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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| 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 * X- p0 U: `. j; V
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    解释的不错9 E- z: v/ H' [: f1 x# @" F/ q

    ! b- P. ?' m' M$ b; G1 W) c5 Z2 p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。
    - ^  m# U+ E. L! G1 _7 ?7 Y% }9 x
    关键要素
    $ \* }5 N+ j; F  v8 {6 |' B7 ~- Q" v1. **基线条件(Base Case)**
    ( s5 O% V  v5 k* |) a, L. k% x   - 递归终止的条件,防止无限循环- J: @. z7 l" ^9 q# r, V% A
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    & E; R0 T6 O; N8 W# C& Q0 q% {* l
    2. **递归条件(Recursive Case)**
    . o2 ?0 {2 X  P3 i/ r   - 将原问题分解为更小的子问题
    - m4 j% W2 c7 \& Q# K) X   - 例如:n! = n × (n-1)!
    # s0 j' i! ^) a  R6 b, P5 @% X+ a8 d; g7 |
    经典示例:计算阶乘0 @, {% h/ U6 d" X' r
    python
    . @4 h( f4 v2 J- ]% V$ R- ], Wdef factorial(n):  y% j* p! f' z4 L+ X) c/ m
        if n == 0:        # 基线条件6 {! i" w$ T4 S: z3 Z& @
            return 1' w& T" D! M. t' f6 J& t1 l1 }
        else:             # 递归条件
      }6 t5 J# `  Z( F& k3 d0 T, C7 ]3 }        return n * factorial(n-1)# R7 x% J* Y6 K4 _; R7 L/ q
    执行过程(以计算 3! 为例):
    ' W6 a/ f  u! \  ~6 q6 s( z# rfactorial(3)
    8 `& D8 c. r$ G) B6 U3 * factorial(2)
    - P/ c# j# d( x8 Y% B1 \3 * (2 * factorial(1))' T, V0 q: E, y) ?  {: z' S# w2 L
    3 * (2 * (1 * factorial(0)))+ z: y$ l4 C# O* _# j2 X
    3 * (2 * (1 * 1)) = 6
    & v' Y. o0 N  V) Z
    & z. w  W: J) s5 `. L4 s: W 递归思维要点
    3 _6 z1 ~% i: D* @2 G0 |1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    % V, e% N4 t8 W% z+ T2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ) ]$ D5 n- A' e( @6 M5 o3. **递推过程**:不断向下分解问题(递)& G  M+ U! a# w
    4. **回溯过程**:组合子问题结果返回(归)
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    ) A: v2 Q1 j) k' B. x" k 注意事项
    # B2 r- h7 u3 p9 t/ Y必须要有终止条件' y) s( `/ S6 l" X) u6 ]
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)3 ^) @) N' J4 p: V. x1 r. N! O
    某些问题用递归更直观(如树遍历),但效率可能不如迭代! S. }7 m2 ~# |9 o* S7 Z2 z
    尾递归优化可以提升效率(但Python不支持). W, E! k1 ^8 K

    . T2 V( s. R0 y/ j: | 递归 vs 迭代. V+ p7 z( H6 L8 O
    |          | 递归                          | 迭代               |
    / z" r0 ~+ A6 Y  e|----------|-----------------------------|------------------|
    & T* y/ F  ^' @" x) ~. |. q: ?+ x| 实现方式    | 函数自调用                        | 循环结构            |
    3 g6 W- `0 o& a9 s( z| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    4 A! Z1 L8 f5 j2 P' d" X6 T2 T, q$ F| 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         |4 X2 s  {1 V1 X) l% S
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    2 i( V: `: s' }8 P; t1 Y9 ?! A; ]0 g$ q/ }1 m2 t' ~
    经典递归应用场景+ q! E" m, M) K1 K" p6 S( g
    1. 文件系统遍历(目录树结构)
    8 C+ }* f* G4 a% O2 `! I( H2. 快速排序/归并排序算法
    / ]4 K2 B( w8 `  K) S, d7 V3. 汉诺塔问题
    8 N9 o$ w% s, W; i4. 二叉树遍历(前序/中序/后序)
    5 }" O' P& d5 p. C5. 生成所有可能的组合(回溯算法)
    % E) {( n1 q& |  Z
    ' Q% h; X: [+ v; k5 |; u8 Y' ^- r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
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    昨天 06:41
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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 S) U- \8 O1 T
    我推理机的核心算法应该是二叉树遍历的变种。# u6 k) @+ a7 L
    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:
    1 v1 ]* H4 _0 |1 G2 h+ L% A6 @Key Idea of Recursion! `  u# G, m6 r% f

    9 B4 X$ X% r7 QA recursive function solves a problem by:
    % D5 H, E1 z' D* h- J- }$ W& l: I% t/ Y
        Breaking the problem into smaller instances of the same problem.
    - ]8 u$ B  e/ Q  f$ H
    5 c+ B# s. D% w! I    Solving the smallest instance directly (base case).
    4 X7 j7 i+ ~1 D% i, Q* v" w9 u6 D3 F! Z
        Combining the results of smaller instances to solve the larger problem.
    - [- T4 p' e' P( Z# D4 Z# i( \% u  g! V$ j# Z1 |
    Components of a Recursive Function4 d1 B6 h8 l) E" S, r, o  W2 p
    ; H2 Z3 T* w  m7 ~2 J
        Base Case:' i& S+ _9 t7 ~; O, N- Y' x8 f
    / F+ q# f2 ]6 ~% ~6 P3 T4 P" y7 u8 k8 N6 ]
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    : P- h# @% |2 Z4 {- C( F6 `
    6 {+ c2 n- p3 {  s        It acts as the stopping condition to prevent infinite recursion.
    ' q4 j7 `1 Q: h' ~( c3 N- p" {% P% O. y1 N  {
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.' E, L% @# s) X  d- M: i$ z, ?
    % s* v7 i7 F. d7 ?' ?0 J) x
        Recursive Case:
    , f* o# Q8 V" X% W1 |" G. t$ `  w6 P  H- O, v
            This is where the function calls itself with a smaller or simpler version of the problem.: }/ t2 w" ~  t# R: }9 F- {6 D

    0 M6 u! H' R& U, r  k        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).' L4 ^* @! R0 ~% O; e

    1 A- c) K; r$ k% [  T" }) uExample: Factorial Calculation% z0 l3 k! X' t; M' `

    3 v6 L# M9 L8 y+ E2 WThe 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:
    4 _7 V3 N( C/ C1 H* [1 t% F0 r( n  W. \( r2 ~7 ~# R, l. G
        Base case: 0! = 1% c+ h' Z* R9 T  B' U3 f+ @

    5 i5 A7 s. `+ u3 d7 K3 |0 q    Recursive case: n! = n * (n-1)!
    ) o+ I( _& H. w, _
    - i/ Z7 e/ p2 {. E, R# `Here’s how it looks in code (Python):
    . j, t8 \6 E5 C. v  L/ A: dpython, |6 u& R5 u8 k5 b- u) K- ^0 R- }
    ; P( T5 ?/ A7 V- t* N- W, }) v) b! x
    * R4 z/ J5 o, N
    def factorial(n):
    ' B7 }2 H) D+ `+ L. W# y    # Base case
    7 y6 p. B6 R, i    if n == 0:2 T' v! Y8 @5 c. v# h
            return 12 u/ e4 O0 Y9 [. Z  C
        # Recursive case
    / y/ p! m  B, k8 `/ }: I. Z: @    else:$ S7 N. @& p8 f
            return n * factorial(n - 1)7 n- u/ Y1 X% F
    6 ]+ `% C. n6 @
    # Example usage) j& R5 d  z2 ~0 G9 Y2 |" I2 t
    print(factorial(5))  # Output: 120/ _* B+ h# V) B. \" ?! I
    - \, X( N4 k7 I5 X# o; a  G2 b
    How Recursion Works) y  V5 C8 J  L' D- |1 ^+ {6 N
    8 e! O6 I2 K$ T5 e! t( p3 L& t/ ^! E
        The function keeps calling itself with smaller inputs until it reaches the base case.$ P. y, Z4 f, D) N6 P1 o2 Z6 w
    3 O5 {1 w, B' L9 N
        Once the base case is reached, the function starts returning values back up the call stack.$ n1 ~! [6 I2 e* q# ]
    , y/ Q$ `& L- T
        These returned values are combined to produce the final result.
    # x( A/ s& A1 \: H- X+ B" N6 e/ l2 |  n
    For factorial(5):6 x% m' M) Y" ]" U) J  k. p4 z
    / _8 g3 m' w# y5 d7 C0 j
    - |) y6 c3 B, s% M, j2 M' l
    factorial(5) = 5 * factorial(4). g$ e; \+ l  e; y/ T9 Z) s- [/ X# R
    factorial(4) = 4 * factorial(3)
    0 ]7 }# {. c: a  Ifactorial(3) = 3 * factorial(2)- ]' Z# A' K+ C
    factorial(2) = 2 * factorial(1)7 U7 v: g; `% K! P' Q7 b
    factorial(1) = 1 * factorial(0)7 S7 h4 W7 H; s0 L3 i4 l! [+ x
    factorial(0) = 1  # Base case. S8 A1 v" a8 {& {9 K$ w

    : U$ C+ V- O' tThen, the results are combined:4 L  @9 ?- `4 T4 Q  Z
    - |) W$ m. T/ _" Y
    ) a$ R/ h2 \9 X. x1 ?1 }& U' I. @; d
    factorial(1) = 1 * 1 = 1; j" N" t  I3 ~" \$ p6 E
    factorial(2) = 2 * 1 = 2! d2 S, S; k1 g9 e: ?3 s
    factorial(3) = 3 * 2 = 6
    4 I- M) o. G+ F9 q- X( R' \factorial(4) = 4 * 6 = 24* t+ e" |% t# |/ m  u3 T
    factorial(5) = 5 * 24 = 120
    , N/ N/ v5 D  L% C2 ~: Z! [' Z4 ?) O: X0 j* [) ?8 o$ m+ p  [4 I
    Advantages of Recursion$ B: t- D: Y$ Y8 F' O% M
    8 N- ~5 c$ G8 w2 \7 n4 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).. r5 \' f( `+ R5 C# w9 D! D" v' c
    + {7 G% F( v  S- ^
        Readability: Recursive code can be more readable and concise compared to iterative solutions." Q7 A( q" }  I$ v0 _

    # s" }  N9 l9 W1 _Disadvantages of Recursion" ~) Z, ^3 f: A: w. |

    0 s8 o+ A1 s( R, c* v5 k. o* ~    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.9 S1 e9 L0 r, h: R( Q
    / ~3 {. H+ S5 }$ H% |. X* \: i5 x& {) ^
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).4 R; h9 y0 ~6 o' e* c

    ! w; m1 t4 }, V1 b" v7 EWhen to Use Recursion# m' _' w2 t. b. M; h+ m( b8 Z
    5 d8 N! M0 D, C( V/ O. [# x
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).; k7 ~# g$ v3 P- H4 k

    / z! Y: ?! }/ c( F, Z8 u& b$ {    Problems with a clear base case and recursive case.
    % P7 m$ b/ J% {2 R2 X$ {8 g5 W0 t! }( n: K; Q5 I) ]2 y3 U' G
    Example: Fibonacci Sequence
    ! j8 G0 }! E* ]) w" z- W3 q7 ^
    " r; y. ?6 ~* F3 \' R4 QThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:$ s% o. H; f, _8 @; T: i

    . z1 K/ A, d8 v# x4 U    Base case: fib(0) = 0, fib(1) = 1$ y# v% ~$ K: s
    5 J' [3 H5 X2 u% e' {/ i! [9 g
        Recursive case: fib(n) = fib(n-1) + fib(n-2)3 Q( C0 z) ?, n' r( p( V
    , H) M  G6 ~) T0 V4 ?9 J( t& o2 {
    python+ Q. F# b" \# n! D

    % H1 n: F4 T5 M
    5 j1 ]. A; N. W& n. fdef fibonacci(n):6 A) w# J  Q& ?4 _+ u* O
        # Base cases
    3 W! W# s3 U! i. _; V4 }1 w    if n == 0:
    1 W+ |3 p, l% @. J+ M6 Y, M" \6 h        return 0
    " `* ~; ?; L, \+ q    elif n == 1:
    " _8 g  [6 a5 ?1 u% u        return 1" b. q2 B. H( Q  b
        # Recursive case
    1 t  X7 o7 a5 H5 g; P    else:! i  e3 w  O: j
            return fibonacci(n - 1) + fibonacci(n - 2)
    ) L: }6 e1 W6 T/ r& W
      T3 ?2 \* a9 P# Example usage
    7 J, r" {6 P* m: B5 f3 D3 ?, X9 xprint(fibonacci(6))  # Output: 8; E( Q5 \) @; ^/ U% J& ^. @' k5 V
    * }& M& l4 c0 a+ p5 U' C2 N
    Tail Recursion
    2 R% l6 U1 x3 q8 X! Z
    " a3 e/ F: m6 O  g9 CTail 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).
    ) I4 L3 [. V1 s, l2 L, x, u% n7 W0 j) L) m: i3 l5 K
    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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