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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
    开心
    2025-9-8 05:08
  • 签到天数: 3 天

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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
    2 ^& |6 E! Q- D) k$ u# p9 s$ p2 V! e* I
    解释的不错
    . \) \" R+ h4 |4 V, |' e% l0 V# O1 S4 h( y4 w7 A
    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。
    * R+ u( }3 h* m9 H6 Q5 D
    + w' t6 L% O9 q( t4 ]% l9 s 关键要素
    & T1 e! _9 x1 S1. **基线条件(Base Case)**0 ?; b6 P. Z7 p* P
       - 递归终止的条件,防止无限循环
    * r5 O. p/ v# r. d, x   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ; Q7 y+ y( Y5 p2 M
    3 A5 u9 m6 K$ m  l$ g0 S, l* M0 z, q2. **递归条件(Recursive Case)**
    ! E9 F$ Y% Y" Y; q   - 将原问题分解为更小的子问题. W0 {- P" }! ^9 ~' Z7 n' D$ l
       - 例如:n! = n × (n-1)!% t! L" N" e9 [  u" f

    9 J+ X0 `; B* k8 P% u 经典示例:计算阶乘2 [2 }* Y2 @0 l( W* T7 U# h: W
    python( u, \, \9 j4 C! P% C% n
    def factorial(n):, f, ?* u0 w% P& x
        if n == 0:        # 基线条件0 H0 t& c$ t! n, p6 |& Y' M
            return 1
    6 i. c  F8 ^; R0 _  e8 ?( e    else:             # 递归条件
    + O( Q# J$ \6 q  x" |$ m        return n * factorial(n-1)
    ( l. a' ?! a( h, j执行过程(以计算 3! 为例):$ t1 L: Q* o* y8 E" C/ p* k
    factorial(3)
    1 B0 P7 @" l6 Y3 * factorial(2)" e6 l) V! `) X* q$ N5 K* M! m
    3 * (2 * factorial(1))$ `7 N2 ?% k) P: z9 T# G
    3 * (2 * (1 * factorial(0)))
    7 Z- M" A( q" N8 ^  @9 b2 l: R3 * (2 * (1 * 1)) = 6: p2 G5 U: J! T. ^1 v
    # n3 _, K. R) ~
    递归思维要点6 g! e" M- @6 e0 v) k
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    , n8 ?) _9 Y. {; G* |/ f& G2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    % h& @& a0 a9 L1 E5 x+ q! v1 }3 q' i3. **递推过程**:不断向下分解问题(递)9 |+ R" q7 L: K6 `9 J' K5 a
    4. **回溯过程**:组合子问题结果返回(归)9 H3 d* u; k) V4 b! C& r& I* d
    - [* ~, U) K' A" W1 S0 u. Q) o
    注意事项
    & W. m$ h' e6 e% h" F+ A0 O必须要有终止条件
    " k* o: X: a! {9 j; g递归深度过大可能导致栈溢出(Python默认递归深度约1000层): U6 Q2 U9 e( Y0 p. Z0 _, ~
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    * q+ \# S' R% w7 r7 f' g) N9 v1 T尾递归优化可以提升效率(但Python不支持)
    3 r4 d1 Q, ^" D* |- ?9 D- A+ I8 U) A" _9 y2 V. `
    递归 vs 迭代
    , ?$ n% W# i( Q8 m+ @7 ]|          | 递归                          | 迭代               |5 U: Y+ s; {; K3 Z
    |----------|-----------------------------|------------------|, G7 u" q( i3 ?+ P% W% d+ J
    | 实现方式    | 函数自调用                        | 循环结构            |6 H4 i$ ?- k5 ?% T$ H* ~
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |3 N$ U) `6 k" g5 Z% 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         |
    $ f" q( l, T1 k  Y" L; L5 M| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    * l' X8 l- T) `% A
    8 P+ r5 p5 T" x 经典递归应用场景6 @/ c! n4 A# R, m3 d4 |8 D% T5 t
    1. 文件系统遍历(目录树结构)
    2 P9 \$ s8 a7 E9 w6 w, `2. 快速排序/归并排序算法
    8 g7 E& I, W& E7 [- |3. 汉诺塔问题: e' j1 [- ]& \+ A
    4. 二叉树遍历(前序/中序/后序)
    % Y# M% e& @0 |, O& u3 f5 Y5. 生成所有可能的组合(回溯算法)
    ; C+ _0 k7 B1 W1 O- f0 R" j1 s& a6 \$ j, U: O; x
    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
    郁闷
    昨天 12:19
  • 签到天数: 3107 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,; O$ K+ I% c" o! i/ g
    我推理机的核心算法应该是二叉树遍历的变种。8 w  A- ^5 v7 ?8 W3 A
    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:
    ) y/ S" V3 P( p' L1 ?Key Idea of Recursion3 }$ g( ]" m. ^( p8 |, Z. I
    9 e3 Q: U! m  v: k( Q# C
    A recursive function solves a problem by:8 b8 Q3 d& H3 I' _) N- u0 k! E, i
    $ z( j3 y* Z1 i5 `7 T% G5 a  b, y
        Breaking the problem into smaller instances of the same problem.6 @1 W! L" n: m# D( S" i

    " C: @" t% P' }# t    Solving the smallest instance directly (base case)." F4 o8 \& O: S% W+ I3 v& f

    2 G; p: Q& S0 g8 Q7 y    Combining the results of smaller instances to solve the larger problem.
    1 @( z9 a* O+ [8 {# f3 U& q: k. J2 o
    , N5 [) k. b3 [" [: w4 {Components of a Recursive Function' |5 u/ `/ ?% Q" b; J
    ) A" P& {# z1 p. M" f) T
        Base Case:- R! R( T; P1 P, N4 m3 t' T
    , e  Y. g: }) l5 `$ r. w- H' D
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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    8 J  I0 s/ i' I) x) ^# c$ T4 y        It acts as the stopping condition to prevent infinite recursion.
    2 R* N' l! w; h4 x
    1 S- X- q: U) H% ?+ s        Example: In calculating the factorial of a number, the base case is factorial(0) = 1., ]( `6 i, Z( b; X
    " t7 S/ e4 U' n" e  R% n
        Recursive Case:. s) G$ L& q; A9 V" h( U
    7 ^) b* E( V- T: d
            This is where the function calls itself with a smaller or simpler version of the problem.
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    7 i, m5 V; l1 p( G# s5 v        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    % G! w+ M- T7 @& P. J! ]7 h" [7 A4 e2 P1 I3 x
    Example: Factorial Calculation  B+ E* A! [  e6 [% W6 J

    . k1 L& S9 J& F3 L$ P9 BThe 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:8 y5 X2 q6 q2 [' f+ s7 S3 @

    1 [' d  |* O& R2 B    Base case: 0! = 14 `) y/ @! C& W5 \0 I# x

    + |4 h) w# K9 }5 J0 ~    Recursive case: n! = n * (n-1)!
    6 e8 t2 @/ b9 q8 T
    $ U7 f" p; h# b) ]; l  M2 ZHere’s how it looks in code (Python):
    1 k- X" L# \8 H' f) W; P+ zpython. P: N1 C3 P. y9 C' W: }% r4 V
    " @$ ^7 V9 z- b& x" O' [, G

    6 |0 d  @3 ]5 y: v4 Rdef factorial(n):
    8 s0 p$ ~( k( L' S: s! Z) B    # Base case9 N! E" D. w4 P3 L) D
        if n == 0:
    . q: G7 [9 V1 z6 ]        return 1
    : d, r' M, A' D, [% w    # Recursive case/ N1 R2 F* c4 x8 t& r/ V) u
        else:
    1 a" K$ p: r8 J" x# D        return n * factorial(n - 1)7 \" A- M! T4 Z! t& }- S4 t
    # v& Y& T+ l3 F5 x) Q
    # Example usage
    / Z9 z, l4 C* {& ]: u( L+ lprint(factorial(5))  # Output: 120) Z% t4 N& E0 b' \; r

    1 i; D( V' M. w; F+ Z. CHow Recursion Works3 c& u% v4 A* N3 S1 I2 x6 u& c

    : j- m0 D5 K6 k' o3 I5 j    The function keeps calling itself with smaller inputs until it reaches the base case.
    1 ]1 E# |! Q+ C: R
    ' g4 r6 I; i. C8 @. C) K    Once the base case is reached, the function starts returning values back up the call stack.9 U$ [* |( G+ Q& i% |* d- d. r

    ; {" D2 e+ m' ~    These returned values are combined to produce the final result.
      l* e6 q9 _! Q7 F+ Q: m
    " I: K4 R) @' x" Z, Q- {& e/ pFor factorial(5):- \8 Y$ I* _5 Z
    3 M9 F" b& r' N6 T7 I

    4 z1 \4 B+ b9 a" q, k7 J" A! N0 pfactorial(5) = 5 * factorial(4)
    0 N5 N/ {1 l4 `$ Xfactorial(4) = 4 * factorial(3)7 D/ B7 E1 ^1 K  l# D6 d1 W
    factorial(3) = 3 * factorial(2)
    , p+ Z& e1 H, V- h7 C1 p" ~factorial(2) = 2 * factorial(1), W( l4 `# D$ T0 s! f/ S$ D6 T
    factorial(1) = 1 * factorial(0)0 h: Q; f9 S; C3 l
    factorial(0) = 1  # Base case- T- s. {8 Y. F9 z
    5 [0 u% z/ o2 E: b/ E0 t
    Then, the results are combined:
    7 Q: t9 `0 S5 M& m& O/ S. Q& ?% z
    ( X; N, M- B& o
    6 e0 E& m  T" T; A7 qfactorial(1) = 1 * 1 = 1
    0 _, F5 h) b, f) O$ N/ a3 f5 zfactorial(2) = 2 * 1 = 2
    6 q5 Y% Y) n& k# C+ J, c' Lfactorial(3) = 3 * 2 = 6: b, K$ x, t: T* _# W- Q) V; R
    factorial(4) = 4 * 6 = 24" u3 T) @+ ^- d( J2 t- z/ G
    factorial(5) = 5 * 24 = 120
    3 |4 t9 e/ v$ v- ~, O+ \! H0 t  X* F- a7 b
    Advantages of Recursion
    ) b9 g$ z8 Y% t( I. K+ d0 a' K8 K8 b' {* z" p. L
        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).7 p  ?& j: @2 p4 k/ a

    9 I) E4 M# M/ y3 n4 e    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    2 Z; ]6 f6 f1 r8 g: ]" V' r. Q4 D, K
    Disadvantages of Recursion
    7 Q; F1 O$ Z7 t" T- Q+ w& q6 K1 G3 v5 `) Q6 H: K; g6 u
        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.
    / j6 P( ]9 ?* k3 M" _9 f& y% f
    " m4 l3 c0 P0 U6 L$ T4 }. D    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    & s) W! `$ P/ p8 k2 i& }
    " {% I/ D/ \' q" tWhen to Use Recursion
    ) p3 R; d4 ]; X; m3 U. R0 h( o* V- n/ s
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)./ v/ x( {/ }+ v( C

    . K' @. Q6 @2 i$ i$ `. e7 J! v    Problems with a clear base case and recursive case.
    ( P- l4 n# x! C. b$ \) J
    0 R/ N" E3 F) o$ I, aExample: Fibonacci Sequence- t" V& A. @/ J* K
    ! y2 V7 B$ \# C5 \
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:0 z5 |1 q% `* a5 q
      I* n: _5 g! R( u9 j3 n3 g4 v' q
        Base case: fib(0) = 0, fib(1) = 1! p5 k: n4 n- F4 I& M+ ?& f5 f. }

    " @4 z# _/ ?4 X1 ^: w7 y) K" b8 E    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ! r! U. X* m* c! v, W
    ( L2 `! w9 {% ]& D4 Dpython9 m& q/ E$ p; h  ~* L" e

    1 `' [/ P% K2 A, F7 ^" M& N6 b; W! z+ ~) ]* D# p; b$ P* x0 Y
    def fibonacci(n):/ X9 i  N7 o3 {! o5 J* O7 v( T& B6 [
        # Base cases7 {  _+ E5 V0 K0 e+ ?: S' W
        if n == 0:
      D8 W6 w: k  S$ f) P7 P5 I  B+ a2 ^4 h        return 0
    1 W+ u" L8 s/ E' }3 i    elif n == 1:
    # B6 h) y; }1 A1 X3 X0 R% n. a        return 1# q6 m' p8 L$ J7 h% b9 W
        # Recursive case
    ; a, b8 [; \: i" C! P8 Y3 \    else:
    % Z/ K) I( E! Y. v- A9 f! \        return fibonacci(n - 1) + fibonacci(n - 2)
    * D5 u% Y; z) ?( c) d
    % ^+ l/ h& x! `9 E& x( O; G# Example usage8 E" W4 \6 f- G, W
    print(fibonacci(6))  # Output: 8/ K5 i5 [1 F8 N: F5 l

    " L& B- J4 Y+ [- cTail Recursion
    0 i& c8 c: C* Z; x5 L- I5 r4 D8 J/ L' b5 [* c8 J
    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).: ]: D/ K. [8 P' p3 A+ V
    1 A& T- q- K/ F* r0 E5 ?
    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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