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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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    2025-9-8 05:08
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    楼主
     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 0 }- T4 @* n+ `$ A0 Q7 Q

    ' a; \% z4 `8 w2 j解释的不错/ Z* b' y3 I; m, x5 O
    9 U$ k( F) i7 N- i; 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。2 `% i9 j. Z' Q( Z, z
    * o  q/ y& n3 r, Y
    关键要素
    $ ?' \7 L5 V8 e( S( I' L; d# l1. **基线条件(Base Case)**
    2 i0 a6 O7 H+ e/ a   - 递归终止的条件,防止无限循环% {, T; {- C5 M4 x& T' t+ ?8 k6 v& z
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1; r) q) [% p+ t6 z: O- H" C

    & j2 `3 I1 B5 F. W' ~- W( V2. **递归条件(Recursive Case)**
    * p; s7 v( i8 _7 z   - 将原问题分解为更小的子问题& A' c% E' L: I" ^  a, b
       - 例如:n! = n × (n-1)!' k$ i; s0 o& b% r. a; f

    ( \5 y+ B. h3 f. ^6 u  m 经典示例:计算阶乘
    0 m7 h, Y( _8 F) h8 s5 d1 {. Xpython  J7 g; c8 Q2 N
    def factorial(n):
    ' @2 @* F  C- o% N7 }; J7 }* i    if n == 0:        # 基线条件
    6 y: f8 t+ z. P- W2 p, g        return 1
    ; j4 x3 D0 D: @8 i" K    else:             # 递归条件6 B& G" Y% f) y0 L2 k+ F0 t) r: h
            return n * factorial(n-1)+ q# O, O+ i  H. N) @, d
    执行过程(以计算 3! 为例):' v) F" C) l9 V- s
    factorial(3)4 I- M% O7 U) ?) j; r/ E8 D
    3 * factorial(2)
    6 n. r; p- P: A( T5 Y3 * (2 * factorial(1))0 P8 O2 j0 V- G
    3 * (2 * (1 * factorial(0)))9 x# L/ O9 A; k- Y2 e$ L; D3 R, {
    3 * (2 * (1 * 1)) = 6# |5 E, ?) V* W1 @0 N" S
    ( z; {9 i8 P, _6 x5 z1 ]( ?+ q
    递归思维要点5 t# a9 n' i7 s# O6 [) ?9 @2 L
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    , `! j# r' h6 }4 p% h! Y2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    9 a/ t' F* y; \% G& s4 N3 p. {3. **递推过程**:不断向下分解问题(递); z5 Q+ S- @% d4 I$ [
    4. **回溯过程**:组合子问题结果返回(归)
    ! _! N- ?: e2 @' p9 ]  Y% H
    % S$ d, K- ~2 m0 p( q# c' V 注意事项, ?) `5 Z+ L2 s" m3 S& `! p
    必须要有终止条件
    / d8 p- l' ?% K) }( i' a! V递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    9 s+ w) g% Q' b9 G  u某些问题用递归更直观(如树遍历),但效率可能不如迭代
    # [  z. c8 N; k. k( X尾递归优化可以提升效率(但Python不支持)7 N7 n* b- R- u; a2 ~) i, K2 F
    ( S- {' g1 j( J( V* |6 c9 h
    递归 vs 迭代; n8 Y- H( x" ^2 R5 g7 R
    |          | 递归                          | 迭代               |9 {1 [' V# ^# X* ?
    |----------|-----------------------------|------------------|8 l, B0 ?& j" C
    | 实现方式    | 函数自调用                        | 循环结构            |
    0 s) p. k/ g2 i- u| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |8 i! j+ t7 G7 R0 p3 ~
    | 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         |5 y, o1 [' s4 R( b6 f5 D. y
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    / u, A: q. V' F5 N4 b0 ?# k: e' P& H7 ?3 j- j* a
    经典递归应用场景
    ( x+ ]: d. l, D( J" y0 B1. 文件系统遍历(目录树结构); ?0 G. F7 |! M8 z1 ~
    2. 快速排序/归并排序算法
    ! W9 z9 T8 c! }% B9 A/ Z3. 汉诺塔问题
    ( k3 f9 e* w  D& ~4. 二叉树遍历(前序/中序/后序)
    + W% Y# d$ i+ P5 v9 d5. 生成所有可能的组合(回溯算法)6 G0 V) h: t+ p$ Y, M

    ( i/ t' r+ s& C: J8 n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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  • TA的每日心情
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    前天 01:20
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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,
    7 z! n, u* R4 n/ f) n我推理机的核心算法应该是二叉树遍历的变种。9 p' a# [8 z, e
    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:
    / q& K+ m2 V; E0 W' c$ \( IKey Idea of Recursion
    6 m/ V3 \1 H2 ~1 I
    ( }# z! e' O. W: d" ]A recursive function solves a problem by:/ i8 b& A7 X: q4 @
    ! [% ?2 q3 l+ T" c+ Q' v- c( H
        Breaking the problem into smaller instances of the same problem.8 H# a  n7 H# i' N
    9 G; I* s& `9 J: }( t/ c: N1 R% p
        Solving the smallest instance directly (base case).8 d* P4 ]$ w) Q5 T3 t( @
    0 ~: j4 u7 G. Q& s- }7 l, S
        Combining the results of smaller instances to solve the larger problem.
    1 Y" G2 i6 h1 J9 e5 N9 I
    0 I" }: \, o  p  s' X7 f! U' r* {5 Z) XComponents of a Recursive Function4 g8 H; x! n1 g5 Z6 m+ N. m

    ' K( m% W4 q8 Z$ Y    Base Case:' W; l+ R: f: G  {5 {

    2 ^. ^. Z8 i1 X4 w: X# A        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.: c& {+ n+ o2 Z% Y0 e% S) k4 k
    9 [0 q5 B, b2 |! G+ s
            It acts as the stopping condition to prevent infinite recursion.- E  ^) f3 R3 O/ j) x# H
    3 J/ b% V$ K2 G0 r
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    6 V8 D7 Z1 j, l0 @: g3 j
    3 U) y: M0 l- F3 _2 y+ J    Recursive Case:
    + E* e4 h; _9 d7 U- ~" m4 ~- h  z" |; w4 M2 F
            This is where the function calls itself with a smaller or simpler version of the problem.
    ! g( N0 j4 k% i: h3 T
    / v! J$ u' F5 V4 E6 ?& N        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).+ A2 H7 s# l- O' [0 G( k" F
      v9 z0 a# z" L7 B. s9 ]
    Example: Factorial Calculation
    1 \  {5 q6 C3 v7 g! ]
    & V; U% F4 R5 h/ d2 n5 e" JThe 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:
    " o9 t7 d: L) Y2 }6 x" G* u0 S/ B: ~: j7 X. j1 Y! X
        Base case: 0! = 1
    ! n1 C0 \: e# S' n/ e$ \& M; w& G( r! i/ a1 O) X
        Recursive case: n! = n * (n-1)!
    9 F' m4 {$ l8 L. S* F, z; Y) `/ ?5 Q" B9 a9 q6 Y
    Here’s how it looks in code (Python):& Y+ Z$ A$ }9 _$ J. V6 U9 `
    python( M, o! D) X" V. ~
    2 o/ U) W2 T- E, f3 L" {
    4 {7 ?& R5 A/ a& ?$ W5 x" W: q
    def factorial(n):
    6 I9 |' v8 \9 V! z    # Base case- W1 _! h6 m6 g! F9 r5 Y( M2 V2 P
        if n == 0:8 V9 Y$ e4 D0 k; m5 f
            return 1
    , O2 f5 Y$ w# O4 a2 V, M$ \    # Recursive case. R8 j8 o$ y3 b  v* J' F4 r# o7 R$ o
        else:6 S! i- l9 \/ `/ T7 X) E+ _
            return n * factorial(n - 1), g4 g# ^/ s+ j% N% u* T; E2 v
    ) a$ E/ L( ~0 q) b9 `# L$ R
    # Example usage
    ) n7 S4 F2 v9 Z9 n1 `% |print(factorial(5))  # Output: 120  Q& Q: o& z- S  s6 d& ^: g8 W2 s
    " V  r3 m; I/ k* O" F
    How Recursion Works. D5 s& m/ a( _0 j# \
    1 c1 r3 e: }* C' T5 o
        The function keeps calling itself with smaller inputs until it reaches the base case.- l2 y5 g. `% n) e- u

    4 g+ s$ h) A% M  N    Once the base case is reached, the function starts returning values back up the call stack., Q- s1 b. a9 D) {: m. f
    ; o7 r9 G4 ~2 h1 N" }
        These returned values are combined to produce the final result.- j) j) i% \5 f* a" {! O/ s. @) u: r

    $ F4 K* {  y& U6 F4 GFor factorial(5):
    ) W5 H# v- x& u. `! ?+ }6 ^! o3 f+ _( Q0 m* d

    9 [% Z$ P/ @* w  F) a" ?factorial(5) = 5 * factorial(4)
    ' c! w) b# Z  Y( n1 w# ofactorial(4) = 4 * factorial(3)
    ' O2 V1 y9 ~: _$ O2 C1 Afactorial(3) = 3 * factorial(2)
    4 w& h- u' N0 b+ H" N% hfactorial(2) = 2 * factorial(1)
    ) u- p$ B: X: F4 F6 ^2 z4 Sfactorial(1) = 1 * factorial(0)
    - |: _5 e& ^, B% U2 Cfactorial(0) = 1  # Base case
    7 p) y: h4 C- j4 l1 b. [& ~% b2 ?2 i4 A
    Then, the results are combined:8 m( ]" g3 N& o
    - \: b6 B8 \7 h2 \" Q; p7 P4 e' N& J
    $ H3 L6 b( ]( J& h3 M, a" q! H# h
    factorial(1) = 1 * 1 = 1; V! c6 w1 a+ [
    factorial(2) = 2 * 1 = 2
    $ `7 ]/ k+ I/ z; S3 O' Efactorial(3) = 3 * 2 = 6- `8 W( X5 h8 p- E9 g
    factorial(4) = 4 * 6 = 24. g7 o% _8 o, K, }* s% t: e& f/ U
    factorial(5) = 5 * 24 = 120/ S  _9 @" p, W2 P) B# S
    : a/ M" {# s8 p2 A6 H% H% \4 u
    Advantages of Recursion
      e# V6 R/ D2 q- ?" F! x4 F6 L3 b% x7 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).& j+ Q4 Y; ]4 g1 h8 l" j- l* s

    ( Z/ @/ G5 ^5 L7 I    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    + e, a% E2 F! W2 ]; |1 y" R/ d& @, O* D6 p: n
    Disadvantages of Recursion
    ! L' w& Y4 V( V1 l
    / f! ?; `: L8 U/ y    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.; T* v8 N  T& g. h) j1 `9 y
    1 S1 [% L1 u6 P+ l6 [
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    $ t) k, e+ [$ M1 C9 W* S& _% Z
      H9 C) W/ P) L$ iWhen to Use Recursion2 e  F) k) j8 j! ]8 N, ?
    . {0 L; F. M; A, {1 y% j
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).3 F: J; L/ N1 j# j! T) ~# F6 W/ `
    $ P6 d' B  E2 \0 H! H6 G: `
        Problems with a clear base case and recursive case.# }4 p) q( ]8 K* g  w

    $ f+ s7 Z. R6 q' m: z- CExample: Fibonacci Sequence
    2 M1 U  c3 A; G% }0 W' F- k
    5 p+ {6 H8 S) fThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:( n; q9 F5 l9 q1 Z
    6 W  w3 X' h: O: c( h6 L- m
        Base case: fib(0) = 0, fib(1) = 1
    6 S! \7 x4 l+ h0 J& V( O( N' F
        Recursive case: fib(n) = fib(n-1) + fib(n-2)  m) x7 y4 D  M, F$ p3 r2 m
    + p/ i" G7 T- L: Y6 ?! W9 l
    python2 ^* i9 m- H4 U6 l6 b$ _( ]3 Z1 J

    4 r; T% w+ f) n9 I) Q- \/ }# I& R. B0 E3 Y
    def fibonacci(n):* ?% b' W3 Q' ~8 _
        # Base cases
    ( x, J! Q4 C* [; z3 i, P    if n == 0:0 S+ [. h% j( _* I
            return 0
    / g: R) Z# `$ D! V$ t( c/ o    elif n == 1:
    , b1 I- |: H, o4 H: _0 R0 x2 f        return 1  @0 z% s; d; {8 c+ A4 A% ]
        # Recursive case, z. Y0 c) [+ U; D3 [8 J( Q- R
        else:
    : g5 Z' l) M0 I- F9 L* A& p        return fibonacci(n - 1) + fibonacci(n - 2)0 x7 G' p* s! p( Y
    8 N7 s- b, N& k+ @
    # Example usage
    # G4 e. L0 Q; G( {, A: A0 rprint(fibonacci(6))  # Output: 8
    * Y0 Y; k% S# r  n
    6 y6 H  p# _; [Tail Recursion
    2 a' m3 c2 J- {
    & e5 x. y* U6 d* }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).
    6 P( o( P% h9 Y! p5 q- O! d- m9 o5 ?- h' g) y
    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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