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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
  • 签到天数: 3 天

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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
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    解释的不错
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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。
    - t) @0 J3 H) h9 k+ U" Q: O: k0 I4 W$ l' x" q) I1 e* Z9 s
    关键要素
    : n* I6 S" u1 ~! e1. **基线条件(Base Case)**+ ^1 ?2 C# Z1 @/ F! H% D/ {
       - 递归终止的条件,防止无限循环7 t( [. L5 S0 \5 y1 g6 z
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    # ~6 p: x7 E* H* S+ O6 }5 Y- e- Z( p( u9 U' R; X, v3 k. o' i% m
    2. **递归条件(Recursive Case)**
    * u& Z! M" ~; }& `! H+ N   - 将原问题分解为更小的子问题
    ) K: Z6 [1 V  A- t  ]% z* c   - 例如:n! = n × (n-1)!
    % r5 t5 T, x4 {4 _: v* V/ E2 a# n
    经典示例:计算阶乘
    0 E& ]: A+ F0 k. T6 e* l5 ~python, u# g8 e6 |5 J. q7 Y! X
    def factorial(n):
      D  F+ Y; ?% K$ t- i: Q    if n == 0:        # 基线条件
    1 ?5 C) H8 ?( j5 b' \* |- E8 H3 Z        return 1
    ( B; y* B  a- ~8 p4 V. d3 S' N    else:             # 递归条件5 b% N2 H# D5 t9 Q, o, c
            return n * factorial(n-1)
    5 {$ A5 `& j+ k( _执行过程(以计算 3! 为例):
    0 T6 B) R4 y+ ^! `2 [factorial(3)
    & m5 D  e  n8 x( x3 * factorial(2)' a2 i% P% Z/ J$ k+ T- {! f
    3 * (2 * factorial(1))9 @/ O, J! P. J7 B
    3 * (2 * (1 * factorial(0)))" |) j1 [1 e' \  n4 c8 k
    3 * (2 * (1 * 1)) = 6
    ! ~3 p1 z4 M7 |0 `. A) |8 S
    2 u) k- R4 Y/ _# u& C( E* ] 递归思维要点
    / c' b. O: `" f: p- m4 u, V7 l1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    0 Q( Q( h! c& f' D! E0 \' Q2. **栈结构**:每次调用都会创建新的栈帧(内存空间): Z% ~" Z# z9 b
    3. **递推过程**:不断向下分解问题(递)
    - i, W" i+ O3 a# a' O% t4. **回溯过程**:组合子问题结果返回(归)
      _; c% o# J. A6 w( t( j( q
    & W) z3 ]$ h( C# B& v 注意事项
    ; Z$ I3 |; q6 B$ a必须要有终止条件
    ' ]% V/ g$ ]- X6 l递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ; M* m# j- a& n某些问题用递归更直观(如树遍历),但效率可能不如迭代; t2 W0 s  H" U  q3 y* F
    尾递归优化可以提升效率(但Python不支持)
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    递归 vs 迭代. z5 w* u8 V8 d9 r. T0 N
    |          | 递归                          | 迭代               |6 {4 \# _* `* V, q0 S) I, {
    |----------|-----------------------------|------------------|
    / x- k! X# q* u| 实现方式    | 函数自调用                        | 循环结构            |+ O. W9 x4 K1 q5 ^
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ' @( b# b: J: W% h7 H| 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! X6 ?' {' L
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |, g0 w# ~. ^' F1 h

    6 r" Q5 E# m' a& T# x" [ 经典递归应用场景( v3 `8 ]' i1 g5 t5 u, y6 c) v. J5 X
    1. 文件系统遍历(目录树结构)
    ' C  v& |4 G' @* K" d& V1 n2. 快速排序/归并排序算法
    ; `: c' u* V: Y" X2 q3. 汉诺塔问题/ b8 g$ f7 B+ T7 ~. k8 o
    4. 二叉树遍历(前序/中序/后序)
    & ]5 Z9 ?4 @9 ]/ N* A: T5. 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,6 j2 Q0 u/ D" m. `9 r+ a4 f
    我推理机的核心算法应该是二叉树遍历的变种。% z, R& c" S% F$ `" m2 c9 o
    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:
    1 n+ i) P! [+ F7 F9 }* s  `Key Idea of Recursion2 J; x1 n/ M$ @; a  {

    ; H% Q" s1 D  q, @$ i. VA recursive function solves a problem by:
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    3 h  A# H: u0 o( L* i9 l5 Y0 R    Breaking the problem into smaller instances of the same problem.
    5 x" U0 r1 b  j5 i3 R# R% G5 N' o3 v9 s# y9 y
        Solving the smallest instance directly (base case).
    2 u" e/ ]' S  ^
    2 t# O  |. b) {* H2 U# W    Combining the results of smaller instances to solve the larger problem.
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    Components of a Recursive Function+ q, i. C( X( s5 c! p  f- p
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        Base Case:
    # Y. y* s' S7 {0 H, f7 x  _6 n4 J) F% T: X7 f$ U: P* G# _5 o
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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    7 y1 m- r8 e* ~% b, O: D5 G; x        It acts as the stopping condition to prevent infinite recursion.7 f5 j* D& i* R* d' d( I1 h2 u" ~3 b, a( r
    $ f- B2 }" [6 M9 Z' X
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    + t2 M$ K# F6 D, L. I    Recursive Case:# |" W& m% O: ]; s7 |* c7 G
    3 X/ k3 A) v+ p* c
            This is where the function calls itself with a smaller or simpler version of the problem.
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    9 y3 ~6 v; B! F+ T3 s2 x9 x        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)., m0 S; q6 V$ P  S5 |
    " Y" [* q7 J( \
    Example: Factorial Calculation7 @3 @  i" t' L# R0 ^1 v0 s9 l

    2 y! C2 o9 \. gThe 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:
    9 u& l2 D( r9 p: M$ e6 ]% s' H7 ?* [' e: W7 h0 q: f8 B2 l# |
        Base case: 0! = 1; ]* Y  K" J1 J. R9 V. K/ W, r
    6 h' `, B! X1 Z9 T4 M: S0 T0 F
        Recursive case: n! = n * (n-1)!* z6 C! j! i  S; S

    1 K- W# x9 f+ _3 x9 @. MHere’s how it looks in code (Python):
    0 |, _& @8 {) q& i" mpython$ |" _0 Q# B7 U+ }) ?: b. H

    0 i8 S  K! ?- I: u# }# j$ h/ C" g
    2 [# \  Q5 b  k5 idef factorial(n):4 o) B/ t- u' i# c' v" q9 B
        # Base case- h; Z5 ]- S, |! v0 S) \
        if n == 0:
    , A: J/ }5 v' W/ h7 [1 k        return 1! H* X2 |' H; R& p% H! N4 U
        # Recursive case3 d& [; a! C3 e  s" e6 P
        else:0 `( V, |1 s0 K/ n; T5 o
            return n * factorial(n - 1)8 H; I9 ]% {- U4 U3 ]$ [' u4 l
    , G) {- E  B+ _  y3 ?) n. K
    # Example usage, x' c; q+ J5 Y
    print(factorial(5))  # Output: 120) N* U" {- r# [

    # Q0 q/ M0 n% u& ~( \. wHow Recursion Works5 J6 w$ P5 X' M

    ' }$ z7 D& F  K' |    The function keeps calling itself with smaller inputs until it reaches the base case.; o+ [$ w0 Y, F6 Y$ j: j' F1 K2 O7 \

    " l  S5 @: s" b" F    Once the base case is reached, the function starts returning values back up the call stack.. P) B7 }( {: t8 p8 g  N  B' E9 m

    8 p, [  Q  i0 y# k1 ^, {    These returned values are combined to produce the final result.+ d# {( W( K# i4 v

    & x. {& V4 d1 _4 k1 MFor factorial(5):9 H% A+ L# ]$ S- p- V: W
    # _4 `% D+ J7 ]0 f, r. I* K

    5 X9 F5 ?" \( f1 wfactorial(5) = 5 * factorial(4)
    - s$ Q, E  P% m* Gfactorial(4) = 4 * factorial(3)
    * F; g) j. _$ L; \: c  [factorial(3) = 3 * factorial(2)
    ! r' T3 ^3 |8 ]; ^7 Sfactorial(2) = 2 * factorial(1): E4 h+ Y1 \; F. p0 t  \: U# t
    factorial(1) = 1 * factorial(0)
    ( q$ M+ I, r1 P6 I! Y5 d4 Lfactorial(0) = 1  # Base case4 J# b6 {7 {* _" i* |, `- b/ u! c# q
    . T4 I5 g) q9 O$ B0 A( m" B. N3 ~
    Then, the results are combined:' c8 `6 A/ i/ o  i

    ) r2 W+ B5 U" \' D0 }- Y% B( B$ ~7 S- i4 P. i" }/ H9 \
    factorial(1) = 1 * 1 = 1  o1 W- F9 Q; N! @( E$ Y
    factorial(2) = 2 * 1 = 2" ^5 a# u$ `( {: C$ J+ z7 W0 A
    factorial(3) = 3 * 2 = 6
    . s: K  H5 q- W( L! N4 jfactorial(4) = 4 * 6 = 24* o$ j' h  k8 ?6 C4 X
    factorial(5) = 5 * 24 = 120
    & {; c9 B* g( ~; E, a3 l4 x- R$ u6 w) E9 Q3 `( V& ^* u
    Advantages of Recursion
    , z+ P1 l4 [, P2 `
    7 F, Z  ?  W$ E  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).1 O  ?$ a' |; m% N0 d3 O: ~
    ! w( E+ w+ q9 o" {% Q  a7 O4 }# q
        Readability: Recursive code can be more readable and concise compared to iterative solutions.. l' j4 N" I# y: L* _: m* x
    ! T3 F+ V  H+ L
    Disadvantages of Recursion
      B$ {* x+ B0 {2 E! t; l: c0 N2 ?
    ; N9 c* O; C- F3 J    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" r+ W6 f1 i) b! l# V  p- z4 A! r) h! b
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    , g7 f& A" z1 u3 m5 t1 v9 r2 y3 U% Q" ^9 l& o/ [& p) r/ J
    When to Use Recursion
    3 U: ^5 R7 S2 M7 Q1 b* S/ U
    - T8 P+ n& @3 @7 F    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ) o( c/ {5 F4 x2 {0 \) Q( Y
    % Y$ o* K% R9 L* Y  R    Problems with a clear base case and recursive case., H3 O; ]7 x3 e. N! g) ?2 J0 A6 C
    4 W+ y1 H/ h3 K6 Q4 d  }
    Example: Fibonacci Sequence! r5 h" x$ W( B# ]

    / T: O1 A3 b, s- z. m9 h$ p2 xThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    3 O5 n2 c) B* o9 k6 n
    ' M9 r% H0 K  X4 L% C1 h; a0 ~    Base case: fib(0) = 0, fib(1) = 1
    & v/ o) }+ r% x$ Z2 ?5 n  U. \; F2 k8 p' K; ]: J
        Recursive case: fib(n) = fib(n-1) + fib(n-2): X7 M* y. o4 U3 u4 R* H5 b: z

    ; \& E- A1 q! c  R# Vpython. i3 V+ Q+ z5 o2 g7 Y  U. m
    5 e9 b1 k$ o) P. n# [3 j- [
    * ^: X& i# E! a5 r0 s; Z! B4 u3 G0 d
    def fibonacci(n):7 L  T9 d3 {, v; [
        # Base cases: X  x, I; w3 d4 f3 ~0 i: {- B
        if n == 0:
    & e  \6 c2 I# d. w8 V9 I        return 0
    ; q- C4 y, e+ H% |: e9 ]( H* Z8 N/ B    elif n == 1:, Z: T  h0 f4 ]# r( D0 u
            return 1: [8 Z% {- O6 d' G& v0 f# Y
        # Recursive case
    ' o  r/ @/ c  N4 J    else:
    . J/ h9 q9 S, X8 k* V3 Z7 I; I+ E: N1 b        return fibonacci(n - 1) + fibonacci(n - 2): s3 T" k" p, q/ w" J
    + W) n: I% F5 o. u% |. O
    # Example usage5 \8 O* L2 y' f# V7 V8 \2 a! v
    print(fibonacci(6))  # Output: 8
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    Tail Recursion
    # g; @6 x5 k" s; ?! M! N
    . m; m; j: {. n8 NTail 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).
    ( r) A- ?/ b. h3 E6 s( L% L+ Q2 B  Z  _/ ~; U5 J6 ~
    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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