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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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    2015-11-30 11:11
  • 签到天数: 2 天

    [LV.1]炼气

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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
    " |: |8 B5 g- d/ L5 _6 C5 ]" f$ O9 F+ M/ s! j
    解释的不错
    2 p7 N+ M6 a$ I" E1 m/ d6 ]4 v6 s7 i0 N3 [6 h  N) \
    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。; }' [1 _( \  ^  D8 H% n# f
    ' q8 Z" A& M* [& `; A! e
    关键要素
    8 I2 a( v2 R8 {# l% v6 R3 G1 U1. **基线条件(Base Case)**
    % T& h) O6 p' {* P0 L6 S3 {   - 递归终止的条件,防止无限循环, U  g- f" J- G* k, f
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1; \' R( m. z' w% x3 r6 s% O
    & x, V4 \: Y1 o* w  g6 S
    2. **递归条件(Recursive Case)**
    - z6 J$ K& B8 U; I4 i( f) o" Z" r+ }   - 将原问题分解为更小的子问题6 L8 ]6 N, N2 Z( W; `4 I$ ?
       - 例如:n! = n × (n-1)!( v! F. D4 _2 K% U8 l/ M

    & ]. e3 l+ Q/ t8 p: k- e 经典示例:计算阶乘+ @& L4 e" S" L7 V* \. r  S
    python
    3 H: \1 ~0 l- q# K. |! |" Ldef factorial(n):6 E8 ]; Q5 F5 X' k
        if n == 0:        # 基线条件
    6 Q" u9 o+ W: g2 w1 v3 z1 @; I        return 11 l0 S( M8 Z' \- K8 I
        else:             # 递归条件% c: Z/ h8 }+ u* l
            return n * factorial(n-1)
    0 l* ]2 t  x  N/ w1 k, P/ `执行过程(以计算 3! 为例):
    6 ~, T" q; a- L% ?) ^; d4 z% N2 Y: `factorial(3)7 M! J! d5 l1 ^! m2 m
    3 * factorial(2)
    3 l: m- H" @, d0 ]0 M; L3 * (2 * factorial(1))
    . }* m' L$ u! f1 N- p. H; b- \3 * (2 * (1 * factorial(0)))
    , }# n; O/ n$ }; g7 p3 * (2 * (1 * 1)) = 62 P/ I8 H+ H% o; a

    ( _( \. x+ Y' j0 F. `/ } 递归思维要点0 m9 b( A( X! D9 e, a; `& X* X
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑$ L) l) {3 j; \% J# {1 V2 V1 Q8 q
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)# h" C5 t+ c* {% L" e$ M8 ]  Y
    3. **递推过程**:不断向下分解问题(递)
    % Z$ o  p0 T6 B( n4. **回溯过程**:组合子问题结果返回(归)/ T; [! f! h% f( A; H
    3 ?2 B: j3 X" W6 ?
    注意事项0 y, k/ X9 p. U8 w# v; G  _
    必须要有终止条件5 ^$ C& I3 w! [/ R0 Q$ a
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    7 S- d. W8 H( P! O( ^某些问题用递归更直观(如树遍历),但效率可能不如迭代" R9 h1 \% _' l1 ?
    尾递归优化可以提升效率(但Python不支持)
    9 D' n% j2 u+ d8 ?' I* @' T
    3 q# ]9 N6 O2 W  e- [5 I% e 递归 vs 迭代' K1 L% ^: u( S7 p  J
    |          | 递归                          | 迭代               |2 p% \7 R5 X) ?+ j
    |----------|-----------------------------|------------------|+ R" W( S: K2 r
    | 实现方式    | 函数自调用                        | 循环结构            |
    ; I0 n! {4 w9 {0 J  K+ g| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    , I7 [' U! J. x8 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         |
    . s5 `! D' g4 |4 ^2 X6 k! }* ~) T| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    2 A9 U4 R  Z; V  z' z& X4 B
    4 \( B: ?7 L3 t2 d 经典递归应用场景
    ) J; M8 A& Z6 b" j3 {1. 文件系统遍历(目录树结构)8 D1 ?8 r% J# u4 L' c& M+ D
    2. 快速排序/归并排序算法
    ! G' b% J* H* y5 l! e7 y3. 汉诺塔问题) D* e1 P0 _9 ^2 O9 |( ]2 X! E; y, Y  P
    4. 二叉树遍历(前序/中序/后序)2 p6 C0 \# ?( u7 t
    5. 生成所有可能的组合(回溯算法)2 ]' _/ |1 j3 Y: @' V  Z6 ?

    ' M' }7 R# F5 K  W7 Y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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    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,
    ! X: I$ {0 z8 L! j+ Y  G我推理机的核心算法应该是二叉树遍历的变种。+ j( J9 N$ Y/ k: ~7 V) A
    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:
    ) L6 E2 l% u* ^" f8 I; t  ~5 MKey Idea of Recursion& M. V" p3 v7 Z% ^7 b

    , q3 p: s- d* Q( T; U* [A recursive function solves a problem by:7 |$ [' T/ K/ {1 z' q5 g# @- P1 Z

    & e8 l2 n# M0 _    Breaking the problem into smaller instances of the same problem.
    ; ~% I/ \3 G$ e" l* i& \1 `4 p! A4 p4 `  M
        Solving the smallest instance directly (base case).
    7 {$ y: g% j2 t# V0 a* w3 t' d- P# r" K3 U
        Combining the results of smaller instances to solve the larger problem.
    5 B, G. A, b( \3 Z
    6 R- }8 o1 }1 X6 e* gComponents of a Recursive Function
    , k7 K4 k, W9 o- L4 t
    ) q6 ]( t5 {0 q    Base Case:1 e' L  s4 @7 \; A7 E( v7 p2 c5 I
    " S% X- Y% g! G' N; l  a" N5 V
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    , l) s+ i& Q' L, ^* ~" b: a# K2 J6 d3 Q" H. U" P7 k2 R7 J
            It acts as the stopping condition to prevent infinite recursion.
    " X) V9 Y2 S/ D% C' p5 n4 L% S# X- d1 I0 }/ D7 ?' }
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.+ ^  h2 ?- r. \3 E& ^

    7 U! z: E2 L+ {9 ?5 h+ {- J    Recursive Case:' g. i) Z4 l7 h. y

    ! ]# K% n+ }* n        This is where the function calls itself with a smaller or simpler version of the problem.* I2 z! y0 w1 r3 t0 f% Q

    ) W! ^/ k( ]  F1 y- z4 `! ^* P3 e        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ) E+ ~! O* w3 B+ U" `7 S) ]- J0 C1 R( d. l) }! I
    Example: Factorial Calculation0 H+ Q2 v7 y0 @1 E+ D% M

    ) G8 C, p: v% v# U9 J/ f+ h# QThe 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:
    & a1 V- l6 |3 ?  U8 i' s3 L( N9 Z* l) U) |9 i
        Base case: 0! = 1
    ( `% Z" R7 m4 y/ V* G  `
    . O) |& {4 O* M8 a6 t    Recursive case: n! = n * (n-1)!
    0 N4 w) s/ k5 _& W, N# z2 y- _/ }5 r5 p
    Here’s how it looks in code (Python):
    ) O) k! J  T' D# g# ?python
    ) w" N, R3 ^$ b2 S! `4 C* ~# }) A- Y; ]' {4 G3 S+ [
    & r6 R) a7 M" T* `6 W2 u7 z
    def factorial(n):% d3 ^4 k1 k$ D& {& M4 W" S
        # Base case; e5 N( ^3 h: G6 L/ h
        if n == 0:
    0 n7 f8 j# x" C7 Y# Q        return 1# p& _0 a" x2 m- P: V% T
        # Recursive case& I3 S. L$ f5 U0 X' B4 y1 C
        else:
    ! r3 |& x0 @2 q        return n * factorial(n - 1)
    8 |# f5 ?6 ^+ K/ N7 N9 c7 M# m# S- P" |: m0 B
    # Example usage
    ' \1 |+ h3 [* ^1 G" Gprint(factorial(5))  # Output: 120
      F0 h; Y% u( p  s4 f9 D9 T5 Z( N7 B+ d; p2 I
    How Recursion Works
    , u  a9 d" t+ e  Q5 j9 F1 Y* _2 g- F" Y: n' L# K0 z/ a' s
        The function keeps calling itself with smaller inputs until it reaches the base case.+ T/ @3 d# T, s% g
    + ^% u' `/ \$ [9 A0 S( s+ j, q1 N/ L
        Once the base case is reached, the function starts returning values back up the call stack.  K9 F# ~. d1 \$ ~& G- W+ |" V+ P
    # O" I- u7 `8 C% L0 Q
        These returned values are combined to produce the final result.  j; A4 |# W3 J0 I- ?" w2 Y

    1 a# _( v8 v- H4 JFor factorial(5):
    4 U: A" s8 w8 V7 o0 L
    9 O1 b4 P1 G  J2 c7 D) v+ ]' l* e. d; u  S" _; c: K
    factorial(5) = 5 * factorial(4)
    ; T% c  M! f, ?% y  k' e: ]& ufactorial(4) = 4 * factorial(3)& d  p9 u1 o/ f+ j* a+ L! r2 S
    factorial(3) = 3 * factorial(2)7 `( V7 Y0 z  O
    factorial(2) = 2 * factorial(1)8 H; e+ n; P' E$ N9 R& I! z
    factorial(1) = 1 * factorial(0)3 c' E6 E$ g& f/ d% p
    factorial(0) = 1  # Base case& S/ Q. u# e/ b2 o! }* t6 ~' E' ]

    & O& f4 P& a8 v  R; N* _Then, the results are combined:
      U8 Z; q$ z& o  ], p$ V9 T  Z5 ]( I

    ; r4 @" H- n" F; J6 M+ w/ b0 Ffactorial(1) = 1 * 1 = 1+ G2 b2 L' G0 P/ x) L2 f
    factorial(2) = 2 * 1 = 2
    ! o/ A2 B( T) B4 F! a( ifactorial(3) = 3 * 2 = 60 c, d' [7 v# w8 O
    factorial(4) = 4 * 6 = 24; j; W# P; j& o' W) q
    factorial(5) = 5 * 24 = 1204 B: Q0 \  ]9 A7 L

    6 z8 w/ _# j  Q, qAdvantages of Recursion+ f- V3 y' t! u! J" R
    / e. K1 p! q- W$ j
        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).1 y+ O  _  y0 m$ y( x& L2 ]
    . e# i: n4 X5 o5 p
        Readability: Recursive code can be more readable and concise compared to iterative solutions.- C) B9 E  _5 g/ _
    3 C* H7 Q2 R2 {3 r: L
    Disadvantages of Recursion
    ; r% t% |, h4 r- p7 J
    5 @! G( j# |' n    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.
    ; P+ A# L8 R% d* _' f- e) f# J. _: H
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ' t4 [+ g1 N9 N8 N/ A0 ~
    / o# J3 r5 p# P  p7 v5 g. ^When to Use Recursion
    - l# `& p+ ~% [" ~$ L
    # r8 h4 L" y7 b9 n, u% Q: y2 H    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ! e( b8 r4 U& A7 L8 j: f: g, l+ H9 z8 ^. ]
        Problems with a clear base case and recursive case.
    " x* M1 {& ?' G3 T' e  b' E1 [
    / d  _+ N+ @$ P. OExample: Fibonacci Sequence
    1 P' ^1 @, r9 i( P) R  R5 F
    / {- Q. [8 _' m9 W3 O6 I; j: nThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    / O8 @& X' G: c& K7 k; _
    ' j( u1 u) u  P& Y" N5 m    Base case: fib(0) = 0, fib(1) = 1- _9 I6 c- w4 [7 C. ~" V. y- T
    2 F- U. g6 b$ J8 f4 y0 g
        Recursive case: fib(n) = fib(n-1) + fib(n-2)( H8 ]8 A$ ^' h. a0 b1 l

    + t% P1 w; l- ?$ t" Xpython
    / \$ x" b& h0 l  |( a# a4 p) Q' n: L6 o' }* \6 i% |( V) \: a

    0 M. V& Z  O7 Sdef fibonacci(n):7 w( b( r2 T! b
        # Base cases
    * U% N5 _1 d, m" o6 f) I/ \    if n == 0:
    5 r1 U4 U; f' \9 P        return 0
    # S; L# s3 m# {    elif n == 1:( Q9 V$ S) S$ h% p# I: Z
            return 1
    ! v2 c/ Q/ a0 N. r1 ~$ Z    # Recursive case8 i7 o! T( M# {& ?
        else:
    3 e; `, D3 E; n& `9 ?9 k        return fibonacci(n - 1) + fibonacci(n - 2)
    # }1 ^4 v# V: r+ w. o- b8 ]/ d
    ' b7 Z; ?# T6 q# Example usage2 |# U# H, Y. ^2 b! v
    print(fibonacci(6))  # Output: 8
      j0 g! y$ l: e3 v* _7 F' E7 c, K% c7 u% ^; Z
    Tail Recursion6 `# r8 E2 m& h. B$ k4 K4 g

    ! _$ K; S8 Q6 k% ?  y% n% sTail 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* {$ H* C- c
      \1 w3 k# N) g$ |: GIn 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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