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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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    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
    3 Y( b9 t# ^9 V2 x" Z2 G3 w6 Q( L8 K& b5 Q2 k8 u
    解释的不错
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    # x2 W! f" R& G0 Q) o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。7 J9 h' B7 {2 l7 Z( g6 m# ^" n$ S
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    关键要素: J+ z6 v0 f- ~  x) V  w0 Y9 N
    1. **基线条件(Base Case)**
    ; s+ s( z$ M( L% s! o1 `5 E   - 递归终止的条件,防止无限循环
    % ]& I9 _& f% P' p4 {   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    " g7 c% e9 @: e2 o0 f- v: n
    ! f( M- Y% B& P5 G$ _2 [$ F, H2. **递归条件(Recursive Case)**
    : g+ s& ]" u% v) L   - 将原问题分解为更小的子问题( e) {" d$ G* s6 X0 n( S
       - 例如:n! = n × (n-1)!. F3 ~9 w6 I" U/ p6 F

    0 h; S- j, s) Q& K% a 经典示例:计算阶乘, I' ^: v3 N) P% N) J
    python+ E" B1 m. |' f1 Q  r. [) C) Y
    def factorial(n):0 v& m( \9 r" M2 E7 G! i5 B: y! q
        if n == 0:        # 基线条件
    1 b0 a8 ~  c8 ^8 B8 \# W        return 1+ O9 m, D+ e: z
        else:             # 递归条件: e  [$ `, s* Y! }/ R% ~3 d
            return n * factorial(n-1). ?# R, a* `8 Q
    执行过程(以计算 3! 为例):
    4 r# @$ ?& A% q5 }& R: ~factorial(3)3 Y1 _% T, |; z0 b" t7 y
    3 * factorial(2)  ]% K  }, X6 k- p+ o
    3 * (2 * factorial(1))
    9 U% }' ^3 G; E- i5 G& R6 j5 d3 * (2 * (1 * factorial(0)))
    9 n6 J: S: f% X: C3 * (2 * (1 * 1)) = 6
    6 e6 Z  f- D6 ]* b9 c; M) E: Q- i4 v" Z% T. I
    递归思维要点* G( n( e6 p9 a
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑8 t/ B% O5 O9 ~/ N" j% f9 m
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)* z1 Y1 `( z9 K4 @: c5 H. [
    3. **递推过程**:不断向下分解问题(递)9 m6 @1 s* `. m+ D! l% I
    4. **回溯过程**:组合子问题结果返回(归)
    % z$ W4 v  _7 U* W* k/ j, u$ ?: ?! I! e) G% |$ k
    注意事项( ?/ w" V* m3 ^+ A. a. J5 g
    必须要有终止条件
    2 j9 L- j- Y. X) q递归深度过大可能导致栈溢出(Python默认递归深度约1000层)+ a3 v' l' s& Q) k4 R: c5 T1 c8 F
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    - k, D/ `0 o/ Z# U8 d" ?尾递归优化可以提升效率(但Python不支持)& |, a  k2 O6 s, x8 Q6 Y

    % E& o" R6 i& Y7 \: L! q 递归 vs 迭代- {: z, h2 U# X4 f4 s
    |          | 递归                          | 迭代               |
    8 n0 H" z5 D5 E|----------|-----------------------------|------------------|' A4 f9 d2 R" q; l4 h- @2 g0 G3 y/ d" x3 K
    | 实现方式    | 函数自调用                        | 循环结构            |
    1 H/ o! ~+ j" ~( v| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |5 C. F7 T) Y0 H% _7 Y3 R9 j6 E, a% Q
    | 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         |
    / o3 @2 t' p) u& R& @, {| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |9 U! v6 }, [9 a4 A; x4 e5 Z% n  F

    % n$ X7 n* a+ n2 d 经典递归应用场景
    ' R; N9 r8 L$ Z, q0 Z1 o1. 文件系统遍历(目录树结构)1 a" Q, D7 X# r. i
    2. 快速排序/归并排序算法
    " M6 b4 Q* R  q! G- K$ ?' I+ F. W3. 汉诺塔问题
    3 I2 z' l5 |  x4. 二叉树遍历(前序/中序/后序)2 r8 n& o7 z" s
    5. 生成所有可能的组合(回溯算法)
    ; w& y+ c! a7 \% f: C! F! x4 j& H5 |# ?/ e: b, |" O/ 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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    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,
    % e% O( G1 |4 {& ~+ `) G) s( O我推理机的核心算法应该是二叉树遍历的变种。, r4 X  v) F9 E  j8 s; ^
    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:* o% Z+ \! c! b* u+ {, m
    Key Idea of Recursion
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    A recursive function solves a problem by:+ Z/ _! t: f( F  I
    $ ~  K! W) ~: z
        Breaking the problem into smaller instances of the same problem.' e9 Y; {) X$ L+ H9 x

    0 c: ~* P4 E( V! ^% D    Solving the smallest instance directly (base case).
    1 A' X5 x* ^) g2 s
    + J, M+ j. H. Y* h5 H: J  v    Combining the results of smaller instances to solve the larger problem.* K# ?: f) Q1 R
    + Z( T  a  o/ ^2 y
    Components of a Recursive Function# U9 e6 N+ m: ^6 j1 W8 l# T8 G" T

    ; E/ h' y8 u7 i1 L* E    Base Case:
    5 p& X' b. n/ O: h" I2 v! t
    2 X5 |" C6 c7 b. h* I! j% H# |! c        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    * O2 h  d' M- O5 K# K. j( M2 C0 W0 B4 W8 ^1 H* ~; [
            It acts as the stopping condition to prevent infinite recursion.# C( t# s- E2 Z8 K
    , W& c2 B, L1 _! Y
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.5 `8 p* `3 W: T& r, h

    6 w5 }- {3 m% _. R    Recursive Case:' g" `6 i) J- U  f, s2 ^
    1 j7 j6 c# y+ w/ R  a6 q3 h
            This is where the function calls itself with a smaller or simpler version of the problem.+ t2 k8 Q1 }+ H6 `
    ! S+ ^2 T/ I% i3 v/ n% q
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    3 J: f" u8 G" m  Y$ h, J7 l
    7 t# a3 r6 z. ^# c" OExample: Factorial Calculation+ h. }0 ]# @. l2 e' v6 y
    : i0 U* ~% G% u! p- u8 q
    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:1 |7 p: c& M- p% z

    : d& `9 c+ B3 [* R, Z1 h  d    Base case: 0! = 13 e  d1 v5 |1 t' w
    4 E: K# m% Y, T3 A
        Recursive case: n! = n * (n-1)!
    1 ^# R) X6 E. w
      M* t; X9 Z2 P5 z7 ^% nHere’s how it looks in code (Python):
    / p0 b7 _6 n: ?* ]* T1 }2 cpython
    + X2 V  i1 G6 Z: U) C& J: T' t) \( u! V: ?) k  r) ?2 [7 R5 j5 e
    & ?0 K% u! X( o8 k
    def factorial(n):
    ( X5 h  o# E1 C0 M+ M9 t1 S    # Base case! ?0 u$ `. ^, N- {
        if n == 0:
      f) o3 L+ i7 P3 x) p8 W        return 1
    2 i7 z/ C4 p7 a9 w! m+ Z; g    # Recursive case
    ' ^, R8 V, k( \' k( x9 q" `    else:& V% G0 k# C# d- J( q
            return n * factorial(n - 1)% g, U8 ]0 ~: x6 p
    ! \2 [8 ]% l# J0 A4 O" H, i. X
    # Example usage& k3 ~! C4 W, R. X
    print(factorial(5))  # Output: 120
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    & w7 q" a, N2 s& A" u& P- NHow Recursion Works- W" N8 B1 i8 u4 V
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        The function keeps calling itself with smaller inputs until it reaches the base case.# V* v. x+ F, ^9 r

    5 C8 |9 _3 e2 t    Once the base case is reached, the function starts returning values back up the call stack.8 d- k* g5 R7 G) _
    3 ^# v' P) U  d# R4 t! y* X
        These returned values are combined to produce the final result./ H0 T/ y0 V& T0 A+ n1 q

    4 k2 K8 ?$ O. x% bFor factorial(5):
      `# j4 q6 ~. |1 \5 C( }; W  R- w, f9 n

    6 B% O3 D' z0 Q& a- j$ A2 h7 K# ]7 l) Rfactorial(5) = 5 * factorial(4)
    ) ]) e5 k9 g  u4 Yfactorial(4) = 4 * factorial(3)
    % ]* M" g0 s, y' Z) |9 K7 [* I3 kfactorial(3) = 3 * factorial(2)* f' o/ q3 X, A8 |1 B$ H/ Y
    factorial(2) = 2 * factorial(1)
    0 R) Q9 `7 ]+ G4 S/ vfactorial(1) = 1 * factorial(0)
    # o" I5 l1 |# F, x1 s, _factorial(0) = 1  # Base case
    ! |* c" Y: b0 G* u! _5 O! E; D' Y* I0 _# r0 c5 L3 @3 ~6 D& y% V- h* L
    Then, the results are combined:( s/ ^9 y# E$ u- a. D
    # C. k* w+ Y* j6 C$ ?
    - k- Y( s  [% K# W
    factorial(1) = 1 * 1 = 1
    4 Z! O$ Q- t$ o5 ^' Bfactorial(2) = 2 * 1 = 2
    , F* E5 o9 E. D5 _factorial(3) = 3 * 2 = 6
    1 e) u. l% r/ b- Vfactorial(4) = 4 * 6 = 24
      t2 t- Q" ^; l' o3 z5 V0 \factorial(5) = 5 * 24 = 120
    9 e5 R* \2 D/ c0 |
    , C, E6 m0 d: k4 v# I" `+ tAdvantages of Recursion& [; ~! `3 [+ @
    + n0 e- t6 w; E6 g  V8 R
        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).
    7 r/ F2 R6 b' ?# K& D, H
    $ T; Q9 O; D8 o* e$ j& Q/ a% i    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    + u, Z1 r4 Z! \) {3 E) U) n- _% @# I7 R' n3 ~! F
    Disadvantages of Recursion. F% x9 m( }& c( q, y

    - n1 `' x( ^+ O. \1 I4 A* H, j    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.  R7 Z3 P& o% Y" |+ r
    . J+ f) H2 R" A7 P/ d
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    4 w3 o( A& l, _/ X9 C) ^1 |3 V7 J; E+ T/ {& N; g% I% {0 ?# N
    When to Use Recursion% V' H1 W4 [9 t5 W* A% v$ U  _
    7 m8 K! p  ?+ ~, r2 S4 s
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).  F( x5 i: u8 `$ K, s' _" P) F+ L
    + C3 {1 s. O( n* J$ f
        Problems with a clear base case and recursive case.: s/ x- B( h* P. a# o
    & z% y9 b4 e8 V  u  k# a; t0 y) y3 m
    Example: Fibonacci Sequence6 I. s+ p1 ]& Y  Z
    5 T/ A" h8 [, p5 `: @/ J
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    / r* _7 {& w1 [; t$ v: {: |; X
    . w" S% `0 S; h: y* A7 E3 `    Base case: fib(0) = 0, fib(1) = 1
    : _: B0 N) M% s0 l$ P+ a5 e$ O9 `- L& ]( G* j/ V  `' Q1 `# S4 n
        Recursive case: fib(n) = fib(n-1) + fib(n-2)2 z; F+ t; }" u, W
    * t1 B" D" E! v( l- R
    python$ Z$ O9 B- N0 {2 I
    % t# p) x3 y5 n0 @
    / x: p3 I1 a1 g! X* h( j6 N7 f
    def fibonacci(n):+ J- r  O# ^& X5 {' c" B9 n* R
        # Base cases3 t# `- O8 Y6 ?; n
        if n == 0:; y  t- I* b) z6 g. I( M/ g
            return 0, V. t; |9 x3 I& M7 [1 I
        elif n == 1:
    9 ~. `, t( v5 V) _' ~. L4 s. Q        return 1; X' P$ ^! `* Z; |' N0 v3 r8 P0 Q
        # Recursive case9 v* m* R+ v- d+ r% A$ u, C  d. C( [
        else:  @" M( s2 g/ P! g5 t8 P! p
            return fibonacci(n - 1) + fibonacci(n - 2)
    6 J4 P! |2 Z- m
    " X/ p/ p) n: J# Example usage
      R5 y5 _( u% o2 Bprint(fibonacci(6))  # Output: 8
    " U7 F* P; B; K( z2 G" k& a1 H5 D7 j; z
    Tail Recursion
    : A" S9 M  i4 E6 s* B. Q- g" R7 a7 [3 Y( X/ b
    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).
    - ^  ^9 X9 m1 ~3 C
    2 r* b0 J$ M) O+ R" W) c  Q/ U$ 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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