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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
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    9 r6 H+ ~. `+ i6 U# o解释的不错; j" `# g3 m7 s  N& h
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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。( I0 h& g4 c" v0 W1 V. b
    6 ?7 E% c' }# \2 P3 `& C' v
    关键要素
    2 J: i, Q7 \6 t) |1 @1 Y7 c/ S6 t$ M) |3 N1. **基线条件(Base Case)**+ C" A% [5 \' Y! h- [
       - 递归终止的条件,防止无限循环
    4 D9 s" }( P6 X4 q  `- V* e' I6 {   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ; _* T0 J/ g/ n2 I* O
    ( ^. I: {/ d" q5 I& a# p5 S2. **递归条件(Recursive Case)**, [- J+ B& B/ d( x
       - 将原问题分解为更小的子问题' M8 M. q) A3 n& ~* z5 t5 q! A. K5 I
       - 例如:n! = n × (n-1)!- w$ |7 e% `; q: E7 |. K) u0 V) _; a$ @

    . m0 p8 m6 D% m. ^3 N8 t/ V 经典示例:计算阶乘
    3 K/ f1 d: l( N2 I1 ?python
    " s( B+ V1 {3 r$ K- F7 S. Ddef factorial(n):
    5 N$ [* s( m7 X/ E# B    if n == 0:        # 基线条件& P& @, P( z% j5 |) g4 Z' O
            return 18 i( u3 O2 R* u1 i; _2 a5 E$ [9 p
        else:             # 递归条件
    , E7 K; m8 Y9 v. ^. d" t" k        return n * factorial(n-1)
    ; m' ?+ o3 P- @: }7 \8 D, G5 V执行过程(以计算 3! 为例):2 g4 o! A- A1 H9 K! }- D, f
    factorial(3)
    % o0 o/ A+ }  Q) j3 * factorial(2). U+ v/ l' Y# d: v
    3 * (2 * factorial(1))
    $ N9 A' H, g, Z% j8 _3 * (2 * (1 * factorial(0)))0 E, Y6 h$ {4 l
    3 * (2 * (1 * 1)) = 6
    & G) C" t/ J& O7 N( {
    ; f' h: M2 b& D5 I 递归思维要点7 G; M. `/ j0 f
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑' T- R2 W/ J0 d2 M0 {' u' q
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间): V6 E! ^- p  i
    3. **递推过程**:不断向下分解问题(递)
    1 [% z! a) G- _& C) @3 l( N4. **回溯过程**:组合子问题结果返回(归)
    : G: N" m5 [/ P2 I% L2 r# D& s% Q+ R- ~1 `; q" _4 L
    注意事项6 O  M/ Z- h* h- U7 v& T/ h
    必须要有终止条件, D. h6 e* Y; g$ l% C6 r* M
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)) C) f2 Y( a6 M* f9 w
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    % h+ _. \5 t$ O) f1 E) b尾递归优化可以提升效率(但Python不支持)' v3 f) t" O3 _! i& a, a
    6 G. y- s! u1 H" W- H
    递归 vs 迭代
    4 _. N* e2 G% f|          | 递归                          | 迭代               |
    ) J$ @/ c1 J4 L% P: ]|----------|-----------------------------|------------------|
    / Q+ X8 |" L6 ^7 R5 [& M, \# V| 实现方式    | 函数自调用                        | 循环结构            |
    ' p4 X: t3 ~. h" s8 e" ^, I| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |7 L0 \5 ?/ d0 b+ l" D
    | 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         |
    " t( l0 T  p: R3 M1 E| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |0 r( o  \% ~6 E/ G: [* Q
    $ X$ X" x+ J% Z/ A
    经典递归应用场景% W6 N6 F0 s% `6 S; i  C* z1 Q5 A$ M
    1. 文件系统遍历(目录树结构)
    . T6 a3 A7 Z! X  b2. 快速排序/归并排序算法
    $ Y) E3 @) i4 b" o5 z& O3. 汉诺塔问题' T  {" B( o, Q) v$ t- n
    4. 二叉树遍历(前序/中序/后序)+ ]5 o  E3 i& O$ d( T3 K) L
    5. 生成所有可能的组合(回溯算法)
    % ]. p) t( h8 U
    3 c4 @. f& M* i3 Q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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    前天 06:39
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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,! N" R0 j* t) @" L
    我推理机的核心算法应该是二叉树遍历的变种。
    ; E6 K! `' L. d7 g! u9 F# s7 W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 g8 x7 X3 Z" N# {4 [2 TKey Idea of Recursion# e* x( y0 s, P& F& N, b
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    A recursive function solves a problem by:" J1 x) W, b( v$ a
    " L; x% ^7 l# [* V6 U2 H
        Breaking the problem into smaller instances of the same problem.1 \1 L$ [0 b7 u0 |5 R: [% ]( ^
    - I2 E5 s& l7 P# V2 s
        Solving the smallest instance directly (base case).
    " N, v0 v& a- _. R* x: K5 X9 ?3 m! h; a$ z1 ]
        Combining the results of smaller instances to solve the larger problem.
    # w; C4 B& r: i7 E
    9 F( J7 U9 E1 x+ R, g9 @& EComponents of a Recursive Function
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    8 E1 _: N3 b& b    Base Case:
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    2 N* a: X' h6 D# `; w        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    0 m! ^9 |- \: N6 h; m3 l- R0 _: o  {; G- L
            It acts as the stopping condition to prevent infinite recursion.& B8 j" q$ G( k; e$ x

    ' c- j) S& S6 f  c        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    9 L* H5 m8 |- W" v/ v* Z& p7 b. D: G8 M
        Recursive Case:
    ! t6 i) ^: C: {- d; U, V3 I4 _. p: o4 z- w9 ~
            This is where the function calls itself with a smaller or simpler version of the problem.1 u" L2 \( ~! }  |$ P7 a

    $ u/ y2 c; K8 ?6 g" }6 Y0 U9 [; E- v        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    7 `9 B: ~0 _- i. e- {
    4 [5 ]+ }" n( A& ~Example: Factorial Calculation
    - l) P$ D/ N0 s9 K/ |0 @1 v9 m# V5 Y0 {! q( g1 u
    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:0 ?! h6 P' d6 J/ k$ k
    & }9 ?8 E4 q/ ^
        Base case: 0! = 1; _! f: J; e& @6 j7 D

    : U7 T% c4 e& j( p. ~    Recursive case: n! = n * (n-1)!
    1 }) I9 B3 |6 U
    2 D/ L9 T2 s. R! X6 QHere’s how it looks in code (Python):  j& `2 L0 `# O
    python
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    6 [5 b+ a  w/ w9 j2 K- J3 U6 a" [" y6 N) I1 M: c& U+ t
    def factorial(n):
    4 y. @  F2 U" T. Z2 z% b% p: v) T    # Base case
    9 e* Z5 O# u) L, ?# K! H    if n == 0:: {1 ?# N5 _; V8 X0 i7 w4 `# @
            return 1
    3 @5 F" h* n( h. r. ?$ K0 o- ^+ f    # Recursive case$ }2 ~+ j( [2 |  N7 g: P, U
        else:: f4 U' E2 R. l+ D% W
            return n * factorial(n - 1)
    ' f. ~# w% i7 d0 n6 w
    3 c: N) e# J" A. a2 l# Example usage: j6 h6 `* `1 k5 B: ^8 l9 ~' \2 t
    print(factorial(5))  # Output: 120  P1 G0 t$ k9 U, c
    / P; F- t0 l' w( V' S9 c
    How Recursion Works! T3 P7 l/ _$ c7 y3 j, W  C4 B% Q

    ' R, {$ v& z! v: P3 f% l$ b    The function keeps calling itself with smaller inputs until it reaches the base case.* e+ [6 p, q* m( g' D

    0 P- c% Y- d. [" j  N/ ^: ?9 ]! j/ c    Once the base case is reached, the function starts returning values back up the call stack.
    ( U4 a' f+ h0 l2 G9 |
    5 a7 |0 r1 {% e9 [4 i% D  `    These returned values are combined to produce the final result.0 w$ a# m9 I' b( ^: c

    2 F* `* O( x, B: {; P8 _For factorial(5):
    . ?: |0 y7 p& x: ^! G
    ' P! T/ F7 I, ?0 M& b$ y
    . a) M6 n" J" lfactorial(5) = 5 * factorial(4)
    ; x6 R* t1 S( W  i. S- d" Ifactorial(4) = 4 * factorial(3)
    - \9 F% x- B* f) C7 {factorial(3) = 3 * factorial(2)# N1 E. T9 _5 ?$ N0 U
    factorial(2) = 2 * factorial(1)% W- }6 c6 C: }
    factorial(1) = 1 * factorial(0): N, S2 o  U9 I4 ?: c* N
    factorial(0) = 1  # Base case
    * @3 d3 L2 B! Q" [! b5 g: l  O8 @
    * s% }  ]( v7 K& U. l/ p4 nThen, the results are combined:$ Q- d* S9 Q0 ?

    : N6 @9 H; s  g5 [5 {2 Y# P6 t; \5 _" x, z! g* L
    factorial(1) = 1 * 1 = 1
    ( w5 e. t* J: o3 t1 afactorial(2) = 2 * 1 = 2* _& l% x( m" L5 y! i
    factorial(3) = 3 * 2 = 6
    + @. }6 e0 c: @+ R3 B+ Xfactorial(4) = 4 * 6 = 24
    ; D  z/ D) g% w! M- W2 ifactorial(5) = 5 * 24 = 120* f3 H, o( ~$ p8 p) h- [
    1 f5 R) [; U2 v  d6 f2 r
    Advantages of Recursion
    2 b. a2 x2 |9 M5 |' w
    0 U; g  Z, [, }5 b- y' o/ k6 W" m8 z    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).
    $ y6 t) }2 U/ k2 y9 w& x) n
    * m1 j7 F8 }7 ]  r5 z  M    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    5 E5 n8 p' Z8 B9 d! ]5 Z  I
    7 Y: M7 U: O: N- |& E0 {0 e) X1 IDisadvantages of Recursion
    2 i& o1 ^  b( \3 R( g9 v8 U* W. H# T5 b- p
        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 t1 X( p$ n; O
    3 l! v6 H. a* ~; C" ~7 w
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).+ ^1 b$ L$ F1 f8 u
      v1 @  F/ |! }. K8 G) z% G
    When to Use Recursion* W6 ?  M, T& Q8 F0 k- K8 E

    9 C; X8 ^, b6 e0 E" U& @    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    * n1 [' G! A9 G7 Z' n. L" a. o' G  Q
        Problems with a clear base case and recursive case.+ a' j0 v& V" J0 f; z% |/ N

    ) F; m9 X, R0 BExample: Fibonacci Sequence' B8 f4 s  ?9 P7 }* H. N- e$ F
    + d3 E7 z2 x% k/ X: Y; P( R9 I" j
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    / D7 L" i  t. l% V" ~- T" j4 e  N) a, \
        Base case: fib(0) = 0, fib(1) = 1
    6 O+ Y" [# V9 t+ ]( ^; y0 y& \' ]6 L# I6 C* c" ~/ h5 ?: `
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    # X' t' l7 o: c7 p
    2 A, g* G' D" L  t" Fpython0 x7 P, F' v/ Z7 a/ E6 F) w

    ( t* @9 L* K; y( `) P7 j9 k+ g8 t+ j: ?8 R1 ~6 h  f
    def fibonacci(n):( Y, w. K( c' \' h" x4 p  j
        # Base cases+ D+ q1 e% ^) x* G
        if n == 0:
    # a/ ~: _) F# ?- v) z        return 0
    , [$ ?, }( o' _% g) d0 R    elif n == 1:- `  V$ h* i/ d! ~! t! q% g
            return 1
    ' d6 P! w- b) k! ^    # Recursive case
    $ D$ Q# _. I9 D; S  s' T) T    else:( F) o, F3 l2 y9 v8 e
            return fibonacci(n - 1) + fibonacci(n - 2)) E: M' R# e$ j, y
    ) B1 q2 O# U5 j, c' F% R
    # Example usage
    9 Y6 j9 ^, j# M2 dprint(fibonacci(6))  # Output: 8  d2 [. g- z6 D: j+ @6 c8 O

    * }+ _9 _7 d- g* LTail Recursion
    & A" ^; G4 X9 ~& R4 F6 V. V' G' z) J) `* e7 ~
    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).( f; a. x' i8 f% \) l
    5 G0 N1 C( i" ?) \4 l
    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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