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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
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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 ! a% M, R+ C7 w3 p

      p! J  @8 t( z4 Q) \解释的不错" J7 G9 [) y5 Z! j2 S

    4 [/ d+ a2 j( p/ ^. f" s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。, c- O: X- }* ~7 _

    ; B2 U& g9 Y5 U9 U: E) i9 ?4 b 关键要素
    4 X7 x: J9 Y# b+ Z+ W1. **基线条件(Base Case)**
    4 K8 U8 I- x! y4 F: l   - 递归终止的条件,防止无限循环
    # e+ E4 \/ @: T; |  s- d8 W& N( Z   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1+ V4 E1 o$ k  H: t
    4 c9 C: d4 s/ l
    2. **递归条件(Recursive Case)**. k- m% y9 [: h, ~1 g
       - 将原问题分解为更小的子问题
    * w/ u; E/ ]8 z; B   - 例如:n! = n × (n-1)!
    0 ~, \0 V& M# O! |5 \3 i* g' T; y  j7 q2 t
    经典示例:计算阶乘' T" ]6 H( [3 X& ~
    python4 H/ Y( m0 {2 M
    def factorial(n):
      y1 N4 M5 _! z8 h" V    if n == 0:        # 基线条件4 k! |5 ]" S2 c% V" L0 I8 n/ @8 @# i' K
            return 11 J8 y* w  k0 k
        else:             # 递归条件
    3 Z) w8 g. d" ?- G# Q" V5 A        return n * factorial(n-1)
    ' u+ b2 F$ @- D. Q# |1 J执行过程(以计算 3! 为例):7 F9 e! `% e& d& R/ j- V
    factorial(3)" C+ b/ F' d- ^3 a1 P2 i9 J
    3 * factorial(2)- m; ]+ {/ E0 S: R8 }3 x
    3 * (2 * factorial(1))
    2 N, K  W. h# @% ?  G& h5 S5 L3 * (2 * (1 * factorial(0)))
    ) X# O8 Z0 ]' ?# k8 ]3 * (2 * (1 * 1)) = 6
    : _, W0 G; _- f  d+ ~1 {
    2 M0 [' [; z1 B; }3 g 递归思维要点
    - O! R- R  z/ t/ H9 |# i9 G, y1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ! M. U9 y0 X; Z7 F1 H% w0 c! P2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ( D2 r- T, P# e3. **递推过程**:不断向下分解问题(递)7 U) u# A  t+ o
    4. **回溯过程**:组合子问题结果返回(归)( S5 x2 T& c" D) T& d

    - @3 ^/ P  L2 i4 B" B" W8 u 注意事项
    * W4 R; l! ?* ^; ]. n必须要有终止条件( n) t! B/ i8 @' ^! m! ?! ^
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ' b, U+ O0 x6 J( |) g$ D2 L某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ' h( A% q( c: e& q/ e尾递归优化可以提升效率(但Python不支持)
    , t- V; l9 h6 L" y5 t
    : O1 e, x* }$ ] 递归 vs 迭代
    ( c) r" ^) y9 z4 b; c3 b% o4 H|          | 递归                          | 迭代               |5 ?( Z  U! L7 h' j8 c
    |----------|-----------------------------|------------------|
    & g$ V5 T) I, ]4 p1 }3 ~| 实现方式    | 函数自调用                        | 循环结构            |
    / b4 u3 w0 b. v9 a# E  C| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    % y+ z3 ?- Q. W* E- 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         |" a$ _" l; Z% e9 W! y
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ) ^: b# o, A4 i: t' U  f/ x* E. c7 [
    经典递归应用场景3 ~( {; {& S- v! y- N
    1. 文件系统遍历(目录树结构)$ i; u- {: t1 h. y/ w0 n& O, Z
    2. 快速排序/归并排序算法; {! O4 M8 x% F8 S- b
    3. 汉诺塔问题
    , |( J0 f. `/ J2 w4. 二叉树遍历(前序/中序/后序)4 }: c# h4 R( ~  T2 h' L
    5. 生成所有可能的组合(回溯算法)5 \5 u$ P+ J8 J' k# U9 x

    : @6 n* j- @7 E5 X: O! k1 {  |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,
    7 ^6 E: D5 R; ~( k4 }4 R我推理机的核心算法应该是二叉树遍历的变种。6 Z" O$ A2 J; Q3 O( 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:% F! r$ v# G3 U' D) u( c
    Key Idea of Recursion9 [" E* C1 G1 ^9 [' {7 ]) V2 U7 p
    3 h. k; F2 [$ @
    A recursive function solves a problem by:
    4 r; d/ `8 E9 K" q7 w
    3 h- A* u- G7 ]9 p, ]. N    Breaking the problem into smaller instances of the same problem.
    7 v: \" l$ U! l. R2 L: l- g, Z% D) u) q# |
        Solving the smallest instance directly (base case).
    ' x; N# s2 g" B# C2 v; U
    $ z$ I/ I# j* a/ |/ W" w* a. c$ M    Combining the results of smaller instances to solve the larger problem.9 M7 r1 i+ E" a: p
    , l6 `; x4 t3 _/ u4 E1 p
    Components of a Recursive Function
    1 i# F% L7 n2 g0 `( {
      D/ H6 \; ^/ |4 W% j( V& k6 u    Base Case:
    ; E! p3 ~- t" B. N% c  z, P+ R+ q8 q9 z, Z9 p
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.9 A  a5 Y4 j5 [3 U( e& S; X
    $ m& m4 F; w4 p) U
            It acts as the stopping condition to prevent infinite recursion.
    0 y3 ]( Q9 G7 c+ E6 r4 V0 S! @8 X2 S1 f4 V3 M/ q) D
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    ; @: x+ g8 P( ^- `- h
    ) E/ F- M* C: E$ Y% Y2 \% w$ q    Recursive Case:
    % N# N1 h* n6 i! y8 o; U$ K/ ^- b3 V& u, w+ O
            This is where the function calls itself with a smaller or simpler version of the problem.) @! S6 `% u6 R5 R" Z# S7 M5 K

    1 Z: Z+ ^+ O* n0 h" R) i! I4 a+ u1 P        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).% X, h; m$ V! A! y( C: v: \
    : v8 ~2 \+ X7 }3 N, H
    Example: Factorial Calculation
    % T; z5 E0 Q; C+ v* E9 I4 [/ l7 o. F: e. ^) f; S& I
    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:
    7 y) ~$ A+ \" o8 u$ v0 ?" q
    # g* ?  J, x! c  B" Q. T* y    Base case: 0! = 1% K* ]/ E, {1 K
    ; ?' M8 V) H; s# I' U' l7 }3 F3 ~; @
        Recursive case: n! = n * (n-1)!
    0 V0 d2 u0 x3 B  @! {7 t* E7 ?
    7 B( J0 E7 a8 A: d" b3 n$ aHere’s how it looks in code (Python):
    5 E; m4 M3 h0 t3 [1 Opython9 s! j! Z8 ?5 b4 A9 L
    : w8 c' `- I5 s
    ( Z3 \: x# P% ]* o6 ?, f5 z
    def factorial(n):( g9 c+ r7 k4 Z# Q7 L
        # Base case' l1 s3 k) i, l& [, y- J
        if n == 0:0 A( [& e/ l3 ^! M6 d9 v
            return 1: I* y8 I* u: w+ I. a3 X: G: M5 p6 s
        # Recursive case
    . R1 J8 {9 O. M0 m0 d% A+ f    else:- H' a% H* W3 ^% e
            return n * factorial(n - 1), I3 L; I% B4 |8 J3 n, g4 T: B

    . W) r& l+ x* B" I7 R& q3 L# Example usage
    , `4 Y$ ]8 r- B6 Sprint(factorial(5))  # Output: 120
    8 N$ [2 N7 L; c0 d8 ?0 R6 C5 Q* Q  E# F" X: Q: Z& k: q
    How Recursion Works0 a0 g- G7 ^, {+ g/ d

    ; u, Q; d! u, [) C3 _7 g6 P% N    The function keeps calling itself with smaller inputs until it reaches the base case.7 M' [4 x6 ]6 [. @9 L
    8 ~4 c( o4 ^3 \3 q2 v- B+ @
        Once the base case is reached, the function starts returning values back up the call stack./ P- R& O0 j0 E, O6 c
      r# C' L6 m: y3 G9 K3 R5 R
        These returned values are combined to produce the final result.1 ~! {9 d+ M% Y
    # i% d3 y; `. @0 n' T! n
    For factorial(5):( a% _/ u3 H5 T( e# \- h" K

    6 y' y% p4 s# \5 f' B
    8 t5 g) k+ K% Q, N* }  Y$ Nfactorial(5) = 5 * factorial(4)
    & X# w' `  ]5 ~1 Kfactorial(4) = 4 * factorial(3)5 j) k2 ^1 M* Q4 ^) O
    factorial(3) = 3 * factorial(2)
    ; p9 L! U  b+ P. k! i# kfactorial(2) = 2 * factorial(1)
    , Q9 `$ ~( m" `6 rfactorial(1) = 1 * factorial(0)3 S9 A  x" K) P
    factorial(0) = 1  # Base case& e& k7 r" g) k: [+ y+ g
    6 }% P9 n  o* E. `: j: h
    Then, the results are combined:
    7 {& M! d+ i( Z4 l! [$ Z; Q4 X" ]4 J6 ]6 x! l
    2 I* p5 ?; w0 w8 a: [. D2 s
    factorial(1) = 1 * 1 = 1
    8 s4 X6 u: I' [' Z) B$ Yfactorial(2) = 2 * 1 = 2
    7 d9 j% l0 V5 V4 w6 O& S; W6 o# sfactorial(3) = 3 * 2 = 6
    # M" Z4 o& ]5 L" hfactorial(4) = 4 * 6 = 247 l/ Y- n/ \: x! _$ {, m8 B
    factorial(5) = 5 * 24 = 120+ v- `& q; r+ C5 Z/ b" f
    6 z# e9 |& e& I/ [1 q8 G& B, U0 {0 E1 b
    Advantages of Recursion
    " _& _8 o. ?0 E0 \+ h6 N- b5 m" q  _5 I8 q0 P. _
        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).
    0 i' l5 s6 P+ r0 `# ]' P, _! i. O* v
        Readability: Recursive code can be more readable and concise compared to iterative solutions.$ J/ ?/ w6 W  y* P) O3 x, p

    " u8 v, R9 @" D1 @- _8 L( F, CDisadvantages of Recursion
    ! {# X' x/ a" X+ g$ Q5 v1 v# w$ L: o" c/ l, ?' Z
        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.  y0 e4 [% J: q4 Q

    # n" R" p: h9 W- D# R    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).$ f* R) B. n( }1 E% b

    ! e- \& u/ [7 L7 D! F; ZWhen to Use Recursion! [, @! A9 C$ F1 L; `8 f: u: Y

    2 y, T$ O1 u7 A/ E1 A$ }    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    - N3 F% q6 v5 n7 y6 \  i& I( Q) A1 v) n' n! K
        Problems with a clear base case and recursive case.0 I$ k& L# k' V9 X

    9 r8 q7 D1 g- |) [* g6 VExample: Fibonacci Sequence
    4 z9 @6 P* Q8 r2 y. s5 F3 R" \5 {$ A3 I  R1 n; B0 j
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:! p' \" }3 a, q! C
    3 O: Q+ a' P8 S# h9 ]. k: L; G
        Base case: fib(0) = 0, fib(1) = 14 h; o: ]5 c4 ^
    $ y/ v# M0 b% [1 z$ w0 R4 u/ U. }+ ]# b6 j
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    / c' j/ ]+ v& w' [* j/ J1 w4 j- d5 t! _9 ]- }; `
    python
    : T- u  C  t. f- A2 K  z* Y- `, I; l8 ~* K

    & @+ |* ]0 d% k3 v1 d9 L3 Ddef fibonacci(n):$ \# ?' V0 E5 B' o
        # Base cases
    5 c8 L' ?/ Z/ Y    if n == 0:4 o* q8 F- D0 M" ~9 ~1 W, W
            return 0) S' k& T3 U' F1 @, B. ?$ _' I3 b7 ^
        elif n == 1:; T" T# S6 i' Y% M$ D# F" |& F
            return 1
    - l/ `) _5 d: I/ z    # Recursive case4 \" d) t) {- {, Z5 }- v( I* U
        else:" h. I2 B. F8 d5 S
            return fibonacci(n - 1) + fibonacci(n - 2)
      t/ u6 x8 R4 F* |4 A2 h2 E& D' s6 f% w
    # Example usage
    " j& k6 X/ Y0 ~print(fibonacci(6))  # Output: 8/ i; D$ [4 P5 Z" R6 I
    + Q( B! _& g8 C# t7 u  k$ q8 ?+ w" t7 f
    Tail Recursion
    1 q8 F% n: _, r( w* F5 W' V: F# W$ Z  g; O/ A' Q
    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).# h8 K' j; n# f; B& I% c. c* x
    1 n7 x. ]7 I- q3 _9 u( V
    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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