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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
    ) {/ J( [' _% b, r! G4 a& K, V5 [+ _
    解释的不错% \/ e5 p2 \. `4 h- a, [

    ; j) Q$ m7 T  A8 D7 J  [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。7 s' }! n0 D: ]2 N1 E
    ) H5 b& r; C2 m
    关键要素
    % q0 x* `; p( C7 P) v1. **基线条件(Base Case)**
    ; {; m) M8 S; \: K5 y   - 递归终止的条件,防止无限循环
    8 u  o" n* S* t0 r   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    9 {9 C9 P- S2 h8 [$ l
      o( t/ T" k- U4 R2. **递归条件(Recursive Case)**
    , R: t- u. ]+ a5 k; A1 |   - 将原问题分解为更小的子问题
    0 c1 Y% y& E, n' j9 I   - 例如:n! = n × (n-1)!
      H! z  `. P+ M5 D0 E( J
    " }& t# H, p* ?- @ 经典示例:计算阶乘
    # f9 i# N+ Q) u& ^4 m6 Rpython# l; C  m0 l. P6 ]
    def factorial(n):
    3 _) }$ S, ~) P9 j4 ^# C    if n == 0:        # 基线条件
    & k0 |/ P2 h' [% X/ ~        return 1$ i# p. W9 s. E
        else:             # 递归条件7 r# E1 N2 P, _7 g  }
            return n * factorial(n-1)+ H0 N3 @/ G$ O
    执行过程(以计算 3! 为例):! A' r" e8 S" C( I7 V
    factorial(3)
    , a$ D3 _8 p( D3 * factorial(2)+ W; L" y5 h2 O
    3 * (2 * factorial(1))
    , k9 X5 ~$ C: z% q3 * (2 * (1 * factorial(0)))
    ) e5 u) U3 `1 J( @! ]8 ~* q8 `3 * (2 * (1 * 1)) = 66 m" Z0 Y& b) U5 O/ \7 K
    0 S" P  x, ]: U8 Z& `
    递归思维要点
    7 }4 |8 n! t9 @6 Y7 r* ]) f1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    $ ?  f5 I/ @# c2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ( N7 B9 Q# b, s' w" O2 H3. **递推过程**:不断向下分解问题(递)
    9 a6 m+ r1 m. ~. C4. **回溯过程**:组合子问题结果返回(归)
    ( K3 Z9 g) D% j. p8 k0 X
    ! K" u* V- O. r% y, E: [ 注意事项
    - Q8 `- D/ E4 d! `必须要有终止条件
    % K, }( S$ y9 i; S递归深度过大可能导致栈溢出(Python默认递归深度约1000层)( c7 `/ y0 z% @& I) W
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    $ ]# H# s% s4 n7 n) b9 y& L+ y* R尾递归优化可以提升效率(但Python不支持)7 z8 M7 S, V- W
    3 ?% r, V/ I4 O. G, H- E& l  P
    递归 vs 迭代
    + u& }4 r8 b! s2 W$ _|          | 递归                          | 迭代               |
    - ~* ~# G- u1 {  _|----------|-----------------------------|------------------|3 I. o% i* n9 h3 N! r  @2 x& U
    | 实现方式    | 函数自调用                        | 循环结构            |
    # L5 a! \2 S' e" h( |. U/ b- t| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    " z+ W1 V# ~1 m- Q3 c  ], X- [' Z| 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         |
    ; J* z' M7 p7 ^) n8 o6 L7 y| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |" K. a# X- [" w

    9 S- \7 _$ i7 y4 |/ x 经典递归应用场景
    / q6 d7 e& c! ?1. 文件系统遍历(目录树结构)3 }$ t5 N/ H8 @1 ]' W6 \3 m; p
    2. 快速排序/归并排序算法3 }  k/ F/ [! w/ _- R7 N$ n1 ]
    3. 汉诺塔问题( G$ R$ \" ?. ~, f
    4. 二叉树遍历(前序/中序/后序)
    ; q/ U' b7 c! S5. 生成所有可能的组合(回溯算法)5 V0 R3 X# K4 |/ t- \6 _
    & B' J( p; }1 s* h
    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 06:08
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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,+ d: O: C! s) H& A* u5 i
    我推理机的核心算法应该是二叉树遍历的变种。
    , Y  f) X/ K& J4 U* m. p* G' z$ Y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:5 a) }+ f4 J4 l0 e9 q0 Y3 Z
    Key Idea of Recursion3 b8 \8 ^$ W- r0 p) r

    - t) G8 a0 `( d8 FA recursive function solves a problem by:
      D7 k$ U, }* E% a) C4 Y' R, _' M% a; _/ P0 c
        Breaking the problem into smaller instances of the same problem.* g! f* V: l: L, T
    8 E! z  x9 |# l& G
        Solving the smallest instance directly (base case).
    ! _* M& {- ~' c) s" o/ W! I( S- x
    6 N* W( W; d0 Q1 U$ {0 v3 a9 A    Combining the results of smaller instances to solve the larger problem.; R+ x3 v* S; [! }

    2 W' M* k( x% M8 s+ X9 g& V! i5 cComponents of a Recursive Function
    8 F- [" e: a' k$ w! ^6 |
    & `$ R- g5 R5 z5 b+ _, K8 ^! y    Base Case:
    ( U: e5 v; t; c- ]& I: a
    . y) S: b& G3 _5 W1 Y9 l        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    4 w7 V* F5 s) c' E
    0 L) j- E7 o: l& m0 _# m7 R        It acts as the stopping condition to prevent infinite recursion.- h* `9 [+ |  b7 B7 ]5 V. j

    ) _  }1 p" S5 o' @0 ~        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.4 Q) Q% L9 j2 @8 f' f2 J
    & X5 F" }3 Y$ E5 h2 Q# Z
        Recursive Case:
    1 `5 c  o3 \! K" T  y- Z4 B$ d; k, o5 b$ B
            This is where the function calls itself with a smaller or simpler version of the problem.5 ^5 }. F  I: U; A9 E* A- [
    6 a9 f) o: p  [& j2 j3 V! H# N
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    1 c' f$ f; A* y( H/ h' O6 q1 b1 h
    6 p% B. T' y' P4 p; tExample: Factorial Calculation
    : P9 {7 J2 u7 O1 S1 E1 o" M, E  Q9 o1 r( u/ I
    The 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:
    6 t, N/ m6 B2 P  D5 {% s8 R  Z; q! _0 n5 z$ i1 s
        Base case: 0! = 1
    & y  K: S+ O2 e( q0 V- r4 p; S* ~4 N# u. p9 v; ^3 `4 ]  e: P
        Recursive case: n! = n * (n-1)!
    ( j! @) T, {* p- [% `' W9 G7 g
    3 E; s+ X/ R/ I5 a8 F# pHere’s how it looks in code (Python):
    # D0 }6 A+ O) K/ K# W# Z0 Upython4 S7 U6 ^( k: E# \; G; Q& W

    ) ]9 c9 O, s: I; ^& k6 p# y9 N, T9 k0 b( L- k, H
    def factorial(n):# ?8 q& z/ [- G$ @
        # Base case
    9 S/ q2 p3 }2 F# i; Z5 R    if n == 0:8 e. u: e# \: p9 ^& }
            return 1. p, w; T$ H( }: t( J) Q
        # Recursive case
    7 `4 ]: T' W& s2 f* Q6 u+ O    else:, d9 k  `& ?  F7 N0 t
            return n * factorial(n - 1)
    0 x1 ^1 n5 C6 U# f$ d
    4 A* V6 c# k/ m4 k& w  ?# Example usage& F2 \' I1 q+ n& w/ \5 m* n( ?% x; P
    print(factorial(5))  # Output: 120
    & H4 c6 }' E% b: V8 `% g$ m6 @
    : ]$ r0 p0 _! Z# K4 THow Recursion Works
    1 ?' ]- g) n8 S
    ! h( d) P4 H' n, e; W4 W! ?# j1 V    The function keeps calling itself with smaller inputs until it reaches the base case.
    2 E( u, ~' \! r2 G* p0 N
    # Z/ v$ K, N1 z2 j    Once the base case is reached, the function starts returning values back up the call stack.5 U5 E1 \# z& v% R6 A' J1 W

    . z- }8 ?+ A0 v) K# r# r7 r5 y    These returned values are combined to produce the final result.! X- W6 U  ?' m  J) M
    ' u# k- l  z5 V3 @' E
    For factorial(5):
    ' e1 o) h  T6 j# _
    / W$ z+ T7 m- P' l4 O  X8 F* N: q5 [& V, _/ Q7 f
    factorial(5) = 5 * factorial(4)  r' a9 D& N& S, O2 v. D
    factorial(4) = 4 * factorial(3); G' V! C4 Q: v
    factorial(3) = 3 * factorial(2), I6 W& x1 z5 @+ W
    factorial(2) = 2 * factorial(1)
    3 l2 ]2 k6 ?5 `( {factorial(1) = 1 * factorial(0)- K3 ^5 z+ v9 N0 Z( y8 L
    factorial(0) = 1  # Base case
    ; E6 c# G+ a8 A( b  G  w
    ) D: Z  T; G4 j: S. z* }' ?" uThen, the results are combined:2 B- y2 ~' u3 a# Q3 ?3 ^$ ^
    ( m  A4 ~" h3 x/ s1 `+ q" `
    1 A1 m+ f$ o# Q# E# H
    factorial(1) = 1 * 1 = 1
    4 ?8 {( @6 E, w3 efactorial(2) = 2 * 1 = 2
    . }6 o) f0 p+ L' l. n7 w1 N" i! p- h8 Pfactorial(3) = 3 * 2 = 6
    0 s# e$ z) \5 Xfactorial(4) = 4 * 6 = 24
    : N- _* c% M" |' _! b& T' Nfactorial(5) = 5 * 24 = 120
    4 S& c7 ^4 Y6 \% r) O. O5 \$ P: {& g
    Advantages of Recursion
    7 m& K/ }: `% p
    1 ~# e: Q. \: L1 i! K    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).
    ) W1 S. L/ F. G1 K, ^" y7 L& k
    ) T8 J( q/ x8 L# U5 `    Readability: Recursive code can be more readable and concise compared to iterative solutions.. _: [' }3 k6 g. W& q- N1 ^1 w
    ( n. A1 Z; v6 s- b3 s5 M3 c
    Disadvantages of Recursion* X8 o5 E9 f9 N; L$ Y; C

    : j% J- M) Z) F& P9 |1 ?, F    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.& w& q) u8 ?5 B# ~- N1 x: e/ v7 `

    ! ^8 q4 ?. f* f: e    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).. O8 x" X% U% M1 T

    , g0 p* b! h$ C# SWhen to Use Recursion
    / a* V4 p0 G9 t" t& l: e* M) e
    % v+ `) n5 }( x; M/ E    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    5 c5 ~0 X9 ?8 j$ w% O0 b7 f1 n5 Y4 x( t, ~
        Problems with a clear base case and recursive case.
    * b. ?$ \! i7 ?; y, W
    6 Q. q) A& @! [7 j9 w; r8 X0 qExample: Fibonacci Sequence8 G7 b" }! E3 p6 Q8 E6 m

    * H) n1 M7 e: Y; _The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    4 |- @1 w, e/ _$ S9 }/ ?5 v1 r8 k0 f1 P9 O+ H. r
        Base case: fib(0) = 0, fib(1) = 1
    4 i% i; ~9 v1 k$ I5 w. b8 E; r! [% D4 |9 Q4 a: x
        Recursive case: fib(n) = fib(n-1) + fib(n-2); p2 K7 {0 E- l& c3 q
    % n9 F- J% K- _+ O% V% a# M
    python/ A; {3 G% u$ l/ k
    # e# S1 J' x5 ]" f" a3 F
    $ O# h1 s4 v6 j3 w1 X1 a$ G9 o
    def fibonacci(n):
      U: M+ R8 |& n! W$ {5 v    # Base cases. W' g# ?7 i* K2 I1 p* v1 f
        if n == 0:
    ; Y$ r. [3 z7 O/ A) N2 ?7 r        return 0
    & r2 o- m3 _" d  ^    elif n == 1:
    # R8 j* g  M0 G/ |( x        return 1
    3 i) d% L$ B5 c/ I; J3 e9 Y! u    # Recursive case' K1 r. ~* o& p. T. `
        else:1 n6 f1 x9 l* i9 m, i
            return fibonacci(n - 1) + fibonacci(n - 2)
    ! v" J1 A1 P4 V: I: Y: k9 A" S& a- |) c
    # Example usage/ h- s2 _2 ~3 K, ]
    print(fibonacci(6))  # Output: 8
    0 }& e) W& c1 e1 q& c$ r7 T/ }1 C( I3 ?- d( C
    Tail Recursion
    ' P: ^9 U) f5 B. T+ d- l! o' M% T  A. r& Q7 u! M4 r
    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 Z1 C7 g! a, ~4 i3 r, G" @) o
    6 ^0 O" z' |0 V: \8 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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