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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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    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 1 z. m2 D4 U5 V6 d! l! C) i9 ]! S
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    解释的不错# f  O8 C- i" s7 P/ S) ^
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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。
    8 {: n. L4 A1 B6 g- x: g
    ) i* p; F5 `( @5 L) y 关键要素+ p& F" g6 j& E( @/ `/ k
    1. **基线条件(Base Case)**
    ; O* t  n/ \+ F, ^  s$ ]% g  W   - 递归终止的条件,防止无限循环
      N2 F: u4 |/ I  i. Q0 G5 W   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1, f" Y  q) Y$ U0 B  u/ G3 Q

    1 U6 W) G; n6 e+ f: Z2. **递归条件(Recursive Case)**
    4 ]% z8 }9 Z% [* H, b   - 将原问题分解为更小的子问题" |  E) s2 }7 ~9 P
       - 例如:n! = n × (n-1)!
    " K  c  d; I5 V0 U# l  O! P6 e; g# z) f6 }! k8 E
    经典示例:计算阶乘' x3 M: @9 r  P1 U; P
    python1 {0 s" o5 n- {* m/ E! o; t
    def factorial(n):, r9 U1 N! N8 u5 I
        if n == 0:        # 基线条件
    ( Z8 N/ |7 [) J! _5 C        return 1
    2 s! L4 \5 ~# G! ]    else:             # 递归条件' _/ F1 r/ T- S- {
            return n * factorial(n-1)
    . n8 Y# o- Y* F2 Y2 v  K" e9 P执行过程(以计算 3! 为例):# ]4 x. T. x# ^: L  ~2 @8 S5 Y
    factorial(3)
    . D$ Q8 r" o- ?5 ~: O9 _+ X3 * factorial(2)
    9 S1 d0 W* t# A5 H# H5 p3 * (2 * factorial(1))
    3 u7 R- H5 z' k" @5 ~: ^3 * (2 * (1 * factorial(0)))
    , j/ t* @- ~  l! P: V' ~( W3 * (2 * (1 * 1)) = 6
    3 S2 Q+ x7 V4 h
    9 H" t# k. C$ J: U' y' h9 Z 递归思维要点
    % \0 @2 I: j! u! }1. **信任递归**:假设子问题已经解决,专注当前层逻辑/ e3 Y0 v* g" C  q3 J. o; H6 Z
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)- F7 _3 C, g- T3 G( O. M, b
    3. **递推过程**:不断向下分解问题(递)
    0 i. A6 s0 |" K4. **回溯过程**:组合子问题结果返回(归)
    / V$ }0 o$ S9 _: w, P
    8 m3 t, D, F% L 注意事项5 _6 D; i) Z& f. z* _
    必须要有终止条件4 p: \" r8 S1 k, {3 @- L! w/ h
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)0 K. W3 q; A# n" K
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    8 x% W  M1 }0 B) r. o0 l: I尾递归优化可以提升效率(但Python不支持)
    . s9 ^2 e8 v$ i; u% ~! u) U4 X! q* K
    , N+ b4 V$ [% w" M 递归 vs 迭代! y- q3 ^5 e0 q# S
    |          | 递归                          | 迭代               |
    , |; W; {, W% E1 }4 G' j7 d! g|----------|-----------------------------|------------------|
    - R& n0 ]! \2 Z5 e/ x| 实现方式    | 函数自调用                        | 循环结构            |7 X: @# a% B  B( l* t, w9 {
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    - b1 l: Y! z  u# x6 v2 y| 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         |# K8 S4 S1 k3 B$ T' I
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    4 y- O5 b# r& P2 M5 K4 z4 u0 G
    $ n: q* A! u2 n 经典递归应用场景
    ( }/ }- t# r  m/ s1. 文件系统遍历(目录树结构)4 {' I, f5 v+ s; ]: W
    2. 快速排序/归并排序算法$ _' P  x2 s+ J0 \# l# J6 S
    3. 汉诺塔问题
    ( f+ N# J' g+ O" _4. 二叉树遍历(前序/中序/后序)
    4 K8 q; z) x0 d( z4 T& F- h* H5. 生成所有可能的组合(回溯算法)7 N9 n, w5 o- @) q& [

    1 M3 _; X/ D' \1 h# @$ W; 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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    昨天 06:27
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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,
    0 X9 w( X) {2 M3 ^, h  m- Y# v我推理机的核心算法应该是二叉树遍历的变种。( B0 X9 p2 F, I2 j( z" h
    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:
    & n- M/ X$ [* j( SKey Idea of Recursion
    ; m# t4 G- R& s9 {" z0 }3 j
    ; [4 _: e. y* f. d+ G7 _A recursive function solves a problem by:  A1 z* D* K3 Q) a. f9 p& {9 {1 h2 ?/ ]
    " i7 B+ K1 W0 C& Z, ^
        Breaking the problem into smaller instances of the same problem.1 H8 v5 t# N* X1 F4 h6 z
    / p( M2 J7 n+ k0 b
        Solving the smallest instance directly (base case).
    ( [2 A0 _' A8 y4 v. `
    ( r: B5 J) A& z) r8 g    Combining the results of smaller instances to solve the larger problem., p! L9 m9 z1 R7 x+ b: a/ T( g

    5 I9 G% M+ P7 J# l2 H) kComponents of a Recursive Function( N" B; u' ?! N8 R! x  r) u6 ?

    / G. Z2 ^# R! y- C    Base Case:7 s+ M) o3 }/ ~. ^6 v" o5 Z

    + s1 O# U. b9 d1 ~- H        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    4 V3 `6 [# Y5 r& D5 Z' z2 ?
    $ Z: x8 L1 W. B% g        It acts as the stopping condition to prevent infinite recursion., d2 X" d. D# y# ?* m( z: [

      f% v0 a& c9 W3 E- r6 Y        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    , A9 A+ j: c% S! C
    - F' b; g& a* V; F. K" [5 F    Recursive Case:
    4 k1 ]. y# ^7 f
    / I5 B- n  Y4 p: K1 b8 T, J% J        This is where the function calls itself with a smaller or simpler version of the problem.5 ?9 i$ V# W0 W

    6 ^8 I$ g8 ]6 N$ n: x        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).% b2 S+ U6 b4 N8 g0 x

    ; |/ t1 [6 W* o3 X$ yExample: Factorial Calculation
    4 y2 Q) M8 D! c0 B% |8 d8 R2 a) h$ g2 G/ A2 z" ]
    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:! y3 i( ~* R: O9 v, V
    + P+ Z% U' r- R; g. w/ S/ q9 t
        Base case: 0! = 1
    ; D+ w" g4 D1 P& l$ r
    9 W5 \- e1 S+ N  Q" x2 U+ _. ~6 R    Recursive case: n! = n * (n-1)!- ~' k  A/ S' |! ~6 F$ P' D

    # i6 a" q. s; L$ a5 hHere’s how it looks in code (Python):! T* i$ K  B0 P1 s; c% p- U! w# I
    python
    , B4 J9 X+ @% y
    . d+ ?$ I2 K* U' }# R7 c0 n0 ]
    , w5 J1 f* f9 mdef factorial(n):! a* o! V9 f! B+ a, ]
        # Base case7 d' M' Y5 R$ g1 |2 i/ E
        if n == 0:
    & }  h6 v$ k, e/ U/ x        return 1& T5 Y2 P& \; C1 R# a, r+ ?9 Q- S% J
        # Recursive case
    + D, o1 u% ]' D6 L% H    else:
    7 \* L2 E  s) l( q9 _3 A3 S        return n * factorial(n - 1)) A3 V8 \, d( E( x$ |6 F
    2 l" v; o# e- K, [
    # Example usage
    5 m2 u4 o4 K+ K0 p1 e; X! J1 Tprint(factorial(5))  # Output: 1203 E: U: v  v3 K7 Z7 ]4 B. I! D

    ) x5 b+ s/ T- s% wHow Recursion Works
    ' n' K9 u# C' ~: p! C, i
    * B  Y3 ]8 k: h) V. p' H    The function keeps calling itself with smaller inputs until it reaches the base case.1 c/ x5 S2 y; x% p+ @
    7 a% n) Y$ g) c" U6 q
        Once the base case is reached, the function starts returning values back up the call stack.
    , c( a+ P1 A& u- j  U; U7 b( ?
    7 F3 L! {. G3 m3 U- \5 A' W    These returned values are combined to produce the final result.
    9 K9 s* m2 D6 ?# @
    ' [' }" L3 i' w! W. NFor factorial(5):- X- L' }* Q# D, D7 Q7 Q

      E, {& X3 l% b; T
    4 i8 \) U( R& Cfactorial(5) = 5 * factorial(4)+ a0 R0 E- {3 m; [2 }9 F
    factorial(4) = 4 * factorial(3)  K, p: d: T! c! }
    factorial(3) = 3 * factorial(2)) I2 y1 R4 X  \5 [. I5 {1 [- d7 t9 b
    factorial(2) = 2 * factorial(1)1 b' g" ]2 C% ?) [, ^  z
    factorial(1) = 1 * factorial(0)
    ( F; r2 Z' U, A( Dfactorial(0) = 1  # Base case5 _% q' F% ?1 e/ d" \

    ( K9 `4 ?# i$ ~' OThen, the results are combined:# ?: P! N  K- }! B

    4 E' y! o2 A3 Q( N2 f( [/ q, H5 t7 x$ \. |& M7 d2 Y" @' C0 e: g% J
    factorial(1) = 1 * 1 = 1
    ; ?; F( u) H& H. i6 E1 U/ Nfactorial(2) = 2 * 1 = 2" @& b& ?: Q. a: `) E& i
    factorial(3) = 3 * 2 = 6
    , ~. A$ e6 `/ ^& d  R& cfactorial(4) = 4 * 6 = 24
    : r: k8 v* R8 b& ~9 Ffactorial(5) = 5 * 24 = 120# n* l3 W# m. {/ H& D

    4 m- z8 g3 i. T5 c' CAdvantages of Recursion, M, q7 e) p1 ~# x  C

    $ O3 p" \, ^, @9 M) ?    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).' f- g$ T- P1 n
    , i0 m$ E; g- Q9 W# ]
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    8 n& n& `% {6 w; R" \6 N5 t& v. U; x; T4 G$ l
    Disadvantages of Recursion
    & e1 B4 c8 T. j- C$ w! G7 l2 ^& o" y/ z3 r- d# x: @4 @1 f
        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.
    : M6 }& z; J& t" q5 R% P+ p1 i" }- h  f. K+ \7 f+ C
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    5 K- W2 w4 P+ S0 _8 \
    ) a5 J0 q# U/ M0 w" tWhen to Use Recursion9 I0 \) U4 Z6 z! q+ g

    / K: O: M$ D. O! M) G! Y0 q! u    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)." X' G/ ~5 [$ l4 J
    0 T2 e* ^1 A9 N+ Q5 P; J/ y. ?0 [/ N
        Problems with a clear base case and recursive case.& m2 P" B* ?$ ~( x- H6 k
    8 R/ P* q+ h* t
    Example: Fibonacci Sequence
    4 j: Q7 F! H( I0 s% E* ]6 |# P7 V! H, v7 b
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
      {# S2 O8 `% E* P8 M- o7 i; x5 R! Y( y- Q) i" O6 ^, N
        Base case: fib(0) = 0, fib(1) = 1
    9 y2 A) g) |$ K4 D" H( c$ ^, x" T. _5 ]5 h
        Recursive case: fib(n) = fib(n-1) + fib(n-2)/ I5 f- B- h) ~6 d2 R7 F7 [
    + ^* d. T% p/ q) j
    python8 e. c3 z9 @; T1 V1 B7 F

    7 X7 ^* b4 W) N& h* h$ X4 C& O0 k  M, c( X
    def fibonacci(n):# I' O2 P. R# h! B$ X/ O; ?0 Q- l
        # Base cases& A: Y; ?3 H# h
        if n == 0:7 ?- J( u" x+ N3 w& A- v& y
            return 0  J: Y7 `" s: ~
        elif n == 1:' R3 D) W$ V) ^" R  n
            return 1
    ! i$ \! t1 y" w2 T$ c    # Recursive case
    ; @; F' Y5 w3 }/ o5 `% E: `    else:
    & v4 }: v1 y  M; f; X- k        return fibonacci(n - 1) + fibonacci(n - 2)) J) o; n" B7 C3 ]; n, ^5 ?
      j& o8 i9 f6 G3 D& G
    # Example usage
    : U5 v1 p; U: A4 xprint(fibonacci(6))  # Output: 8
    # h7 q  f* x4 i; p& R" Y, m% z9 l, A+ d: L2 f9 W. r7 v, v
    Tail Recursion1 O# b* |; v' o7 _! A

    8 Q% ]) f4 t* n- TTail 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).
    $ I% W2 c, U( ~5 x1 W4 n+ T6 e! A( Z4 s$ s; k( n. u
    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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