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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 # _9 N/ x4 I. U, E/ K$ j
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    解释的不错! T! h/ S3 u2 I# P, T' y5 o

    . z' `( N5 h- {3 k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。
    1 P+ p& T# }- U; e+ a( {; m0 A+ s) T  K6 W  l( F) K0 X3 _1 i1 z# E
    关键要素
    ) K1 ?" ~  J% @( U- |1. **基线条件(Base Case)**$ }) P$ G6 B- M! w% ]9 N) v' S+ ]& }
       - 递归终止的条件,防止无限循环
    . J! ]0 o; L4 N   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 10 v: I: l0 ^3 Z- j1 i7 f7 L. t

    3 n* @4 s! q- [9 _2. **递归条件(Recursive Case)**3 F* y9 ?5 [% `: g( F
       - 将原问题分解为更小的子问题2 l# n/ p* t4 J
       - 例如:n! = n × (n-1)!/ }" r( u& Q9 F3 C

    0 [$ x1 Z: \0 s! ~' y 经典示例:计算阶乘
    ) m- ^+ Z. |/ B# }% x: _' b3 q( npython) k9 J* W8 J- k
    def factorial(n):+ [* g8 c: g+ o* X" A4 o* n
        if n == 0:        # 基线条件
    4 ]2 a) M0 ]; F& N) y* ?        return 1
    ) H) g- |& ~  v$ ?; S% K9 j    else:             # 递归条件" l- C. S( `+ B) ~' i
            return n * factorial(n-1)
    + P: {& t: A, v9 c- z6 y执行过程(以计算 3! 为例):; y; L" a( B; T5 P2 K
    factorial(3)3 T; M- ^; y4 k* H
    3 * factorial(2)
    % U" ^7 I: L7 P0 h3 }9 h' m" p3 * (2 * factorial(1)): w5 X% m5 h8 L9 V
    3 * (2 * (1 * factorial(0)))
    + f% U/ u; K3 [. c, X' v' v7 O$ S3 * (2 * (1 * 1)) = 6, N( E7 G! {6 i7 j% B0 J! {0 V& {! R; C

    , A- U6 V: R0 Q 递归思维要点
    , K/ v- ~+ Q2 ~% ^" k) O1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ' V1 j2 g/ O/ n3 l2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    , w) _# h1 S9 D; \( n) `3. **递推过程**:不断向下分解问题(递)
    9 s9 v* S5 t* g4 t& I. [# v0 y# C4. **回溯过程**:组合子问题结果返回(归)- U+ W7 l0 U' n( ?5 f0 J

    0 }4 @7 x# ^2 P) K' s% } 注意事项
    6 m* [& q; X+ y) B/ r必须要有终止条件; H2 k% ]* H/ [3 F. [
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    : v" C: k) E- O5 l某些问题用递归更直观(如树遍历),但效率可能不如迭代
    . m5 K7 |& e: }尾递归优化可以提升效率(但Python不支持)
    ; b: h1 [% z2 M/ }; G
    + o2 q, i2 q# A7 T3 c0 s" q# O. L0 \ 递归 vs 迭代5 C7 c3 a$ b: V7 n, u, z9 ]
    |          | 递归                          | 迭代               |
    / o1 R% L' m* W/ F- \5 A|----------|-----------------------------|------------------|! m2 [4 w& X* h5 g' {. M% t5 f" x( O
    | 实现方式    | 函数自调用                        | 循环结构            |" o: }' y+ h2 c; I
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ; n# `3 o# D! {$ q) \; ^) {| 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         |
    2 e* L5 i2 U: V- h) A1 l7 \| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
      Y0 ^1 B6 W- m) N$ k8 |6 ]" e2 Y" E5 D& M* H+ t1 ~
    经典递归应用场景
    ) S' L8 b: k% o6 |+ w6 i5 |$ T+ `, W1. 文件系统遍历(目录树结构)9 e1 T* Y3 T) D$ e0 X2 `. f
    2. 快速排序/归并排序算法$ k! x" P& F$ N9 G+ s0 ]; ~- |
    3. 汉诺塔问题8 t; N; ~+ ~0 s, l7 I# E
    4. 二叉树遍历(前序/中序/后序)' G/ A+ `2 J. V# m/ G; o" ^
    5. 生成所有可能的组合(回溯算法). F$ v" P) F: C8 k' m" C0 p
      @: a6 r" J5 {
    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,
    ! @/ q. o; L2 T# E/ O  |我推理机的核心算法应该是二叉树遍历的变种。; _7 u' x! K. {. \* u+ ~
    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:8 Z# w. H% k9 r( ~* L7 G
    Key Idea of Recursion  x/ F: k6 }" F! [9 @
    : g  r" c" |* P. j1 g
    A recursive function solves a problem by:+ R; j. s$ r5 j, m
    ! L8 C& A" v) l4 k6 {9 I2 {
        Breaking the problem into smaller instances of the same problem.
    8 u) ]3 a: J( P9 C; r
    2 g' g9 n8 x$ X' ]  f    Solving the smallest instance directly (base case).+ R7 b' ?3 g8 V  ~$ a7 T
    : k9 j$ ^( f' [
        Combining the results of smaller instances to solve the larger problem." B0 I! l* h& @) k' w3 X2 c, c

    4 F2 V" z( B& u) z; J3 G$ L1 r3 ]9 jComponents of a Recursive Function
    ' V3 g' u) k. Q6 z' |, X" ~: V4 o, X$ b5 Y% U  I6 {: n+ h& S- g# z
        Base Case:
    ' q/ r, X; e6 m: W6 X  y4 p$ v2 i; z7 R
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.3 W: V% t( e$ y6 L1 [

    0 f# l+ b. y/ c( E+ `        It acts as the stopping condition to prevent infinite recursion.0 l5 w% i: H* O( |0 B6 g( J
    # q6 {5 C- T" k) i" u8 B
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    ) t3 M$ Z, j0 P( O; j
    0 I# F# a' z+ Z5 z) c' ^& C    Recursive Case:
    " z6 X: _# m+ K4 L- n6 E1 E3 |5 K0 ~3 q+ d& `3 w
            This is where the function calls itself with a smaller or simpler version of the problem.' n$ q) u/ n& _& _7 ]: F
    1 D6 K& Q1 N9 Z' u$ k. D
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).' L* e8 `$ j# O7 b' J- p9 ]
    2 A( |8 U# j8 K7 F. E  M
    Example: Factorial Calculation7 r( a' `2 V# [8 j; w

    5 R/ I2 U8 a- I8 d* Y& q5 mThe 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:
    ( b8 E2 E/ V( e! E( z4 J& C- N. P3 m' I
        Base case: 0! = 1& {; Z! x! N( i" `" g2 y# T

    * e# O$ [8 b( u. U    Recursive case: n! = n * (n-1)!
    - J$ j1 D$ t. p3 Y8 r
    * i/ C- Z  c) I/ }  j- T' j. ^Here’s how it looks in code (Python):- G. v/ u% _9 w0 `3 q
    python
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    def factorial(n):; o  J1 [0 h( Y
        # Base case4 K/ N6 j7 ]1 S4 b
        if n == 0:- b* J, E! C" b9 n8 ]8 }
            return 16 \4 J5 E4 d* E2 w* |
        # Recursive case
    8 i2 W2 }% F& S1 [    else:! D2 c+ j0 V" E. r, I
            return n * factorial(n - 1)
    1 R: ^8 y2 j; x/ L% t( Q9 x5 D. [( N( \2 K8 |7 e1 ]
    # Example usage
    # f/ L  X' {  Y( t, Tprint(factorial(5))  # Output: 120; A$ [! P) w, f4 d

    , `( E5 K' h2 `3 lHow Recursion Works
    . k' i1 x( a6 e  ~. S- q4 {* M
    - M) n2 q( `6 k$ s% J5 i    The function keeps calling itself with smaller inputs until it reaches the base case.
    , p: z, l+ L5 }; p! i- w! J4 ]5 e) T  S+ q9 m( U+ h% |8 b
        Once the base case is reached, the function starts returning values back up the call stack.* C; y5 B. H2 h8 ^* {' b; {
    3 L" q- {1 [2 d* f
        These returned values are combined to produce the final result.
    8 N! J, f& E: c4 m4 w* }
    : j' O5 w1 h3 Y9 BFor factorial(5):
    " O/ z. p; ^8 M9 G) ^9 Y4 e1 u
    & c' U) m6 y! \- q6 u( F& S) I8 L" J2 }
    factorial(5) = 5 * factorial(4)- K: `+ W: G: }: k7 s8 H
    factorial(4) = 4 * factorial(3)  C/ x9 r2 }2 `! z- |
    factorial(3) = 3 * factorial(2)+ G# a7 e/ ^$ J6 j
    factorial(2) = 2 * factorial(1)3 W7 T$ ]) K% ~7 y3 K
    factorial(1) = 1 * factorial(0)
    3 l3 r2 P, u& f. t+ b% |factorial(0) = 1  # Base case
    + ?8 A* D" w, v* U4 O
    , e, I6 O7 x3 a6 s, l6 Q3 @Then, the results are combined:  D) G1 P) Y. b" |3 T) ~4 k$ a9 Q6 v
    ; m3 t$ k7 F/ R
    7 e2 k- l3 y# z0 ^1 a/ j
    factorial(1) = 1 * 1 = 1
    " g* H6 e. s+ T6 @factorial(2) = 2 * 1 = 2
    * ~: b. r% H. P. @4 ufactorial(3) = 3 * 2 = 6* S" W- R& y( r0 a
    factorial(4) = 4 * 6 = 24
    ) i0 g/ K3 U: b+ Y) ^- Rfactorial(5) = 5 * 24 = 120! J* }9 T. O6 T# f/ N

    5 r: N2 c( P$ s7 k% B( j8 wAdvantages of Recursion
    2 S+ e8 H# w2 C5 J0 f: f
    ! E$ C$ V: q0 P" M; S$ @. q    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).7 e& M0 R0 R9 X8 l0 j

    % {0 W! y4 o% j5 ?+ `6 w    Readability: Recursive code can be more readable and concise compared to iterative solutions.* Y: Z& n  @& p8 w
    . Y& f+ S7 s, s) k5 v
    Disadvantages of Recursion$ i2 ~, {: e" L: @7 P# ^. Q& M" s
    + r* l% A* @- g: l9 e, 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.5 D  W0 R1 Z/ t+ {

    ) q& q7 p; y) ^! m6 o) Y    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).0 U, t! N' C% s( j3 ^' t' C$ E

    3 i; a' o0 q6 i& w: _When to Use Recursion
    5 J8 M: O+ K# |' }3 P, u# z
      |0 [1 i+ e# M1 o; y! x' i; b    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ( Q- _0 h) s1 ~5 v* J
    ; l$ w/ M" ?5 B* |8 ^7 q    Problems with a clear base case and recursive case.
    % _' e! _9 G1 U& z3 g0 G$ ]- {, [6 F( L: N& P5 b! z: W4 m
    Example: Fibonacci Sequence
    6 C( F/ f3 U- a) ]2 b! t0 N/ t% a& p1 [" n* r: \: I" l" n3 K: w
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    2 K" Y. L9 L  v3 @) {  P+ Z3 a- f
        Base case: fib(0) = 0, fib(1) = 1
    * n# v0 U9 u0 H& y7 Y: R0 A# Z( M1 d+ N
        Recursive case: fib(n) = fib(n-1) + fib(n-2)7 h# o) S1 W0 {3 D# Z/ h
    3 ~# J: [# p8 ~' [
    python
      S" w( W* ~+ h% N2 d4 q( C9 r( |" |! j9 z
    0 m, C3 b7 X) B- l
    def fibonacci(n):  d' G$ O. f: c$ w5 y; |% x
        # Base cases
    1 t4 K4 v: S1 }4 h$ M    if n == 0:
    0 A, k& Z+ D+ Z; n, A) n, T9 b        return 09 ^. W- c" t) K: G& G
        elif n == 1:
    " c" x* s5 q4 N        return 1
    4 f; `. S! b9 W- t$ W    # Recursive case% M6 e* g' {+ c- A
        else:
    + i/ Y* ?" l* b- R' U  V        return fibonacci(n - 1) + fibonacci(n - 2)
    / X' q# a! `/ ?0 C. K- @5 J  [, T; V- Y
    # Example usage' @$ Y9 s6 T3 Z0 F0 o
    print(fibonacci(6))  # Output: 89 m. F- g! Z0 w% E3 N& y

    % N4 r* ~  x- x+ t  s0 cTail Recursion
    1 @4 A! b3 v' ^' g- B! g4 {- Q% s. E' {8 P3 j; g( ^( D$ 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).6 [2 S( P% ^9 W

      f, ?/ s4 ?9 ~  c6 M  J- ~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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