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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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    2025-9-8 05:08
  • 签到天数: 3 天

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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
    % D& M3 h8 S! d5 l0 q4 o
    - u- U- f) t; m; u: a- k" @解释的不错: V/ l& p/ V9 F) P% X
    1 T8 I, Z9 f6 ~3 D
    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。4 U. H4 g' ^/ r% [% x
    - F+ g; Y  v' `9 i
    关键要素: q$ Z2 ~3 K% d9 S* N& D! m
    1. **基线条件(Base Case)**
    ( L  V* }( ^5 B; o  `   - 递归终止的条件,防止无限循环
    3 K& J" T, v2 W" l. H6 V2 m; H   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 14 m, f. D1 l# B& f4 ^9 _3 `
    ; x: ?8 _/ q1 {& U; \
    2. **递归条件(Recursive Case)**8 }' }& D; ]/ O) {" F4 e
       - 将原问题分解为更小的子问题
    , [' o, {& T' Y+ }8 @8 d   - 例如:n! = n × (n-1)!
    : j, J9 P5 c6 X3 A5 e. R
    $ {4 \7 w' p' k$ ~ 经典示例:计算阶乘
    : H$ d# p4 w: N  opython" X  P) W& j6 n5 A+ r; ~4 c
    def factorial(n):
    . I; n+ Z2 G8 c) [, c4 ~8 U) V( b    if n == 0:        # 基线条件
    . K3 b2 U: |( J# ?4 g* J( O        return 1
    5 [* P" x, @/ J* {, N7 `    else:             # 递归条件
    9 Y7 {- ~# e; {" P        return n * factorial(n-1)" c, k9 e/ s: j. o) W* c, G' v/ [( w
    执行过程(以计算 3! 为例):$ T4 X" ~2 x& V8 _" `
    factorial(3)
    9 p/ F8 t# j- `& F3 * factorial(2)
    2 J) t: |0 p, y9 x9 @* H" v/ b3 * (2 * factorial(1))
    8 `& Q. j$ Z, `5 l7 o* J' t& n3 * (2 * (1 * factorial(0)))
    " w8 {7 Q/ P& I2 t1 F* `' m- K3 * (2 * (1 * 1)) = 6
    $ _2 Q  x( u; q' ^/ S8 }% g
    * _' z8 n" U* q) _. P4 P 递归思维要点) {5 b- Y+ U/ H0 T4 p: d
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ) P- v  o  U& \. ~+ a, k2. **栈结构**:每次调用都会创建新的栈帧(内存空间)  \% e9 U* n  U6 V
    3. **递推过程**:不断向下分解问题(递)
    * z9 R9 P7 t- O% x' A4. **回溯过程**:组合子问题结果返回(归)
    - `. d6 s( k! n9 @2 v8 p7 e, {) ^2 g: `
    注意事项
    2 J5 Z/ ~: h/ U3 Y$ @# A必须要有终止条件
    $ k0 |! b8 k- C0 `递归深度过大可能导致栈溢出(Python默认递归深度约1000层)+ h% f% g+ x" g% B
    某些问题用递归更直观(如树遍历),但效率可能不如迭代) Q$ k& p; d9 ?# q: t. Y& c
    尾递归优化可以提升效率(但Python不支持)
    4 N$ E) Y3 }1 P/ H& `, Z- j8 K8 X: a/ E
    递归 vs 迭代
    - W3 g, w8 o! p4 p! B|          | 递归                          | 迭代               |
    , m8 R* C( _. l. q1 v+ Y|----------|-----------------------------|------------------|5 I0 B" A1 i* u: m/ r9 O& y
    | 实现方式    | 函数自调用                        | 循环结构            |
      M+ J( H" z: h* [: e| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |+ f$ \8 ?& G; Y) b5 g2 Z! C; b
    | 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         |
    ' ?) e0 m7 ~5 R, v0 I5 o8 {| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    : \/ b2 s$ T" B2 L# g
      A7 l8 |4 E/ c. B7 H& @8 A 经典递归应用场景
    . {; s% ~' `5 E8 Y7 I+ y1. 文件系统遍历(目录树结构)
    $ }/ f- P8 o. Z8 X" A2. 快速排序/归并排序算法
    9 N- y( _2 f9 v" T* S9 f! h3. 汉诺塔问题; v+ W+ s2 t* p" d) ~" ]# N
    4. 二叉树遍历(前序/中序/后序)
    + S0 ^/ @! V0 }' E" P5. 生成所有可能的组合(回溯算法)
    8 f# K- L+ ^+ V; D: p0 q1 r" V6 w6 H3 E# E7 h5 C
    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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  • TA的每日心情

    3 天前
  • 签到天数: 3217 天

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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,5 e3 N) T+ f( W: D+ N
    我推理机的核心算法应该是二叉树遍历的变种。
    ! B8 S$ z, R: ~$ i* b6 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:
    6 p0 x, L3 O# R5 DKey Idea of Recursion  C& s& f5 f* h" w8 a+ w

    % i' b- P6 e4 a1 ]A recursive function solves a problem by:# ?: G' t# Y, g
    ; P* e* T* n* w
        Breaking the problem into smaller instances of the same problem.- p# t. b! P2 v! w4 f
    - W. }# I- j) ]& H: |- y  r# T
        Solving the smallest instance directly (base case).5 m  q4 z- j. f; H3 \5 u

    ) e% o. f" |, O9 u+ P# m    Combining the results of smaller instances to solve the larger problem.
    3 i4 a9 Y6 i! E
    ! ]/ U: b# ]. R& U2 r" e# ~/ D* lComponents of a Recursive Function
    / d5 o5 e7 l1 Y& ]1 h: J3 k
    ! k) L& N, ~  F2 x" K    Base Case:
    . A8 Z7 Z) P) ?3 N7 u
    ) W# w1 v, c6 l) [/ {5 H        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.& C4 l. e# f' a& V8 ~) _% f

    / L* J8 O" L: I) U  S! ]        It acts as the stopping condition to prevent infinite recursion.
    " Z, ?, X: D0 U4 K% v" O3 b6 v; I
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    9 J( I, C2 G* r: t# U
    ( n4 o/ \0 ^) t$ R5 `) v    Recursive Case:
      p$ w0 o5 W4 q$ m* z) X! i" @7 \: r- C. r0 j8 \* y
            This is where the function calls itself with a smaller or simpler version of the problem.
    $ e$ |/ |* ~1 N3 ^8 o
    % _) c2 @" J0 ?* a, K) W        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).7 W8 _1 m# Q, Y

    + K- w1 W. K; j) i- v! @) r. Z4 EExample: Factorial Calculation8 [  k7 x8 w  w/ F( v9 ?
    0 f5 k' h/ y; Z/ E* R. d
    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:
    ' l+ v& m: a  f% K" V3 ~+ _. Q) @$ d0 [6 W% T" K! T5 b# _" {( J
        Base case: 0! = 1
    3 m" Q; Z$ O2 s' Z7 [. N! N4 a* I0 v% W
        Recursive case: n! = n * (n-1)!
    * U' A9 g0 `. t4 e# r# I2 b$ c7 Z! ^( e$ x$ t
    Here’s how it looks in code (Python):7 P: a' I+ t7 x+ e
    python
    # m' N7 v$ s; n% z* s4 n" u8 }  ]# n

    ) s! ]8 Q7 O/ z0 L3 K, Y9 m$ Ddef factorial(n):5 u) r: k- q3 O2 q
        # Base case
      F  h6 e9 ?+ N# m3 u2 H    if n == 0:
    - W- x/ q) P) g, q        return 1. e$ R  H; M2 u5 R
        # Recursive case
    5 ]( d9 w! R& r6 @5 X# l8 g* t' T$ u    else:
    5 y( }, i% ]  ^$ E! U$ t        return n * factorial(n - 1)
    1 I  P- j4 Z5 L5 }& Y4 N, ?' u$ ?4 Y' ]3 _% e& U
    # Example usage
    0 R" \( g6 o. ~. kprint(factorial(5))  # Output: 120
    . g" q7 }; X2 j
    ( c6 p& H1 w8 d2 YHow Recursion Works! a& k, f, m7 i8 ]  s& |  z

    $ S! J. U$ B% G$ @/ B3 K    The function keeps calling itself with smaller inputs until it reaches the base case.
    ( u' J# W. {% c
    $ r; r6 m' O7 e1 B    Once the base case is reached, the function starts returning values back up the call stack." R' k0 f& Z- B  e" h+ \% t

    $ F& `# a$ j2 t8 \# t+ p    These returned values are combined to produce the final result.! a9 N+ ~# c3 m) s  C. Z! z

    9 G" T* e; Z2 [5 xFor factorial(5):
    $ I* c& b3 E0 F7 ^5 q+ k' ~" P- x4 p9 ^: X

    # J- y7 y/ j0 T; ifactorial(5) = 5 * factorial(4)# A3 u( r2 a; A  ~6 |0 ^3 I# S5 w1 T% d/ v
    factorial(4) = 4 * factorial(3)
    ; u% H/ E% I2 Z- Tfactorial(3) = 3 * factorial(2)% Q- Z  D4 v# n% y, Y
    factorial(2) = 2 * factorial(1)
    . @4 q6 Y. k. Z3 |& w& K1 x, gfactorial(1) = 1 * factorial(0)0 q6 ^, x4 X2 R$ F7 p. H
    factorial(0) = 1  # Base case
    / k# Y% P: ^" O# M- W
    * P' s9 q0 C1 H% y" E& |Then, the results are combined:+ q: @) m7 }  c  s
    $ q/ B* x+ t7 w0 I6 C% n

    " _5 e7 X% k7 o4 k4 ^' I( c" M9 Dfactorial(1) = 1 * 1 = 1
    1 K# ]/ K# [6 l$ I  C( q) q% E: y2 I( v% ufactorial(2) = 2 * 1 = 2
      p" N$ v$ |# q% q; K6 \2 i& Sfactorial(3) = 3 * 2 = 6
    " k; S3 E0 @/ ^; ]% efactorial(4) = 4 * 6 = 24
    0 j3 f; `" t/ y, Y) K1 ]* `; Kfactorial(5) = 5 * 24 = 120
    ; i, Q+ Y0 c5 f5 H0 \2 K$ Q' Z) v' X0 \1 D( {7 n, x
    Advantages of Recursion
    & f, Y3 ]+ {4 C( E, T% E
    $ G; ^; q: g0 v1 `. h    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).$ G) X# G- h4 j: K

    5 z" Q9 I. U% j. S6 l; k    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    5 ]1 N0 R) K. w3 ~: }6 P4 J' m- d6 c7 A$ \0 L
    Disadvantages of Recursion8 W% u9 u$ u  H  l2 C

    6 ^" L* h4 M7 i" m5 w4 e; A    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.1 _& {; l" D' }
    ! L8 k; O9 I, a: l
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).( ~3 m9 ~" ]- L. G0 j+ N
    ) _5 [3 v- D. I3 B1 ~/ u2 N
    When to Use Recursion
    ( V; {, ^3 }9 G
    * I- x4 l& X2 O$ q- d1 d% G    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).2 X& L, X3 R6 Y7 T4 q+ y% C
    8 J* J+ R- Q8 _; T/ w! t5 v
        Problems with a clear base case and recursive case.
    0 p& Q7 T: _+ z2 N3 o
    1 ~- ?# v* y; @0 XExample: Fibonacci Sequence
    2 A' N% x* d: E! I3 H
    / P# f4 e. V4 g* Z3 L+ g* ZThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:# K" g$ Y' k7 O3 y% u9 @' Q

    ) e% t: u* Q+ Y- G* v    Base case: fib(0) = 0, fib(1) = 17 f/ v; m3 I8 W
    ' M0 W7 X/ }! e; ~" l9 }
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    5 e5 S( K6 I0 [+ }1 a! h8 c) I6 I& z/ @& L4 e6 e
    python
    4 Y9 P: H) a. w  R' L8 m
    9 F2 e& z( l! A- x8 A) k: y  W7 N+ c" m9 c% `% t: L3 D3 u5 q
    def fibonacci(n):6 W# ~( b5 K. l. x& Y: K% `
        # Base cases" w: d8 g" V! G3 ]# j' ^6 O4 K
        if n == 0:2 l6 t, Y3 _8 y0 W6 s  E* o, a
            return 03 z/ D6 U5 t0 G1 d. W
        elif n == 1:) ~; C% S3 b  l2 T# x$ |7 @
            return 1
    % H' M' \! F- D+ B4 t    # Recursive case
    ( m% K& R% B' m3 L    else:
    ) g, ^. P4 ~3 Y) a: W7 F        return fibonacci(n - 1) + fibonacci(n - 2)
    % G8 S0 D! l! x1 \  ]& n2 U1 [# X/ Q7 m$ U. Q+ u
    # Example usage
    - R. O* f, f4 k' ~print(fibonacci(6))  # Output: 8
    ' @+ \5 b) P  U8 I8 `, x9 `: b% C: k. K) |
    Tail Recursion2 z9 L& o- h! ]+ K9 r
    1 t/ J' w1 k3 u. e( n! C5 R
    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).9 p# C% J3 o. q0 _. E

    0 W+ p  X9 ?9 i7 ^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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