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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
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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 ) s% _& C% r. ^- _$ E

    - H  S5 V6 L5 Y7 x解释的不错
    - m  r1 D) t( s7 b( ~# i
    2 r0 p& L: }1 b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。$ M9 T7 [( k1 W  S4 H) Y- y
    $ b* R) D! `! b# W6 n/ X8 m
    关键要素
    7 ?! j/ L# ?5 D, o1. **基线条件(Base Case)**' r' q# l1 m% E' p
       - 递归终止的条件,防止无限循环* v, D) C! U4 x
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    3 o- A* E; S+ Q. L
    # W! i0 ^6 W  q$ e* ?( T0 t" L1 \2. **递归条件(Recursive Case)**
    ( t" p( }6 l! ?* A# [7 Y# D3 B   - 将原问题分解为更小的子问题% p" d; \, p4 T* o+ E& d8 I- a
       - 例如:n! = n × (n-1)!, x- {5 S) O+ y& E

    ! h' s! r& l  B 经典示例:计算阶乘
    . l' J  Z& r/ Wpython
    7 Y3 U% M: a: b9 I2 p9 I/ j7 Vdef factorial(n):
    % |- M7 p' C5 h# g* l    if n == 0:        # 基线条件' \1 N; v7 U  [5 h
            return 1) G" a7 Q8 q7 f* g* H' s8 X+ u$ F
        else:             # 递归条件
    6 B1 }2 Y1 p1 D+ I* Y0 k        return n * factorial(n-1)
    , {: j. n: }) e+ I执行过程(以计算 3! 为例):& s' I, o/ `6 @% }5 F! H
    factorial(3)' q- |& G& T0 \8 j! \/ y; i
    3 * factorial(2)3 j6 C2 z& ]! I) z9 d" Z0 g) d
    3 * (2 * factorial(1)): G- }+ C" |' F
    3 * (2 * (1 * factorial(0)))/ Q% y8 z, T! k/ k9 i
    3 * (2 * (1 * 1)) = 6$ e4 L& s: @; T& ?0 g0 C: y

    ' V- `8 i! Q6 h/ d4 m0 a" A 递归思维要点+ d7 P9 D5 {; z" W, _% u
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑: H8 V' K( i% O# |/ L5 X. O+ z
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)' Q% v- z( F! r* I1 y
    3. **递推过程**:不断向下分解问题(递)
    : w2 K+ ]1 j( a/ R( o1 k  q0 G4. **回溯过程**:组合子问题结果返回(归)
    % l+ M/ x, d: _7 z1 e( v; a" f5 h  ~* v$ S# j- Y5 `
    注意事项  x1 n. f2 H; r$ C0 F. z
    必须要有终止条件7 ~" ^5 u8 ]/ s8 g' F$ D
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
      }. X8 p6 g* v, d3 A某些问题用递归更直观(如树遍历),但效率可能不如迭代! l& J3 u9 i; x0 W* ^; O
    尾递归优化可以提升效率(但Python不支持)
    3 D; o/ K' m% h' y/ A) [
    * t" E% P% E/ p+ P4 X" q 递归 vs 迭代
    + I  T8 t$ n! y# m' R+ X! w: ?|          | 递归                          | 迭代               |3 _/ s- J3 c) s; _$ J& S: i
    |----------|-----------------------------|------------------|5 M1 x$ R6 x8 Q/ }+ m* Z7 P) w
    | 实现方式    | 函数自调用                        | 循环结构            |9 T% j6 |- |" K" T' j1 S9 T' @
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    9 M5 e- P8 t  \- s& u| 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         |8 |# N3 V; R% m6 ^9 _: u& f
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ) q2 Q) M3 L$ g8 t4 D1 I7 J+ u- [# Y" O* Q
    经典递归应用场景
      f' X+ I. H- ~, B& r9 V8 B2 J1. 文件系统遍历(目录树结构)
    " `" F) e$ X+ O/ i( E) M, [# n; ^/ @2. 快速排序/归并排序算法
    7 c. a, o: A: t/ d3. 汉诺塔问题  a6 o' l1 s* E
    4. 二叉树遍历(前序/中序/后序)
    " u$ F+ b6 R3 {1 z7 {6 t  y) ]5. 生成所有可能的组合(回溯算法); t# a4 Z7 ~( e7 \

    ' m" N: y9 I  v$ F/ 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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    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,/ r+ x* t9 b" a9 D' C
    我推理机的核心算法应该是二叉树遍历的变种。; K7 K" ]/ g) x/ t
    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:
      \& U# E2 U: m0 d; @3 s8 s9 oKey Idea of Recursion
    # e! H% i+ K2 {* i6 o
    * S8 }, C$ i8 c- J( |* F5 n$ HA recursive function solves a problem by:- b# w0 F' Q# c1 Y

    0 S  R; b! u7 U$ @8 L$ d' W/ e9 w    Breaking the problem into smaller instances of the same problem.
    8 Z1 D4 O2 A1 d! H1 ^) n- o# k: a; ]
        Solving the smallest instance directly (base case).
    1 n' z8 N% M' n1 ^9 U9 X# c7 k5 q/ O
    6 l% }3 J8 F7 S) B* k    Combining the results of smaller instances to solve the larger problem.# w1 S% T) R) \- z# Z! q$ R* K

    , E; Q& d) j9 Y0 `* Y5 d3 iComponents of a Recursive Function% M( n" ?! B4 `7 S# u' O

    + Y& m8 Y3 K3 `- e* w. K( B+ B0 l    Base Case:
    8 Q9 Y% f' N+ |$ C/ K; }+ o0 P" J
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion." f* u7 q- {: b9 N1 o$ p1 A% e  S. _

    : C! p, n, N* F. m3 Z        It acts as the stopping condition to prevent infinite recursion.
    - z8 N) m9 a# j
    4 b1 }9 c/ J# D# G4 o        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.* M* @8 b. J- e$ G' L( a

    5 Z0 ^. I) V. G0 p    Recursive Case:
    2 A' n- Q9 c2 [. z9 [- r8 J3 E9 z* o5 t: K  n$ g! [1 t
            This is where the function calls itself with a smaller or simpler version of the problem.- L7 a8 Q2 a- J. g% e( U  A. J
    7 V) I9 [1 v& `4 ]  I; P
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).6 ]% i, c% v" q* a2 m; e

    1 G  ^! E( S" X) {; rExample: Factorial Calculation/ i% T- w8 _- ]

    . h" v' j9 y) ZThe 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:% Q# Y1 i  S# o" J& N! y' @5 I

    9 \% u4 O& K0 ~8 b  R6 L* ~    Base case: 0! = 1  e6 {" {) ^/ K* o) n" W
    8 [. B+ K/ |7 I# t# @6 L$ n
        Recursive case: n! = n * (n-1)!3 F( j5 X# \; r+ g, L$ F
    + r4 x! P. x' b' u' S7 o
    Here’s how it looks in code (Python):: S) D) Q0 ]& J1 A
    python
    5 k; t# ~+ U  Y1 B$ @
    ; C- B* f9 v& \- \7 c; b& h# |
    ( G8 H5 ]3 b- ?" |2 r2 W  @2 ]* c6 pdef factorial(n):
    ' F7 J/ D: Z# m3 a+ _    # Base case4 w, {" w6 f" @# k0 h3 M6 A
        if n == 0:7 ]: t) v5 h3 ]+ u( a* R2 D
            return 1! m+ W) Z$ E. G; Q5 z1 E: q7 e2 ]3 G7 h
        # Recursive case
    % @/ M) F$ ~. w; [1 o+ Q    else:+ I- n2 r( B- i7 S. l! D
            return n * factorial(n - 1)
    + o# X% @. b' ]# V) z, \3 B$ C2 y, j# o4 g4 [! y* W: N2 v
    # Example usage- {8 J. |+ }( {9 ~8 y  S( j8 _
    print(factorial(5))  # Output: 120
    / [8 l, F  n* J: G3 S
    6 s( b' s' c  {: n  A5 E1 [How Recursion Works9 F# z& y0 C1 @0 c9 Y7 A
    8 ^. d7 b! ?4 ~- z6 f
        The function keeps calling itself with smaller inputs until it reaches the base case.2 N" o5 n6 `0 b  P. y2 A
    7 N" ?1 y; V4 T" G/ |/ U% n
        Once the base case is reached, the function starts returning values back up the call stack.. X$ z/ k4 o, N/ f' g# l
    * s3 d  @. O3 U) k7 f2 G5 c
        These returned values are combined to produce the final result.
    5 w; x+ y" d8 `, i, @. v+ d2 c' Y2 Q) w
    For factorial(5):  V5 M1 v( n* p2 }

    6 T8 i5 V- Y) B; X4 D. `  e
    ! p' Q( W, s7 |- X+ _factorial(5) = 5 * factorial(4)6 @" ^/ j( B1 Z: v& G
    factorial(4) = 4 * factorial(3)3 @; _/ ?$ d  k  ]; J3 [3 ?- C
    factorial(3) = 3 * factorial(2)5 M5 f' |' {: `! r5 u6 R8 j" }
    factorial(2) = 2 * factorial(1)
    ) C0 b( w" y; F0 @1 m( Cfactorial(1) = 1 * factorial(0)9 b, V. ^% K; d% ^3 `
    factorial(0) = 1  # Base case; |/ F5 P2 z: b! ?. i) Q
    7 k- U: d1 o2 t2 Q$ U' X* e% w
    Then, the results are combined:
    ' W2 b* ]* b; s& v9 A+ P* O( r7 z( }6 }  A$ ^! Q6 z7 d0 J1 y) ~/ J

    8 [& a: x/ n) m( ~, F- V% Qfactorial(1) = 1 * 1 = 1
    % ^. D+ O' D- W; z8 vfactorial(2) = 2 * 1 = 2
      T0 _( Y" V6 M4 Lfactorial(3) = 3 * 2 = 6
      t; a# A: R7 @, A; s% N9 h% M) Cfactorial(4) = 4 * 6 = 24
    % d7 Q$ u3 }) s1 H7 k! H9 rfactorial(5) = 5 * 24 = 120
    " k/ b$ b$ G* c" _5 f' Y0 w; V7 M- p& W9 X0 k- b8 w
    Advantages of Recursion
    . ]; ]7 E/ q# O) D9 t6 u+ ?) F: ^% {6 c
        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).
    ' R4 b+ L: C0 Y: y# ^0 w, p; m* N
        Readability: Recursive code can be more readable and concise compared to iterative solutions.% F, V6 s* t5 S

    # S  M' r' t  b/ U9 HDisadvantages of Recursion: v5 V4 n/ E( B( g+ q# u: u

    , {9 }/ k" t& a- |! L: e* i  v; x    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.
    & G$ `7 ]6 X5 u' L5 c
    0 i# \  f; e! I0 E7 o: p    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).  P2 T+ W% ^6 d: F0 B- I. A

    & _" C0 w; e$ G. [9 n, `9 e9 oWhen to Use Recursion
    3 g% B2 m/ I3 y4 w
    ! Q1 C% P' T' u5 I    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).1 G) }3 X6 }% G$ y2 |
    / v8 E- T1 ], V& D( N" f( H
        Problems with a clear base case and recursive case., Q# k# m" i/ V2 H
    5 _( k# g- {0 N5 y
    Example: Fibonacci Sequence. d! f9 K" K0 ]. n" V. s

    3 X8 N; j. t# ^. f  gThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:" k- Q! |; J' y; U, G# U

    - y# o! ?1 s7 h& ]& N/ r    Base case: fib(0) = 0, fib(1) = 1+ ~9 ~; b; i( l& C+ q* B

    + B; i+ z/ O  F    Recursive case: fib(n) = fib(n-1) + fib(n-2): v+ p7 }9 f/ V  {  y
    , m: y0 b2 g+ d; w
    python* p$ y8 s0 Q0 G2 u7 @5 h" [
    " n6 r" f: u9 n: ^5 e2 H( c
    # S; j3 ~; G" U$ i* k. T/ X
    def fibonacci(n):- e# y* ]* n6 y4 A) Y4 G
        # Base cases0 A+ r$ k3 A; J& n8 c' x
        if n == 0:
    ( k' I6 N; l+ b1 @  k" X        return 0
    ; y' I( s* G$ e6 h    elif n == 1:
    6 ~( u* v1 H. L7 s- `) I; G        return 14 C. v/ u9 W% ~$ \; B, Y
        # Recursive case
    7 s6 J( y- \& c* O- `' e    else:
    + `6 i  V& D" X! F        return fibonacci(n - 1) + fibonacci(n - 2)+ @& t# X6 r6 I4 }& y

    % S- d2 O% J& g4 c# Example usage, f& T# J* S. M6 c$ b5 j0 b: y
    print(fibonacci(6))  # Output: 8- \# H5 s; x2 M1 |! o5 [5 |
    : `$ E. W! [+ R% D  z: b; |
    Tail Recursion
    8 H# [) N1 p0 L
    9 T) p$ y2 G* }) OTail 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).7 }6 E' [6 C; w3 l( o: l
    * T" y9 i6 C$ j3 k
    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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