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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
    : r( H3 W! U- j1 y6 e. K( c" t# \1 x# @* l
    解释的不错
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    ) p$ ?2 @) F( E3 C; g3 W$ ]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。
    9 n6 A/ v" v' I8 t" b  P  E6 g4 I# M* u
    关键要素
    3 g2 s7 S7 s3 @: H1. **基线条件(Base Case)*** o8 u" _  r& R3 \: i8 ^
       - 递归终止的条件,防止无限循环- b. T, Y2 E, P8 D
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    * J& `& H9 R' T
    ! S4 c" j, e8 R! s' y5 X2. **递归条件(Recursive Case)**
    - S" I- U- e% h* W3 \4 T7 r7 p   - 将原问题分解为更小的子问题
    1 `' n5 l6 T; O1 k( L4 U4 J   - 例如:n! = n × (n-1)!
    ( T; t' O' l: a$ R0 S5 v. M+ B
      E& r& z. }/ m! F. e3 {! o8 L 经典示例:计算阶乘
    + R3 K" B5 l: K" x2 Qpython8 B1 x7 p' W! @/ S% t6 ~& \
    def factorial(n):
    4 [7 W* g/ ?; |' R% D/ [! i    if n == 0:        # 基线条件
    " U$ d9 B+ M- C        return 11 N6 e7 H( t! R4 _
        else:             # 递归条件
    : U" |% E8 C% [, c        return n * factorial(n-1)4 q* c9 V# O' y% \- N5 |+ @
    执行过程(以计算 3! 为例):2 S1 M+ W6 p) q$ }
    factorial(3)
    % h1 K0 \9 `6 J7 N3 * factorial(2)
    9 J& H$ W. d/ l& f# R5 P/ E3 * (2 * factorial(1))3 V1 w  P: B/ \* J
    3 * (2 * (1 * factorial(0)))! ?, C  E/ d9 t& s1 J: P3 O
    3 * (2 * (1 * 1)) = 68 x; q, S% U3 m6 ~' g
    2 ]5 C+ a9 Y; b( V7 R: u# U
    递归思维要点
    3 X: v6 ~( O+ s: a8 V1. **信任递归**:假设子问题已经解决,专注当前层逻辑' e( l; K( C& I3 \
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    . o$ ~) k$ Y' ^: n9 a3. **递推过程**:不断向下分解问题(递)5 i& b( k  E0 \: H4 ?' [
    4. **回溯过程**:组合子问题结果返回(归)
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    注意事项3 \) o% a/ g1 E) C0 G3 _9 W$ @
    必须要有终止条件* Q* P  ]1 H# i; M" f; [& i
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)* r$ v$ D1 \' h
    某些问题用递归更直观(如树遍历),但效率可能不如迭代2 |: V/ m; V* h
    尾递归优化可以提升效率(但Python不支持)
    $ R, z* Q! C, ?6 S# v. e& {9 O1 B1 Y! j! }
    递归 vs 迭代
    ' Q* l- U# H+ K, r9 h|          | 递归                          | 迭代               |! R" I: W2 [) z- D
    |----------|-----------------------------|------------------|+ z) p# q4 @- ~) V6 ~# W
    | 实现方式    | 函数自调用                        | 循环结构            |
    : I1 |- N) L: E" G  p& e| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |+ M- ?5 C% R2 B
    | 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         |
    # S. c& s1 B3 d3 o7 p4 I5 _8 l| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |/ R- N0 K! M1 L. Q7 S! o
    1 k  ~) W5 @4 _2 K
    经典递归应用场景5 s8 [5 z, R3 Z7 B7 e( m! e5 p: A
    1. 文件系统遍历(目录树结构)
    . {% g0 {7 ~9 M$ o# h8 j) l2. 快速排序/归并排序算法9 |1 x1 [, K. Z! B1 G' n. m9 R& {
    3. 汉诺塔问题, ]" z7 O2 Z$ m% z4 \( Z1 q* r$ M
    4. 二叉树遍历(前序/中序/后序)+ G1 V; ~( p8 M, e3 d( ?7 p6 J# O
    5. 生成所有可能的组合(回溯算法)
    9 Z3 m" o% ~/ G9 ^/ f2 H# b/ ^2 S) n3 `3 i
    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,& c; U/ \* i) {- K) T  H
    我推理机的核心算法应该是二叉树遍历的变种。
      [8 h" _$ k) x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:9 W# M+ Q. i) b
    Key Idea of Recursion) [- U6 W* B5 I  u/ o/ ^0 G0 w
    3 u# G& _" s( K
    A recursive function solves a problem by:! i, b5 o, g6 e0 b6 R: C
    1 L- A, r: g8 \' E2 d( u/ g) C7 v
        Breaking the problem into smaller instances of the same problem.
      K6 I6 V' x% C) N" r( b' @% U5 G: U  }- Z; n# h
        Solving the smallest instance directly (base case).+ x" s# V% e8 H9 ~; x1 z

    ) y- `) T! e1 l' @. L1 c& H    Combining the results of smaller instances to solve the larger problem.( \. ?+ z; y- O" p  @1 q& L* E

    6 \5 X1 y. X! \8 N  |$ _Components of a Recursive Function
    . I6 B  d  }- E% ?$ V% Y
    5 ]. o* j) q+ v+ }: z8 W    Base Case:
    + @) y9 p& G% T# d- P! ~
    - S$ z; s. q) v, v+ Q        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    6 J4 k+ A. z' S
    , Z$ n6 A* x9 E1 `        It acts as the stopping condition to prevent infinite recursion.$ y! b- Y: J" A4 m' f! M; x
    & |/ v( x$ z$ `9 ]& w# P2 O+ o
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    # @" X: n9 v1 v1 V9 _- }- `! D/ N- n& C' k
      O( J9 W  U+ E    Recursive Case:
    & _6 V& i0 q, ]0 B$ N/ z5 r. u3 N: [+ h6 F* i/ i
            This is where the function calls itself with a smaller or simpler version of the problem.* c& h/ \- H2 ?1 {. B

    7 \5 S3 B' ^2 q( \5 j7 d& R        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).3 [7 E- J, O6 t! B6 e6 E' u& L3 u

    6 T) e" N9 j: d% g/ ^Example: Factorial Calculation( v# l. z8 `; z# t  X0 Y

    - {* Y. d+ K% o" B1 s, \& Y. q5 Z2 Q1 VThe 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:
    0 ]) y  ?6 X4 h+ ]9 W* g6 S% z' s" h0 Q7 _2 H* e/ e; L8 N
        Base case: 0! = 15 G4 S+ N9 x; f) u: L* G# J
    * C- z$ v! V* Y
        Recursive case: n! = n * (n-1)!/ {) q, C. k2 n' F" o0 c$ s' ?+ s
    & r1 Q) A2 X" ^
    Here’s how it looks in code (Python):
    ' \9 M& V: @, C/ L+ M+ npython
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    2 o' r+ {3 @0 F- {
    # ^% c  M  c2 ?. Idef factorial(n):
    8 C) j9 W% F( w# i7 ~5 {    # Base case1 J* d- b3 w7 Q
        if n == 0:
    - R/ |2 J! Q8 \- C. d5 o; }        return 1
    9 M0 I0 g* r) E4 u/ [2 _    # Recursive case8 F3 c9 k" Q6 e
        else:8 ~/ j; v3 p& Y# r9 I7 C  O" k
            return n * factorial(n - 1)
    1 J: V8 F# u& ]- b, D% @+ H; @5 G, C" ?# r
    # Example usage
    9 q. }+ m. t! nprint(factorial(5))  # Output: 120$ g9 E! d6 }8 X' Y2 @. ?; F  f
    9 w0 b7 D* j; S" z2 w# m4 k7 \, z: f
    How Recursion Works% m  R4 A8 I0 `% q! {
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        The function keeps calling itself with smaller inputs until it reaches the base case.3 k4 s) ?# W; g
    " h! v  s9 I. n; _+ w- n/ Y
        Once the base case is reached, the function starts returning values back up the call stack.- l3 v  ~; N# d$ f  l) W! n0 r

    * Q; l# |& r! }% ]8 n/ q: O    These returned values are combined to produce the final result.
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    For factorial(5):
    8 T2 l, I- G9 D; @2 T1 T) g9 g5 f

    1 O( L4 i  g) V* ifactorial(5) = 5 * factorial(4)
    0 c4 h) y1 O# g+ ~( ^( xfactorial(4) = 4 * factorial(3)% j& f7 F' }6 r, E4 n
    factorial(3) = 3 * factorial(2), t, P* e5 R: K( N4 ?
    factorial(2) = 2 * factorial(1)
    & z& I" o2 P' gfactorial(1) = 1 * factorial(0)$ n& b  W4 r4 Z* t* n( d
    factorial(0) = 1  # Base case! L, G6 C( D2 X0 s. {
    : j6 O7 ^" a/ q! N, t. N0 t
    Then, the results are combined:: G- ]/ T' N" T  F! ?

    ! g7 i7 Q4 P6 D, J/ V" p8 x  J1 `. x6 X. m* s2 V2 _
    factorial(1) = 1 * 1 = 1
    # k& P9 ?1 q( D+ |factorial(2) = 2 * 1 = 2
    , F$ I) ^# D+ Z0 G3 `2 k# efactorial(3) = 3 * 2 = 66 J/ t# y& L7 s; l
    factorial(4) = 4 * 6 = 24; H7 ^5 y! J) }
    factorial(5) = 5 * 24 = 120# B% i  Y+ r+ r0 D. \! J, D

    # k: T1 d0 j4 X& V9 lAdvantages of Recursion
    ! q! c- n/ {* O$ m7 ]; Z$ X4 [
    " \' l: A/ i' S3 E/ `    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).
    6 q/ S2 h/ ?" [9 W
    + d9 J. ?. z" }0 }& @* w    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    & G( _9 v* e% s, h" i  e
    : W/ B* g* i4 i3 Y. }Disadvantages of Recursion5 l+ b( W6 @- V2 Q1 @

    ' U6 q) s: k( f- k& h5 ^8 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.
    . n) ^) o" K! ?4 u* p/ Q! g$ b5 b4 y" i+ K
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).# f8 i1 k2 ]/ I4 j  a0 r

    " Y- ]* y: z# U3 q1 e* ZWhen to Use Recursion
    8 L, j- h6 H# J1 {$ T! R6 X, c" q8 v0 I
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).9 m. M! X  o8 h% Y

    5 T" C4 K% C; S- l6 ]; M    Problems with a clear base case and recursive case.
    4 p: f0 I2 ?* M& j9 B
    4 Y! @& B+ C  a' C" }Example: Fibonacci Sequence
    0 n- ^# N" Y6 S( D. Y- T! a3 {: Q3 v/ b1 k6 l- y4 b5 A- Q# b
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    3 L- \9 ~; H6 N: {' y
    $ @+ x6 J, p# t$ m" R+ @    Base case: fib(0) = 0, fib(1) = 1- D* V9 P$ t# E0 i

    5 m; f  A( P6 p    Recursive case: fib(n) = fib(n-1) + fib(n-2), j. S' l3 ~& G7 g( V( J
    % A/ Y% {+ D% b' q2 c
    python
    % C, e$ q; R, B" y! u2 W! S8 W7 Y; G
    , E3 u6 L% _: U
    def fibonacci(n):; n+ B3 o' e; w! a! H
        # Base cases
    $ f& l" r$ M8 e- n8 |/ E    if n == 0:
    0 \! r2 E5 s, }        return 0
    / y, L% C2 K' M; R* f    elif n == 1:3 o0 R0 V/ j0 g0 t4 Y! Y' E
            return 10 r7 N4 |& ?% P2 U: n, c0 r
        # Recursive case
    5 e7 t9 M# Z! y. U# L% r$ I2 `0 u% w4 q    else:2 y: d& u0 @( K
            return fibonacci(n - 1) + fibonacci(n - 2)' P$ @; `/ N. U- L- n8 W$ M

    5 U, u" ]% o: g+ U4 o. t) A: C: V# Example usage2 u; i' s8 u7 G' N1 t  \
    print(fibonacci(6))  # Output: 84 b7 _- N8 \- b# \1 O
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    Tail Recursion# h+ z2 C6 o! T, v% p+ p

    0 S8 j1 ?) E7 YTail 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).! c' b( u# ~' d9 Q. ?+ O

    9 i7 Q. N& T' qIn 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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