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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 " ~: O: _+ H0 E, Z  O
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    解释的不错
    ! v; p" A' D8 o8 V8 Q/ Q' g. i; V1 X
    / b, q" C- o; {- W7 c' X( v7 m  q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。
    1 d9 Q' E4 w$ j2 N: B6 O0 k: Q5 b. s* s% n! v0 H) h; ?* _5 X
    关键要素) d; D( W3 f- @/ X  }
    1. **基线条件(Base Case)**2 v- g8 Q. {' ^$ _- t! c
       - 递归终止的条件,防止无限循环) x+ Z. `2 a' H- L# a
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1) x: m$ L% ^% m. {

    ' d7 x% `" c6 ~. e2. **递归条件(Recursive Case)**
    & c- R' y! y, _0 D   - 将原问题分解为更小的子问题
    2 F- ?3 G. H, [+ Z   - 例如:n! = n × (n-1)!. ?/ e1 S! \# I8 ~7 Q
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    经典示例:计算阶乘
    " L% U; @% F  I% c: E8 H. ~python
    ' O7 K0 k1 m' \/ ?- bdef factorial(n):
    ( f8 h* V( L  E* m$ ^    if n == 0:        # 基线条件
    1 U. y% j% n3 \% ^        return 1* c% m2 H, B: E
        else:             # 递归条件4 t2 S6 _. b5 [9 W# f, Q9 ]. \
            return n * factorial(n-1)
    6 W" t1 L; q; L# j0 ~- ~执行过程(以计算 3! 为例):
    $ Z/ Y: H- p& A6 @7 Y3 V  q/ Y! Ofactorial(3)" B+ C& Y( B8 f& Q5 `3 H9 w
    3 * factorial(2)0 }8 v: I( ]4 \1 P% _: J
    3 * (2 * factorial(1)). @% [+ f- A) D$ c+ a
    3 * (2 * (1 * factorial(0)))
    + o2 L$ M7 {) U7 c  ?8 f- W+ ?3 * (2 * (1 * 1)) = 6
    ! }/ U# E8 S/ Y3 r7 J2 [! ]& a- l9 Z; h! [8 g( y
    递归思维要点+ Z4 E' @/ u: K7 b5 e$ V
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑! [6 C3 O5 C7 {8 S* q" @
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)8 P6 Y* v% V/ C! O/ A0 d, I: G
    3. **递推过程**:不断向下分解问题(递)
    & Y0 N5 v3 G' _- q4. **回溯过程**:组合子问题结果返回(归)" G. [& \8 [9 n  }: U& e
    2 Q3 e( s! G4 d* q2 ]- [" @  Y6 I
    注意事项
    ' l2 g; V; w1 l- }. l* v* y必须要有终止条件7 y+ ?+ F9 X/ w$ R3 d
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)8 ?3 J: K) j7 \  E
    某些问题用递归更直观(如树遍历),但效率可能不如迭代9 h- z* }6 H: n: s
    尾递归优化可以提升效率(但Python不支持)
    7 c4 f) ~- U6 |/ R+ b. S3 X5 F0 g  ]* z' z& b
    递归 vs 迭代% J" R) s3 r; F2 h5 ~! H
    |          | 递归                          | 迭代               |1 N$ U5 T2 E$ ^$ T- M
    |----------|-----------------------------|------------------|
    * n3 \2 _) F3 Z& Z| 实现方式    | 函数自调用                        | 循环结构            |
    7 T. G. g4 a. l0 c6 j8 J/ Q| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    5 o# k; ~; m/ Z5 f5 T/ j| 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         |
    & z( k$ w5 z) G| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    5 b0 i, s4 ~1 c% r5 T- I+ X& ^& d8 m9 b" ^  y8 {8 p
    经典递归应用场景
    ) Q2 J8 {: t( |  F9 I  v1. 文件系统遍历(目录树结构)
    ; v5 a8 J* ~! b4 j" U- g% z2. 快速排序/归并排序算法8 g5 n) c8 t' U
    3. 汉诺塔问题
    * x3 e! _. I+ d$ @8 B+ }! T4. 二叉树遍历(前序/中序/后序)! O$ `" h  O  Q3 _6 V: i4 Q3 x5 t/ y
    5. 生成所有可能的组合(回溯算法)- W, B: N2 r! r+ q$ C) W, [
    % z4 b; e4 g3 u7 P
    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,
    - }- i! M1 m1 W, V, X& J我推理机的核心算法应该是二叉树遍历的变种。
    # S2 E$ P3 P2 O5 [9 e; C9 m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:; I; ^; V% R+ W4 l: O
    Key Idea of Recursion9 a" r2 T. g9 ], a% ^2 ?
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    A recursive function solves a problem by:
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    ! [! \- N* G- |& n    Breaking the problem into smaller instances of the same problem.
    ; U& @2 I) z6 h  f% P) D- [5 ]. q, }) a& a6 g4 D/ b* ~
        Solving the smallest instance directly (base case).( P, {" Q, H& [9 m

    $ e1 V8 k' N; z! g9 ^' _    Combining the results of smaller instances to solve the larger problem.4 t  _% y  h# y. c: Q

    3 Y2 K5 G8 i+ w: I+ uComponents of a Recursive Function
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        Base Case:' K& X3 W, v8 z# w7 t, e# X
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.8 w6 E9 O( K: P6 u0 Y7 h
    & n" ~! m* \$ c8 @3 a' q) Z" o
            It acts as the stopping condition to prevent infinite recursion.
    7 G4 U; `" g; q+ D2 p0 E
    ' M1 {( H! `9 E* J        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.5 _4 ^3 Z( f2 B
    3 O% A2 w  c. N& {$ n
        Recursive Case:
    4 t4 S! b* Y5 C) d
    2 a! p: x7 v. Q$ S        This is where the function calls itself with a smaller or simpler version of the problem.
    % d( g1 V* J. T
    8 Y; \' }, K- e6 |5 [1 U        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).2 o3 X, L' {1 S- G' p* U
    & s2 [7 z5 k+ B& |& R
    Example: Factorial Calculation
    7 ?  D$ N% V1 D8 Q1 U1 A
    ! z' ~5 c8 L2 ~) c' U7 nThe 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:/ L: W# j: M7 U+ ]# a' S! q5 q2 [
    : B- d' \5 h7 s  ~9 `
        Base case: 0! = 1
    $ H$ U: p6 I. X5 t  R
    % }' t0 M1 N% y( @) V    Recursive case: n! = n * (n-1)!
    . ~2 N# ~  _4 z0 n1 Q$ S
    1 U$ B6 x; K, g& h: H/ y: vHere’s how it looks in code (Python):
    4 I% l4 h; c1 a; Lpython
    ; d- [- Z( A5 e
    + e, T6 k! ]4 }
    $ T6 a8 ~$ u9 ?def factorial(n):% M7 B. Q9 c1 a, ?2 G
        # Base case% }" u& p- N4 j% r4 h4 B' J
        if n == 0:6 q9 _: j1 J# e; Q
            return 1
    9 o* y  A- ^% `3 [# C    # Recursive case4 E- f. P% E4 N% O, I( c" `
        else:- f* j. y+ M& `: Q" y7 R
            return n * factorial(n - 1)% ]* N& ^& |3 t/ \

    ( ~" c* T% o+ H8 ^# Example usage% ~7 G' h! x8 g! B1 C1 }
    print(factorial(5))  # Output: 120
    ! z0 ^8 i! ~  m) `! e9 v
    . ~! S* d1 d. b8 J1 F. LHow Recursion Works
    9 W+ @6 k5 ~' d: f/ R7 s! p! g7 J$ t- R. [; U! k  w! u  r
        The function keeps calling itself with smaller inputs until it reaches the base case.- c& h5 g* j- n. s" D
      S/ m* e! u7 A' T
        Once the base case is reached, the function starts returning values back up the call stack.
    , _3 t% R9 \3 q* B: a1 @4 s6 T6 L
        These returned values are combined to produce the final result.
    2 a! I' E7 u0 h) a& o2 B# w+ T) G. B% Z9 P* a4 Q. z; h
    For factorial(5):
    / ~& G, D0 }6 o* s2 e# B
    0 f6 [1 x) y: Z+ D2 _! `! i" F
    ) b2 S1 R% L# a8 c2 {factorial(5) = 5 * factorial(4)
    2 {: O6 _! g$ v0 E% Y! cfactorial(4) = 4 * factorial(3)' U8 q# z& f; y( A6 f, _! D8 y
    factorial(3) = 3 * factorial(2)( ?, \) T2 e" O! u( @" [! @
    factorial(2) = 2 * factorial(1)! ~$ u3 r* b4 x$ q
    factorial(1) = 1 * factorial(0)
    9 S' j( U& e0 u0 ffactorial(0) = 1  # Base case# @% J; _" u4 x
    ; b  b$ E# p" t, |( F- V* y9 g" l1 n
    Then, the results are combined:
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    ( b5 i2 c, f& [. L% k7 u& F( s, d4 P$ {7 C
    factorial(1) = 1 * 1 = 1; Q% L; x3 m+ l+ P
    factorial(2) = 2 * 1 = 22 N. j1 i6 W$ G
    factorial(3) = 3 * 2 = 6
    / W7 E5 [& J/ P) R8 p$ l6 sfactorial(4) = 4 * 6 = 24
    - N3 N5 P* u7 J: [+ o5 Rfactorial(5) = 5 * 24 = 120  c3 }1 c2 f/ ~$ W

    " Y% E0 z7 X6 BAdvantages of Recursion; ~1 M4 M- i5 k
    4 y2 F4 b3 n& Q  d
        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).
    8 {$ @2 d" J% g
    - G3 R. g# s: p& l2 V5 z. X: t    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    : N) @, V4 [7 M- S, d; a9 |* w, x, D9 }3 U! ~! t
    Disadvantages of Recursion
    5 W" l$ g6 l5 r
    ; O  U; _/ O" _1 m0 n6 w  w/ l5 g* O    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.; |9 ^- V# w; B
    " w, }8 n! T7 l- W
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).4 v  B1 n8 [9 V

    2 ^+ L; @% w. P$ i' ?When to Use Recursion
    ( t3 G6 K! v2 j5 D7 F3 f% v; _7 U  x, t3 i/ c$ X# z; Q
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).& g4 e! E' t) a

    - e" @! p, J/ c& X4 g* x    Problems with a clear base case and recursive case.$ Q* N2 ]/ n& W& ]- a9 P% \) _
    4 H/ O  I: v  [0 ^1 T2 m
    Example: Fibonacci Sequence  n* V. H4 d9 ?3 i+ I

    7 B) a% X  U* l/ Y7 k3 W& l* u( jThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:& g0 Y$ S# g+ F0 L5 \9 s
    7 @# W' N8 |; h) k# T% C
        Base case: fib(0) = 0, fib(1) = 1; ~, k: X1 i  R* ^- G

    - g) P+ b3 D4 u' n  t5 S4 Y    Recursive case: fib(n) = fib(n-1) + fib(n-2)
      c# y5 C3 @2 Y8 s2 h2 f) H  n0 {: V" g8 J4 {
    python
    4 G7 k) _2 B' O0 ]% |" \+ p1 I+ t( ~" J1 u* _  }! d+ O9 k* B
    + N" A8 Q/ W3 ~( {$ a" q
    def fibonacci(n):: F$ ]- I& z9 a8 r, Z
        # Base cases
    1 Y0 l8 X8 c/ o* I    if n == 0:
    4 `) K6 q/ U: n+ d        return 0
    7 d" m) |  D5 n1 q* j    elif n == 1:
    4 T" V) M$ e0 p) B3 q/ k        return 1
    . ~8 B* x0 g& i% S+ q4 b    # Recursive case
    & W! X2 k! W+ H6 v( ]" o; r) q. b* a    else:8 l/ c4 n% t, s$ D( [3 ]; j$ A
            return fibonacci(n - 1) + fibonacci(n - 2)
    % C0 H# x: ]& p3 q8 a8 E0 m
    # X5 E& i( T# a' N4 s7 H* s" A# Example usage
    ) n, V4 H- X- j0 S% c* w! dprint(fibonacci(6))  # Output: 8
    ; }: T: m% q' q- P' Z  f  z- c7 E9 H) J9 S: n* M8 _) A/ o+ ~" R$ E
    Tail Recursion$ ^3 b1 f  b- a) }$ U; _/ d8 Y
    ; P0 s4 c# j: O+ A- {& V
    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).8 |- G5 H, ~8 G- H4 O
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    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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