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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
    # y: s& |3 Q& V" b7 @8 D+ I- p% d6 o+ ~" Y7 B1 ~, q$ u
    解释的不错
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    2 h; q) f2 R' ?) s' [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。
    4 P+ G; G) Y4 _; H; V
    . t$ z7 y% n: K9 a5 Q1 L9 y 关键要素
    9 d5 |$ j, v/ \# u* Z. a; C- O/ q1. **基线条件(Base Case)**6 t7 g- G! s+ e: H% S: M# `: q4 s
       - 递归终止的条件,防止无限循环
    + X3 }$ i' M$ a   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1* K. ~2 R; j- }+ Y# i

    5 G5 x, L& r  Y2. **递归条件(Recursive Case)**
    & x) {/ D7 ]" ^! P# I- f8 b) o; L- A   - 将原问题分解为更小的子问题
    ' _# u  d$ J; H+ @9 O! E! ]   - 例如:n! = n × (n-1)!7 l+ J: K# m9 h5 S+ h. @

    7 b' ^5 V3 i( {8 E% s 经典示例:计算阶乘" M. Z$ ^1 ]- n) Y, Z! E
    python
    ; H4 B/ ~$ V2 y8 Ndef factorial(n):/ u* I' w$ ~5 e  Z5 x+ }% e. W* X
        if n == 0:        # 基线条件
    6 w: _/ A2 a4 [2 b  C        return 1) h8 H( {6 b# ]3 l% M$ v
        else:             # 递归条件
    8 t% R' k, d6 Y" r6 i        return n * factorial(n-1)
    9 e$ @3 I) f1 f5 C% @执行过程(以计算 3! 为例):
    + @# v+ r- A. T. ?2 c( H! nfactorial(3)
    ( @4 _8 W+ ^0 w  r5 B' V3 e3 * factorial(2)
    % T+ ~9 j$ y; i# D% w% U5 r: V1 F3 * (2 * factorial(1))& A- U: _6 V; @
    3 * (2 * (1 * factorial(0)))
    : |# e( j4 e9 U3 * (2 * (1 * 1)) = 68 X2 X! q+ }, C; z. c1 |) s* P

    6 w7 o; V* Y# \9 h 递归思维要点
    & ?6 k. g) T9 d1. **信任递归**:假设子问题已经解决,专注当前层逻辑8 W! k% b9 A# I% j- S6 f
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)& W" L$ P+ l. z  Y9 z& C4 J
    3. **递推过程**:不断向下分解问题(递)
    ) b0 x  x3 M3 w+ |+ g* r4. **回溯过程**:组合子问题结果返回(归)
    4 d+ i+ r% [' j" q" k# m; J  k% c% [6 t1 x/ d% t! {3 i1 f
    注意事项% |2 M# O& U& G) Z; g: L6 }
    必须要有终止条件( E) }$ e7 D" m
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)5 D) ^. n1 F7 L5 H' B
    某些问题用递归更直观(如树遍历),但效率可能不如迭代( b& O8 M, j0 D2 J& Y& a! y
    尾递归优化可以提升效率(但Python不支持)' a6 l5 v: ~0 N6 H- B2 z0 i6 b" x

    - ?5 r. d) S3 H4 J* a& r& {! ]2 Y 递归 vs 迭代
    2 t* B+ Q0 S1 i, f; L|          | 递归                          | 迭代               |
    ' L+ g5 |6 `$ I* k. D. Q. V|----------|-----------------------------|------------------|  S- ?, _2 v1 K8 B4 C8 o6 Q
    | 实现方式    | 函数自调用                        | 循环结构            |3 `9 K5 ^7 t' L
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |, _  {! }4 E2 r( b
    | 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         |* f4 Y8 H( i; O( C1 Q# l& L8 \8 C
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |% R" z4 P# R$ A( r) Z3 E

    / O' A5 `3 k( f- s8 ~( I6 {1 w 经典递归应用场景* v3 M  R  b# b/ }. C( b( y/ v+ n
    1. 文件系统遍历(目录树结构)/ X0 I. ?' \* f$ I
    2. 快速排序/归并排序算法
    2 h6 R, L) V/ {& J$ D7 S& q$ i3. 汉诺塔问题
    2 I% q: w* k5 q* }4. 二叉树遍历(前序/中序/后序)* G+ M+ ]8 }+ }% B9 I6 }7 K0 O- Y
    5. 生成所有可能的组合(回溯算法)& O/ t' j2 d% i% I

    3 v/ ?2 A5 S9 O( U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,
    / \$ t* V0 s- N( I我推理机的核心算法应该是二叉树遍历的变种。5 M% u$ s8 P) a" w
    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:6 Z7 |( f) ]+ _! h* {
    Key Idea of Recursion
    5 {, L6 [, B1 f1 `9 Y3 U: w8 R8 l8 N" Q5 g% I) H4 d/ S0 u+ K* j
    A recursive function solves a problem by:
    8 B# H% T0 |, [9 r, l4 Q3 Q1 ^& ?( B/ N2 e5 g3 G1 D$ Y
        Breaking the problem into smaller instances of the same problem.. c& H% K, J+ V" B9 T# ?: P/ P
    $ H7 b# I2 S) A) Y8 N' B
        Solving the smallest instance directly (base case).$ G3 W3 ?! P1 J' z! f" I

    8 X0 e" }+ w/ y7 Z% S5 i" B    Combining the results of smaller instances to solve the larger problem.6 U& A0 o8 h$ d; ~4 J

    $ F" F' y& s0 [3 _* m! Q, Q  f) mComponents of a Recursive Function
    ! t& p% R, _0 {$ d  q. w" |8 D8 _6 h! c- ~$ Q/ w: D* W
        Base Case:
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    + s" u7 t+ t7 l7 d1 E3 [3 A$ Z        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    & W! @; ]) E0 i" M5 [
    " y  L7 P0 [7 s0 C0 O8 R  J        It acts as the stopping condition to prevent infinite recursion.
    3 l9 F6 i4 q# _+ y! K% H4 [& @, z! Z. j3 ~; w+ q! j
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    ; u# ]/ I7 H% ]6 s0 x/ o+ F7 |, t; H2 H2 o5 Z/ e1 k
        Recursive Case:
    ' w' K( a7 o. V' N$ c1 q6 ]& z: M1 ]) g& U  o5 m. K9 Q, ?, |+ s
            This is where the function calls itself with a smaller or simpler version of the problem.) p, B1 y9 Z' [- ~7 z% f
    5 J% q* l6 g$ a! U, ?
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).( X" L" D  ~, p

    5 ?/ {/ [+ `) s  ]" m! @/ FExample: Factorial Calculation# V4 J% U0 H) J( v# Y1 X

    & Y; I9 C6 z; T  q; J2 c6 @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:; j% |' z8 W. q- |( Z, m

      V+ L4 H  M- Z8 ^    Base case: 0! = 1
    ( i; }: [7 a6 r. X$ |; T; `+ W) u; j: x1 j
        Recursive case: n! = n * (n-1)!  u7 F  b. d& H( }" O1 |1 K
    ; m5 h2 [6 m/ O" _9 O
    Here’s how it looks in code (Python):
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    ) B" H, \  r0 q  @  ~& G
    0 e( H6 \  L$ |def factorial(n):
    ( ?: {6 L# x" V' v4 I2 ?    # Base case
    5 n* H+ o# K$ W, a    if n == 0:" i/ O. c' c( L
            return 12 a$ o6 z  x& R( o
        # Recursive case
    ( x. k: q# M- f    else:
    6 M7 R8 }3 J2 W2 }+ k0 V- y) }        return n * factorial(n - 1)& x2 J7 o2 x6 N3 f( V2 x

    ; \, b  ~1 a: F! i! R: {# Example usage: c3 N, b& G% \" j7 y3 e' {* d
    print(factorial(5))  # Output: 120; ~6 C. r- K7 w# |- a% x

    2 f- z2 u! @2 S, U  BHow Recursion Works# f, b, x5 v2 h" `9 v% a; d1 \
    & t4 j5 Z2 V+ \% k$ c4 [+ l* r
        The function keeps calling itself with smaller inputs until it reaches the base case.
    % [  ~& U6 r% a
    5 F& M9 }! z# `( I* S( N6 A  |    Once the base case is reached, the function starts returning values back up the call stack.) S7 z1 P: ~! _& N0 V: u( S

    $ ?, d6 G7 ~) {5 ]    These returned values are combined to produce the final result.
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    For factorial(5):
    ' S, J0 p( H4 W/ ?# m) e& z  ?: B" C( u( l' z; \: X9 Q
    ) [# |8 v4 I- p- O" K
    factorial(5) = 5 * factorial(4)( A/ X; w: Q+ ?1 B$ E
    factorial(4) = 4 * factorial(3)
    & C$ n0 F7 _7 Q/ v5 ^factorial(3) = 3 * factorial(2)
    5 y& b+ e* Z+ h4 c! F! Hfactorial(2) = 2 * factorial(1)$ e, d; d( ~7 `: Z" w% ?/ X
    factorial(1) = 1 * factorial(0)* E, v8 j5 y2 R# P8 T$ d
    factorial(0) = 1  # Base case
      t$ ?9 ?  n  R5 X
    " m4 [  ]8 j2 d. k  sThen, the results are combined:
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    ; S/ K6 s% A8 v9 y) }# X
    - x9 x6 u) w7 P$ e4 Nfactorial(1) = 1 * 1 = 1
    ' ~0 Z2 {  r% `% E$ d6 y& cfactorial(2) = 2 * 1 = 2
    8 z! n/ C7 I" p% r* H5 `factorial(3) = 3 * 2 = 64 b& g: q2 j+ x9 T2 d1 _: v
    factorial(4) = 4 * 6 = 24; [8 M3 _* X/ [% E: Z9 }6 A
    factorial(5) = 5 * 24 = 1207 q8 p0 _1 L) z- a

    ' r2 \0 E8 J. z) G* \) w/ i, ?# hAdvantages of Recursion
    2 k3 [  e7 n, n2 v' O- B& M2 ~# ^3 ?' o( f, u! L+ X
        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).
    0 D$ _" t& @# u3 A
    9 S4 r) r6 M" O; R* z* T& x! x    Readability: Recursive code can be more readable and concise compared to iterative solutions.7 T/ X  R. n; [7 D# l

    3 Q( Y) M/ U* }$ T" g4 r( ^: H& uDisadvantages of Recursion
    $ S, Y- n! ]! N% z. X+ I
    ' H& `* [9 {. _  t: R1 u" p    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.
    6 u0 p8 O* `2 d6 |( z8 t
    3 a: w' @+ k) v) T4 q/ }    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).. @! i9 z( M/ J0 @! ^, g
    : [! Y: F- B3 E
    When to Use Recursion4 h( h8 i4 e! J1 O2 k' G& k% ^
    + x8 ^; [- P2 g9 I# x' X) a
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)./ G; {$ K9 g) q6 B& Z) m

    # D3 b/ ^  a  N$ Y0 z9 C    Problems with a clear base case and recursive case.  r8 o) V2 h1 L" E
    4 ^4 B: @8 ^9 g2 R& p! d
    Example: Fibonacci Sequence
    6 K7 X2 e5 y1 z7 G7 \7 L1 v$ x! e: z4 ?0 P" f3 [( R+ G
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    7 p0 J! U. H: r( k  g  M6 |" R' o7 [7 y+ W6 f
        Base case: fib(0) = 0, fib(1) = 1
    $ r6 ]9 l% F  S% i
    0 Q! L" i( o: a# M  G) P! h    Recursive case: fib(n) = fib(n-1) + fib(n-2)1 O2 s" t% ?1 t$ z# w7 D& s
    " i' J( m9 V7 @) Q6 z' W  _' K
    python' W& ?' O( P' h# m2 X: I
    $ h3 A4 o! @* o/ G' t$ p
    $ |0 L2 F9 u3 H
    def fibonacci(n):6 G) W& L  W8 g* C
        # Base cases
    0 T4 S6 K5 h; u8 U    if n == 0:) ^9 W; W+ }5 ~- |% t* T3 \/ E
            return 0& N' _5 b' f* @% X4 C# M
        elif n == 1:
    ! b/ K5 Z# Q" A, C9 ~        return 1
    % X/ T. M8 |9 v. E1 E& `; v* |    # Recursive case$ R9 d; H5 p% I2 c9 z6 y+ ^1 u! @
        else:
    - v0 ^2 X, P' ]! L8 v5 ~        return fibonacci(n - 1) + fibonacci(n - 2)' k. h0 P: P, |' _/ W, g

    3 f) f4 E4 f0 _  Q$ \3 S0 q1 O6 T# Example usage- N. w( N5 I" K+ p
    print(fibonacci(6))  # Output: 8/ a' o4 \& D7 W; i1 J+ C0 D
    0 M$ i- g  E. `; L8 `, `2 D+ Q
    Tail Recursion! F0 G) R* d# S
    ( T; c  O, Y6 G+ |; H* S9 l1 `9 R# q
    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).
    . p3 ^+ a7 \/ A  V# @  k" Y9 n5 V2 L+ `3 U& Z! }( f
    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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