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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 5 C6 l, W2 z, j/ l7 W
    0 l3 V# X' R! I% G) j. T
    解释的不错
    2 K! d, V; |& w) A+ [$ V/ o' O: r( N$ Q- S. @  u
    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。' R. E- ?. }2 T5 O8 q

    , u9 X9 W0 y4 t* M& C7 [ 关键要素+ O/ K. ?6 t7 C) G* Z, D  }/ z
    1. **基线条件(Base Case)**
    ; |0 Z8 b- @$ ^6 _   - 递归终止的条件,防止无限循环
    1 b1 o3 A3 j" t9 E1 j   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    " g9 J) y/ T2 W2 ?& t
    4 `, c, l6 h/ `6 x2. **递归条件(Recursive Case)**
      o- @# N( }4 t+ B, p   - 将原问题分解为更小的子问题- C6 N% m. n7 ?# j& C
       - 例如:n! = n × (n-1)!
    2 }% K3 I' c3 m$ ?1 B( j  Q3 p0 y% P# d5 o. a2 ?; `0 V5 U
    经典示例:计算阶乘
    # B( S4 v( `1 x2 ]  Spython1 O8 q2 `( j1 x  {% S0 w; p
    def factorial(n):
    0 L1 z5 K( V1 f- @) V    if n == 0:        # 基线条件
    % D8 O3 X3 p# h/ }7 k" [        return 1$ q# o3 V6 E7 ]0 S
        else:             # 递归条件
    % c) H; _4 I2 s. Y* e5 U) M# f        return n * factorial(n-1)) i: X, f) o1 d# N: Q# @
    执行过程(以计算 3! 为例):
    8 T0 E" j' Q4 e7 Kfactorial(3)/ l( x/ a. Q$ p* t9 {
    3 * factorial(2)
    , S# |& r" i( K) h5 v) {7 Y3 * (2 * factorial(1))
    ( V2 M( L( X! B! }- [3 * (2 * (1 * factorial(0)))6 B. d$ N. c$ i) n. [3 {7 a
    3 * (2 * (1 * 1)) = 6
    ; a# U. z. L7 g
    8 z) K! W( _5 N3 t; @ 递归思维要点
    + G' X) B4 p, J4 u1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    # L- x! c$ N' l% M2. **栈结构**:每次调用都会创建新的栈帧(内存空间)1 q( P% ~- d. ]& {# I- B# n. s
    3. **递推过程**:不断向下分解问题(递)
    2 s, A: \) ?  q/ `" g& `: `4. **回溯过程**:组合子问题结果返回(归)4 H+ H& O! H( i1 p  ^

    0 i# I9 p) u  y; Q1 d/ q 注意事项+ q2 i1 i! ]3 i$ T6 }
    必须要有终止条件
    9 a" q$ f. W" m' E7 b递归深度过大可能导致栈溢出(Python默认递归深度约1000层)& J2 f  ]) M. V& j9 w7 L  T" _2 X
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    6 ~- w0 o0 {+ b9 _! a5 a; G尾递归优化可以提升效率(但Python不支持), I' k, B( F+ y) V6 F$ {

    5 n1 }$ H7 f5 K2 L" q 递归 vs 迭代$ @5 P. D6 m' e. o6 ?2 p2 M
    |          | 递归                          | 迭代               |6 a/ Y7 d2 W+ q$ N5 ]1 I
    |----------|-----------------------------|------------------|
    ) n' \8 f+ X8 _| 实现方式    | 函数自调用                        | 循环结构            |  X! ^3 H1 w7 ^& ^, }/ S
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    - q- t! d! D$ W" Z5 f& 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         |
    + q/ z+ f3 P. m& Q) {! Q9 z3 [| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    - D0 W; v: t- `6 e' g9 G
    , S1 A" |' \9 z. F+ w( \ 经典递归应用场景1 I3 h+ B- |& j: F4 _
    1. 文件系统遍历(目录树结构). A8 u" ]/ P" X" w9 g
    2. 快速排序/归并排序算法
    % x/ T3 H4 C4 d! \3. 汉诺塔问题/ q2 T. e/ w6 E" c. n
    4. 二叉树遍历(前序/中序/后序)
    " |+ [# S! h, L8 V5. 生成所有可能的组合(回溯算法)1 B+ @5 E4 {$ q& l0 P/ T7 T

    7 m9 @% k! I+ o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,
    * m0 ^* s$ i" B0 h+ b我推理机的核心算法应该是二叉树遍历的变种。
    & F9 X3 ]9 |/ Y+ a0 g$ k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:
    2 T4 C" t# o& ?' e2 k7 MKey Idea of Recursion
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    - s! V/ B- D& I: Z; f& A+ d/ {8 PA recursive function solves a problem by:- ~" |% I/ Z) }( u; R! A9 p

    5 y" u/ V! J6 U+ t& w    Breaking the problem into smaller instances of the same problem., i6 z6 T) K% C7 P, N8 {6 C0 \. X
      ~) ]# v3 U) r5 _) ^
        Solving the smallest instance directly (base case)./ G0 D% L" h/ F# V  w) P1 f$ S
    3 H% U( V- n% d$ {" h
        Combining the results of smaller instances to solve the larger problem.# [: y3 u, K3 \* f
    6 H: Y  {% b' h: T" B' g- P
    Components of a Recursive Function
    , l* t$ s% ~) U5 O8 H' b: l4 \2 e# P" y
        Base Case:0 j" c4 O. w+ T
    / W/ p( g( Y4 |' |4 V
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    $ M0 Y8 t3 k' ?9 Y/ s9 g1 y* ~- T) b6 V6 @9 t% c& [
            It acts as the stopping condition to prevent infinite recursion.. [: i# h5 G" S- Y+ W% ^1 X

    ) T  |& [7 S$ ^* ~" Y        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    ! x( Z$ B  `1 ~. F( X7 |, l. C2 v/ ?8 X% Y, D
        Recursive Case:( d" }$ V( ^5 }! Y  \4 m  I
    5 k, r1 h! M7 v" }
            This is where the function calls itself with a smaller or simpler version of the problem.
    + h% ]; o% b" g; S/ l+ V; m) {: O; g* v) L4 l# m7 K
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    4 f0 x+ R% `. [/ }& N8 f0 C! Z" b* `. [$ D* {
    Example: Factorial Calculation
    4 `& i& e( q3 A! ?
    / y2 V, Y1 y) m* E; Y& YThe 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 @1 t: h$ y3 k! P$ U) L3 A: h# d' ?# z5 U0 U. U
        Base case: 0! = 1
    8 {! ?& C0 X0 [( v
    4 J0 \% r* t! J7 T* s3 i: q" Z' V    Recursive case: n! = n * (n-1)!
    4 Q9 u; P1 W) q9 L
      [5 }+ R9 g6 y0 u- J) yHere’s how it looks in code (Python):1 ?2 x! _5 L: s9 e  z
    python
    . |8 O2 P3 ~1 @& n1 m8 l- e* I- w# p% r) K) B& @
    0 C# f1 |6 V2 l# \$ W
    def factorial(n):* o7 r% l# w- q/ s! e
        # Base case  @( T8 u) O) x) Y, _& G1 l  j* ~
        if n == 0:% u. W3 `& Z# M9 r/ [  I4 A
            return 1
    / @+ C/ p* F. o2 Y3 P    # Recursive case# [& F: Y/ l  w" M7 s  s
        else:! @& M, P" E5 l8 h7 c2 @  v. r
            return n * factorial(n - 1)8 z/ r  z9 E. R3 o+ p
    % s0 e1 h# g  Y4 W- s
    # Example usage
      a2 v! X. r* H3 Y- b4 T3 S" Tprint(factorial(5))  # Output: 120( v3 ]9 b1 ?4 s

    2 x8 b, R4 o8 u* q  R: G' `How Recursion Works
    ( q8 s/ L0 _+ }0 B& K1 E- [
    : e4 u; j1 ~/ y6 ?  c6 w) I. D    The function keeps calling itself with smaller inputs until it reaches the base case.
    * d" W) p4 {/ U% U: O# d; Y/ u5 u: w+ g. }- S
        Once the base case is reached, the function starts returning values back up the call stack.! V3 x' b& s5 i2 J
    8 L5 w- X1 Q  ?; Q
        These returned values are combined to produce the final result.
    0 J. R& A3 |8 ]' G
    , C2 H3 S1 g8 L8 WFor factorial(5):
    + T* d2 x& j: D' D; M; L  j- |3 s  x% k; |# h3 P& y

    . \8 _& }4 v8 tfactorial(5) = 5 * factorial(4)
    ' A, C1 p* j1 q/ yfactorial(4) = 4 * factorial(3)
    + Z) }! \. L9 _8 z1 w* l% S& _factorial(3) = 3 * factorial(2)% t: M2 j! Y! a9 F' I! a
    factorial(2) = 2 * factorial(1)+ ?0 f) x5 S  D# Y9 ^
    factorial(1) = 1 * factorial(0)% C$ S# ?2 _! h9 E& u
    factorial(0) = 1  # Base case
    3 L8 O9 z$ N8 m# o3 F/ [: E# T$ o3 ~. e5 J
    Then, the results are combined:
    - T1 k8 \/ {. @+ t% C, b1 C% o
    7 U, ]: h% {( Z. |
    ) s9 R: h$ y4 ?( r/ O! H: ifactorial(1) = 1 * 1 = 1
    . c0 Q5 X& H$ v% S6 Afactorial(2) = 2 * 1 = 2  E! O' q( b4 u7 ^* O
    factorial(3) = 3 * 2 = 6
    0 ?. F1 R! Z. `" k, Ffactorial(4) = 4 * 6 = 24
    9 m6 E2 Z0 u( T! J; D9 x) vfactorial(5) = 5 * 24 = 120, p- Q* w  ~5 A: h) `" U2 ~
      u4 v& k8 p( h8 C
    Advantages of Recursion
    6 _( P2 o3 y) h+ c# h# J
    8 x* r8 H7 V7 |    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).
    0 s( @: j- H6 S% y9 L7 ~7 r6 u* r, C0 m; d  T/ q& w' \$ V. m
        Readability: Recursive code can be more readable and concise compared to iterative solutions.% _- ?$ k# t5 t

    8 @' B( x, A1 [' r. M& YDisadvantages of Recursion
    / Z- K( s$ f2 O/ B" S' T
    ! [( \5 ?4 t2 U8 L2 {3 i    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.
    % D7 t4 x8 E$ U3 k) K& a+ u1 h! G' d4 m& P, v7 j) p
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    6 R+ Z+ j8 W. {$ m: q; `( R3 r/ V8 d. e5 @4 I1 I+ T
    When to Use Recursion( X4 o; o) g) R+ {- }; C+ k. L
    7 w' P' j4 e) U" q( I
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    6 h% T' b' O& V0 q7 F% N' `" ^7 b( k2 f
        Problems with a clear base case and recursive case.
    8 C3 v- n  @5 }
    ) T4 O% L1 \: e7 RExample: Fibonacci Sequence: m2 U# d; F* ?/ ~- j
    $ W- P+ Y1 F+ X( U; ~5 P/ o. }
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ) P' Y/ b$ H) D# y) M, s4 `, s* e. Q- z( e3 V( p- ^5 T  `% E
        Base case: fib(0) = 0, fib(1) = 1$ c) e/ F  \9 N7 S4 w. t2 y

    & S( U6 x7 c  H6 ]    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    * i2 T, c! L% r6 B0 S$ W5 c9 X' J- e. L0 S
    python
    " Q: \# k/ |' {- \4 E: H8 I" F
    , q+ ?6 B% p/ e4 S. }: ~( U- n( E+ H5 p! n3 ^9 M, ~4 X
    def fibonacci(n):  R& Y3 M( J8 O) M# z$ e8 ?
        # Base cases
    + L7 m# z+ j/ G9 V* F/ B+ }0 o    if n == 0:. z) {+ s9 }7 P, |( x
            return 0
    5 T% m2 \4 N# C8 J1 b  t    elif n == 1:+ N2 U% \- n5 N  Z9 C% `# s! H
            return 1. D: G3 u, ?; q! {
        # Recursive case3 D0 Z$ F, z: x/ C$ @( [
        else:
    2 R1 _( [' @& I5 B' {) |# x        return fibonacci(n - 1) + fibonacci(n - 2)
    & K4 T# G9 ^+ {' o: p/ s7 B0 I
    % n" h) {" k) ?: b( i9 x+ y# Example usage
    " @, c6 g7 w1 Cprint(fibonacci(6))  # Output: 8
    & t% ]' ?3 z6 `) X& v/ R  u, L+ }$ A# N& l' b% J
    Tail Recursion
    % }% {6 N# b/ _0 n$ i6 ?+ u! k6 P( B! a0 m8 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).
    " W7 X; s+ C9 u( e4 J/ Y% }
    ( L, B+ y# a9 W; Q" vIn 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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