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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
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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
    " s3 w5 |2 y- e1 g4 R, R, o* c% v
    1 }5 Z! `# m$ g( ^( ]解释的不错/ {+ k! E8 I! N& D" M; u! y% F% s

    6 g7 t# B5 m* B1 F+ 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。
    - {6 {4 Y7 J( \8 d( [
      T7 e) N+ y0 j* T0 k& D 关键要素
    1 x" X( E) t; v' E/ L4 S1. **基线条件(Base Case)**- P3 t( v% O1 e* @' {( }' q
       - 递归终止的条件,防止无限循环
    $ M( C4 b. f3 w* A$ V# q% @$ w, u   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
      u' ^6 S: f) T" K
    ( E! i  O/ i3 j1 u8 T2. **递归条件(Recursive Case)**
    ' ^+ l0 ^$ s% q/ q/ V   - 将原问题分解为更小的子问题
    * f" O" s0 D$ r0 x. m   - 例如:n! = n × (n-1)!5 `% e4 M  L$ ?, |- A
    , }) m' M+ j8 h7 J7 f4 _( r
    经典示例:计算阶乘
    ) O0 y, |+ E- M0 `+ g" c+ wpython
    1 L( m  G: s( h: Ndef factorial(n):
    * m9 Q6 M3 B1 F) V  P% i    if n == 0:        # 基线条件; T, v' B# U+ F8 O. l/ d0 c
            return 1
    $ b; L! T  v7 |5 [9 x; K    else:             # 递归条件
    # f2 b4 q  I# f$ G1 J' G6 D  b( p        return n * factorial(n-1)" R9 B3 d% d7 Y5 M
    执行过程(以计算 3! 为例):
    8 [, j) E) w: e1 O4 N6 pfactorial(3)
    9 `1 o$ z: H7 Y* J3 * factorial(2)
    6 D/ E; T: ~% h3 * (2 * factorial(1))4 n, u# Q- B8 l9 l
    3 * (2 * (1 * factorial(0)))
    8 f! G& c# j6 L; d7 M0 O3 * (2 * (1 * 1)) = 6
      H+ f) u9 f* f/ G' j+ S- P' g  w' E% p
    ' l. f, Z8 N8 H$ } 递归思维要点
    " R* ?# \, `- m+ x9 R9 |# z1. **信任递归**:假设子问题已经解决,专注当前层逻辑8 ?" X, ^9 Z& V1 Z) O1 z) D
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)! _4 f; |& Y% T% P
    3. **递推过程**:不断向下分解问题(递)
    6 I# d4 E4 Y2 y! J$ X2 ?6 _% w, S2 O4. **回溯过程**:组合子问题结果返回(归)
    9 L3 Q* o; o( U9 |4 Y- r6 Q/ E' M2 }5 v/ \
    注意事项
    , p; j0 R4 h5 t- `: D+ f必须要有终止条件
      U! Y( E2 O$ k; h& a递归深度过大可能导致栈溢出(Python默认递归深度约1000层)* K: {6 J) x0 O1 V- }0 P1 c
    某些问题用递归更直观(如树遍历),但效率可能不如迭代. X4 x7 r- r2 [0 ~* o1 h* `& [
    尾递归优化可以提升效率(但Python不支持)
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    # G) r. P7 n! x7 h4 x3 Y- @, _ 递归 vs 迭代+ A# E$ c% p4 ?+ c! Y) L
    |          | 递归                          | 迭代               |
    1 \( f  \% h& K: e|----------|-----------------------------|------------------|3 h6 c, k. j. x; l" [
    | 实现方式    | 函数自调用                        | 循环结构            |
    9 J/ S" W1 T7 N  n# K0 H* q| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    3 T8 w& r7 ?+ T, m1 w( v7 ^" @| 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         |
    # `' }5 T! |# C# Z| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |. D; z7 H3 p5 }3 t2 j

    4 C" M- H: F& E6 M! } 经典递归应用场景
    ) S6 y4 i; N0 n8 U, Z1. 文件系统遍历(目录树结构). e. h1 m3 s" R5 J
    2. 快速排序/归并排序算法
    # O; ~) m2 k* Z7 L3 q9 h( n/ B3. 汉诺塔问题
    6 H3 Q- d7 X' U/ G) {4. 二叉树遍历(前序/中序/后序)
    2 N1 o' @# j" w- W; V, z5. 生成所有可能的组合(回溯算法)6 ^  P! G/ Z1 B' L

    , s8 }4 u8 s+ l" T* r6 C* ~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 15:11
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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,
    % ^. Q; d: a; q: x0 [我推理机的核心算法应该是二叉树遍历的变种。
    # G0 [2 L% U' [0 p- ~) Z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:
    5 |0 M, H9 j# O3 f, m0 @& RKey Idea of Recursion
    ' ]" ]5 e& ~) k$ Y
    * `$ H- U4 {. X, _. J1 ~/ O4 w5 |A recursive function solves a problem by:) f1 A& N6 n; \4 p3 L3 [: Z
    3 D4 W2 S% _. ^7 G: l' P( C& J4 A
        Breaking the problem into smaller instances of the same problem.
    5 T0 F0 z  y! H* c5 |
    9 \( i2 }0 d6 `) n( W    Solving the smallest instance directly (base case).
    $ L1 T% U' N/ |5 s, C% U3 G& s1 [, f$ f! b  y
        Combining the results of smaller instances to solve the larger problem.+ c8 t+ R$ h+ ]) d
    8 A* s3 c) b) W/ |4 V
    Components of a Recursive Function" W# W2 v8 I" G% r, d
    / A+ [4 n4 z& P
        Base Case:
      g; ]' V! T4 ]$ ]- c* P$ p. L) G6 J, V: E- Y
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ! s7 v0 x4 p4 R5 T  A4 v7 C* N  b& S, o3 r
            It acts as the stopping condition to prevent infinite recursion.
    $ s6 o4 m0 `* t) i" d/ E( h( w7 u3 V+ I8 q0 r
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    2 s6 q- ?' z0 ?, u6 j& |# K% r+ b+ ?7 S2 X  I5 S' f1 A
        Recursive Case:
    ( \7 @/ `7 p+ ?4 ]8 B1 [8 M
    - M7 M1 x% f; |/ D5 L# |, n1 O( f        This is where the function calls itself with a smaller or simpler version of the problem./ ]0 ^" M/ b6 d9 `; [; c2 r

    ! C6 ?" L5 g# c4 q6 ^& A$ T% r, [+ z2 w* |0 ]        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ) V3 D% a* O$ i# [# h& Z
    , B: O$ T1 E6 ZExample: Factorial Calculation$ X0 ?* Q; \! u# I+ T3 }
    ; K1 M( Z& K. N* D4 V# x" u$ m
    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:
      ?6 j1 M" F! g1 u) U- W$ F8 o7 V, z* `4 {) o. L
        Base case: 0! = 18 Y8 V( T! p- _* q! d2 t6 ~; X9 D! U

    # M. B* x  w- f9 c    Recursive case: n! = n * (n-1)!
    ' j9 V! B' k# ^+ i3 E
    ) w" Z% l% C1 \/ E4 z7 U/ q# YHere’s how it looks in code (Python):! w8 Y4 h; c+ `: z
    python+ T/ m- b3 k& l: I# i' R

    " b( k( t8 m" `# @# [/ N0 n& S
    * [1 l: P# i9 |, |def factorial(n):
    5 r& }* S" I9 X; P    # Base case
    ! G: f2 t8 x$ }. g1 h8 F    if n == 0:5 L: ~% r, D; D" f% F$ L& a
            return 1* b% b2 ]2 Y1 X
        # Recursive case. E2 R% Q7 l" O8 ~$ j4 M
        else:3 ?9 B& a: a: A- A
            return n * factorial(n - 1)1 L7 E' O7 y/ q$ m5 h
    / b8 Y# y- f  N& W
    # Example usage2 Z4 c9 |7 r( a; f; g/ D, c4 k
    print(factorial(5))  # Output: 1205 u! X2 [; ^; u! M5 @

    * f9 _- U0 u4 Y9 HHow Recursion Works) G! c# A9 d+ U

    " J: z; f( i+ G" W% B8 J    The function keeps calling itself with smaller inputs until it reaches the base case.
    # {$ i7 s+ Y; u* E0 ~" F: j" w( I! z5 u1 f
        Once the base case is reached, the function starts returning values back up the call stack.
    $ ^& l: h- e/ d7 R  V  U$ i) r
    ! [+ `( }2 Q. @- J  Y    These returned values are combined to produce the final result.) p, T! t& L8 z4 ]3 o2 P! }6 M7 s" ?* u

    8 {! M  x5 U9 H3 ]2 EFor factorial(5):
    $ _3 [# D% S5 d; I& ?, P: e" F: }' `5 ?$ c

    3 I! @5 |5 n/ K: _+ Kfactorial(5) = 5 * factorial(4)* y7 \8 Q5 o6 H$ L+ j. d
    factorial(4) = 4 * factorial(3)
    7 F3 h5 e  [8 [! a  l* cfactorial(3) = 3 * factorial(2)
    , Q6 b4 R# B* J, Y2 b+ L$ ?factorial(2) = 2 * factorial(1)8 }& X+ B+ p2 u" F0 A3 W7 O* _  O
    factorial(1) = 1 * factorial(0)- J9 G3 b! u% [. @
    factorial(0) = 1  # Base case6 t- A  o3 v; z
    $ k3 g1 X  K* n. e! S7 v% U
    Then, the results are combined:
    1 g" o1 Y4 @; i# z" d/ |& }
      T' B) D) _9 o# d3 O) G- }
    7 O# f0 l' \( i6 [factorial(1) = 1 * 1 = 1: F+ B$ n) F; j0 D2 m! |* u2 y: i
    factorial(2) = 2 * 1 = 2
    " X" k5 G) b* W8 K% w6 I. Rfactorial(3) = 3 * 2 = 6
    ; x( Z- m; x8 a: L! C/ ~factorial(4) = 4 * 6 = 24( D5 K9 [' v0 ^  Z
    factorial(5) = 5 * 24 = 120
    3 y# r0 w4 a* n: @* r. x# w9 p* y8 F% i' G
    Advantages of Recursion
    0 I0 ]" j2 |! ^8 q& k7 J  a8 Q
    , C2 _5 T) [9 @( |5 p5 P0 {" Z    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).
      A6 t6 R/ Q/ D. `3 E0 L8 r( U7 M. y( D6 h  l( P* n
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    " Y+ k% \  y5 A1 J5 B, O: o  V- V) N9 Y( h) Y
    Disadvantages of Recursion
    4 V2 }- Q8 I- s
    / E& x- D  t7 w& E& o' D    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.
    8 J' e; V, V4 @" V9 ~: d  V" l4 o! _
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).. m5 \# _+ W- g& J0 d+ @* V7 ~
    3 K  h3 Q! J0 e
    When to Use Recursion
    5 [$ s( t+ e4 V  V. A
    - |" [9 g8 v3 N7 j& K4 F9 c6 n    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).' J, c/ a( u+ t& L) w- |

    . X, N, D$ f: e: k2 `    Problems with a clear base case and recursive case.6 B# ~* P9 t# P% b

    # l. r$ _- h1 C2 N% W& S7 uExample: Fibonacci Sequence- ^. h" h# O  A- o
    ; g& U4 b; [0 a( ?; s( s3 w- z+ d5 P
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:9 T- x5 G* Q# [: l/ p

    , D5 b! T- @4 ]( p7 ?& N: f/ W) {    Base case: fib(0) = 0, fib(1) = 19 `1 f5 O) D5 {& }7 `
    8 a* O4 p; [' [8 Q. ?
        Recursive case: fib(n) = fib(n-1) + fib(n-2)! u+ _. _" h+ g# A
    - b* C8 g/ f* K. ?' b. w
    python$ z+ r- K6 `" M" O) [' f

    9 N0 A  y% w  `: v7 K2 ?
    2 y1 K- w/ F) m2 B: [- }; d2 Ddef fibonacci(n):
    4 l# t3 y+ ^- r/ ^    # Base cases( |2 Y5 X& }( @; L
        if n == 0:, T' u4 D9 U3 K" n
            return 0
    * h, r; y6 S' y: O6 d    elif n == 1:
    * ~8 K3 ^5 t: E- Z6 _" o        return 1
    / W3 d1 h2 X! H3 w3 `  z. ]    # Recursive case+ w5 j* E) R# k; Y' |$ A
        else:8 ?' z  b; S3 b6 z# Q
            return fibonacci(n - 1) + fibonacci(n - 2)# N: I/ K5 K; h

    5 c* Y; @. b8 s# Example usage4 j$ v( [* I2 _, Y
    print(fibonacci(6))  # Output: 8, H8 @" u" _3 G# h' u

    ) W3 T! R  x8 C* L8 n9 c; r/ F0 `* |Tail Recursion* X) W  ?% {5 S- ?
    9 Y- F) Z: U1 K; M/ V
    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 |6 I, H% K2 M# O
    4 j- _4 f) K' f
    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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