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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
     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 ! n4 G6 m7 N& x: E, @

    , B1 c, @% A% L" G; n) ^3 x解释的不错0 L" ~+ s7 P# p9 S( ?9 k
    ! {0 R1 n  ?( G% t- s- W' h
    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。) Q8 S. z0 L. O( H) q
    * x0 F) B: U. e; z  [% M  ?- w
    关键要素
    3 d- c: D/ u7 [' n1. **基线条件(Base Case)**' P: d( d/ V& C8 Q7 R* p$ K
       - 递归终止的条件,防止无限循环9 ?9 D4 n/ s9 x) a, Y
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 10 F4 x9 i. R0 v$ ]9 z

    ) C( u4 ~% r& y* I% I0 ~2. **递归条件(Recursive Case)**
    ; a8 s, v; r4 b! i7 C  V  `   - 将原问题分解为更小的子问题
    + |' |* l. B! @3 C$ z6 n   - 例如:n! = n × (n-1)!
    : O, J0 v( C- R" E' X
    ! G, q; n8 R1 `  |8 o  A% N0 k 经典示例:计算阶乘- p" ]$ i$ Q  C0 h2 n
    python* `% z5 }: O  @) F3 [2 U
    def factorial(n):
    , A- g& k' A' W5 X    if n == 0:        # 基线条件3 P. ^* P% Z3 Y4 B: s4 p
            return 1% A# ?3 w/ B1 N
        else:             # 递归条件
    & N' `2 `. `7 x5 o        return n * factorial(n-1)8 d% w% v; T; y+ L4 D; _
    执行过程(以计算 3! 为例):
    , u3 r. Y$ a4 T& j( @2 V. M5 mfactorial(3)
    + R: T/ Q, O  b; Q3 * factorial(2)$ a" W0 P( R, a) Z5 y5 i4 Y
    3 * (2 * factorial(1))
    $ ^4 |/ W9 x! |( _+ p( a& {) S3 * (2 * (1 * factorial(0)))
    4 h7 ]9 J( `7 b3 * (2 * (1 * 1)) = 6
    / D% k- c5 i2 q# l7 s
    / v, Z; i  C- M  e 递归思维要点3 }  w2 G, h7 q" N
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑: a, y' |5 [& y" V& H. ]" ]( k+ G, u
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)) f- x  a7 E( ~* g* S
    3. **递推过程**:不断向下分解问题(递)& Y& h3 d9 G; Q( e& e, [
    4. **回溯过程**:组合子问题结果返回(归)
    ! f+ P$ m8 w# G3 w" h# G# m' H9 R3 u0 ]+ [- l1 l
    注意事项
    9 |: S% D* ~7 Y: m8 \必须要有终止条件( r) w; G0 N* W0 F5 a! R; E& T
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    3 s( d* i1 m" I. B' U0 n某些问题用递归更直观(如树遍历),但效率可能不如迭代3 ]* S3 H6 o; F8 V) O# }8 m
    尾递归优化可以提升效率(但Python不支持)
    6 a5 k8 D6 W0 P$ ~9 w1 ]5 C6 I: F1 R( N  g0 h
    递归 vs 迭代0 L# c& ]3 H7 w/ o! W3 o( G
    |          | 递归                          | 迭代               |
    ; ~! D" k: U" X: [% d|----------|-----------------------------|------------------|
    $ t5 n3 C, U7 a% q% L0 w; b; Y  m| 实现方式    | 函数自调用                        | 循环结构            |
    1 y, y6 y/ T( Q* V| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ! ?* {4 \  ^8 S/ |! G- R4 @- L| 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         |
    7 G% P& c8 x/ i8 k% m8 e: i| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    : l6 I6 n! y  {6 f' B
    , r  b! \% ], T* S. I 经典递归应用场景- T% x% t; z) g. [* A9 U/ {0 z
    1. 文件系统遍历(目录树结构)
    6 h# g6 A5 h% n! n7 }8 n2. 快速排序/归并排序算法/ H' l* Z; z9 @/ s0 S
    3. 汉诺塔问题3 |  G: w' W0 L) N& {5 T  e7 Z; D+ w
    4. 二叉树遍历(前序/中序/后序)$ O1 N! j6 I7 H9 i1 O
    5. 生成所有可能的组合(回溯算法). Y! T, e( P" x6 e9 G4 d
    9 I9 X- Y7 o6 z- ~
    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 06:43
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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,/ {! W% n; o0 O+ P
    我推理机的核心算法应该是二叉树遍历的变种。7 t9 |: j% t7 L- l  A/ q
    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:
    7 {6 ]( }# l1 a6 YKey Idea of Recursion1 C4 O  q* ~) l) G& p) K

    9 d4 G" c' n- o5 }A recursive function solves a problem by:
    9 d( ~1 j* S& c  ^) d& _+ J% @5 K
    / g7 o0 U) f! @- P( o; y0 K    Breaking the problem into smaller instances of the same problem.! g9 c/ y/ h. O9 S& q

    ) K* }' f! j% j+ v: u9 D* D1 p    Solving the smallest instance directly (base case).
    . S0 L; k9 m# d, q6 @  E* a; _# _2 X7 ~: s' o
        Combining the results of smaller instances to solve the larger problem.
    ! H. K* }% w( [1 y9 J" O
    2 d6 }( \0 i) X! x$ [9 L% dComponents of a Recursive Function
      t5 b; r2 f9 \6 a
    7 X, A* g/ r, U  B. o% X    Base Case:
    ! r2 a% k) d' m) x1 b% ~4 X# C( }: K# t; A$ v7 ]$ V8 c! [
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    1 e  c) a: G# Q; j+ l( Z! j" l$ Z; s1 t7 h' y
            It acts as the stopping condition to prevent infinite recursion.
    $ T. R3 W. J  b2 p4 s3 D, p* \& y- |2 W7 _7 P( Y# _& ~2 m
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.4 I& r$ w' g) J% b5 I5 y+ Q

    , d4 u  I  r# ^; c7 o    Recursive Case:
    ) }/ U* r$ k7 P4 a# k( ], D$ o# J+ ?7 z) u( K
            This is where the function calls itself with a smaller or simpler version of the problem.
      W4 J, B% D0 |, z
    . J) X' B/ f5 S- S: P; F# K) H        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).7 X! z1 [+ x$ l! J4 N1 a, n/ Q) X
    2 U0 A/ H$ c  X; u/ a1 h2 L& l2 K
    Example: Factorial Calculation
    ) h5 C  |% P# R
    1 |2 h+ k# u) c% @5 BThe 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:& r7 @! z5 g3 U* j

    ' O7 O' C. ^: u: O# e+ D4 j    Base case: 0! = 1
    2 x, ^5 B  f; g, N. G9 ?( p" F# J
        Recursive case: n! = n * (n-1)!, R3 {9 T% W' j
    ( O- u8 A& p, |* ?" K
    Here’s how it looks in code (Python):' c# t) d3 l- }$ E
    python
    & l& B1 E  f' z, ]" ]; X
    4 m9 \  @' j2 Q# ~  N7 \7 P. \3 v8 I* a7 i; Q7 I
    def factorial(n):. W. @+ a0 B1 d- l4 p. ]2 R
        # Base case- ]: ]/ s3 ?3 K2 K! z. ]+ m+ q8 Z
        if n == 0:4 R0 G+ b: d1 T/ M
            return 1
    8 q8 R, ]9 t5 I    # Recursive case
    3 J; w6 i$ M8 N! o/ f    else:
    & y1 f* Z4 i  h+ U4 L% K        return n * factorial(n - 1)
    7 O' i" F2 v2 z6 D/ Q0 h: M* ~& l( s! I! R) B7 P2 Q
    # Example usage
    * i8 B. U- R% u. A; E  b: ]print(factorial(5))  # Output: 120
    ; ^, R% @+ g& B. z  ^6 A
    : {/ M- A/ q7 o0 S) O6 [How Recursion Works
    - i0 J# W0 Y6 m6 {  Q+ L! P
    : y" N+ L# w" {0 F    The function keeps calling itself with smaller inputs until it reaches the base case.
    . i, h# F9 U+ e8 T. O( v5 G& F6 z$ k& @) r0 ]
        Once the base case is reached, the function starts returning values back up the call stack.
    - ?! {% n9 c( v1 @9 J  R' _! }3 o. B, h+ \) Q( o6 u
        These returned values are combined to produce the final result.
    8 I2 X$ c% o- v1 T1 [) k; W7 }: O; q) U
    1 \" W5 |7 A( t8 ~% @2 b$ q( N0 T% YFor factorial(5):
    % ?# z: Z. T: o# R$ L5 i* O9 Z8 V* H/ r/ T# {5 z; Z

    # h. X  t  d% [/ C( `factorial(5) = 5 * factorial(4)$ r& r, X1 U6 |& C$ z- ]% a. R
    factorial(4) = 4 * factorial(3)8 n; E: ^4 o" D# _
    factorial(3) = 3 * factorial(2)
    6 l$ o$ d' W  x2 H( sfactorial(2) = 2 * factorial(1)
    3 B/ c. E+ K, y- ^factorial(1) = 1 * factorial(0)4 w8 A0 G7 X; J( {5 ~1 c/ D1 U
    factorial(0) = 1  # Base case
    ( j; b1 T9 ^( F3 d6 N' A6 s6 {. i4 z# W3 X* L* s5 b' @
    Then, the results are combined:
    2 X/ z. [' @8 Q8 R. ]; L: k
      F& i' K/ p! y0 t, U& O. ]. i0 R; N3 }( R* ~- z- H
    factorial(1) = 1 * 1 = 1
    5 Y; y) y& K1 s" @factorial(2) = 2 * 1 = 2
    & Q8 P$ m  L% R2 n6 wfactorial(3) = 3 * 2 = 6" ~" }" }8 E, e( D
    factorial(4) = 4 * 6 = 242 a# `- C" M$ n- j$ Z
    factorial(5) = 5 * 24 = 1209 U* L! d) J0 v% g
    3 w3 n. a2 r% w/ u2 U
    Advantages of Recursion
    ; K# d+ @9 O0 j: m7 ]4 |+ J1 B3 Y3 A" \
        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).
    ' T) O, w8 }, o5 a% J
    9 }/ L- ]3 N6 |3 ~    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    9 M* n: H) ~0 Q  Z& e# B6 j2 z' e9 ?8 s% j7 \1 S
    Disadvantages of Recursion
    7 [3 O) A4 U. K
    / z6 S3 g/ P; O% i* i  z  \    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.
    ! Q0 k1 a% I$ K/ j8 _: c- m0 C
      \+ t* b" K5 |, _% e5 J    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    / P3 G- B! D0 L, Q) N
    + K% |5 o% \/ T( u- x; e# ]6 B, X! zWhen to Use Recursion
    6 q7 c+ h+ w# n/ \7 v% m/ n( x. k# X8 ~, j1 V
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).; C* `" K4 p8 f" f; }, ^# e/ _
    ( p# d6 K% j+ Z9 q
        Problems with a clear base case and recursive case.
    4 e* p0 J+ l1 G. a% z) e0 k/ v" e& A! \7 \6 D' l) @% O# a6 R% a
    Example: Fibonacci Sequence
    0 |0 M4 n* s3 |+ z% N4 y6 |" i( q* W6 ]- W! R
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:% l3 j1 g0 B% w; d: M7 m- }

    " w' R. i/ z3 S$ {    Base case: fib(0) = 0, fib(1) = 10 p2 L3 [8 g6 C0 x3 z

    , m% z6 k5 u" P" g    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    & m; k2 E# W3 Q6 J& W' L; ^
    2 Y. F" d8 E' Qpython# h4 u9 u8 |$ q5 V

    $ C% [! @, j& ?- B* Q  z' O3 Z  N, |/ e  q
    def fibonacci(n):
    * H+ I2 v2 n2 J$ r! R    # Base cases  c4 o3 L3 |9 U$ Q. j) l
        if n == 0:
    * ?$ [0 M6 w1 M- w" A; [1 @$ v        return 0. I) O/ ]% N( A
        elif n == 1:! h0 V6 V% T. E
            return 17 H4 F& C) l8 T" W6 p% a3 L6 T
        # Recursive case
    2 T: t2 x* U$ m7 U2 v    else:# B8 K# C0 Z8 i/ V9 v
            return fibonacci(n - 1) + fibonacci(n - 2)
    ! _, i- g+ }# H0 {# k, Z0 q) U, c& W1 W2 p4 O9 [3 _
    # Example usage( V* I; ~" u. L$ J1 `( P1 A  e
    print(fibonacci(6))  # Output: 8
    ' L/ j; }; I; M5 d0 B1 ^  U% \7 d) t% G/ m) M! V
    Tail Recursion; I4 h3 |* {% m& r/ G4 k: l# K
    2 ~; A0 ]4 Z7 p5 V& x; D) G
    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).2 H% N' E1 _+ ?3 K- i

    * \1 {, ?! N% K1 g8 R$ ]' v8 AIn 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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