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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 " U: r. P/ B! q- L9 x

    " U; M; l" Z8 _0 D: w, C解释的不错
    / y# E4 D6 a2 u, r6 ^7 O2 N# U; `! H* a/ Y4 ?
    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。# q8 u" t% m# g. E
    8 R2 a$ i7 p. W5 G
    关键要素
    & w2 }' A- p5 v5 b) `" p1. **基线条件(Base Case)**
    8 q8 L) J7 L  s9 @1 X' U! z   - 递归终止的条件,防止无限循环
    4 F8 x4 @7 u" G( S  y5 M   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    4 E8 a& M' H  h! |
    ( r. _, {1 R2 f( A, M* j1 z* l2 }2. **递归条件(Recursive Case)**, B' G4 R% X+ _8 V) B
       - 将原问题分解为更小的子问题
    % ?/ F$ h7 S: q+ x   - 例如:n! = n × (n-1)!, f9 ]& _" n: N0 `7 u
    ) `( B# n' L/ `4 @  J2 p5 ~1 b! b6 W
    经典示例:计算阶乘
    $ `4 V+ }8 |' \$ \3 [/ e1 Q$ tpython
    ' v3 R4 v- P' K: S8 N1 @9 u! wdef factorial(n):4 \1 K' K2 ?. L- X3 d
        if n == 0:        # 基线条件9 n! `  _; l3 V$ ~
            return 1
    1 o6 p6 c. ~& a* s    else:             # 递归条件6 H9 ~0 u8 W# @- o
            return n * factorial(n-1)
    $ N! t6 \& @# g* \2 s执行过程(以计算 3! 为例):
    , _6 n6 A! ~- J! ifactorial(3)
    0 |) g3 D+ [7 H1 ^7 ~3 * factorial(2)
    ; E( y1 J# E4 i+ A. N$ K  y3 * (2 * factorial(1))
    ) p  y1 w" f8 I  H3 * (2 * (1 * factorial(0)))
    : b/ v: L4 K0 ~! d( i/ w& e9 q( g6 A3 * (2 * (1 * 1)) = 60 S+ x2 J5 d2 l# E
    % V2 H$ ~8 x3 h! D1 p# H
    递归思维要点  T  W. u, \8 L2 r
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑* O- _' m0 Y' ]' o1 t
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)* G4 a, x) h% U( d/ w0 O
    3. **递推过程**:不断向下分解问题(递)0 T* b3 j7 d; n8 m: N' E% |
    4. **回溯过程**:组合子问题结果返回(归)
    ' u* H' O' J  H% c! W2 f& Q! x
    ; K% R+ o8 v3 _ 注意事项
    2 M1 z% \, |$ @1 b必须要有终止条件
    2 y2 X, U9 ?" G递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ( b6 O$ W7 `) P3 h/ E9 R" Z5 d某些问题用递归更直观(如树遍历),但效率可能不如迭代' R% M1 C: k! Z8 E4 L
    尾递归优化可以提升效率(但Python不支持)7 i0 o7 u- Z8 H  H" |) i) H

    5 \' W. A" z) E  ~ 递归 vs 迭代
    8 B/ l8 l1 b) o( a& n  P|          | 递归                          | 迭代               |
    * u* R$ V( f/ M& a2 x0 E, w- g- {|----------|-----------------------------|------------------|
    * [# M' z9 V; G- W6 [% }3 }| 实现方式    | 函数自调用                        | 循环结构            |& @& I) P7 z& G- g& V  t6 C( y5 g
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |; j$ i) n; E. j% R
    | 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         |% M3 k" M4 P5 j( B
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    3 R9 o; r# e4 y: n7 S+ y3 o4 ~3 d  ^% M: g+ \
    经典递归应用场景( }; U/ h) a  p9 L; v+ k+ H) L# v
    1. 文件系统遍历(目录树结构)
    0 S4 m+ r% z! c7 s; o0 i2. 快速排序/归并排序算法5 S2 n/ P% d- C  \; i; n( t) p6 X
    3. 汉诺塔问题7 f( {5 }% N/ L+ l7 x/ T% T
    4. 二叉树遍历(前序/中序/后序)) _; D  |: ]0 ~- S, ~$ ~' I" D
    5. 生成所有可能的组合(回溯算法)
    7 g- B5 U3 C8 A( ^% e
    8 S' b/ `5 ?) s$ _- j3 J: h5 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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    14 小时前
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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,7 w+ ?8 d8 u8 {( O" P
    我推理机的核心算法应该是二叉树遍历的变种。* c3 S& Z3 {" H7 C
    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:
    , [* m: `! N' X, T4 _% y1 u1 UKey Idea of Recursion' g- ]% p& W3 J! X. N- g. G1 |

    $ U2 |% H/ j% K: gA recursive function solves a problem by:# C: B! t% T! Y  ?. J
    ' }# y; f8 o0 i- G4 Q9 d! Z
        Breaking the problem into smaller instances of the same problem.
    4 J& X( p, L9 Z) g; e+ E9 ]% Y8 t3 {' Q1 U/ x' J" d  [  E) k$ I! X; a
        Solving the smallest instance directly (base case).
    ) ?# _& }, D$ d
    ( ?' @& ]& F- R( v1 D. A  b3 _    Combining the results of smaller instances to solve the larger problem.
    # m" C2 b8 `/ A- Z4 s
    6 M; \* ^9 {' `6 ?0 T3 b- oComponents of a Recursive Function/ [" I' q$ p* n7 S, C5 e/ [

    ( F0 i. b' W! ]( _: M    Base Case:& v- J2 z; H5 Z' X$ [* a
    9 f9 L5 _9 e2 l/ @, \9 U
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.& F0 G4 ~, T  Y" ?5 {

    ( t) U, W' n3 J7 g" V+ G        It acts as the stopping condition to prevent infinite recursion.. }0 x; @$ Y* r+ Q
    & p4 P- w. l( l5 N- E4 L
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.1 d' n* R  R8 I+ `5 D

    7 P  b8 h  K% M0 w: p" I) y3 R    Recursive Case:
    $ r( C4 w9 a, q- [4 C4 |7 h/ M1 F7 H7 M  _- o8 [
            This is where the function calls itself with a smaller or simpler version of the problem.  `# A) A) L7 v% t. B' {
    , W0 t8 u& j; F
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).* S0 A/ x7 f5 ?1 w

    ) J% e# V( E  v; p( `Example: Factorial Calculation& i1 c) I" v5 _  C) w; U# q

    $ q: i4 _. p  f6 TThe 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:* t9 O' n7 w& a* J. ?
    & e9 E$ z3 V) O. V  ?
        Base case: 0! = 1+ O$ ?7 P5 H$ B+ E  Q
    5 t; K- D. ^8 \
        Recursive case: n! = n * (n-1)!
    , i* [) w$ Y/ ?3 Q. [7 m1 l. u" A
    2 l" a/ X) t/ _3 uHere’s how it looks in code (Python):
    4 @# y9 n: G" p2 M$ P9 i5 G2 i" @' Epython' K$ z% C  z3 e
    & n, y9 Z7 h# L! ~
    6 a1 T3 E' q7 o5 M4 M
    def factorial(n):
      Z! W+ V% a& o    # Base case' R4 E, u5 j$ ~+ o/ i
        if n == 0:* I1 \- z. q% S$ W5 O7 k
            return 1
    # D% |$ N, c, o    # Recursive case2 O2 }! D9 J: ~: A" o( E
        else:" f1 C$ G/ E8 G1 ^& ^
            return n * factorial(n - 1)
    - P# h& Z: |& i+ [$ p9 N+ Z
    & z4 m4 u+ [& _9 Y# Example usage2 y1 p( K' z/ |
    print(factorial(5))  # Output: 1208 c8 b( j! W5 t8 _. U* U

    : p* v- j$ V* N( h+ qHow Recursion Works
    2 y; ~3 t/ x. F- w9 P+ ]" O0 x7 U$ q" z. o  {0 X8 {
        The function keeps calling itself with smaller inputs until it reaches the base case.
    , [. I% \6 |" V3 I/ r; V+ _& F& p1 Y( A" B2 T. S6 g3 [) W8 ?
        Once the base case is reached, the function starts returning values back up the call stack.
    , ~  }  u& w2 d" i( v! k' T% K% B% x1 e) @% S0 v
        These returned values are combined to produce the final result., l. U5 s& s, C! a2 |$ b. d

    - Y, G  g2 D7 X+ KFor factorial(5):
    ) c9 N! s, y1 [4 T2 a& N- S* u" Y1 O9 l6 W8 T7 q( L
    1 D- n) b: e; r- t* u% s
    factorial(5) = 5 * factorial(4)
    2 S% H+ n4 s; C, |) H2 a' K+ |2 d5 ifactorial(4) = 4 * factorial(3)4 ~8 v( d4 Z/ n3 C$ R# W4 x
    factorial(3) = 3 * factorial(2)
    / W  M: i( R+ L7 u$ Efactorial(2) = 2 * factorial(1)
    4 X4 b! o% X1 W: L- Rfactorial(1) = 1 * factorial(0)
    ' ?9 c8 ^7 }6 g8 Rfactorial(0) = 1  # Base case% p1 p6 F. I' _: e
    0 w) Y) @$ ]. u; d) s! K0 l
    Then, the results are combined:. B# ^5 l8 v7 l1 I7 o; W

    . f% k5 N) _5 s8 B  s0 N# ?
    % g" t+ G5 s* Lfactorial(1) = 1 * 1 = 1
    9 D1 v/ c) A% Wfactorial(2) = 2 * 1 = 2
    7 e& H2 r. H' s; r& nfactorial(3) = 3 * 2 = 6' O* Z, R! [1 w# l7 D. E
    factorial(4) = 4 * 6 = 24
    $ a5 D- P/ @5 i( c8 n- @; J6 W* d3 Jfactorial(5) = 5 * 24 = 120
    ' P* x2 o9 F* v! [
    ; C% R) t. a( S% i9 v, jAdvantages of Recursion
    - F& Z; k4 I/ k3 }/ V! t6 v9 Z
    0 N0 m* Z8 p) r: T) |% E. C. t    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).
    9 S% z" c0 v% o/ m2 j8 n5 `" P# h7 O/ P' b5 k# F% S' I/ O
        Readability: Recursive code can be more readable and concise compared to iterative solutions.6 H& }/ A0 n% N8 f

    - y& Y" f9 H( y# C7 nDisadvantages of Recursion
      o% M+ m5 ^6 |& G/ r
    4 z* ?$ P* s" s3 ~: j    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 G* _2 |( }0 a7 V
    * ^3 ^7 B! a$ @9 Q    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    % p4 P  z& Q& P
    ' z" q, K1 H. f$ M3 u# IWhen to Use Recursion
    9 B5 [2 \3 j8 D% k2 Y% _
      w0 k8 K2 W6 y$ y2 B3 B  G    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).0 M6 X) t: A: Y( U! p4 U! ~4 U6 J4 M6 {

    ' Z! X% O; F4 j! H  c5 d" F0 K5 Y    Problems with a clear base case and recursive case.
    6 F5 ], ]8 j+ r! I; t/ B) U$ J: q& @7 \  @
    Example: Fibonacci Sequence
    9 G1 i$ K0 H: e1 S6 e  S
    $ X' Z. O9 f4 O' jThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:1 e0 V; V' e  w" c
    $ f0 K1 ^. a+ w' U
        Base case: fib(0) = 0, fib(1) = 1
    * m, s. D' C; w4 e2 h. i/ [9 E0 M4 o
        Recursive case: fib(n) = fib(n-1) + fib(n-2)5 Y* Z3 E0 d4 P  ]

    & R3 I6 ?& `# h' \6 T- ypython6 L  }$ Z; }( y* f1 R

    6 P8 W- M, z1 D0 r; D, @! a+ p. O3 f" T8 ]+ t* B8 w7 n
    def fibonacci(n):
    3 x( ~" x# s1 f( D0 d9 D; o/ b    # Base cases* Z4 J0 p6 _; F: e
        if n == 0:9 V2 a0 q& }' r9 t0 o
            return 0- C9 ]7 G- {/ z
        elif n == 1:
    1 [0 }8 w1 Q; f* b4 ?3 w; u        return 11 H0 I' W$ b5 O% b0 v6 Y
        # Recursive case
    + m6 ~1 l3 j5 c' a7 @7 z8 H    else:# Q7 {) P5 j0 R( H/ }7 v
            return fibonacci(n - 1) + fibonacci(n - 2)5 J8 f9 a/ a; z" _

    $ V0 k* Z; n8 V+ v) x. |; H# Example usage
    " p3 E: F! w* H6 ]! M4 C+ f. Vprint(fibonacci(6))  # Output: 8
    " m  ]! ~! N0 U: v7 `5 }7 d
    7 `! ~# D4 e& Z0 B% a/ NTail Recursion
    5 {- J7 P. w/ t- j. f, N' A0 i. o6 m& B+ @4 z
    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).
    ( g9 A; l( F. p! h9 y' ]6 B
    % T3 t, N0 G- E0 d, O% ~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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