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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
    $ \, p4 A% r- F# X# I4 j$ U
    - B& ^7 ?7 h/ W% X, [* B7 Z) \解释的不错
    1 E. a) c+ }$ {2 ~3 y2 V. Y0 Z6 M
    $ v+ O( x0 k7 L# H/ w+ ?4 f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。+ x( g4 q6 w+ q, d
    % A3 g1 [& t3 u0 t
    关键要素: F4 p, c& G, _2 P5 r
    1. **基线条件(Base Case)**
    ! o- g8 T1 ]5 {7 n! d3 {; U7 X2 `   - 递归终止的条件,防止无限循环9 U  G$ G- I6 `7 E
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    5 e; z( G2 c: L1 t" @* V
    8 v- L" L& U# b! U2. **递归条件(Recursive Case)**. S- x% g6 |6 b/ n6 N' v/ ]: H
       - 将原问题分解为更小的子问题
    9 |  g3 b; H. b6 R5 u$ ]   - 例如:n! = n × (n-1)!! z  c" o5 n) K- N+ l

    " t- A% ~* }- t: f3 p- i 经典示例:计算阶乘, f( f& u0 Q1 V2 b7 G( p5 [3 g
    python# v6 l. @" l) N# Z. T9 O
    def factorial(n):( t: C3 c" b' X- v# A
        if n == 0:        # 基线条件# M  A0 U  l5 Q! M5 \6 X' p
            return 14 }; m) t. `  W# V  b" z1 G/ x4 {0 r
        else:             # 递归条件( w  E7 c6 n1 S
            return n * factorial(n-1)0 Z, G5 P) g  {
    执行过程(以计算 3! 为例):* T; W# a% q  K1 C- k% O
    factorial(3)# Y! K! W, U# Q7 ]& M9 Q
    3 * factorial(2)
    & H8 i' c+ a# r- U; E3 * (2 * factorial(1))8 \$ L& l$ Z$ h  G1 K
    3 * (2 * (1 * factorial(0)))
    5 B, K1 A) W& \  q: m% r: l# B0 r3 * (2 * (1 * 1)) = 64 I  q/ s/ O) U+ S! Q

    ' I- h% F! ^. q( d; m; c/ \- d6 [ 递归思维要点4 m" \2 \; q3 z5 ]5 K! T/ f
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    - D, k) E. _, e2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    + ]" W! y  b, W; m3 d3. **递推过程**:不断向下分解问题(递)
    $ K* j6 G/ S/ d' l6 }: f" p$ g; p! g4. **回溯过程**:组合子问题结果返回(归)
    0 Y- @0 v; {0 t  _( C( t% C* F; `" t# B( ^' q
    注意事项
    6 D6 L+ ]& x8 T  N% Q' Z必须要有终止条件
    ' A$ ~& L" D. C* I; q% H3 X递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    , ]- k" h' y9 e6 O& g( k某些问题用递归更直观(如树遍历),但效率可能不如迭代; R+ f" g8 G: w8 j) `
    尾递归优化可以提升效率(但Python不支持)0 h$ s' b: b8 ]; `0 u

    ' Z( k0 n! y* v4 N( C 递归 vs 迭代
    . r" U8 i! L6 `# p& _: w! P|          | 递归                          | 迭代               |
    & ?; b* ^+ a4 d4 K* }: {6 i|----------|-----------------------------|------------------|
    / ?5 }# |5 a  n3 o| 实现方式    | 函数自调用                        | 循环结构            |1 ^5 `3 V$ x$ o5 I1 n& Z. A! G9 W
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |* D6 a4 ~, `- }7 c& B# ^4 |  y/ o4 i
    | 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         |
    # M$ k% V8 _& R6 R' d, \| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    7 E! E7 Q% B. }- U0 e  Z) y! k# l  f
    经典递归应用场景
    % ]0 H; `% S" ^1 i( H1. 文件系统遍历(目录树结构)
    5 @& X  U: k/ d- P+ ]! M8 p' v2. 快速排序/归并排序算法
    7 w% C+ M4 i+ \" d6 _3. 汉诺塔问题9 v9 u0 l" ~; B
    4. 二叉树遍历(前序/中序/后序)) R" v5 E1 ^4 S- h5 w, N& m: k) T
    5. 生成所有可能的组合(回溯算法)8 ~; @8 c) c+ y7 U+ F! W
    0 M7 Q5 E& r: l( Z) |& r
    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,
    6 H# K; M# a' j* v我推理机的核心算法应该是二叉树遍历的变种。
    + D" W6 b3 Q; e1 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:
    & k. T6 h# k; [6 [; uKey Idea of Recursion# ^' o7 S3 L/ C8 c/ R7 O/ r

    . T% V7 D  \: nA recursive function solves a problem by:
    & \0 Y1 w8 p. z; z* ^  J9 C. C$ R& G4 l$ e  ]* w0 O% \9 Q
        Breaking the problem into smaller instances of the same problem.
    ! c3 g$ K; U6 _: K  J, s8 \
    $ D/ U( d; G, G  M" r+ Y7 j6 n    Solving the smallest instance directly (base case).
    1 k- j" L  O* x- E! V& j8 e
    & H! r. r2 h' V' G3 x    Combining the results of smaller instances to solve the larger problem.$ h( S# u# F5 E9 O
    $ I, c; y0 E/ a* X3 \
    Components of a Recursive Function9 S4 T. b/ {  j( I
    2 O5 [/ ^: T- u- R5 z( k+ j
        Base Case:
    " Y" d4 U; H( G2 p
    # x" W, e7 o0 D  z        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    4 c2 m. r: n2 {$ `
    6 U1 c7 \1 m0 _& N) }# Z0 z        It acts as the stopping condition to prevent infinite recursion.
    5 Y/ f  k# f0 y1 O+ K6 A% Z9 C( ~, D
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.% p8 [1 ^. f$ y, l3 c. Y" c
    . S% k) Y" X0 P+ P" P+ {+ ^
        Recursive Case:
    0 ]; z  Q/ `. j) _- l" @9 T
    ; ^& @- c& W7 w$ [1 x        This is where the function calls itself with a smaller or simpler version of the problem./ W8 P% I  e/ M# B8 T
    $ W" T% [1 z1 B/ I
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).* B7 j9 [" |$ {/ k0 n) A5 u4 x! W
    2 g8 C) x& W7 ^+ N. D+ u9 ~
    Example: Factorial Calculation
    9 ?  x2 o7 @7 M2 h4 `
    " g4 H2 m: t$ Y  u( x, _# UThe 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( Z9 F- a" _0 X5 Y( L, L
    % V% L# f( x) e6 @; R4 ?
        Base case: 0! = 1
    # A  M  @, ~. B  E6 M. v$ u! N/ K' n" |& |8 Q
        Recursive case: n! = n * (n-1)!
    * u; y7 c$ s" o+ e* I9 t, R: L
    ' |. E9 ?( K& z( ?+ \. y, k$ Q; `4 qHere’s how it looks in code (Python):6 S% T4 r: m1 W* I: J
    python3 D, ?$ I9 Z$ k  [! e! k. O' ]9 A, b

    ' v+ ?# F3 Z# K7 J5 `$ ^: D7 r( L- s, H- ?% v' L* `
    def factorial(n):
    8 e2 N- N# z) X+ |' {9 c% V    # Base case
    2 t9 D/ j0 ^7 X    if n == 0:
    * d& z9 z: Q! z* a5 V; b        return 1
    $ u% l5 s3 O2 x% K; P7 m+ T    # Recursive case+ \: p& j+ `  R/ O8 y
        else:
      z( M( Y  A: `        return n * factorial(n - 1)) E* R6 B- n# ~6 }

    " S1 |3 u3 N/ o' Q# Example usage
    8 g$ M0 I* e& h* @6 W! w6 o4 ]+ rprint(factorial(5))  # Output: 120
    5 }" I. [9 G# F! o- T  P0 X7 h7 h" y6 V' @- ~0 b$ w
    How Recursion Works
    + U4 c, J$ x7 |* K8 b! H0 H/ Q! Y: A9 Z4 }- ^$ E( d! f
        The function keeps calling itself with smaller inputs until it reaches the base case.
    ) J7 F) {& n) T2 ?
    1 ~; o/ E, G) Z# M# k    Once the base case is reached, the function starts returning values back up the call stack.
    " }6 ]6 E1 y0 ?% w2 [& V) V" |! _6 j
    + ?0 a2 P8 s4 S' v% A& Q8 L/ G3 f    These returned values are combined to produce the final result.* J! b- y, V9 w: c% m

    - z- _6 O4 x4 uFor factorial(5):
    % H5 S  g- Z! D' h; D8 _
    3 m! H$ h, `6 l1 G4 T
    0 r8 V  c  D* n$ r1 g" ?factorial(5) = 5 * factorial(4)
      `, |9 d6 g6 h0 O  H* z( vfactorial(4) = 4 * factorial(3)
    6 U* S/ z6 l5 Kfactorial(3) = 3 * factorial(2)& L& X( M4 }2 _* k
    factorial(2) = 2 * factorial(1)+ B7 V0 i& n6 i) u9 G% f, V% M
    factorial(1) = 1 * factorial(0)
    0 u+ J7 _: l' g+ k- ~factorial(0) = 1  # Base case
    6 h6 I& \5 B+ Q- r! }1 ?
    ! D  S0 n$ T( t. wThen, the results are combined:2 N- _: w  z+ _% w- B" o8 {2 }3 }
    % |, Z7 t  ]2 w6 h! X

    ' q5 C6 r* {( o0 A/ mfactorial(1) = 1 * 1 = 1
    3 H" [  }; J2 {6 ^factorial(2) = 2 * 1 = 2! n. s. F% b' m+ t; u7 D& N1 ]
    factorial(3) = 3 * 2 = 6% p. F1 b: Q9 P
    factorial(4) = 4 * 6 = 244 {( ~5 }" }: `. H0 I" z  m8 E9 |
    factorial(5) = 5 * 24 = 120: o' k0 `  w8 x6 s1 b
    2 s0 O+ _2 U+ O2 r, @0 `, `
    Advantages of Recursion
    / u# V- I/ M2 K1 t6 k  ]1 ]0 d
    & Y, O# d% p' ^    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).
    2 h/ t! o+ Y" n/ R& j1 H5 r: _9 p& }" l
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    % `2 y5 U6 @! h! E
    ; ^# H% ]! d) a; }Disadvantages of Recursion6 m! ^* w; M+ R! e& p
    # }  F+ Q4 V5 A+ V
        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.3 W/ r! K; t/ Y2 E, R7 X

    9 \- M/ H4 m( T) S) R! `. J    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).: p( g6 j+ _; H

    1 n! e( S' Q' [When to Use Recursion
    ' Q& g5 `5 F/ e, Y: H  ~  A' j, f3 B/ q
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ; @& W$ Y# V1 [4 \6 @  h9 m  r2 m7 b9 Z6 o% u
        Problems with a clear base case and recursive case.
    * _7 O* h3 S) t; h5 f
    9 W) N, s! a' b7 [5 tExample: Fibonacci Sequence
    ( n  O. O' }( k3 m2 \; y
    2 a$ V) y) v9 ]: b9 TThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ; p, v( h; h" n. W# t* }6 g6 I8 g# ?
    % g1 {! C( ]. x    Base case: fib(0) = 0, fib(1) = 1: b# q5 @6 d+ G% x* I
    2 u& }7 f/ w& M1 h& v4 b' H
        Recursive case: fib(n) = fib(n-1) + fib(n-2)! H' v% t! C( K

    8 P, P" i2 [5 l, ~% v8 a% {, d1 U: ]# Spython
    / _  `0 [. R7 A. b4 y3 }) i( G
    8 m6 S1 v1 I% R! `; w/ p! i: b! `  l  r
    def fibonacci(n):
    ! W; t/ l$ a6 }0 s    # Base cases
    / a1 W  D) I: x1 N: R& q    if n == 0:
    4 J7 e3 ]% n3 L! n2 F# V1 p+ r$ I- T        return 0
    : l# u2 K+ S% [  \9 r+ G    elif n == 1:
    9 W. D" l% l5 y        return 1( u0 y/ L5 A) r/ K  k8 t5 c8 c' E
        # Recursive case0 z* T; F7 @. n1 c4 p
        else:* s( {) c9 f5 E  m! L
            return fibonacci(n - 1) + fibonacci(n - 2)
    & Q1 @  a0 c2 y. `
    1 }' N6 N8 h/ L# Example usage
    4 R5 |, C) g+ dprint(fibonacci(6))  # Output: 84 H0 o% Z& h$ |# k6 ~

    % m0 G1 u$ i: s' |1 BTail Recursion  ^2 Q( h; T( U3 P
    3 Z* S: H' a3 a; x) E
    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).
    0 H5 k( d, H( m. u7 s7 C2 u; e8 b4 u. D5 C
    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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