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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
    6 r5 H& T  g5 J, c  n6 v5 O7 L$ ]9 T/ I
    解释的不错
    ) h0 b2 g* E  l! ^+ i
    ' j# o0 b. K' }8 R. y( e6 e: |: t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。6 q7 ]7 l4 f' F+ M0 U! s& V9 a
    # H& Q; N7 h- s5 t: N- _, R
    关键要素) H0 x- t+ ]  |1 h5 n0 T/ ~+ R3 A9 M
    1. **基线条件(Base Case)**
      ?; \1 G* I1 \/ _- T. E9 v   - 递归终止的条件,防止无限循环+ `; {% ]. s; W
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 16 d, n3 N4 A- b3 @

    ' g; R/ L7 p+ c) C; R2. **递归条件(Recursive Case)**
    7 H- _* g& d0 K   - 将原问题分解为更小的子问题* ^( i$ i6 T  r& m
       - 例如:n! = n × (n-1)!
    , T! |3 R" b+ Q& c
    + H6 d3 n/ s8 k4 f 经典示例:计算阶乘
    & J# u5 _: v8 Q/ w4 j; spython; C; A! [8 S) {; K. |1 Z5 a& ]3 N
    def factorial(n):
    * y% h+ y. d  ]5 I2 y8 Z2 A5 i    if n == 0:        # 基线条件6 K; O; [) J! A/ a8 D0 l! n0 u
            return 1( E' j/ x3 A% g! `. ~/ M; I* |6 J- r) I
        else:             # 递归条件
    ! ]. c: K5 z& s' f; y1 B        return n * factorial(n-1), O0 c6 R) r5 {! N' p4 e5 n# J
    执行过程(以计算 3! 为例):7 e; ^+ `+ a0 E) {
    factorial(3)
    7 C; \* @$ P: b3 * factorial(2)& {/ `( v7 U$ D8 V8 |2 n
    3 * (2 * factorial(1))9 X6 c! n5 @# u, r3 k
    3 * (2 * (1 * factorial(0)))! ~+ a4 [* v% K2 _; ^
    3 * (2 * (1 * 1)) = 6! _- u2 j9 O" N7 }9 d8 w! T9 u
    8 N1 w& i$ b* l& m
    递归思维要点
    # E- @# m, `; U$ n- A6 v: O1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ! Y' D* c! f, Y! _2. **栈结构**:每次调用都会创建新的栈帧(内存空间)% [6 s0 ^# a0 _
    3. **递推过程**:不断向下分解问题(递)3 |  v$ u0 ^* ^$ [! L2 ^
    4. **回溯过程**:组合子问题结果返回(归)
    ' {9 }/ C4 D/ u1 P' `
    . S: L0 g, \( s- P: S 注意事项# ]* d& K/ V# x* ~% r5 X
    必须要有终止条件& ?; I& O9 ?% ^/ d5 e/ h7 B
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    , X  x& r  t7 [# h某些问题用递归更直观(如树遍历),但效率可能不如迭代1 \" L! J% e2 T8 H* ^! P
    尾递归优化可以提升效率(但Python不支持)
    8 Q- Z9 x1 w- X6 B" {$ n8 i) |5 k+ Z( U6 Z& L
    递归 vs 迭代4 g3 ~7 [4 }, |$ k7 v
    |          | 递归                          | 迭代               |
    ) Y& N5 }) s$ g4 h% d|----------|-----------------------------|------------------|+ N% C5 P0 \- Q+ x4 p6 u4 V$ i4 B
    | 实现方式    | 函数自调用                        | 循环结构            |& v* r! r" q4 c
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ) Y+ k3 @/ Q( Y  Q8 t| 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         |% V0 q  h9 h2 G/ x
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    : O/ i8 F( |' i9 J8 U; i. _) V3 s/ Q7 f3 G
    经典递归应用场景
    3 y% Y; v) Q8 G% q  _) z: ^) R* r) w1. 文件系统遍历(目录树结构)8 F. ?$ U+ Y4 R: u/ r" k6 }
    2. 快速排序/归并排序算法# K3 T/ ~7 r9 v/ m( L# q4 b
    3. 汉诺塔问题' U! ~8 v, \. T
    4. 二叉树遍历(前序/中序/后序)
    4 q+ U8 M2 l" O5. 生成所有可能的组合(回溯算法)
      ~  \& {" T" m3 g( c) h' |. x( s# L" b1 G
    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 06:32
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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,% d2 y5 C, j/ i1 t
    我推理机的核心算法应该是二叉树遍历的变种。+ I! k" M( ~8 H5 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:( m* j( L- @9 w" |) \# d/ x
    Key Idea of Recursion4 d, H" m8 h/ i% `5 a# Y  v

    8 ?% L& O: _3 R5 O/ ?A recursive function solves a problem by:" Y& ]& a# v( A' F3 |

    4 b% a' e1 \+ B5 g6 z2 W& {9 C    Breaking the problem into smaller instances of the same problem.
    0 z1 V) c+ L7 i3 G7 [& D0 e. c
      o" n9 c3 U( ]& C3 N2 N1 j    Solving the smallest instance directly (base case).
    , j0 H* _) J' }' }2 y
    : ]' R5 j+ s% o% Y4 b    Combining the results of smaller instances to solve the larger problem.6 D+ R4 m0 c* _) J" n( F/ w

    , s: Y# \# ^$ x2 vComponents of a Recursive Function$ N" Q( L! E: s& _' c+ Q# q/ T
    2 Q( u* `4 y0 o" k
        Base Case:
    : I/ o  @, W6 T0 `1 H1 y' L9 p* F' u6 S' @& d
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    . d7 x* ?; @$ M7 t5 h$ L$ \5 A) F8 G( j+ R3 a
            It acts as the stopping condition to prevent infinite recursion./ ~. X2 R% D$ h0 L3 J# [% e8 b
    3 T) ]2 x3 L9 M) C3 m
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.8 R; f5 F+ g1 y* F* p
    # f% }& q9 z9 O! M; k7 B
        Recursive Case:
    ' S4 b8 L3 }$ x/ _! Z2 a5 E0 P  ~$ w! |1 [: @
            This is where the function calls itself with a smaller or simpler version of the problem.9 [- `( B9 F' F/ S: h- g
    % ^$ `* m& O' Q) }4 E% ^
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).! L/ \$ |* p5 F) Q. n

    & ]0 E) E* t4 l: e% i! z7 L' gExample: Factorial Calculation2 K! Z3 U6 c1 H  k2 G5 B9 @

    * @# U/ H1 ?2 D! y9 ^" U" A3 `9 RThe 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 n' Z/ S- b  M0 U7 N& h- a" L/ ]/ I7 X3 c5 G' T
        Base case: 0! = 1
    * g8 t3 r- f1 J/ n+ e( d
    3 J2 y$ V  r: L9 R    Recursive case: n! = n * (n-1)!
    - w4 g5 i9 [! T4 f
    : O: E* H# N! [+ m( xHere’s how it looks in code (Python):9 H% {8 K2 z- k6 v
    python
    : [9 d! z1 H" T/ B: o) F3 X0 N8 ?# S# I9 h+ K- G7 q; U# c; t5 g

    5 |0 s3 b4 [+ n% s1 E7 ], v0 Odef factorial(n):" J5 s0 h% I( q* g, X) E
        # Base case8 o2 U: I: J9 i0 X7 O9 |$ L/ K4 Y4 H
        if n == 0:
    : Q$ Q1 v. V9 @+ _! A/ o        return 1* e; P( ?7 q% L1 L
        # Recursive case
    - f2 {6 [) y3 q3 M/ R# w  x    else:
    ! M6 T# S; W; [3 G0 e        return n * factorial(n - 1)/ P: n% j' ?# w  r
    4 u; m1 k, ~  W
    # Example usage$ O- w/ W+ @' p
    print(factorial(5))  # Output: 120. v" S8 G9 J* ~; |( V5 E* l
    : U9 R; L9 i, O& [
    How Recursion Works; O; n1 L+ X7 i0 x3 K5 n

      ^. W  y9 ^7 J9 I1 m    The function keeps calling itself with smaller inputs until it reaches the base case., ?1 M2 e9 G+ E* D5 N, \
    * O" V% V/ J' ?( W6 M1 A
        Once the base case is reached, the function starts returning values back up the call stack.8 ^: O  m8 N- I

    " \0 m& q4 F: V    These returned values are combined to produce the final result.
    ' h3 [! R+ `3 w6 Z
    ; h% b& y( ?5 L5 o% m3 [, [) u% JFor factorial(5):
    2 c8 h& g7 `; j' P" m- h; i+ o/ o. ^: Y5 t' Y
    % P( ~/ Q7 {& i1 }
    factorial(5) = 5 * factorial(4)& v. U7 X. Q- L5 p$ i
    factorial(4) = 4 * factorial(3)
    4 q7 u, ]& X, r* f: C. ~8 Kfactorial(3) = 3 * factorial(2)
    3 q9 I; p: X4 x1 Jfactorial(2) = 2 * factorial(1)
    ; s5 Z0 Z: t( X% q- W, dfactorial(1) = 1 * factorial(0)
    4 A8 T& e) g+ O9 L7 k& S5 ofactorial(0) = 1  # Base case
    $ a: h, j1 }/ f4 g, B) ^. @
    6 f4 X- A; v. {5 v8 u5 h. `Then, the results are combined:* I3 i) l6 a  j4 o" ~& H

    5 p" h4 Y$ H  Y: U& d; ]& T9 m5 p9 x: n3 j% W+ s
    factorial(1) = 1 * 1 = 1
    + U/ k$ [* w; W& ?# Y& kfactorial(2) = 2 * 1 = 28 n9 D! R( G! J2 Y
    factorial(3) = 3 * 2 = 6* u" O- v, w! e3 I) e6 E" D
    factorial(4) = 4 * 6 = 24/ \8 r2 \  @" j% Y  S! o2 H, S% m' f
    factorial(5) = 5 * 24 = 120
    1 w5 g0 A6 U$ L7 U0 t! [/ l
    ; T# ]9 I5 Z- q& Q/ ~" ~% NAdvantages of Recursion
    : k0 U: C/ E5 p1 r* S% L
    9 z7 B2 ?  o: f7 b    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).) Y$ P/ _% H7 ?7 e" K
    3 Q# Q- l" Z2 [0 ^# ]% O4 n( \
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ; Q: ~& `, l  K6 }1 h8 j1 B0 ]: e& u/ s. O$ N$ {1 C+ A+ `, ~
    Disadvantages of Recursion
    / _' P' O* e; g! {8 y
    " N0 ^- t3 q& t* }    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.
    / o/ Q. U* Q( z+ y. @; M4 F+ _/ \; P
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).3 l  D. d8 B  h3 a! N+ q
    ! ]& S* l3 }5 W! i' d1 p  Y
    When to Use Recursion, v9 q5 s& @/ v& X) c* x% d  t1 e
    : V1 W; a* P' V( d* H
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).; z9 k* e0 }8 n

    ' U2 U( Z5 w* G! Z% U4 ?    Problems with a clear base case and recursive case.: ?, I  h; ]6 E9 T; e

      [  t% ?- i4 v$ W" n( M! lExample: Fibonacci Sequence( U0 m( Y2 i* l' g2 u) N- `5 V
    : k( ^; g8 C5 ^% E0 B5 b
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    * u* U7 c9 @- ]- }
    ; P0 b/ \% ~+ I9 j. ?% C3 Z2 T. B    Base case: fib(0) = 0, fib(1) = 1: ?$ T' @* |6 A
    & ?* f/ }8 l4 k9 F
        Recursive case: fib(n) = fib(n-1) + fib(n-2)4 \1 }' S+ Y! \! ?

    ; o: z3 K$ y' c% `0 vpython
    . s" k" J3 D% O, ?! c; W. ]
    & E9 _# W, }6 H' `3 Q- i! a- b' E: d  @# G% A0 ]( [1 ^
    def fibonacci(n):1 X" Y- a7 t- Q, z
        # Base cases( x# h/ q$ ^* U% n9 s
        if n == 0:
    4 U3 i' k, E' Y/ r        return 0* ?+ N! e9 g5 L
        elif n == 1:7 P, H+ v2 t0 H% r9 f8 ~0 q' F
            return 18 c3 I/ C; _! I' m# `
        # Recursive case
    # f% _1 ?+ Y$ Y    else:% f1 k0 h: F3 y. e0 u6 _* }
            return fibonacci(n - 1) + fibonacci(n - 2)
    5 J0 r6 u; {# z9 W; c  E- z9 U; @( _/ Z& f- E, i; E$ }1 A
    # Example usage% D- ]  G( h. G$ f" f! a$ M$ \
    print(fibonacci(6))  # Output: 8( ~0 g* I$ v" f) I, w, H
    3 C# D/ D" `+ J$ C3 n; B) t
    Tail Recursion
    1 [/ y5 [1 `4 N/ O% \
    # U' P7 ]2 X0 Y; w" I$ W7 ]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).
    " F+ [1 h1 H3 g
    3 h+ d% k  a- c9 Y* I8 }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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