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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 - k" I( k( s! L
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    解释的不错; I2 L" E! x  I! S) u
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    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。5 S$ u4 v. p2 }  ]0 P

    # g1 L# V; E2 R/ d- c/ v5 @ 关键要素
    0 B8 B2 Z. T6 Z- U" g1. **基线条件(Base Case)**) k. _% G1 m6 _4 ?0 ?9 e& ^* S
       - 递归终止的条件,防止无限循环  f; ~. f9 F' v
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1# U- _- g( ^! ], V; ]% K1 Q" [
    * p( n; j5 L0 s
    2. **递归条件(Recursive Case)**
    + Q) v' n/ V- o: x1 j, C   - 将原问题分解为更小的子问题/ n8 r4 t( V# K. R3 l2 r, `0 v
       - 例如:n! = n × (n-1)!
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    经典示例:计算阶乘$ V, }  ]0 u6 m" X+ K
    python3 P0 g3 C& Z" J8 T4 T) [5 G
    def factorial(n):
    0 i* n7 x, c: m: c1 Z. r9 K# t6 H    if n == 0:        # 基线条件! `7 G: L' Z  q
            return 1) S* Y0 q% Y" @) W
        else:             # 递归条件
    * z, L% |  ?# r, {  Y        return n * factorial(n-1)) w4 c% n* @3 o  f& ]( L; k+ U5 y
    执行过程(以计算 3! 为例):
    & s$ @" N! h& efactorial(3)
    ; K1 @, M/ a) b3 * factorial(2)9 F- ^) z% Q4 v2 y7 Q$ p6 v/ Q! X; G
    3 * (2 * factorial(1))) @- j" s8 R4 T& K9 t
    3 * (2 * (1 * factorial(0)))
    . v% ~; {( Z9 {; q3 j" Q3 * (2 * (1 * 1)) = 6! ?8 u4 t' M7 ~) h$ z" W! E

    0 z7 Y6 R; D$ W$ N/ D 递归思维要点
    * y5 J0 {% R' P+ w3 I  U1. **信任递归**:假设子问题已经解决,专注当前层逻辑. y& S) B, m: q+ W! S* ~, H
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)8 e# e0 i2 F7 G4 L
    3. **递推过程**:不断向下分解问题(递)
    % ^. \) |" w' }* `! o$ y8 b5 D4. **回溯过程**:组合子问题结果返回(归)
    6 l6 Y3 p7 ]' |4 t: q& C8 O
    % S% k0 ?/ n9 h$ F( z; n4 z 注意事项! d1 N2 U8 b" i: Z: B% G. h
    必须要有终止条件
    7 n9 u% {0 H' g/ ]: M% L递归深度过大可能导致栈溢出(Python默认递归深度约1000层)$ g6 V7 G+ m+ K- `6 Y/ N8 ~
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    $ c9 _$ R% E/ O尾递归优化可以提升效率(但Python不支持): f' I& n: I& l- K+ O

    . y* j) x& T1 r 递归 vs 迭代, U8 Z8 V6 x6 {+ O9 j  ?% g1 `
    |          | 递归                          | 迭代               |$ d  u% ]8 @2 l% ^
    |----------|-----------------------------|------------------|
    # o! O5 M/ J- z, Z; J| 实现方式    | 函数自调用                        | 循环结构            |* E' Y: C/ z. S- T7 a6 N1 X" p
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    + h2 |- M, b, b: {2 v* j/ P* o5 `# A| 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         |
    * W% F" K' ?7 {' l  y9 J3 m7 _0 e| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |* r; Y& G1 ]4 Q' |. z8 B
    " }. \" ^, ]% Y+ }
    经典递归应用场景
    8 K# k2 i7 ~  u( G( h1. 文件系统遍历(目录树结构)
    ( E( B; Z& h9 a3 `( C( t: L2. 快速排序/归并排序算法
    ) t2 O. i& V0 r# E8 g' [8 W/ N3 |3. 汉诺塔问题+ L* D8 r* M" e
    4. 二叉树遍历(前序/中序/后序)$ M0 J+ x: l6 r
    5. 生成所有可能的组合(回溯算法)
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    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,
    1 Y* Z+ K# T. j0 s我推理机的核心算法应该是二叉树遍历的变种。6 V5 `8 P3 v" p, y4 o* {: p
    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:+ q3 |) m; Z6 G: l
    Key Idea of Recursion
    , }: Z% e+ |0 e8 C* x; m  M. a
    A recursive function solves a problem by:
    " T3 T) s& V9 w# l0 s# M  G/ F  I
    " R: p6 c9 [! q7 \+ P    Breaking the problem into smaller instances of the same problem.7 i  M4 J' z) ~$ O% {, g( O: d

    1 b' z) |( V9 p% ~. R    Solving the smallest instance directly (base case).; f4 a5 h4 `- a" E( {2 R
    5 p' i5 Y/ k) |# v8 T
        Combining the results of smaller instances to solve the larger problem.
    3 K3 M2 e! u. }  |% F+ U. h% `# }) _- O( Z; |: r! T( V
    Components of a Recursive Function
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    4 f9 \& l& R  V2 F- C4 s    Base Case:& g8 ~3 L$ D- D% G
    3 ]' v4 `! i* b* \% ?
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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    $ w) n+ u( a: {/ S" {        It acts as the stopping condition to prevent infinite recursion.
    2 n: U! u3 e7 N: j4 S! Z) ?7 \3 r( j- l  K- u3 X* o& t
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    : \' z, P+ O. @9 ^( ?8 a8 k4 V# F. r* j7 p7 l
        Recursive Case:5 l, y1 g, i6 n2 a

    . d2 j& j4 o$ h  ]9 ~3 F        This is where the function calls itself with a smaller or simpler version of the problem., T' A! {  c& t! K2 \, I# B5 X

    ' ?, q1 w3 @7 {+ P9 h0 m/ h        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    8 y( O  t3 g$ r' D  J( Z6 V9 I6 u& \2 V$ }
    Example: Factorial Calculation0 Y' F+ E- b0 Y

    & t2 P5 m* U$ e3 W8 G1 Q) 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:# P( }1 [2 K4 t; c

    . O( Y% A' {- k) t, B4 W    Base case: 0! = 13 Z8 G' ]* E* Z. P: {2 m: T/ N6 C

    ; W5 O; S9 h" F% {- `* j    Recursive case: n! = n * (n-1)!
    0 b5 ]# C- S6 w1 _" I
    7 }5 N: j0 ^/ k0 C. B. d7 nHere’s how it looks in code (Python):9 N/ C1 M7 o/ I& e# \  @" a9 z- g
    python5 E8 k, P5 @7 z+ }3 a
    / ^1 r" j1 y+ _& z3 E; ]- m
    5 e: i. B5 v. _+ Q* j/ x
    def factorial(n):; G4 Q( k) v- @1 G* t# b' t4 ?
        # Base case: n) V/ U2 Z7 h5 h+ ~, w* ]( L/ K
        if n == 0:" z4 A9 o  v& @0 F
            return 1& e5 t" T5 \) F9 j& }  H
        # Recursive case
    $ L/ s) Z8 d: G: ^& b    else:* J( V5 t- K9 Q; K' p% D; K7 J
            return n * factorial(n - 1)- t; K' B! C4 v
    $ r4 ?- \* Q  M  U- {" E
    # Example usage
    ; I3 O* Q, t/ u* I$ aprint(factorial(5))  # Output: 120* Y  p- k; ^; ?' v9 s2 q. W
    / ^' z( {7 ]# D1 K% q
    How Recursion Works
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        The function keeps calling itself with smaller inputs until it reaches the base case.
    : c$ m; _- j7 u8 ?) H& I+ E) L4 U3 B; r; g! w" [
        Once the base case is reached, the function starts returning values back up the call stack.
    # A+ b# y: a4 d4 f' B4 f8 i  M2 K3 B3 y& t/ U4 X
        These returned values are combined to produce the final result.
    % n" W( E* ?9 X7 K: k& X, L5 Z2 y1 o3 X. K7 t8 J5 Q
    For factorial(5):
      T( W, e9 C: L5 x# ^2 A2 J9 Y) V- o) O0 g# B& d( w$ n2 ]
    6 Y+ e$ p+ m4 p
    factorial(5) = 5 * factorial(4). p/ X$ ^( X/ E5 \" \$ S+ s
    factorial(4) = 4 * factorial(3)
    ; e  o( U+ o0 C" O* q, W. Lfactorial(3) = 3 * factorial(2)
    2 W+ d" K4 B# k3 V3 g& t. `factorial(2) = 2 * factorial(1)& R2 X  n) ~/ F0 }  ^5 C6 E. q
    factorial(1) = 1 * factorial(0)
      }7 {" J6 @5 K. wfactorial(0) = 1  # Base case; O5 P* G/ ~8 s' l6 W0 G
    . s, B" e2 Y1 Y9 J
    Then, the results are combined:
    # Y* H6 m* R) b& H
    7 {! v2 ^" N9 G4 N1 ?+ H" _; N, X3 B/ l  `8 V8 f% X
    factorial(1) = 1 * 1 = 16 E+ R8 D/ F. X) @8 v. b. J( V3 o
    factorial(2) = 2 * 1 = 2& c. x9 ?  V3 E7 {1 w  d! v
    factorial(3) = 3 * 2 = 6
    1 ?/ ?* Z( Y# Lfactorial(4) = 4 * 6 = 24
    2 m& V* O% J+ j9 \# Z1 J' zfactorial(5) = 5 * 24 = 120/ l, i& k, {" _$ E

    2 P5 Q. ?1 j( J5 f" t9 QAdvantages of Recursion
    & {7 q' E( o, |8 k
    5 F" X* L) F/ L& r! C& ]    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).+ U  o, m. ^+ m0 A0 [3 b4 `* V

    " u+ E8 b$ J& b5 D1 n    Readability: Recursive code can be more readable and concise compared to iterative solutions.0 k8 \6 ^; b" }
    8 H) M4 k( q9 D4 D& s" B
    Disadvantages of Recursion) c# }8 L" q5 a( z( q2 Z

    ; n2 o$ e& Z; 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.7 e: k- @3 I  }( f! J! d. [/ y2 f6 G
    * g: U! f4 \5 G- y$ g
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    + E0 p9 U& W* b- `; i# E4 u
    ; Z6 A- g# X' S- _; eWhen to Use Recursion% q" J7 M, C! e! x# |# K/ B

    ' j% l" e4 ]9 c/ v  o    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).; r( F. {6 a8 \5 d
    $ H3 S+ \- ^. q, C
        Problems with a clear base case and recursive case.! D* j' w% g- Q; i$ O9 x. I9 n( ]

    . r# T6 \+ d7 g3 |& f7 k. ?6 PExample: Fibonacci Sequence
    9 j5 ?. `% y7 r: y6 |5 C- A5 @- k$ E# S6 Y& G- P' K4 Z
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:+ j2 c+ O0 }; m, x2 P8 M

    ! ~" P9 F5 x5 @    Base case: fib(0) = 0, fib(1) = 1
    5 ]* y: L  E/ j% N- v9 k
    6 z( R0 c  q' |, Q# P    Recursive case: fib(n) = fib(n-1) + fib(n-2)7 Y5 H& b0 `4 u, A

    * u0 A3 j% v0 p$ k7 }python6 m8 s: K/ T0 A1 p. Y9 x

    9 P. S6 Y& E1 r' `/ X% A5 }
    # c/ }$ b( H: odef fibonacci(n):
    % i% g6 L/ A8 J1 @# }    # Base cases
    3 [) `2 ?- C7 e% A) F    if n == 0:2 F+ m( ^+ E$ }
            return 0
    . a/ u8 k0 x' d% ?- r+ ^7 P    elif n == 1:
    3 w$ V/ b* S" ]! V! k' z7 ]        return 1
    $ C1 _, N  u" c% e9 V3 Z( x9 x5 B! @    # Recursive case
    ' A: V1 x9 ~' S    else:5 b) U7 j. t+ _' U& g0 k
            return fibonacci(n - 1) + fibonacci(n - 2)
    0 U9 k- g0 ^# \" W6 O  M* Z" F
    7 t5 N: F# {6 l3 a0 t: X- ^# Example usage" o( z. W1 j% y5 P% ]' S% e$ |4 R
    print(fibonacci(6))  # Output: 8
    $ L+ s. @/ G; P5 O
    8 q# b; A; p  F+ |7 Z; _Tail Recursion& x( j. k2 }0 p+ s3 x$ E* }
    * a) D1 c# ~& a8 t! c
    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)." x5 }, O# D) D, g

    ! W( r4 b" Q$ `6 p  E% I) p2 FIn 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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