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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 2 {2 O5 @! {$ Y7 k, Q# d
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    解释的不错
    : s8 _0 ]0 P* l' w5 J% n! t( i0 s3 a3 [) j4 U, H0 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。9 b% i- h+ O5 W! @# }

    & u. Z2 b3 E3 A" _8 u 关键要素0 x5 p+ L* D8 O+ U7 E
    1. **基线条件(Base Case)**% ^' e4 [0 H4 H# K( h. |7 W( m
       - 递归终止的条件,防止无限循环' R1 A! W/ n! d7 o3 w
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1& v9 L1 N3 A. w# A
    ; Q" y$ ?1 j3 U8 b. `" s
    2. **递归条件(Recursive Case)**( M  O# b& I* n  C  v5 y  K! c
       - 将原问题分解为更小的子问题+ O8 E0 W& T3 p7 H: U- s
       - 例如:n! = n × (n-1)!
    6 W; y" l' p4 L( L4 [+ A
    5 Z  H/ Z+ m1 j8 C 经典示例:计算阶乘1 w+ r9 S. g5 j' `2 U) c
    python
    6 Y6 X) X+ R3 y# U& xdef factorial(n):4 O9 \5 n1 M& P" Z
        if n == 0:        # 基线条件" U. B! t7 ]3 w% e' v: Y7 H0 K
            return 1
    . F% e, R: {( J- c. A    else:             # 递归条件
    $ W1 n8 D+ |" U0 l1 C        return n * factorial(n-1)
    ; ]( Q' [+ \) a执行过程(以计算 3! 为例):
    ! v/ T$ `* ~& m1 u( y# `factorial(3)) ]1 K  s' b. k
    3 * factorial(2); X  M# n$ l% a' P6 ]$ `
    3 * (2 * factorial(1)): d% g+ f8 @! W' R9 |0 n* |
    3 * (2 * (1 * factorial(0))); r% M2 x! ^' k; P& d3 V
    3 * (2 * (1 * 1)) = 6" G6 h* n5 }8 w6 n6 Z

    & c- X. L% C- @* l& l 递归思维要点
    4 F5 F. W7 i! T" L- E6 _1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    - u& r5 l( g/ Z5 r9 c2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    / F8 t/ K: I. r7 o" \3. **递推过程**:不断向下分解问题(递)
    ! r& j3 S) S( {6 ?- y- r8 t4. **回溯过程**:组合子问题结果返回(归)
    ; U3 V- [3 u9 a# Q3 `) s6 `9 Q
    1 u' U1 g: Z5 d% c 注意事项
    $ T5 K; z  o+ o5 ^4 D! T+ o- D必须要有终止条件
    6 l# e; }9 m5 M' a. z4 [递归深度过大可能导致栈溢出(Python默认递归深度约1000层)3 i9 V: `  [; M% m+ ?8 u; k
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    9 s* o+ J% c5 s' x5 T尾递归优化可以提升效率(但Python不支持): Q8 W4 v6 o# C" p: D; y5 t- v
    - L- m; g9 `% t! ?1 W/ M& \# }
    递归 vs 迭代
    0 ~1 j1 J5 W. S$ Y+ q2 {|          | 递归                          | 迭代               |
    5 ?$ \# B, r! o|----------|-----------------------------|------------------|
    % B! I6 }- A# x5 Z* {| 实现方式    | 函数自调用                        | 循环结构            |% m: _: r* {8 i/ D/ n' Q: o
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |: h+ c5 r/ E2 B7 Q6 C
    | 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         |
    1 D' ^! i: F8 N| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |% n) v! o& h5 s
    . R  T% u% t( h
    经典递归应用场景9 u: R$ U; L" a3 i
    1. 文件系统遍历(目录树结构)
    7 U8 L7 r3 ]. P' A8 c" ~% F$ H2. 快速排序/归并排序算法
    - r% I+ d9 q1 c. @3. 汉诺塔问题& U# O0 P& i2 x1 u# `
    4. 二叉树遍历(前序/中序/后序)2 T: E4 H% j/ U! H
    5. 生成所有可能的组合(回溯算法)
    & M4 T- u; D/ \! s# M; q9 [+ Q. f* A' Z; v$ t$ T0 a
    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/ \/ [: S! @9 N
    我推理机的核心算法应该是二叉树遍历的变种。0 g1 x6 b1 u$ z3 ?
    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:4 u, I7 f% o7 |: q' V( o) T
    Key Idea of Recursion" ]2 `8 w- r% I/ p! b/ Q6 ~- F1 |

    5 G* W: A0 G3 S; ]A recursive function solves a problem by:
    7 W9 i7 L2 @1 u1 X/ G
    5 h- P: W2 m! a    Breaking the problem into smaller instances of the same problem.
    5 ^; x3 g9 x* P% I
    + E$ `" r3 `( M; |. z' _9 P    Solving the smallest instance directly (base case).5 m# J: z: N9 _1 ^' R0 Y( d2 \9 d

    ) Q# F4 F) y. u& r; F, a3 E9 f    Combining the results of smaller instances to solve the larger problem.
    1 ]! G7 _$ q  t
    8 j9 _# [( F. \; A! `6 _+ o4 jComponents of a Recursive Function- G9 t( D0 `/ O  O

    1 J9 L! S" {1 [* m    Base Case:
    + i, F; g" D7 q; u1 y( U4 I, \5 S' D- r
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.$ o2 B) \1 ^( }! J/ Y
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            It acts as the stopping condition to prevent infinite recursion.
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    . l. K7 e* l0 n. a        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.+ i- t, g6 S2 v# |! r. L8 W4 N

    $ O* Z' E3 q! r- _$ U: X" w" n  L4 p    Recursive Case:  J  k  L' m1 c7 F# {- \
    , d9 l9 Z5 D4 p0 p8 l+ Y
            This is where the function calls itself with a smaller or simpler version of the problem.
    ; n7 L: }$ z3 r! `# O' @( N0 ]
    / E2 _* R7 O4 w/ a4 U4 S        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).5 |$ v0 F* U9 e) A9 K& g

    # k0 i0 i) A6 e- v+ C, Z9 RExample: Factorial Calculation
    2 ~" L& h7 y4 Q% z& ~" i' s9 N# @) K
    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:1 b9 q1 O" N) Y0 G& n" d

    7 A  g! q5 I$ l% i$ i$ @1 ^    Base case: 0! = 1+ r. H. b4 [, z3 ?9 l; a0 I
    , F" i( r# Z: }
        Recursive case: n! = n * (n-1)!
    . }& {7 ^) i* t0 E' [/ ^" _( @: W" _  V$ t6 z
    Here’s how it looks in code (Python):
    ' N  C6 }% S6 U7 P  Q2 w1 Ppython
    , ]+ [! `' q7 O8 \" |' ?% p- A7 A3 w9 ^4 W
    ; k# Z6 v% O; W- q5 E, h4 t5 {
    def factorial(n):
      h! c7 I, \- |: o3 _5 b    # Base case
    0 a! P+ {3 B4 w% Z9 S$ N    if n == 0:1 q/ F! w7 U$ Q$ _6 H3 w, \* i
            return 1% @5 X# I. n2 s% [
        # Recursive case1 W4 A: P+ G* j
        else:2 X* ^. _- F( B9 W, Y! Y  ^4 L
            return n * factorial(n - 1)
    ! A! @) f' l8 W  h8 K, J
    ! W5 c/ E1 T- }+ T# Example usage9 ^8 C  {/ I3 e, {
    print(factorial(5))  # Output: 120
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    5 ?0 Z+ E- G- X4 Z# d( NHow Recursion Works4 G% C4 e# X" y$ l' c+ b+ ?
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        The function keeps calling itself with smaller inputs until it reaches the base case.* X/ D; M/ X8 K2 s3 [3 j
    1 z3 x0 H1 t: L  ^7 r# j6 j
        Once the base case is reached, the function starts returning values back up the call stack.
    " r5 w% H% T5 n9 `! h- ~! c0 s9 o
        These returned values are combined to produce the final result.5 g- I4 T* k$ C7 r+ P

      c' H% Z* O: z% b4 v: a3 AFor factorial(5):
    6 B8 D1 X! V8 O" s- W1 w6 b
    , x' n; D) A$ M; [, F: c; d0 F. E9 b! x+ ^. i
    factorial(5) = 5 * factorial(4)' f% X( ~5 j7 o1 \, g: ?
    factorial(4) = 4 * factorial(3)
    + x3 m- X: {' N/ Q' T3 |factorial(3) = 3 * factorial(2)0 C' Q/ x( j2 w7 H$ e6 s# L+ H" i# M
    factorial(2) = 2 * factorial(1)
    & Y0 `1 w9 G; b* S- ~9 V3 Nfactorial(1) = 1 * factorial(0)! n* Z- G) r7 ?1 L# Y
    factorial(0) = 1  # Base case
    6 V+ z" P9 C- h* X5 r' U4 }# a! r
    Then, the results are combined:2 g  N% ]5 F8 b

    ! k, V4 Q$ ?" e+ w+ q0 s3 N3 d, R  a( f- b9 e; ~& l' E# ^% x  x
    factorial(1) = 1 * 1 = 1/ S7 y! C6 x1 N
    factorial(2) = 2 * 1 = 2" s( V- @( a  K( X, C  U
    factorial(3) = 3 * 2 = 62 B( ^/ O5 @2 ~0 Y/ X2 x  m% _
    factorial(4) = 4 * 6 = 245 v% E/ H  h, l) N! K' Y; T
    factorial(5) = 5 * 24 = 120
    ; g) d# }& T7 r% X
    7 D! \$ b7 r. B0 x& Z& MAdvantages of Recursion9 G& D/ d6 t6 p. C/ G
    , J2 g3 C/ _* j- A2 A
        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).
    , J4 x9 B/ d  ~; s$ }' N' @3 T8 d" D0 ?1 M1 N
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    5 T# `$ r  W% M! r, I( @% ]/ X* e# |1 ^, L
    Disadvantages of Recursion
    / y' s4 Y5 o- u. D( U1 _& u6 n& @4 q
        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.
    ( X' U8 w3 u5 P% Y) W1 v  c7 Q' ~" u' C
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).  j" i* e$ H! I9 Z0 Z7 G5 ~1 {
    5 @+ Z. K& x# ~% Z+ a, l0 `. m3 E0 P
    When to Use Recursion7 E8 Z0 n2 t7 [% s  E4 }0 |

    ( u  N6 Z: P4 }" z    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    9 r6 R( y3 ^+ G, E1 \  c4 O: p/ t; O5 e0 n8 f7 S
        Problems with a clear base case and recursive case.
    " D$ f" y' `7 c1 ?4 b
    6 m# N$ @* O  p+ ]Example: Fibonacci Sequence
      f* v. [" X$ U8 i" a8 {+ \! ?7 |! H) O
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:) j9 _4 o. U8 d7 m6 L8 ^7 ^+ Q8 j4 I
    9 r: t% L% N( e$ o' B/ H- U3 ^
        Base case: fib(0) = 0, fib(1) = 1/ J, {! Z+ T" I6 [/ @9 c4 M9 X/ n

    9 c  b) E9 h3 u! h2 Q: I' z    Recursive case: fib(n) = fib(n-1) + fib(n-2)3 G, g2 E- `' x) [1 d
    7 S9 H% y( g! r6 ^$ v
    python
    : v; M6 K* j+ R. I+ h4 h/ B' D4 j1 m% A# w
    % Q  C& s- e2 S# ~. \5 L
    def fibonacci(n):
      Z2 _% H$ E( P    # Base cases
    , D$ ?: @9 L- _7 T    if n == 0:
      e3 a5 V6 W3 g& r' n        return 0
    ' y* w  T" e+ h- D3 s# Z    elif n == 1:
    - u" I' Z% m3 \        return 1
    ' F. U8 M% ^; A" ]( ~8 U# F    # Recursive case" p1 k3 u' W: T/ q& X7 ~- X. b+ W" L7 {5 M
        else:# R1 _# o3 O7 Q. X! e: N  w
            return fibonacci(n - 1) + fibonacci(n - 2)
    # x; z" f. Y/ M3 a
    + t- f5 \( {0 b6 b6 |7 o# Example usage
    & F7 k5 s; O/ q5 e  Kprint(fibonacci(6))  # Output: 80 c9 T9 b# J7 J- D8 l8 R

    ! |8 {+ B" I# qTail Recursion
    * D8 p& c) D6 ^& d) B) U( N. V  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).
    " J. Y, ]( S' L  V1 T" }  a3 x$ E; w* j+ b: ?# k% r$ ~( s/ ^
    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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