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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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     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
    ; }: G; c7 |" H; M$ w' ]' x# d. c5 h) e* w* a, g
    解释的不错* }' M- \2 T' H- Q8 |. I1 Q

    + a5 g# H( T# R7 f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。
    5 b  S  j, p+ U, q8 O# B2 l2 q/ G8 M" L# Q
    关键要素# z. e9 y6 F4 T9 O
    1. **基线条件(Base Case)**- }  r6 z9 a5 h3 U! W
       - 递归终止的条件,防止无限循环
    - k" c* L. W% |8 E   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1: e1 ]) L; F: D" ^0 E& n

    & r! b, k* l2 }2. **递归条件(Recursive Case)**7 c# K4 w. }# B/ L$ R2 |
       - 将原问题分解为更小的子问题
    1 K$ n2 H  C& P/ r' {2 ^# [  k   - 例如:n! = n × (n-1)!, z7 ^$ F( o  u- T) h0 [" E0 x( L
    , v4 X, }6 Z) |* _& i2 e  y
    经典示例:计算阶乘0 |# G$ W9 p- F+ j
    python
    ) X3 u' ~+ e: Q8 ^  i0 h" [def factorial(n):6 Q! F! L! |- N+ C' i
        if n == 0:        # 基线条件$ X/ B5 B% D# K( ]
            return 1
    - [2 l6 _: q1 g. @/ ?4 p0 B: V. ]* H    else:             # 递归条件
    ; S, @/ y' A9 ~1 C        return n * factorial(n-1)
    ) B5 L! ~/ c: M执行过程(以计算 3! 为例):
    0 h, ~. A/ h, t6 r0 v, Nfactorial(3)  J2 Z9 F+ b% e2 p: P* X3 c/ M2 L
    3 * factorial(2)0 u) H0 [1 g% O
    3 * (2 * factorial(1)): `* g# |/ d* X9 W' q
    3 * (2 * (1 * factorial(0)))
    6 J/ V8 t, N3 g6 y8 G2 x. Q3 * (2 * (1 * 1)) = 6
    2 d: ~0 E! n+ z. E+ ~& B
    / g' c( s6 _2 r  j 递归思维要点
      \( R0 H" q6 j1 w7 P1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    1 K) F( a, c  `2 _' b* ?6 ^2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    / p4 ?* U9 j# Q3. **递推过程**:不断向下分解问题(递)! s' |& @6 B  g: L" K
    4. **回溯过程**:组合子问题结果返回(归)% j+ H/ s6 {  q. t
    & D# ^5 Q+ V5 d  }) f/ f+ U
    注意事项
    # {* T9 |. n- C/ B必须要有终止条件
    7 m! |! F+ Z: a! P% |; d- K" E递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    : n+ A$ l8 @8 H- X某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ) {, `+ E8 |5 a. X+ Q3 i& C8 ^# g& P尾递归优化可以提升效率(但Python不支持); W. _, V. I* A

    , b+ J( `) x% N9 B0 h5 r9 ~6 [" f 递归 vs 迭代& A  @$ d2 R+ L* ^4 P0 b
    |          | 递归                          | 迭代               |
    0 p) |+ r1 t; z) a/ I$ C/ Q|----------|-----------------------------|------------------|
    8 y$ @6 p, V$ ~| 实现方式    | 函数自调用                        | 循环结构            |' _9 h: M* Y/ Z- P  K& n. w4 X2 C
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    4 {+ @* \! c4 s  o8 u9 r# G| 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         |
    6 k& `: L$ I# o) ?| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |  m' ]) c3 v( }8 C0 R
    5 R. h$ w6 v9 t5 F/ W+ I' ?/ ~0 z
    经典递归应用场景; z2 V: O- K# k* c9 u: D: C
    1. 文件系统遍历(目录树结构)
    0 A* A' @& b- @+ z( R+ R2. 快速排序/归并排序算法
    & ]1 B& ~( e4 g6 d! z, \3 z3. 汉诺塔问题5 n  q. d2 e% G0 S
    4. 二叉树遍历(前序/中序/后序)( ^+ s9 e) y9 L2 s# u7 x
    5. 生成所有可能的组合(回溯算法)
    4 F5 {2 j% ?' }- V
    - v6 V; @' `9 A0 ~% 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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    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 x! C) B- c9 t2 @, u" q
    我推理机的核心算法应该是二叉树遍历的变种。
    - ?4 P9 c& H. \4 @! Y. h# M# d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:; P: O0 w9 Z) p6 ~; q
    Key Idea of Recursion
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    A recursive function solves a problem by:
    ! h) e0 j" A( E2 P- S6 D5 J. t* n& ^
        Breaking the problem into smaller instances of the same problem.% P% R9 q  o& e4 e% y7 I8 l4 N- t

    1 \+ r, E8 M* Y6 s; s    Solving the smallest instance directly (base case).  F. v! j6 f" j% `0 F6 J
    6 j3 X7 F9 a9 C# D
        Combining the results of smaller instances to solve the larger problem.2 R6 @- R; m, @) i2 }3 J

    : S+ Q2 Q4 @7 z! T$ C9 xComponents of a Recursive Function" w/ s" ^" `9 R: t

    5 f7 G" `8 V) C( j    Base Case:
    4 u# s; s  X9 w5 @4 y( }/ z0 `
    2 p. P) M7 R3 E3 E        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    : b* g7 |- i; W) H9 d
    7 j# V& u( _8 |8 A        It acts as the stopping condition to prevent infinite recursion.
    , ]% C' o- j3 z' A- u3 j0 y
    , G! K& j+ a0 m- ~, e( J  Z4 V        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.7 Y( y. n" Q5 j0 k' x# U& A: l2 l4 n
    # c  }- [: ^2 S9 ?8 n5 ~
        Recursive Case:; Y) n# ?7 ~8 c# N4 M% {4 ~
    2 Y  W/ i3 p0 T6 v% t8 {
            This is where the function calls itself with a smaller or simpler version of the problem.. y3 m( J5 L9 H4 \$ ~2 n

    . C5 b7 X& j7 j% Q( o        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).* u) ~/ F* H9 Y- p$ _
    3 j; a$ J1 K3 |5 e" g& R
    Example: Factorial Calculation* M2 _, u7 G- v8 ^7 k. |' l
    ) w6 U3 G: m) ~0 j% ]5 ~
    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:
    " _  B) E0 ?0 [5 ^  ^1 n* U
    ! M4 [8 J5 J1 @* `) J$ O" Y' h    Base case: 0! = 1
    ' e2 P/ A/ O: ?0 n- }, B5 t3 E' X! {! [' q
        Recursive case: n! = n * (n-1)!
    2 U& s0 F- ?, s  S! t- J8 i/ p
    ) L' @* n) ]  Z- S& X1 ZHere’s how it looks in code (Python):& I# o. Y. ?4 i" \7 G0 T! A# {
    python5 u# M$ Y8 q5 ~  v9 C) W4 P. ^8 u8 j

    - R( u6 n  ^0 G1 s" y; @( R" D% m) z7 D& c7 L* s( W4 ?
    def factorial(n):* K( {6 w7 h) m! k3 b
        # Base case) L. C# v' w2 \( ]5 c" [) g' O: E* C
        if n == 0:7 Z/ B" T# R1 @) a7 D% [
            return 16 v. }& S2 Y# L& c0 }, d
        # Recursive case
    3 v  [( L: v" U7 y7 \    else:2 k: ^/ W4 i  }# v' j5 I
            return n * factorial(n - 1)
    # R, ?3 N( A6 k# e7 x, z8 \6 m4 c" }" [2 c, k3 a/ Q  q
    # Example usage3 z! d# v5 B5 ]+ r$ R3 w
    print(factorial(5))  # Output: 120; m! ^+ I2 m/ o; b& }  s/ O

    9 F! v' C1 m' c& q4 Z. iHow Recursion Works
    & a: `: o* r: o- g
    2 z  n9 @" X$ t1 d" u7 _7 [    The function keeps calling itself with smaller inputs until it reaches the base case.$ ^3 i4 |' J: I0 v( ~, P$ G

    0 v: {0 N2 m; w' F+ u6 {    Once the base case is reached, the function starts returning values back up the call stack.
    1 Y$ O& v* u6 Y  H1 M. B( H( z' P9 E& e- Z6 R. C# L
        These returned values are combined to produce the final result.. h# O' x3 m% q+ S9 f, w# O

    6 Q0 `6 S7 j9 f$ d8 nFor factorial(5):
    5 b" |2 K+ h; p/ ?1 m4 q' f. }  ]9 |; P/ C: e! E, K
    " e; g4 _3 s3 ^" I  p
    factorial(5) = 5 * factorial(4)* V) y0 }& e6 H0 G: Y) c
    factorial(4) = 4 * factorial(3)3 e$ N6 n" p- n- C$ c, ~, \
    factorial(3) = 3 * factorial(2)5 N5 c" w* |9 R+ Q  G4 ^" _0 U
    factorial(2) = 2 * factorial(1)
    6 P- B7 c9 N3 l9 l3 ?2 G; l: ]% Jfactorial(1) = 1 * factorial(0)
    2 [  F! h" s0 t- |factorial(0) = 1  # Base case) {3 U3 a3 q/ d' X: I

    " @4 s7 Y8 t& z9 e2 Y# d' cThen, the results are combined:0 J7 t$ c( ]: h

    ( p& `. |3 q1 X6 U; R8 F/ W: f* L/ u. [
    factorial(1) = 1 * 1 = 1
    - M. Q7 X) p5 ~) ?9 Ufactorial(2) = 2 * 1 = 28 v0 N5 D5 C: ]5 j' J; ~
    factorial(3) = 3 * 2 = 6! X( D5 s2 K& h0 J
    factorial(4) = 4 * 6 = 249 r9 i1 `9 H9 N& e+ k9 D
    factorial(5) = 5 * 24 = 1208 y, F6 J1 B% V1 u

    : F3 }- r# j& T, d, qAdvantages of Recursion
    " k* p' Q- A9 X" R! E, P5 F& N6 M5 h
    + l$ Y: F; J/ d8 u    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).
    , p$ a: H/ Q; I
    ; ]1 x2 o" q, J* k, [    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    , M) t( A, C4 s: Y6 Z9 M! X
    # w- Y; b1 J# dDisadvantages of Recursion1 o1 l* a# E1 V9 B$ l+ o

    2 I. I" c/ t. h4 E9 }    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.
    ( ]( z3 h+ Q' J+ ~1 ~7 m
    " w5 z# F1 q& O) U. s. ~0 N    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    - |+ t* ]1 N! e8 N# J& [! t8 S+ N$ _- U6 F" T# O. |
    When to Use Recursion
    + J, u/ x; }* L, x# H( O; }  [# z" z
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ; U; t5 c: F" T& `
    & {' F+ C. J1 Y    Problems with a clear base case and recursive case." p/ N$ k- [# T% {7 y6 s
    4 Y$ b+ |3 o8 f' q
    Example: Fibonacci Sequence
    2 O7 p( Q4 L% l0 \+ D% ~: V# g
    + x' @+ \9 V; s; y, i8 [6 iThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:- A- u. H3 x- u  K0 O/ ~6 C' E
      w% E+ H& v. U3 z; D8 D* Y
        Base case: fib(0) = 0, fib(1) = 1
    8 U2 O& i; a- P( f4 b% k# V# X' f2 ?% k
        Recursive case: fib(n) = fib(n-1) + fib(n-2)% u( S4 H+ L7 y: e) p, v9 N
    ' R# M$ q5 K; `3 n& o7 \) e
    python3 T) v5 ^4 f7 h# ~3 e1 Q& N

    # a  \! \) x0 D1 {/ E  j2 r: ?& G/ C& `
    def fibonacci(n):
    7 K  u+ q, {+ `4 e, s    # Base cases
    9 j  ^' L" ?- ]+ H; o% t. \$ {    if n == 0:
    * }( t  a2 a0 Z6 Q+ I! R7 F) }% b        return 08 x3 V2 O2 v) {/ I9 J
        elif n == 1:. [' O+ ~' H8 Q# o
            return 1
    2 ~7 W5 B( [, x: O    # Recursive case
    $ {1 x8 {0 n' x3 f0 Z    else:
    - K: h- H0 ~  F$ d6 M! |" Q        return fibonacci(n - 1) + fibonacci(n - 2)9 T3 ]- g: r; [1 k' K
    $ u  w) I- N# _; \( {  g
    # Example usage+ g" f  M; T/ q
    print(fibonacci(6))  # Output: 8$ T$ m9 b: I3 t8 X- v
    ! g: w  T' L2 Q
    Tail Recursion( X" C- W% i; ?: c( h  ^
    $ @- R# C* S/ p
    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).
    3 c" x. Y' Y5 N9 i: o/ u2 X- ~5 ~* t9 l4 A+ j0 j
    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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