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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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  • TA的每日心情
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    2015-11-30 11:11
  • 签到天数: 2 天

    [LV.1]炼气

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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 : I- K. j6 A; C* T+ K8 i; ?* l  ~

    . p3 `( l7 C( \6 ]8 G解释的不错
    % X  U/ }$ \" M" _& t3 _% {4 _
    5 ]1 T1 x" u2 i4 Y6 c8 S3 o- {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。
    9 U5 p+ c% d1 i9 f# p8 m# ~% l& y' k4 a6 h
    关键要素, |! q1 E2 {; M% x
    1. **基线条件(Base Case)**
    ' D  ]# f% p) n3 \   - 递归终止的条件,防止无限循环6 y1 Y& M: m" O- x! H) V4 g
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 16 ~9 r- h+ B$ g0 w3 l- H4 E& l% Y
    : O2 t& Q% E" n
    2. **递归条件(Recursive Case)**0 g, Z! a' w+ P# }
       - 将原问题分解为更小的子问题
    ( S/ z! v& Q0 t1 I& F   - 例如:n! = n × (n-1)!4 R0 x9 b6 h2 {3 f6 i6 m' \' f$ p

    ' H) b. i! R- _. p, T 经典示例:计算阶乘
    6 z: ]/ o6 `! {' C6 Kpython
      _1 z  g4 T; k6 s% X# Odef factorial(n):
    ) @+ s! b2 I0 _' N    if n == 0:        # 基线条件, p* k4 q4 _7 W! P0 _+ w) q
            return 10 l2 X! J* `, W" R+ ]2 m. C0 Y
        else:             # 递归条件
    : ?1 n6 U, o0 I; N5 |0 a) G        return n * factorial(n-1)+ y2 k- t* _: \+ Q* z9 _
    执行过程(以计算 3! 为例):; l. X( x& O2 b$ [
    factorial(3)4 u9 O5 I% O* N8 Y4 U9 V
    3 * factorial(2)
      j9 b  v4 m+ ~+ f  c9 T' E0 e3 * (2 * factorial(1))  y. j8 r5 L2 R. A* x6 I. ]3 s; ~
    3 * (2 * (1 * factorial(0)))
    ' g. X! q- W8 I  }0 e0 f$ f3 * (2 * (1 * 1)) = 6
    & S; ^- |: z2 h& ?0 J
    4 i5 Y& K8 K3 H5 n1 ]. U5 ` 递归思维要点) V( G, Q. E1 `% Q6 {
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑' u6 D; M2 ~9 B; E: H3 ~/ A
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ; G8 _% l8 \7 t0 }5 K1 l$ ]" ^3. **递推过程**:不断向下分解问题(递)
    # k1 m0 S" o9 C! Y4. **回溯过程**:组合子问题结果返回(归)
    2 c! O( M, E4 k& L4 o* k9 u# d* X: |$ y
    注意事项6 ^  i' l9 T& w8 W6 |! Z  q
    必须要有终止条件4 `" i  A& h# W6 j
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    : l9 j9 g# i( ~, D某些问题用递归更直观(如树遍历),但效率可能不如迭代/ Y5 h- X8 I  |& N" `7 b
    尾递归优化可以提升效率(但Python不支持)" V+ I1 S2 }% V0 N

    + w2 a+ Y0 V& O% V/ P 递归 vs 迭代
    : m- t* v* s: F5 \|          | 递归                          | 迭代               |
    , w8 y' Y- Z+ j8 `7 \|----------|-----------------------------|------------------|
    1 N+ C7 Y. I& a" m& b: x| 实现方式    | 函数自调用                        | 循环结构            |
    " R& c7 y6 d3 k; o# @+ t| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |9 f2 b9 K8 O3 V7 U$ f1 x
    | 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         |
    - V. P& u( q1 X/ h6 b| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |, B& M/ f, H9 v3 q- @: e

    $ K6 s4 J" r3 C- o) g 经典递归应用场景
    7 N8 D  y" `" L1 z) M, n1. 文件系统遍历(目录树结构), l$ h) `- w& Z2 P9 q
    2. 快速排序/归并排序算法
    , e1 R: W* j% ~! k+ U3. 汉诺塔问题
    7 u: V; B' L4 T" w+ K4. 二叉树遍历(前序/中序/后序)
    * q; i. b0 I: n, y, i5. 生成所有可能的组合(回溯算法)
    % F( w' T! W0 _; s' O; C
    ( e" j  O- ]8 w, u! `9 D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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    2 小时前
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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,
    7 c8 A+ g" [$ o! U4 k9 d9 `) f7 _我推理机的核心算法应该是二叉树遍历的变种。2 A( ~  l# E8 i+ M
    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:
    # U1 _0 {  S  D9 m- V9 `8 MKey Idea of Recursion
    ' F; u( O! l+ Q
    * l* h& q; |1 z1 L+ x' S" B" X3 nA recursive function solves a problem by:
    0 a1 c: h$ m# R# i* Z* O8 |$ w! {! g5 \/ H5 [" @7 c
        Breaking the problem into smaller instances of the same problem.
    ( A7 ~. C; r- c, G5 H4 v/ x0 n8 z9 d1 `
        Solving the smallest instance directly (base case).. X; h, V! Y! Q, w, `8 I: G# X" J

    , f0 U, D* x$ t' c* Y7 W: f    Combining the results of smaller instances to solve the larger problem.( z, ?3 z) w# d4 k; @" t7 [/ F

    5 C6 i: F1 }% K3 @0 WComponents of a Recursive Function
    # {8 P! M  o/ q$ U
      Z2 x/ `% x# k" _* b    Base Case:- Q7 `2 M$ i/ r# @
    . b7 j' b- l" U5 _
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    4 ?4 o/ e& g' W" {4 W
    ' V) r) g/ F- G: Q: }6 U7 z        It acts as the stopping condition to prevent infinite recursion.
    & j- `8 W5 T+ t8 ]/ u( `- e0 X, T: c+ W" ]* u* e- w. c
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.1 H0 d6 w5 y5 R& H" C% }& G6 j& B/ ~

    3 N3 L: b. Z' a! ]; Y% g    Recursive Case:$ K" l$ s6 ~) j
    0 B. G& {' h, V% @
            This is where the function calls itself with a smaller or simpler version of the problem.
    ) l3 I1 k# j% v' B/ Z& h; Z/ K8 ^$ M
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).7 B  H. _- C$ _- m+ c

    " P/ h6 Z0 f" a5 mExample: Factorial Calculation
    " y$ l3 Q: u" B2 T" T# Q( s7 S. N( W4 W& v: p- P
    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:
    : |  d; V& ^/ R7 l; ^/ m. V3 Y" U5 Z( J) |; }% `# j
        Base case: 0! = 1
    1 d( `6 g# f% q/ u: j  i0 m. s  O1 R/ X9 w3 W0 w
        Recursive case: n! = n * (n-1)!2 a7 I! k; Z' \
    ' r; ~5 @. d2 }( L$ Y2 q+ n- `
    Here’s how it looks in code (Python):
    2 u+ U- |- n" [" H# apython, M3 }( l3 k  L0 G
    ! ~! _1 ?# ]! ^# f. I/ v0 @
    ) d; a( D- H# j4 D* ?% c2 [" c
    def factorial(n):: A# [0 F0 O" o' p  @
        # Base case
    ; X& `  S5 j' G& b7 f    if n == 0:9 I- B+ S6 X" p/ L1 K- u* `
            return 1. Y3 Q7 D, y# A
        # Recursive case( \- j* g; T& Y4 {2 M
        else:
    ) h  m5 X  _6 ~7 F7 W+ I2 I        return n * factorial(n - 1)4 T: P1 \& C* d( c& [

    & r2 H3 W# B  s5 m7 }' r7 I5 R; r' A" _# Example usage# J) E9 ]: s. i1 O& @+ Q' g3 a
    print(factorial(5))  # Output: 120
    3 a! }6 P8 C8 e* u
    3 S/ X% Y0 i) r+ i" u: j  j2 lHow Recursion Works1 B4 u* d  r3 V7 G# A% M
    - }+ \& Q: v7 L% k
        The function keeps calling itself with smaller inputs until it reaches the base case.
    ; O( v8 u  o& e9 n
    : v% m, H  u3 S5 t( B( [    Once the base case is reached, the function starts returning values back up the call stack.
    * ]/ d) [# }/ @# M) g, W
    - a: ]6 M$ x3 V! Y    These returned values are combined to produce the final result.; }* q# d! U7 T, D9 ~3 h& y4 |
    / Z( \" i4 F0 ?* p' a) B6 A% y
    For factorial(5):
    . u( |0 y8 d- Q/ W" B# \, @4 J0 M  D
    ( c. n5 n! [. P7 d
    0 j( Q: V/ {! ^. Dfactorial(5) = 5 * factorial(4)
    ; L6 l: O% [* X  \" Ffactorial(4) = 4 * factorial(3)3 K! f  K5 g, q! {
    factorial(3) = 3 * factorial(2)
    7 @3 I5 i% u1 D, wfactorial(2) = 2 * factorial(1)3 J7 E: C5 b( S  ]# O
    factorial(1) = 1 * factorial(0)
    - N' D0 e  i, n2 l2 i6 r0 zfactorial(0) = 1  # Base case
    0 `; F3 m' j  |) ~5 G% F* V$ |! X8 a% u2 h+ K/ I0 d2 z
    Then, the results are combined:9 _3 {7 t' N3 q) s
    3 q; w5 O  F( c0 g
    : m$ O+ \. w2 X: {
    factorial(1) = 1 * 1 = 1
    2 u0 r' m" V3 s* V7 X( Cfactorial(2) = 2 * 1 = 2
    5 t: @/ x* @# n/ T% Rfactorial(3) = 3 * 2 = 6( U. E3 [) w5 }6 s! L3 t7 x
    factorial(4) = 4 * 6 = 24+ J1 u' I8 {" I7 M" a+ {2 V
    factorial(5) = 5 * 24 = 1202 c3 p7 r" ]! X0 j; H1 q: X" r9 i
    % Q) ~1 o; Y3 C* a2 c$ y' i
    Advantages of Recursion- [7 G; g2 Q6 z- j7 }

    ) L* [' t7 N' Q1 R    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).
    5 K  L3 n2 V* E' I1 f2 d1 O* s
    9 P7 Z- {/ ~& M9 W( J; b' E    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    9 ~) ?+ U) F( I6 c) |
    9 P) F3 ~+ k; y, ?2 b$ h: l3 _; B, }Disadvantages of Recursion
    1 p) S. n( x; T" n2 z4 `, f9 O1 V$ B: K9 {: D6 A& t! l" p* `1 `
        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.0 {7 H# U& a! A7 U, i9 K
    8 K9 J' Q7 d$ L: w$ [+ m  h
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).7 q. N! G9 t- N  l" b" m1 y% @
      R5 X! \% d* G
    When to Use Recursion
    $ D; |- h" m, v2 }4 R: d5 M' m& x; {9 Q! d, ~; ]
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).: A) q. A! U& @" j- n& s
    8 n0 `" m6 Y% e: L9 g1 X) x
        Problems with a clear base case and recursive case.
    ) Z5 ^& @: Z9 l- `1 _4 O6 `
    , t) G+ \% U* i& p- }Example: Fibonacci Sequence; H9 ^) b6 T* j1 I' E
    # A/ D' Z. O7 T  _5 R
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ! C% S7 _# N/ ?1 A9 |
    # o# f. |% ]4 Y+ e) @    Base case: fib(0) = 0, fib(1) = 1
    6 W3 n  g+ x" y+ n  z+ {, v6 f8 f* U% x) ^  i+ O
        Recursive case: fib(n) = fib(n-1) + fib(n-2)4 Z' N* L7 R! Q% O4 q9 @1 {. I0 K0 f

    0 h) B6 ?' h, r! ~) g: e8 F  p( ]python
    * H/ O1 a: v, A* C$ W+ }  l: t& I5 o! p8 F  a9 r5 f. J

    ' V' a. C9 r0 z& X# edef fibonacci(n):
    8 D$ l1 w" k  S1 S# d; P    # Base cases
    9 P9 T7 u; o, E2 W+ H/ l7 R6 v    if n == 0:
    . Y& k5 h0 A, l/ d+ `9 d        return 0( j8 n. Q! S6 H" C
        elif n == 1:2 Q( A; B, l7 ]8 j5 ]3 U
            return 13 _. }: S/ t5 q7 {6 {7 K; E& g0 P8 W
        # Recursive case
    # \4 m+ |+ p* l' ?    else:
    - X- r" O$ b1 T" e8 J( s& z        return fibonacci(n - 1) + fibonacci(n - 2). ~/ l: E* ?1 k( ]" [( P% l5 m

    / K, ~) S/ b% r" j4 A1 @& X1 w, J& `# Example usage1 L/ y' y' u; g5 {! O
    print(fibonacci(6))  # Output: 87 A5 g  w% k) A1 R9 k5 S+ m

    , Z8 R* C% z" K" a0 {' i+ FTail Recursion
    8 ~9 q- J) \, R
    8 Y9 v9 ^2 g% d9 P& m/ M  R& bTail 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).
    $ p3 R% X) L$ ^5 u) P. M2 B' I% d1 K9 A
    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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