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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 % r% s4 s0 N/ }

    3 B; o8 U7 D" J0 W$ \解释的不错* v( c, X/ B; _9 Q' e7 L) ?8 i" \

    " `9 U# c5 E* c- S" J" w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。
    # e( l5 p- {6 U, x: |+ Y1 L9 o7 H1 P  S2 d
    关键要素! c& Y4 K; K0 Z$ B
    1. **基线条件(Base Case)**
    2 X( I4 H" n. c" ^   - 递归终止的条件,防止无限循环
    ( }1 t+ n" J/ Y; B4 e   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1% @, k4 P# R. m% l! b
    ! C1 y) H& l, a  f1 [, C
    2. **递归条件(Recursive Case)**4 k$ n# W: ?  ]/ \, r6 P2 \5 `
       - 将原问题分解为更小的子问题7 G: F3 F( {* y% k' {
       - 例如:n! = n × (n-1)!4 N$ u" I9 i% U' G
    # O! A& f# Z- S. M2 p
    经典示例:计算阶乘. p2 X% N/ o! f0 o; m, a6 K9 K1 D
    python
    , T* P. R1 R1 F" n5 }  E7 }: fdef factorial(n):
    6 {+ _5 H% E; x) |    if n == 0:        # 基线条件
    . S4 g' @8 {: d+ k4 c5 a7 f& ^        return 1% J5 ~, |; l) X+ a2 S7 z; f( c
        else:             # 递归条件
    8 n8 D, d, U% l7 I. g        return n * factorial(n-1)
    * P4 h$ o0 k+ e% {- ~2 K执行过程(以计算 3! 为例):
    $ G9 ~/ i8 f! g, T2 Lfactorial(3)- A+ X  J2 S( `2 E# O
    3 * factorial(2)! t9 U) {- g: @5 o; x, M; Z
    3 * (2 * factorial(1))
    ) Y. [  ^/ x: h2 T$ K3 * (2 * (1 * factorial(0)))9 E" L9 F4 t* ^1 q$ \, Y
    3 * (2 * (1 * 1)) = 6
    - U3 E& u# S. H' b' |+ L6 _5 v2 J8 e) k1 Q9 M& K
    递归思维要点
    & J: _  L% y# v/ G& \3 @: m. t, f3 M1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    5 k- o0 J1 \, L, U2 C1 s# Y& R2. **栈结构**:每次调用都会创建新的栈帧(内存空间)+ y* A! I; s+ H9 c( \+ G- i
    3. **递推过程**:不断向下分解问题(递)
    , a! E, r) v- o; t- w4. **回溯过程**:组合子问题结果返回(归)
    : k, N4 z! d  R, T2 F, N3 L" o  H
    / Q/ a& a: F, ^ 注意事项1 h2 A. D% E( c5 k0 f) m
    必须要有终止条件0 w" `( ?& d, U  O8 ?
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)1 S1 |" R$ J! {9 J& B% R7 S- g1 M
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    & g: L% }2 X5 i% C) j' S0 P& q. P  |' r尾递归优化可以提升效率(但Python不支持)
    5 D! T) V) M. x: }9 G) d% Z) R$ c4 ^7 \  `1 M0 u
    递归 vs 迭代% z( R) c  ]" H0 o
    |          | 递归                          | 迭代               |% H  `( N# R) U! j, v
    |----------|-----------------------------|------------------|
    # ]+ M# I0 `- p$ ~1 v. T& G| 实现方式    | 函数自调用                        | 循环结构            |
    ) e8 d1 B% d9 `" c6 m; q| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |" e5 w0 `3 X2 q* F3 j! c
    | 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 r6 |  P* e! V/ |6 Q% F
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    0 }& R0 P' m8 A0 G
    / G; q" Z+ U- V/ ?  c: A! @' V) d 经典递归应用场景' p2 \& T$ R" {6 |! {  ?
    1. 文件系统遍历(目录树结构)
    0 o) d/ c5 A, ~" {+ m4 P2. 快速排序/归并排序算法
    / S+ C* n4 K: o3. 汉诺塔问题- {$ b- x) P; X- C; h' q1 W! G
    4. 二叉树遍历(前序/中序/后序)
    2 L0 D8 a. |% Q$ c) l/ H& a5. 生成所有可能的组合(回溯算法)2 B; C3 ~+ X1 q2 M' ?
    . x. p2 ]! d8 B& \
    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,  B1 u) T& J& y# |/ T
    我推理机的核心算法应该是二叉树遍历的变种。+ X: Q* C, m1 P' B
    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:) c" @1 O) g" b5 J' D: _& O, E4 D/ J7 W
    Key Idea of Recursion2 Y6 i% M9 \" q+ U$ l0 U

    ; y; K; }$ ~+ s8 w' Q% M. S3 G- kA recursive function solves a problem by:! Y) O$ ?6 |( x6 x& N

    1 ]3 g# Z+ _( d1 a    Breaking the problem into smaller instances of the same problem.
    $ u5 F. e( x: ?5 q; E. A. E4 H( G( P
    ' p4 Q& l. H1 \; L' v' x! c    Solving the smallest instance directly (base case).
    ; v: }6 P$ E. P1 M" e9 ^
    . c- I: ]3 W8 r: p    Combining the results of smaller instances to solve the larger problem.
    3 a. r$ {. @9 g7 O. S- M
    " c5 @* N/ H4 U6 `6 v& MComponents of a Recursive Function: v7 Q, R/ y( i2 J0 d4 f

    $ g1 z" _9 y/ l) o: K" Z    Base Case:7 s4 w8 i* T0 Y
    + I- N6 }3 H9 y' B; ~
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.8 F  `% T: G0 o! k) ]: E
    2 d, v9 s( O8 F+ W+ D
            It acts as the stopping condition to prevent infinite recursion.& r0 c. Q! \/ f) k/ g  P0 ?

    9 b# S' R) @6 i* u9 u        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.9 W$ [: C+ M# T# I
    . H5 ~8 h6 S8 Z* w
        Recursive Case:
    / f( I, \. y+ s9 Q4 E% g' w* z$ H: s
            This is where the function calls itself with a smaller or simpler version of the problem.$ d8 M$ O6 |: F6 l; h

    / X' a; B2 f" w! s8 k        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    6 C' C8 K& ^5 u) V: I2 s: m) u4 m4 u  }
    Example: Factorial Calculation3 V6 @. V# ^  {- G8 q- }0 B1 Z

    0 M) T( E% e! S/ |/ bThe 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:
    - u* y) @7 X) z) I# Z0 i6 F8 `6 U- W: K4 @# v. o0 k
        Base case: 0! = 15 }$ i7 S1 L; w

    6 G% k/ b) t5 l4 V    Recursive case: n! = n * (n-1)!
    ! X! x9 I& f, G" \$ C9 i! J
    ; w9 ^" Q3 v  V% [% b% w9 r. FHere’s how it looks in code (Python):$ J7 Y2 B) q9 f8 F+ J* J: n9 x! R5 Y
    python
    4 g( o8 r  v, u5 h, J% Q, X! S2 n" c: m5 o1 g& l

    2 c% K; e, Y/ B- F1 z( q" j6 G% o+ _def factorial(n):
    6 ]& h9 T. k. M  p+ Q( K. V    # Base case
    % ]$ M, I* B8 d    if n == 0:
    9 U3 X$ i- B1 u2 n6 P* \, |        return 1( n- T* ]3 h! P/ F
        # Recursive case
    1 m' A5 F3 v. Q6 {6 z- u    else:& [: b4 |3 i, X7 O: C
            return n * factorial(n - 1)* |6 `4 E  u8 t1 y* U  f

      @+ X( I* l( N: i+ A# S* V8 i# Example usage& P! u6 x% L. Y' J
    print(factorial(5))  # Output: 120
    ; c8 j% C! h" I6 e! {0 S' e* \7 Z) ?3 e) ]2 w3 u
    How Recursion Works
    " b. H: p5 v  K. S! V3 J5 b$ _6 d  x& v! r0 c2 _
        The function keeps calling itself with smaller inputs until it reaches the base case.9 T) I1 A1 Z( q8 f

    , W" [: c% d; o    Once the base case is reached, the function starts returning values back up the call stack.
    5 K* y+ M4 O# E9 ^
    2 F: k2 A/ h) F8 j' b( c    These returned values are combined to produce the final result.
    8 e4 X% U+ Z+ ]  A. y
    " U6 I. h( Z& M* S1 C* X0 Q) u# n( qFor factorial(5):
    * G& J* r' X1 L2 g# `) w! B# L
    # o6 g- @* e( a3 V
    - |8 V, p- s4 m) I- t" l$ A" ufactorial(5) = 5 * factorial(4), J2 W6 z( d1 |+ q0 K- @! n+ H! H
    factorial(4) = 4 * factorial(3)
    0 G8 g: k: ^0 efactorial(3) = 3 * factorial(2)+ ~6 I4 ^( M. M3 c
    factorial(2) = 2 * factorial(1)* U6 }" Z# K. W, p# c0 A  r
    factorial(1) = 1 * factorial(0)
    8 S# Y' I" w8 tfactorial(0) = 1  # Base case* V5 [7 Q7 Q- j2 A) ~

    , Z9 ^  Q& O9 zThen, the results are combined:' Q# {4 K+ d9 {& W/ a9 ^

    7 \/ U6 _4 O+ G5 D& q$ v; u# B2 E) R/ t; I$ o1 J, T
    factorial(1) = 1 * 1 = 1" J# M6 @1 G( f/ T, r+ M  [  r
    factorial(2) = 2 * 1 = 2. a+ V. y; f5 y2 v" x4 v/ Z5 p
    factorial(3) = 3 * 2 = 6$ Q3 q+ b0 |  X; x6 r& S
    factorial(4) = 4 * 6 = 24
    - x& _  f; E  O1 l0 Yfactorial(5) = 5 * 24 = 120( O+ F% D) K, u

    7 u2 g( }! {: D% h0 B1 V$ b2 ]( ]! N7 XAdvantages of Recursion6 S; A9 P7 q3 X. n! W5 U
    $ E& q  n+ @3 B) o, O) I  y$ j
        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).
    & k1 E' U5 ^, `$ z% w/ q
    4 h9 `1 l$ _7 Q    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    % R' R! f3 ^: C2 |
    6 c* ?* V/ v" B2 `. p; ]Disadvantages of Recursion
    4 [4 v& r% Y0 F' ]( ]$ S" i( x  X) y: p, u+ i/ X
        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.7 ]/ x: G2 y, I. w: M( V
    9 ]% v/ k3 P* |2 n' ~# U: m0 g
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    4 T0 ], {8 G+ B6 l" ]4 M$ w: L+ r, L7 ~2 H, O
    When to Use Recursion
    4 r9 ], U% x6 x8 e: g! b0 O: e. |% r: X3 \) T3 g2 n; ^
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).' C* g* W; M% J

    * G- C. C8 H9 \: F    Problems with a clear base case and recursive case.
    ! }2 ]7 C+ g: @$ D6 {/ {4 s' B  b
    Example: Fibonacci Sequence- T5 a% N7 [; e
      ?- A3 B) o" u5 F- }/ U% B5 Z0 S
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    9 D' u& s' ]" I, r  t! |2 {: `3 }1 D# @6 ^; e2 Z5 }$ q
        Base case: fib(0) = 0, fib(1) = 1; X0 _9 x; E0 \
    4 S6 @6 z; \4 b
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    " s  y3 _2 w* Y: p6 O, @5 O' W3 U) ?7 l! E9 k; `
    python/ p! e* D4 k3 A6 B" g9 J1 b( {) ~
    7 V- l' H/ }% ^; l& s( i
    9 Y$ i1 k+ O+ k9 J1 k" p
    def fibonacci(n):
    ( u6 R. Z6 b' D1 k' }' Y" ^  I    # Base cases
    9 k" I1 b* z3 ~    if n == 0:) A4 b8 b2 S& C2 r0 w* Y
            return 0
    * [  h5 Y1 Y* |/ ?    elif n == 1:
    8 T  L( f5 v  j" `/ h9 H        return 1+ E& n! P, M& X6 ^6 P
        # Recursive case9 T! N8 E5 W0 ^" b
        else:
    2 M8 ?( M: a: R+ D, z        return fibonacci(n - 1) + fibonacci(n - 2)1 T4 S0 `/ K( P3 B
    # K  I4 [4 c) t3 A9 N+ z0 N
    # Example usage
    ; f+ r9 N0 L! Z8 N( aprint(fibonacci(6))  # Output: 8
    ' z& i6 z) x# A# X
    + b& p* T; A+ z9 QTail Recursion' h2 B( }$ w( I5 X! s
    . E0 _4 A2 d/ @/ U6 j
    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).' \4 J& {/ q# @
    / L4 V9 W! y5 T% {0 n
    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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