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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

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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 ' q; y7 C) l4 V8 b/ f' q" V
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    解释的不错+ ~: Z( q+ [( C9 e4 M- l) \3 R
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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。( z$ @! B  b, a
    ; O( W) W; Q) B- ?
    关键要素
    " Y- V2 ]! R1 a* d- ~4 r+ a1. **基线条件(Base Case)**
    1 U9 Q: f3 a- x- r1 v; H   - 递归终止的条件,防止无限循环
    / J4 [: V% ?: K   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1  C* C2 S: ]2 C  O
    ' ?! a8 y% Q) n5 l; ?4 }4 {" x
    2. **递归条件(Recursive Case)**5 w( G3 a& o5 o  O2 Y
       - 将原问题分解为更小的子问题
    + C- ?- g$ W) N2 E- P% c1 @  y1 P   - 例如:n! = n × (n-1)!8 M, }3 V' ?9 D

    9 T/ J( n% {1 S" H$ O( c 经典示例:计算阶乘6 L4 S; H  Z5 A/ Z6 p) a) H
    python# a; b* l: @! _7 ?' h
    def factorial(n):
    2 i* C& M% {" H; m* Z3 i    if n == 0:        # 基线条件
    * ]2 [; r7 i0 l: h        return 1" x- U/ Q7 H/ F, E9 M4 `" J
        else:             # 递归条件- Y7 a& {& Z( b+ }8 E; L& S1 v: z  i" |
            return n * factorial(n-1)
    2 h2 c: \) d  U& v: q2 A8 t执行过程(以计算 3! 为例):
    ; `) J9 `! f: V9 wfactorial(3)" z! \8 L* R  V" X4 k
    3 * factorial(2)
    5 V& P% h+ j' P3 * (2 * factorial(1)). {' G1 `1 ]* C1 E( L
    3 * (2 * (1 * factorial(0)))
    2 y4 i0 n1 b" `% c* ?& e3 * (2 * (1 * 1)) = 6
    - D) n1 V/ O2 B
    & k" r. |; i/ J: m* z, U0 o 递归思维要点
      y* i  \5 C* H1. **信任递归**:假设子问题已经解决,专注当前层逻辑) v* ]# [3 L- ~9 n! G/ Y4 B
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    2 z* s$ k8 f, A3. **递推过程**:不断向下分解问题(递)* T4 v9 ~" m4 _  i/ P& r
    4. **回溯过程**:组合子问题结果返回(归)+ w# g! i$ X+ b1 ?- W
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    注意事项
    6 T1 p, @  |9 B* {必须要有终止条件! d5 P8 m- T/ k0 [0 R
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    7 y9 O) P9 m; ]; H! z! `( s某些问题用递归更直观(如树遍历),但效率可能不如迭代* I& P" G, l1 Z' m9 U
    尾递归优化可以提升效率(但Python不支持)  ~1 T3 b- u6 z4 C
    - T4 f  g; w3 v$ @7 ~" K
    递归 vs 迭代
    # P9 ?6 D! k! M3 [+ J|          | 递归                          | 迭代               |  E4 G' U' T0 c8 R$ J3 N2 N8 x
    |----------|-----------------------------|------------------|5 f4 T& R) J2 B8 J9 ?
    | 实现方式    | 函数自调用                        | 循环结构            |: q% Q& E/ A- i) p. r
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |0 S8 v- |, p0 C" 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         |/ m4 {+ }- Y  Z8 L3 C6 |
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
      y' F- V$ @' U: |) r/ @- U
    3 H) g4 s9 g% H9 A- [ 经典递归应用场景2 G3 T' G# F4 K; G- T8 }3 b; c
    1. 文件系统遍历(目录树结构)7 |% v  D$ ]5 _; c6 g8 z
    2. 快速排序/归并排序算法$ E" Y+ w. n) l1 N: S
    3. 汉诺塔问题, x# t+ R$ Q0 L/ g- {% y- D  j
    4. 二叉树遍历(前序/中序/后序)- k0 ?1 t% J  c" E4 b
    5. 生成所有可能的组合(回溯算法)$ m7 _8 W4 S& \% A* W4 o
    # n7 \5 y: _+ a! ^
    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,5 K5 h7 v8 N8 ~0 E( O
    我推理机的核心算法应该是二叉树遍历的变种。
    4 T" @8 u+ v8 n1 l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:, x; D9 h! @- }8 Q2 F% W' u
    Key Idea of Recursion
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    A recursive function solves a problem by:
    5 _1 `' t9 S* y1 U0 a
    ( v4 ^% L9 S% @/ X) a4 P0 I    Breaking the problem into smaller instances of the same problem.
    ! n) x. \* `, j( D. [6 O/ @
    ! ?& ]; @( I2 y) V1 k! A) s    Solving the smallest instance directly (base case).
    4 H0 S* U; @: E. O8 D4 v9 s0 y- y) ?$ l. l: g2 ~, {) `' Q8 n
        Combining the results of smaller instances to solve the larger problem.
    # U! a; U! ^  o2 y  P# b5 k7 s3 E3 T: O* S0 Y0 ]9 s
    Components of a Recursive Function# B( h; f4 I5 F/ u
    6 x7 g+ G8 }) R2 d
        Base Case:& }, z* z1 W/ [/ D. w

    + ]+ h! X3 m; T+ i% V        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.$ N+ Z  f$ \! x; a0 C( _! L

    + ?6 E& ~8 o5 e; V, y1 Q        It acts as the stopping condition to prevent infinite recursion.
    & @0 d& [( v. y! @  P% a( Q# |
    4 {& J1 q. o. P, _1 G2 R, z        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.; i/ n! L' {6 B" Z
    . v6 A7 V( u' B! m' X
        Recursive Case:4 c1 F1 m0 r& s+ c8 M# @* E
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            This is where the function calls itself with a smaller or simpler version of the problem.4 d: ^% A6 s, k7 h

    # V  A& o' d: T! S8 w% A3 p        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).2 c( f7 B" q5 Y+ _
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    Example: Factorial Calculation
    $ r' U& Z" _' z$ l0 n
    # x6 {- p1 w% HThe 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:3 m1 a7 t) x, N8 [2 q5 G
    & P6 D# S" L0 f
        Base case: 0! = 10 Z7 g/ C& h9 H; w1 J
    7 t/ c' z$ H$ L0 \5 {
        Recursive case: n! = n * (n-1)!& f+ u2 S+ n) O# {* @( W
    6 y* C, s9 i# v1 ^8 {9 Z
    Here’s how it looks in code (Python):5 d2 L6 ~$ T$ \# [3 X
    python
    3 W: Q" o$ a, ?. {
    8 l$ H+ q+ _4 e0 W: v8 e9 P9 y1 `1 e% e6 p5 G# f, p9 v
    def factorial(n):
    % F  {! G# }+ c7 G. g    # Base case
    * R, L1 u) L; ]7 r* q  B6 u    if n == 0:
    2 N, ]# \$ |; I& c, a( `' S) u        return 1
    - V! u7 l# m8 u( [, @7 \3 v    # Recursive case& @) y8 s4 z* F7 z7 N
        else:
    1 Y+ k" I8 h6 V5 Y* }( J6 w) W9 l        return n * factorial(n - 1)" f9 {* x: I+ B+ Y
    ' {1 r6 |( L( O/ k
    # Example usage
    : Y, D: S8 s' i; R+ l! o* Lprint(factorial(5))  # Output: 1202 A- D; v4 D; }/ e7 V
    + x/ V. ^, E$ }5 C
    How Recursion Works
    2 `5 ?# z5 g6 M) j8 m
    ! Z1 L$ E. a5 U1 i8 M3 Z    The function keeps calling itself with smaller inputs until it reaches the base case.9 i) D( k* Y; @! r  M# H0 |
    ) m* J  a$ I" b# j6 H! B
        Once the base case is reached, the function starts returning values back up the call stack.9 ?6 V) q( t3 x* ~

      @8 ~# v+ b; b: ^8 [  N    These returned values are combined to produce the final result.
    " Y* J% d2 D5 E% X1 k3 D& s5 n
    . k; ^6 ?% L* O1 x8 TFor factorial(5):
    1 f. K, ?8 [6 T, s( F6 }$ ]1 e
    * X# S& f' C: H3 c3 u1 i' I& S) D' ~4 L/ \9 m
    factorial(5) = 5 * factorial(4)7 `7 V0 z% c. u4 O) i
    factorial(4) = 4 * factorial(3)9 o5 }# K# v1 m& K. c7 h  w
    factorial(3) = 3 * factorial(2)' a5 p2 d/ q7 G+ I: J/ r# c0 y0 n
    factorial(2) = 2 * factorial(1)0 B2 M, k0 I6 ?: s$ F, l
    factorial(1) = 1 * factorial(0)4 ]6 w% H5 I8 F, a: B! ]! t
    factorial(0) = 1  # Base case6 d  M5 W- ]: L" @" h  a; o5 k2 j

    - ~, U7 n$ G3 [. y. J( HThen, the results are combined:
    % A( K$ V% c1 n5 a3 Q4 H
    . {, P9 f6 m: o. e6 J. k4 x" k/ M) Z8 r9 p) `
    factorial(1) = 1 * 1 = 15 j- W1 h7 h; I3 z- i
    factorial(2) = 2 * 1 = 2
    - J8 G% i1 g1 F1 F% Z9 B4 g6 A9 qfactorial(3) = 3 * 2 = 6/ O. B& Q0 w  y3 Y& `
    factorial(4) = 4 * 6 = 24& g" q4 \+ w# D2 I. P
    factorial(5) = 5 * 24 = 120/ q! k! |$ E3 I+ `3 s* h
    / I% ?" A9 l% A" t( y
    Advantages of Recursion
    1 ^4 \% G7 ^# G# i% b! C+ b! i+ s8 T$ f+ E- [" L6 K% f. Q; ?/ T) K1 A
        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).
    9 R! z. }. T8 z5 R) e  u5 K) U9 H
    ) @  Z6 e  `# i  e3 i) s    Readability: Recursive code can be more readable and concise compared to iterative solutions.2 v. H" B, {( D% i8 E* O2 E- u

    # V( U) s( f% L7 }, \# ^8 w9 KDisadvantages of Recursion; ]' P4 Q0 Z- f  Q" I% T# w8 l
    1 R$ ?2 d, t2 b, |/ |5 @3 t& t4 W  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.1 d8 y" S5 ~' f7 T9 I" i  `

    9 E* D6 I7 }1 O' o    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ( K4 a7 r! X0 q2 a6 q
    4 r. ~  \2 b1 I" ]When to Use Recursion
    ; d: H$ s" \7 ~$ {% t6 H
    + `$ y, Q' x- ]; D    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ) H1 j% j" m. p: |& g  ]
    . S: v- v0 Z8 T$ P: `    Problems with a clear base case and recursive case.
    $ N2 I$ [& \8 t$ k1 ?5 Y
    5 Q# d$ b. ^6 c3 u6 P# @% XExample: Fibonacci Sequence5 i1 t9 q. ~9 |( @
    8 [8 t" L# c3 _# I' H8 I
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    : T1 H& K& N" d2 M, Z' e5 d" H6 J0 H2 g" T: ^9 N% v
        Base case: fib(0) = 0, fib(1) = 1
    $ y4 J- B  C+ X9 N$ l) B5 U
    . O) n# d' _2 a9 k+ Z! i. ?    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    / O; j0 J9 `- a5 T
    . g3 ]9 D2 a: e) O* d. @python
    ! u, u3 ~3 Z5 L. R# `( L8 S4 }+ q9 I$ _4 `3 J/ x1 |

    & L1 W: r- n+ j+ I  l1 Mdef fibonacci(n):) U2 Y; V* z2 k! Y# [) k
        # Base cases5 p- w9 b$ e  e" C% A+ }# l
        if n == 0:' h: P6 N* Y' ~
            return 0( S# r0 h' p$ K" P* w
        elif n == 1:/ D) o9 D# D) u
            return 1
    1 d* {7 p0 t0 M: e    # Recursive case. I  Z+ ^  _* @6 d4 p" R: [
        else:
    & a' ~5 w$ w/ v4 V. o. b7 {        return fibonacci(n - 1) + fibonacci(n - 2)
    " L6 t$ g( V6 p0 P/ l$ B: i
    7 V7 R+ o& X' p  }' o% s( b% o# Example usage
      ?3 G% |+ ]& g7 Eprint(fibonacci(6))  # Output: 8
    ; R8 D2 n; E# q. e
    7 s3 ]2 D1 `: ]: n  ^Tail Recursion
    / n" R! x8 `& u$ U8 _. y) I" a3 l' W
    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).% \$ ?5 C" ^9 Q5 c) _) b

      S9 x" s2 }  q3 yIn 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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