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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
  • 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 1 _' [6 a) ~. D( k% J0 _

    / s( R. i* \5 y1 o! i8 X解释的不错% J) k8 ~7 ~  N+ G* I1 L( i
    - ]3 p" Y# i5 V; A! C
    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。& H" |! x7 g& ?: e/ G

    & T- [& E( Y" \2 O/ c& y) S 关键要素+ V5 L" ]2 u4 r2 i% s8 X
    1. **基线条件(Base Case)**
    5 L( V; q$ B& P" ~( p   - 递归终止的条件,防止无限循环
    & H8 p" x/ Y  ~9 y/ ]( u/ b   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
      Q7 [  C! E6 s( t0 f
    . ?; t5 F! _" c7 B- p2. **递归条件(Recursive Case)**% S* H4 n7 I* y' H) {4 ^
       - 将原问题分解为更小的子问题8 r. ?; I; y- q3 [7 R: A3 o
       - 例如:n! = n × (n-1)!
    $ a) p' Y+ N5 n
    # i' T1 u) |7 _0 t3 }0 D7 Q* a 经典示例:计算阶乘
    3 q. \) `" E4 [, w0 F6 y$ ppython" b% H5 i+ }& T  K$ f  L6 N6 i
    def factorial(n):1 n, j2 v# b( m, n+ k! _' {
        if n == 0:        # 基线条件
    1 ^* k1 S3 a+ W) p! z        return 12 O3 @. _* K9 t& U* I
        else:             # 递归条件7 W( E& S$ ^. i- B
            return n * factorial(n-1)9 C# V% }' p' U, r, J
    执行过程(以计算 3! 为例):3 S8 B. c% ^; E& P5 H+ ]9 f7 C0 ?
    factorial(3): N! R4 U5 M0 z3 o
    3 * factorial(2)3 U, V7 z5 Q* o7 X6 X, k. v
    3 * (2 * factorial(1)), Y% C7 u- x8 S+ u& _, L
    3 * (2 * (1 * factorial(0)))
    1 A/ o( @5 w. a9 I5 _0 w3 * (2 * (1 * 1)) = 6
    . [9 E5 R# Z, g) Z9 |( R% m- m; i
    ! t5 x5 ]5 J" I3 d: z. A 递归思维要点% h5 I1 b$ ~3 w7 z* \
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    , l/ j- _( ?4 O: t8 A2. **栈结构**:每次调用都会创建新的栈帧(内存空间)( t" w) J; _2 l* m1 J! o+ R: Z0 |
    3. **递推过程**:不断向下分解问题(递)0 c- ^8 Q# d8 r2 a/ F2 q, F6 g
    4. **回溯过程**:组合子问题结果返回(归)2 N4 _1 z; D! `: F9 M- M8 d
    4 k- J! O! x; }7 z
    注意事项& B: r! r' D* z
    必须要有终止条件9 K0 Q0 t7 s' J; J) Z( R) j; y' {
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)6 ^0 W$ ]' z6 [
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    - D6 }; H$ I/ t% C8 @尾递归优化可以提升效率(但Python不支持): Z2 B# J1 y+ a$ s( N. a% `

    9 i7 ?( m/ N1 C& d  r5 Y8 H& E 递归 vs 迭代$ r2 f+ O( w4 a, b
    |          | 递归                          | 迭代               |, m+ B( l& H- x) l! o( U1 J
    |----------|-----------------------------|------------------|
    7 y' m& x. \# L. Q" D5 j| 实现方式    | 函数自调用                        | 循环结构            |' P' d" `! s4 e/ K+ |
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |% D  m/ ]# m! R3 A, E% V
    | 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         |, `2 [) I) I2 M# h
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    9 F6 o" U# J/ _! B
    5 e& b$ d. ]% Z! J 经典递归应用场景
    + I3 I5 f# Q- E5 X  g" t% o% a1. 文件系统遍历(目录树结构)
    + ]3 P& }) R( b6 \2. 快速排序/归并排序算法( A9 b% {2 D) B( b# [
    3. 汉诺塔问题8 U9 w% s! s% C
    4. 二叉树遍历(前序/中序/后序)
    4 G0 C. c6 n& |3 W4 k: p5. 生成所有可能的组合(回溯算法)
    - k% @# }3 X$ x; X9 K/ p3 _+ 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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    郁闷
    7 小时前
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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,
    * l* o  X1 \# q7 s. J我推理机的核心算法应该是二叉树遍历的变种。# j$ @6 W' R. J0 h7 t' F. |. d" P
    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:
    / j" c, Z; h1 b# y: p: vKey Idea of Recursion( D; L, {+ T6 k! A

    ; D) d  O$ G5 q! i4 u5 ~+ K( E' zA recursive function solves a problem by:
    ) [  p) w5 x- t' j2 t2 R; u1 C: k: I7 h2 ~% D8 `, \5 M
        Breaking the problem into smaller instances of the same problem.
    / |3 N) w% U) F0 V2 W# }3 `% t* p( f0 ?* l. q
        Solving the smallest instance directly (base case).
    ! J1 Z2 N" j6 \9 c- f4 D
    , M1 V/ L& _" R    Combining the results of smaller instances to solve the larger problem.
    2 f! I9 c  h9 K+ \4 P0 q; L" q; [5 d  l- ?% s7 W
    Components of a Recursive Function: @. U' r9 D1 y4 ^9 T

    7 [) e# h" [; B+ v: n3 n5 f# y3 ~    Base Case:/ z' g2 z: S" l* ], {" b. G

    , w) R: O" Y4 H! j( p  g7 \        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    / I4 b6 T! X3 g. R
    5 K' q% Q. Z9 P% P        It acts as the stopping condition to prevent infinite recursion.0 w' x( u3 |9 C& n/ A/ `
    5 L; E  H( E# Z& @7 Q) q9 w
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.. _8 t5 U) V4 Y$ ^
    * v+ \/ f! ?$ t$ \8 V
        Recursive Case:" D! w& C' @9 f; ~/ i& \

    1 u- x; x0 w4 E" r3 _        This is where the function calls itself with a smaller or simpler version of the problem.- h# d8 c+ _1 f0 C. M/ B- U$ ~& I
    / W/ v& Q4 a. C6 n. q
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ' {, C7 L* `5 g7 t0 P& Y( r# @5 _1 D0 w- r
    Example: Factorial Calculation
    9 u, a, ~4 B( ]/ X
    ' g" F1 w/ v4 d+ SThe 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:( s0 s+ y# J, y* o$ V+ b( [/ `) _) Y5 X
    . ?! c* t/ ~" [- O; h
        Base case: 0! = 1: x) m1 S- i" m5 m
    * v+ W1 J# p1 k6 e8 `* r
        Recursive case: n! = n * (n-1)!4 f* \8 j" l# G3 }
    " i# R' t* z0 T3 Q" w1 P! m( h
    Here’s how it looks in code (Python):
    : Z3 G' t( D  o$ Dpython
    ' ~3 b: O* i& N+ _' Y9 f* s
    6 l' B: z& x' Y3 B3 n5 X( U$ _, l3 @
    ! }4 n5 s  W5 W) Ydef factorial(n):3 u0 x& h1 t5 j
        # Base case
    5 A6 q9 D) b0 X( K! V) K    if n == 0:9 k3 \2 [) q# M
            return 1
    6 u" [+ n' L5 X6 F( ^6 [- \3 }    # Recursive case
    . t1 N; J9 ^& m0 O0 {* g2 G' \    else:9 T7 h7 H  l3 `  O$ V( z
            return n * factorial(n - 1)0 ^( _- Y! L) V+ t* u; L7 v/ D5 `! _

    5 u; b/ L, p' _# Example usage0 g3 v/ ~9 U4 w. {8 }  [
    print(factorial(5))  # Output: 120. v4 f1 d4 |0 u, v& m7 H& {: p

    & ?% R3 L: }# H" t( }7 |4 e" W" q1 _How Recursion Works. ]. s9 L$ u  h

    & ]6 w, v9 D& _( R7 f. n    The function keeps calling itself with smaller inputs until it reaches the base case.
    4 t8 j! p( F) ^: k! d! v5 |+ {" I" F' [2 [( q, ]8 @) Q( u
        Once the base case is reached, the function starts returning values back up the call stack.; E; I$ W* B/ y0 u& P+ _4 n

    ( \; K; @9 l% r! I" p1 T2 r$ L    These returned values are combined to produce the final result.
    ; @0 ?1 s' ~" Y! D$ _& ?3 m0 l% o2 V6 N& ^& b% s5 r
    For factorial(5):
    5 ?$ N* D: a% g: \! h. j
    ) Z& _6 o: B1 @/ A1 y2 r* Y3 ~& x3 m" v, s0 c
    factorial(5) = 5 * factorial(4)
      H+ k3 h) w3 F2 W. y: Xfactorial(4) = 4 * factorial(3)3 A6 t+ B% H' J& d# v
    factorial(3) = 3 * factorial(2)
    & K* P: N$ f' R: T7 C6 A! ^factorial(2) = 2 * factorial(1)6 q- i" D3 T+ \; F4 C; z% |" b
    factorial(1) = 1 * factorial(0)
    8 N9 _" C3 W" H9 m) }factorial(0) = 1  # Base case( H8 t" ~) d6 x( H) U
    ! y$ I; i9 R- n) z3 W& p; w
    Then, the results are combined:- {* D) [! e* N
    7 M, S  u. W) x3 m/ F# K  W
    5 l- y  u7 p, Q' X2 k
    factorial(1) = 1 * 1 = 1  M  Q, x* G- s
    factorial(2) = 2 * 1 = 26 N* u( H- e$ t
    factorial(3) = 3 * 2 = 6
    . s  U6 X( V/ Y0 Nfactorial(4) = 4 * 6 = 24+ P. p/ m) b. Z5 `1 ]
    factorial(5) = 5 * 24 = 120
    ' ~( X5 n* ^6 ?" ^9 ^; z0 c! l  J+ p! \  M0 \/ |9 Y
    Advantages of Recursion. j1 ]( A- d$ W% Y, e. m7 A

    + {! P/ m. p. G, |9 B' T    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 Q+ s" o* s8 ^
    ' Z" }0 H! `3 C3 S$ h8 W  U$ A    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    - l# l# n8 A# T- J3 K2 V2 p2 ^
    9 x% o; ?  r! Y* d$ l7 FDisadvantages of Recursion
    , Q" P1 \0 H" |  \% T3 f
      C( S: J$ d8 B% N3 F' x$ N$ m6 C    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.
    7 b* I! y; x1 a; A2 D6 P5 ?; i) P9 `- @! n5 T, g3 Y
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).+ \: t" ^% v9 q, X5 F' ?/ t, n6 x

    ) ]9 K; {% \9 R  S+ b, ]When to Use Recursion% O# q% j. x; Y5 H5 i6 N3 U. e$ l
    $ e5 A+ U  K# T( w' N4 P
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    * |, Z% e9 ]/ S2 v6 U: b' q' k; C' d4 L5 O) w
        Problems with a clear base case and recursive case.- m8 ~0 X3 L# p# Z! \, f$ a
    9 E; b& ~/ y2 [9 G
    Example: Fibonacci Sequence# M( z2 n- l- V4 a
    & o8 a& L+ t2 V4 G: \
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:3 u& H- d3 f  w5 r  g
    2 O0 h& B" h  m7 F- P* ?' p
        Base case: fib(0) = 0, fib(1) = 17 O3 Z/ E2 T3 B) z8 f& q

    ( M" @, t5 }7 q& g2 ~' b! ]    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    " u2 l* n+ ?4 z" Y# N2 {6 x' V
    ) ~6 M7 _) J, z: O, I1 ipython
    3 N8 B; |# W! n* y9 m3 f) E+ `% \# p% F% f

    + M) I3 e+ Z) P7 R' T- g. Ddef fibonacci(n):0 V3 V6 l% ^3 @: M' V5 P  z" k
        # Base cases
    : k8 F9 H7 d! s) k* `) f3 U    if n == 0:
    2 D4 T  B% B" X! G3 ?( O        return 03 X  q# S! W+ k; D, u8 w0 z) x
        elif n == 1:6 L1 [% G. A. i; G- g/ B
            return 1" s# z  r7 h( N% t& r+ U1 W6 \9 m
        # Recursive case
    / e1 X2 K; |, W; z" J& o    else:
    & n/ i: q- u  L' w( p7 P        return fibonacci(n - 1) + fibonacci(n - 2)# `& `/ i5 B) W; J/ `  e
    1 R; K7 I9 U$ |! m1 H
    # Example usage
    + S' h+ u  o1 V& R# \5 iprint(fibonacci(6))  # Output: 8* z5 O% D1 A5 X2 D* D3 `
    $ [) m( f) Q2 |- f1 B! P6 K
    Tail Recursion
    7 d$ w+ y# T2 m: u' p6 U: J+ D& r; Y+ h; O3 t; D: B: s4 ?
    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).8 r3 X# n  ^3 [4 h5 W
    - {' Z( b  b! |8 A; o1 Z1 n9 F4 y
    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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