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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
     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 1 x( z* l4 {& A+ [
    # v. j  `& G. I1 L2 N
    解释的不错
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    - n& Y9 Q6 D; g  @5 a& ?! x. t! p8 Q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。
    ! d: v0 k, t% I' ~/ M: q, V9 x8 \( p/ u2 b9 h
    关键要素
    . f+ R$ S5 a" ]0 e) o0 E1. **基线条件(Base Case)**
    / d; k: U& R4 w4 a' Y   - 递归终止的条件,防止无限循环
    % X: l( J0 Z- g% Q3 z5 @2 R" t   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1/ V3 g( z5 v/ `; x, x# z$ m. m, s
    * T  l9 s" I1 U; ^& w1 C8 b$ n  ^+ F
    2. **递归条件(Recursive Case)**
    ' j: k7 `* N' ^: n4 z   - 将原问题分解为更小的子问题
    3 v: v! I3 ~  I1 m   - 例如:n! = n × (n-1)!
    ) d0 n" ~9 V2 p1 U
    ( l9 F( |. R. V) ?+ s% A+ A3 F! a! ^ 经典示例:计算阶乘8 [/ v+ [* M9 W
    python
    6 B+ e3 x2 s$ d$ Edef factorial(n):6 ^: A! ^0 [) P: P. T
        if n == 0:        # 基线条件, R% p; F7 _" ?/ Q  g/ K: S
            return 1! [8 I4 \# V' o. r: k
        else:             # 递归条件  \( }& K7 R2 j8 u0 @' C: ]
            return n * factorial(n-1)
    ' ^5 s; U" j: K/ x( d6 g执行过程(以计算 3! 为例):
    # Z# o4 ~3 Z. e, {2 Tfactorial(3)" Z; }/ V/ w0 Q1 \
    3 * factorial(2)
    / c1 w0 ?- D# d9 f( h3 * (2 * factorial(1))* U3 h! D$ h8 p/ X/ L% D( f
    3 * (2 * (1 * factorial(0)))
    # K- k2 T; I! b7 `& {& [3 * (2 * (1 * 1)) = 6$ |- ^) b  Q3 P9 m& u
    & Z8 ?5 Y3 j5 i7 T, E' H
    递归思维要点+ X  h0 a) T# H' P
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑8 v- M+ C$ {' x8 K
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间): H0 m# Z8 c% y* p+ U5 P
    3. **递推过程**:不断向下分解问题(递)) {3 c: |" b/ m3 o: H& M. u( c
    4. **回溯过程**:组合子问题结果返回(归)) t! a, Y9 B8 |. n% y
    9 L5 K# A# @, B& [8 o* @, T
    注意事项9 {) |6 W' ]1 S3 [  N( Q
    必须要有终止条件
    # Z5 v7 }0 x0 Q& W4 }9 I+ H, N递归深度过大可能导致栈溢出(Python默认递归深度约1000层)9 [- p5 _1 f& k  U0 d
    某些问题用递归更直观(如树遍历),但效率可能不如迭代0 ]4 [" |6 e& Q5 a3 ~8 J. A% L
    尾递归优化可以提升效率(但Python不支持)
    " A6 l, R4 o) Y/ j9 [1 a  m2 b2 `$ X
    递归 vs 迭代
    * u/ a9 n! R$ Q$ H! l' j. F|          | 递归                          | 迭代               |. S4 E8 p% l. {! M
    |----------|-----------------------------|------------------|+ M: {% }1 B7 N7 {& u
    | 实现方式    | 函数自调用                        | 循环结构            |' V, d4 }; z: Q( u; U) s7 K; Q
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    " S8 f8 A: h1 d3 R$ @7 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         |5 z0 w! U) ]0 N3 W5 V4 F) o% _: N) a
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |4 e: E* D& ~2 t8 i8 a: j& f
    3 ~% Z, ~$ H9 ]- v$ l' r$ x
    经典递归应用场景3 p' Y/ M- n, e" H* |
    1. 文件系统遍历(目录树结构)
    0 @9 H) m7 |% n, o2. 快速排序/归并排序算法1 x2 k+ G) X- R6 D
    3. 汉诺塔问题
    ; P( u* G* j, e3 P) a4. 二叉树遍历(前序/中序/后序)
    2 t, `* b% A6 T6 r) i- I9 ~5. 生成所有可能的组合(回溯算法)1 @4 ]( H3 ^7 o" e7 Y/ d7 R- S

    / Q2 s2 ?" m  P2 Q% G0 ]% Y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 07:13
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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,; l: t3 u0 k2 `& \! \1 G
    我推理机的核心算法应该是二叉树遍历的变种。
    8 @+ p7 n8 @  r1 \' B8 G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 d/ ?( m0 Z  qKey Idea of Recursion3 [, D' ]+ i2 p2 Y3 T$ o
    : W! c9 q& [/ o7 n# E( }" {+ |
    A recursive function solves a problem by:% s/ X( |( s9 l7 n+ p: N
    + y$ ^: o+ Q% L( @7 B$ [
        Breaking the problem into smaller instances of the same problem.  u2 x1 `4 f; [8 x4 n4 y9 o- {
    - s+ b3 G7 V- N9 \: t$ W# B2 B( V! m
        Solving the smallest instance directly (base case).! j( M7 E, S7 o# N0 t

    / z# E1 V3 {+ C% `) `5 v" [    Combining the results of smaller instances to solve the larger problem.
    2 P; z% z8 Y& g5 J! x
    , R+ P; X( C: }2 A+ ~( x! u- \Components of a Recursive Function
    9 ^* Y5 _  i& P% B: d
    8 Q$ Y. _( |3 j+ m  O6 g8 k4 ^& A    Base Case:. C. }- Y. O* U% M
    3 c% `) o4 u6 A9 g0 j  K% T6 j: [
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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    ; z; H2 @( p$ K, {        It acts as the stopping condition to prevent infinite recursion.6 v! Z( ?! t, N; [
    ! Q& N% u$ v9 n: }
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.1 q: A9 I  n5 D1 v! j- f

    4 P$ h8 c9 T$ Y7 m8 Q2 Y    Recursive Case:" L; C& Q  m2 a

    8 I+ ]; \6 b& Z+ u        This is where the function calls itself with a smaller or simpler version of the problem.3 s0 i' R+ i8 V" v

    2 K7 v: _) B; K6 J+ {, |        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).  Y# Z6 a. c* ~

    6 I6 r3 [4 |" X. }9 m- @& L2 YExample: Factorial Calculation1 J4 k% y1 K& N" e' r

    * K3 `1 ?. N5 Y$ r- oThe 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:. U) y  j+ c; L$ F# d

    # Z7 q7 F7 W4 u3 U% y/ s- f    Base case: 0! = 1
    8 h$ f0 b# r, s+ X, Z! z( H' m
    - Z: F. v; \# f8 s- w    Recursive case: n! = n * (n-1)!6 [+ G( ]% V/ f1 x" ~- r
    ; E3 f0 J4 }. e3 F# M
    Here’s how it looks in code (Python):; A; o4 D3 D3 Y  Y  W- n9 G
    python# L+ G( K- Q0 m& D
    0 y9 C$ b3 }( k9 Y1 w# C

    ) W; A* |+ e% z' Tdef factorial(n):
    9 J: G6 ?9 f: E0 u, ?+ u. q    # Base case
    ) T- _+ u! x8 }& o+ J    if n == 0:
    / l3 Z5 g6 i! j" `  |6 E$ i        return 13 F1 |5 S1 e& Q* W+ L5 Y
        # Recursive case
    & Z4 U; t# f0 t* r  J$ @    else:
    . l3 z, w$ N) B, A! x        return n * factorial(n - 1)5 P% N5 c7 ?" R# U' ?* `/ Z/ C
    9 a0 z9 D2 x/ P. H
    # Example usage
    9 |- j6 i9 _# U: K) C9 o+ Lprint(factorial(5))  # Output: 120
    . V6 _# y% y/ ]7 p' |( f$ @2 \$ i. w, g0 k
    How Recursion Works
    7 B5 s" L* X! y$ Y- |! b' H8 c, a! L7 [& H! |
        The function keeps calling itself with smaller inputs until it reaches the base case.
    + o8 u2 Z7 B9 T9 Z# S, B- k2 z
    . z; Z$ u" h" W, C+ a' J    Once the base case is reached, the function starts returning values back up the call stack.
    3 _5 ~3 U. ~, P. O3 l' h/ Y- x' b; K, T( P
        These returned values are combined to produce the final result.9 ?! o# P7 ^- ^2 I

    ( f+ F; Z2 j: C) v5 S% hFor factorial(5):
    2 F  p, B- a$ c0 a* J" W+ C3 t
    8 @, Y) `2 N8 V; Y) T$ X# u; y6 A# z8 t. B' b* E6 r% b+ y) d
    factorial(5) = 5 * factorial(4)
    - d$ o) p1 q2 r: Z* F4 y( }- S, mfactorial(4) = 4 * factorial(3)' M( G) W5 @) A, c! H
    factorial(3) = 3 * factorial(2)  B, j& Q7 [! W: F, ^5 R3 F
    factorial(2) = 2 * factorial(1)
    " E+ R! w) j) i: z1 G8 r* bfactorial(1) = 1 * factorial(0)- y- I. S; \0 o5 L3 Z, |
    factorial(0) = 1  # Base case
    9 X0 t, w7 d# R1 r0 i1 v" j' N) T, E1 j) Q! [! c3 i
    Then, the results are combined:% v! m) r2 s, \$ f

    ' p: k$ ?7 ~7 B$ M) T
    + `" z6 H4 a. m9 x+ i# ufactorial(1) = 1 * 1 = 15 G$ P6 X( j' }6 L0 J
    factorial(2) = 2 * 1 = 2* r/ G* m- L- |9 A1 L! D8 n& c5 r
    factorial(3) = 3 * 2 = 6
    & z, k* X6 A1 r! t; h. O' ~/ gfactorial(4) = 4 * 6 = 24
      Y2 ~1 z6 ?$ z0 S8 }5 jfactorial(5) = 5 * 24 = 120
    4 m( N2 f. ^& u! X4 b; X) s2 Q( w- K! ^, l2 o7 h
    Advantages of Recursion& \: K; A. M1 t. z

    1 Y  ]8 [, \8 _! P( y    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).% p% Q% U" Y+ Z$ K( g7 }( @* Y$ w) i+ \% B

    , @" [8 r5 w3 @% B/ p6 S    Readability: Recursive code can be more readable and concise compared to iterative solutions.( y: P* Z& x2 A3 \( U) Q4 c- Z4 t( e

    , V: u! n" }9 l/ nDisadvantages of Recursion
    3 T, S% @4 ^- {* ?5 a% M* b& `6 a2 @7 O, 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.
    2 l4 }# N" H6 r3 T/ q! h& t! T" J  f% P8 a; S9 x( i
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    + C3 k+ \  P, s) J( p( A; r& ^6 k# s( K8 B
    When to Use Recursion; \* F/ D/ l* C
    1 w% s  |' M/ b! U1 N! p. t9 U
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).$ b& [  U; b, i# m* R/ w( h

    % D- [. ]4 H! C9 j+ |1 \    Problems with a clear base case and recursive case.5 A4 W! a8 j+ e- U+ n! B7 Y

    ( k& D/ j! i; w' Y1 T5 b1 Z/ kExample: Fibonacci Sequence% H3 O  m: c' H8 q7 B& u+ c, p3 K
    6 m& I; e; i) B- ^& u+ N; P
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ) x; J9 d9 I6 n6 [2 W6 L
    3 \' f* I0 r& E- z    Base case: fib(0) = 0, fib(1) = 1
    . r$ O" m* u% t' w& U: I  U3 H) M( M% `
        Recursive case: fib(n) = fib(n-1) + fib(n-2)6 e+ l, C6 x1 K8 F( Y3 ?
    % h) m- r* e& p3 B% I3 m) i
    python9 Q! G# s- ]& r* j  p( H2 P
    1 J: f6 n. p$ a5 I4 U' n
    3 ^, k" a9 r, U0 e* K
    def fibonacci(n):
    ) a# R, F3 D% q, r* a; H    # Base cases8 j' M5 @$ P3 E- c  i
        if n == 0:
    % e% y: }' E' ~" p- q        return 0
      y) ]+ H5 P- ^# x* v8 d" E( J0 U    elif n == 1:
    3 {/ r. d/ |( P% p, z* _; f4 e        return 1
    # D* Z  M) T) m    # Recursive case  U$ c7 W; A2 N; H. J6 i( P
        else:
    2 M3 ?, ]) [- R6 J        return fibonacci(n - 1) + fibonacci(n - 2)6 I: e1 U# L4 ]8 A; q

    0 |2 i+ F/ y/ D, ~" ?# Example usage
    ' K2 b+ V7 x7 _4 aprint(fibonacci(6))  # Output: 8$ S& p+ M) Q6 b3 F" M- J# P( k

    / d3 E4 q9 I6 {3 c) _* c0 Z4 @Tail Recursion' \2 ?, g) P+ q) s
    * f7 G- {5 U2 Z- l
    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).# M4 N" Z: P9 t1 d# E
    8 Q, Q0 C( l3 E; L5 q6 ?+ s5 m: J9 ?
    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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