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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
    ! J5 W' U- f& ^1 R. |$ O/ {7 Q& l( Y% v2 j8 S( K) y
    解释的不错5 \; e* q  H/ ^- g* D9 M  p7 e6 i" R
    0 C6 C7 [9 b) {  @4 T$ P1 _# 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。
    , B0 W, b) j% x1 e# n& O6 I- _& ~$ N6 D1 i" }
    关键要素
    9 H! Y% v9 B; j3 o# [0 H1. **基线条件(Base Case)**
    ' m- K; F' U( E) N5 t# u5 x   - 递归终止的条件,防止无限循环
    1 _7 @4 X* e2 v% W* E   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1' @3 w, z, ]  W  L& a4 f' i7 D5 `

    : M9 R& O/ a& o& V! X; D3 P) w2. **递归条件(Recursive Case)**0 ]- X: h! Y9 M; Z1 q/ C: R
       - 将原问题分解为更小的子问题# `) R# ?9 h+ p: [# _' ]& s: ?
       - 例如:n! = n × (n-1)!
    + P4 ?1 G$ O. J" x8 g% ?  O; |2 q" Q5 J" |, w
    经典示例:计算阶乘2 F+ E( K! T' \* }( q
    python
    & P6 Q8 [' |" v, g2 u; Adef factorial(n):
    3 n1 ~& C! K+ B+ p# Z+ d8 X6 H  J( i* K    if n == 0:        # 基线条件: N: S' P5 P0 d' a- u
            return 11 U# w6 g  y( B/ j/ m/ `
        else:             # 递归条件( N: q- n+ R% j" w) ~" Y5 _( R
            return n * factorial(n-1)
    1 J1 Y: z, ?. ?执行过程(以计算 3! 为例):$ s5 J8 r9 M5 |
    factorial(3)9 ], P. p4 a; V8 e6 @; ~* y8 ~' s
    3 * factorial(2)- |; ?7 R- x) b9 o$ E  ?
    3 * (2 * factorial(1))
    + p8 V7 \. `, n& Q% w! h3 * (2 * (1 * factorial(0)))3 ]4 Z% {' I  L( a
    3 * (2 * (1 * 1)) = 6
    ; O6 I# N# _9 Y3 R+ m8 t, _  d; ^8 L" n5 k. D# Y8 a" S
    递归思维要点
    % S1 i( S. j. b! T3 l6 @: C; ?' d1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ( C* a) @; w, \2 h3 O2. **栈结构**:每次调用都会创建新的栈帧(内存空间)1 \- K; Z( f+ z  n& ?# ]: N
    3. **递推过程**:不断向下分解问题(递)& L# D' h) u1 V% K  ]) ]
    4. **回溯过程**:组合子问题结果返回(归)# Z+ F' [* Q  ?. [- {8 Z6 u$ h

    . |$ |0 X; [& d( |' G 注意事项
    * z5 G; O& X5 S! A- x2 l8 F必须要有终止条件+ i" b0 x. @+ j. m- R
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    * W" m! B7 \8 a! ~9 a$ n某些问题用递归更直观(如树遍历),但效率可能不如迭代& p; u+ f+ B7 R& D6 k, _
    尾递归优化可以提升效率(但Python不支持)
    8 x2 v  i- d8 \# H  B/ z
    6 ?8 |0 {6 E7 w$ X9 n/ ?4 w" `. A 递归 vs 迭代
    ' j" s' l0 u0 u1 O2 _" s7 o|          | 递归                          | 迭代               |7 @; S9 S, b0 \
    |----------|-----------------------------|------------------|
    . Z0 }$ Y8 a# I1 z3 E. v| 实现方式    | 函数自调用                        | 循环结构            |
    4 m2 U, H4 Z& v6 U9 || 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |: i: }: {7 J8 V8 o0 y) \5 p
    | 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         |
    : o# J1 @7 R; q| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |# P, t; x1 S$ V  u/ j, k3 A
    + u/ e3 C+ G& ~5 V7 ?$ V
    经典递归应用场景
    + n$ ~1 b7 m0 F+ Z2 H" b% [2 H1. 文件系统遍历(目录树结构)$ n. ]2 L. ]  X& e3 j5 G1 T( P+ `
    2. 快速排序/归并排序算法
    6 ?  _" _4 i0 a* c: }$ Y$ C5 D; y3. 汉诺塔问题
    : {1 b* y/ T5 O8 A4. 二叉树遍历(前序/中序/后序)/ d2 b5 J- V; H$ x+ J, l6 i
    5. 生成所有可能的组合(回溯算法)* m4 l( v8 q1 H: }. K, p- K! `
    8 [  z0 D$ z# }; J5 n. w! p' s. ]
    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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    昨天 07:20
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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,; w# G* F& U& m( _. M6 [
    我推理机的核心算法应该是二叉树遍历的变种。6 @& v* W+ L4 J' J7 V
    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:9 ?% Z% L+ b0 C( @% I
    Key Idea of Recursion
    ; O% S$ A3 q- y* P+ ?# a) D2 U; n) E% z' m1 m& w
    A recursive function solves a problem by:
    + Y% [5 K& V7 k, W0 l5 f& Z6 {( `' k, w9 u, e) _- C: l
        Breaking the problem into smaller instances of the same problem.
    1 U2 r. u+ b- x
    * a. B/ o/ l/ b5 Z    Solving the smallest instance directly (base case).
    , @- }! C9 ~# D. c) I1 Y
    5 y+ I" D, R. T    Combining the results of smaller instances to solve the larger problem.
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    " n+ S2 }$ A9 Z3 ?. b3 y  Y" zComponents of a Recursive Function
    2 {* q8 o2 ?8 H( ~0 y: q) |, }/ C/ h# X7 V! G. R( I4 \! B4 l
        Base Case:4 j  c- S, m, E, a. W7 \' o
    9 W1 y' D/ H5 Z
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.9 H# ?  ^9 A! t, ]+ g

    1 H4 T2 Z4 ?& y* U7 n3 ^% l" i" \/ p        It acts as the stopping condition to prevent infinite recursion.
    0 j* W# F* b7 |1 K/ V, L0 w9 f, v9 v" Z4 J6 b
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    ; w" C) r- Q( F6 R1 f5 \/ ^    Recursive Case:
    - h- D/ s% g. W# j! i) {
    5 P3 k3 A  }: i$ K6 Q3 M. @        This is where the function calls itself with a smaller or simpler version of the problem.$ J3 W+ ?  u& c9 [) c
    ( T% q; E( k8 J4 v  y8 G
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    2 L5 i4 K/ M* m# R* [2 b# [7 Q  w, R5 X& H! d' w! p0 q) h
    Example: Factorial Calculation5 U9 _/ y# |" s, g- k6 ?

    4 k+ b$ B- l. t2 a1 B" 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:/ d/ V1 g, q* i; l6 K

    5 r: o$ I5 y% E+ \/ E; u$ U( f    Base case: 0! = 1
    2 h7 {1 L9 ]3 ?7 p1 f
    0 P2 V& d- b+ V* i# ~. n+ ^    Recursive case: n! = n * (n-1)!7 n9 c% c" p; U8 @, @% P
    6 s1 w* Z# q$ K+ l" w
    Here’s how it looks in code (Python):
    9 Z5 h; p5 H" E# M) o! W+ {+ kpython4 S6 v, A" i5 v* c& V+ R: U

    5 o+ N" {& d% _4 x; M8 [0 q7 v9 T$ N, L) _
    def factorial(n):
    2 H  z' x( m5 X3 O& u    # Base case( t3 J7 F5 v8 q) \  B# ?& s
        if n == 0:
    ) B! d/ W8 `  j' @        return 16 i: x6 u2 n& @! u! h
        # Recursive case
    + T+ `2 P- }8 m- W    else:
    6 I- }1 S# Q: T- X9 F        return n * factorial(n - 1)" j& B( Z% U0 A- R) }

    % N3 V/ y7 j/ o0 c5 [+ t% T# Example usage
    5 Z: O; a6 S$ w7 w: B- Sprint(factorial(5))  # Output: 120
    / \5 H" L; L0 I# V4 t6 \; g
    & ~2 J1 b) v3 `9 BHow Recursion Works
    . I0 [3 P- p' b: w* }( `5 `$ v/ \1 Z: t
        The function keeps calling itself with smaller inputs until it reaches the base case.
    8 U6 Y. G5 P# W+ Z0 x& T  @  c7 V) B) K! Q2 w
        Once the base case is reached, the function starts returning values back up the call stack.& W  V' ~  f# g; z8 j
    5 c: h2 K  E: \8 ~2 e2 \) q) w
        These returned values are combined to produce the final result.; h8 J& X. g) z& H6 a/ O" t5 `

    1 q: l9 ~/ n5 j1 K% v9 y$ YFor factorial(5):
    , @, ^* o# a( f5 y4 ^& d% f# \0 e) \4 C' i  u5 L

    . j1 U4 l& _6 ~factorial(5) = 5 * factorial(4)
    , U8 S9 O. e* Rfactorial(4) = 4 * factorial(3)
    4 C* s9 j8 q6 Nfactorial(3) = 3 * factorial(2)
    0 f6 O+ o" P' J+ Z7 C1 R0 A& M6 {; hfactorial(2) = 2 * factorial(1)
    2 D/ i! m& U2 C) {" ]/ ^. Jfactorial(1) = 1 * factorial(0)# ~1 m0 {1 T+ I! |
    factorial(0) = 1  # Base case
    + H9 i( F5 f0 Q: d) N) B4 Y/ A8 [* S1 A" Y
    Then, the results are combined:
    9 q' F8 k( C+ j; @$ {2 s& G  T3 Z8 C9 ?

    1 z* m1 Q: T% b, f$ G" D( efactorial(1) = 1 * 1 = 1
    ( P3 u, ?6 ~% N2 y  @3 ^% _factorial(2) = 2 * 1 = 2
    - ^% k- w8 n1 }  o. Ifactorial(3) = 3 * 2 = 6
    ! m7 J( N7 V5 ]+ `factorial(4) = 4 * 6 = 242 L+ D% F5 S7 n# L( d' h
    factorial(5) = 5 * 24 = 120, j, z' t; ^3 B4 y
    4 S0 o  c8 T& D9 k$ Z3 }
    Advantages of Recursion
    , O9 |& f3 ]- u9 P; C- O  k( T, k5 J+ Q: Z/ h: @
        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).
    6 _- j6 E( V5 r; e
    1 c6 l# V3 Z3 {6 `3 R+ b    Readability: Recursive code can be more readable and concise compared to iterative solutions.8 y$ u% L2 r0 ^8 v0 t

    / F# P- x8 {# K  w$ ^Disadvantages of Recursion
    $ Q5 f2 \' V: G( n3 N6 L5 ^3 }  w
    + t% N! u, Z) k9 X6 K    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.5 i' y' e1 G9 n

    & p8 @8 u3 W! S0 P" r/ G; X    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    1 G( k! T3 P- w: s# g6 G: A  s+ Y$ ?
    When to Use Recursion8 |4 H, U9 M7 g3 {* P+ K1 U, b
      m0 C3 E  j) u- i0 d
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).8 v( R5 I8 z+ r" A: N- A  V

    4 @% [7 q1 U+ W) ~, I1 {$ H  E    Problems with a clear base case and recursive case.
    / d/ Y8 M2 F3 w5 v, F& N" }9 \9 {4 U$ b; X# ]: m
    Example: Fibonacci Sequence
    7 x, M2 E! Q' u. K" R
    : d" r1 M. K. j3 P( c9 \The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    * X& t6 }, l! e% i
    % H- b0 U6 j* E5 m! K9 S    Base case: fib(0) = 0, fib(1) = 1
      f7 L- m, c5 K! l- I" ^9 L; d9 k/ Q: W9 W2 I2 G, @
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    + G  S: {/ O! w& R. C
    4 H2 J. Q# C9 ?) epython4 c7 U4 ^) ~6 G2 m
    * a9 i* Z  U) G

    ; `, f) D( N1 n9 }. K! d4 udef fibonacci(n):
    , y& ?; s- J9 k1 d    # Base cases2 B/ U# x" x5 u/ y3 e. t; S& Y: M2 Y; L
        if n == 0:
    % u  g  T0 W, N* v1 R        return 02 |$ F7 f2 t, U% y6 S8 P
        elif n == 1:
    ; _& G$ J# n4 \' I1 W0 t5 V, w        return 1
    + I! q/ G1 H& h: Y! B+ x: C) K    # Recursive case
    8 a' V; P$ E* K  A# s    else:
    2 ~2 w7 W3 P& W6 u- ]$ W        return fibonacci(n - 1) + fibonacci(n - 2)- R! {" S$ g% f; {7 W

    : X1 A4 g3 v7 `; R: r# Example usage) V* F" H9 z7 a; L9 i) }6 m3 Y# k
    print(fibonacci(6))  # Output: 80 t" Q$ \, H1 Q. l- {: O8 \

    ) U0 t( U' h$ u# PTail Recursion% @# k  j: \# Z* S! R; g1 @
    8 A, O# Q# I) e9 L* ]8 _
    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).
    ! ^- U( c1 H) D' y9 v- i, x3 s* `' Z* [% t& |6 U# Z0 k# m5 P
    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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