorial(n):
    " A+ ^3 s$ J4 s$ E0 e! q    # Base case1 {. h! ~; A9 b( {+ A( Z
        if n == 0:2 i) N! z$ u0 n+ k+ C1 Y& L
            return 1" Q) P% i% W  P! `$ I
        # Recursive case6 Z9 e, F3 H0 u. t6 Z
        else:
    3 H6 N! n* Z- ^! i: g  V        return n * factorial(n - 1)+ j: a6 c& \9 T: f

    9 M6 Y% g/ r& ]; i# Example usage
    ) ^" i; q  q  O  ?% l6 sprint(factorial(5))  # Output: 120
    9 W) I& r. P. m# j  {0 r
    " [. \3 X  d, D0 [$ r& L! SHow Recursion Works0 n. G2 Y% F& k; f3 u, e
    5 e6 u1 z+ t! W) t$ N
        The function keeps calling itself with smaller inputs until it reaches the base case.
    7 V, _: Q/ W$ [( j$ ~" M0 M8 g% ?+ I' T4 P& }5 @# C; d, v# q
        Once the base case is reached, the function starts returning values back up the call stack.7 w, y) r3 q3 z
    & e0 g+ G1 J( U- |0 g4 T* X' t/ Z
        These returned values are combined to produce the final result.
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    For factorial(5):* S% _" H  u) n- C6 b. P
    5 T2 _/ f+ V7 t6 j, R( S

    + K- C! {6 @# ~  C7 n& Tfactorial(5) = 5 * factorial(4)
    8 ?3 ~: q7 _+ U1 _0 zfactorial(4) = 4 * factorial(3)1 e' l$ ?* S1 T( ]" O! q5 V
    factorial(3) = 3 * factorial(2)
    ! I" ^1 J; k- c' afactorial(2) = 2 * factorial(1)
    8 U! L% c7 v8 d3 Ffactorial(1) = 1 * factorial(0)4 `( O% S, O* Y9 T* Y( Y
    factorial(0) = 1  # Base case) _& e, L7 }1 v# `
    * M6 c# }. r) O/ O% j
    Then, the results are combined:% N( Q+ D* q' ?0 ~' o- ^  s

    * e; B' q2 H* Z6 k3 q9 \( B
    / t; z9 [+ I  A' f4 ^8 Kfactorial(1) = 1 * 1 = 1
    * D% T# S: y) ]8 ^# t9 mfactorial(2) = 2 * 1 = 21 l' j& o2 ~  s
    factorial(3) = 3 * 2 = 6
    0 |" Y6 q& S5 b4 j- G6 S1 \factorial(4) = 4 * 6 = 240 c1 r2 D( \6 K) t
    factorial(5) = 5 * 24 = 120
    / `3 A( `1 u$ `
    0 W0 D2 p5 B; C0 h+ x1 YAdvantages of Recursion' a5 C( S/ L# j% U* N

    : v/ Z) F, z3 p- y" D8 i. 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).! H4 |3 H& a0 \6 ~
    * B% b7 V& @* D+ _
        Readability: Recursive code can be more readable and concise compared to iterative solutions.& c8 q8 ~- H3 o# n  _

    5 B5 p7 G1 N6 w, p3 P! IDisadvantages of Recursion+ N. x1 @' j+ x3 y/ `" C

    ' Z4 Y9 \% i( V+ y4 J/ N! ]    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.
    1 _7 [! U) G! g9 N
    ) c  N1 P1 R6 l* x# k- Z' I    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    6 _8 q( m: Z6 z2 \& c$ Z4 D) @! a: Q
    When to Use Recursion& X7 W" z3 J2 A4 h

    & y+ J: f+ B6 M  b6 L: `    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).9 K% b9 m: z( H2 ?
    ) N) C8 W1 O8 d) J0 H) V3 F# L
        Problems with a clear base case and recursive case.1 f7 Z+ i( O& @5 l1 ~
      G; O+ }3 @4 H7 L$ v$ L
    Example: Fibonacci Sequence- I) x9 n+ ?" Q( u! z* E6 X( \

    & F: B! t& Z* e6 {The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:/ g% M5 R7 j, I; ?# H) u# ^# H

    & ~% d% A# A. m0 {+ s    Base case: fib(0) = 0, fib(1) = 1/ L. E. C! W$ U- d/ |& m; M4 l
    6 U# t4 l: Q1 w% I+ Z- y# v( }4 s
        Recursive case: fib(n) = fib(n-1) + fib(n-2)3 j. E$ H1 [- T4 g4 U9 Y3 d
    9 U9 k2 N, l2 [! A  K5 f
    python- T( @: K3 P0 m: f9 o+ }

    7 w/ }: B' O, M* Q* I- S
    : d8 q% z: @4 J9 zdef fibonacci(n):; @( p* o9 J$ J  [) r+ W
        # Base cases* }( Q5 H4 {  G4 B
        if n == 0:/ R3 C; \/ i& r- s" l/ \, k* J
            return 0
    + ~: E, @0 u: I4 ?0 ~0 l; V    elif n == 1:" D8 c% B% x- G' C  _! k0 f
            return 1
      [# b. V$ F" p" a" X    # Recursive case$ j) E. R; \2 e- Z
        else:
    / O: }8 t: t/ k( ^( u1 N        return fibonacci(n - 1) + fibonacci(n - 2)% u2 ]6 h0 S8 V- e; T

    . ~7 i3 D& x  @3 Q  ^  e/ X. e# Example usage
    $ u/ W/ q% r. S+ \$ @  E2 uprint(fibonacci(6))  # Output: 8
    2 q8 w0 ^. r" h
    ; h& Z+ ]% W* g8 r0 x" oTail Recursion
    # T1 \/ Y3 H* {$ D: r3 X4 \2 p0 p6 B8 Y  ~, i0 M
    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).% h4 E# N) l# e3 T
    . H& u* K* U  E' a' A6 w
    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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