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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
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    & p: d) E! a8 P! V# G0 I2 T! [( E解释的不错
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    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。! C4 ]% Y/ z; H# O$ s* r
    4 w2 @6 g* o1 T
    关键要素
    ) W- w: L7 n( O) I8 j4 L1. **基线条件(Base Case)**
      g' C& e5 m) S   - 递归终止的条件,防止无限循环7 E7 u5 f; h7 R  f! d1 h* E
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1' b+ i' j. Y1 ~* Q& L/ {5 P

    0 t; ^- `( v2 I. l! F: r2. **递归条件(Recursive Case)**: M) S% h3 |  g. W
       - 将原问题分解为更小的子问题# |" c# k4 V' b- Y
       - 例如:n! = n × (n-1)!
    * b) ?& N4 C+ K: l: k2 i  t9 Y
    $ u% e, h0 t. z2 A, D" |0 t 经典示例:计算阶乘* N5 [1 g; l3 {( n8 S* v% J4 K
    python, I% r9 n( O9 Y  D% k
    def factorial(n):! U' X4 g# B; d. v# P  R6 Y
        if n == 0:        # 基线条件8 J6 L) Z1 g0 K8 ^! `, ~
            return 1
    : K6 z6 M: d/ P$ _# E; M    else:             # 递归条件6 Z6 X, Q' v+ ^% X4 Q( F2 v
            return n * factorial(n-1)$ }; Z% X8 ~4 k
    执行过程(以计算 3! 为例):- ~7 K) J3 L7 Y' L8 C6 W
    factorial(3)% }1 y* c" r) l# c' ?7 z
    3 * factorial(2)
    ! [4 O& C) F4 y9 P' Q* J% _7 _3 * (2 * factorial(1))) \0 Q8 Q* u( k  G& P+ Q
    3 * (2 * (1 * factorial(0)))
    ( E0 y, q$ z$ e3 * (2 * (1 * 1)) = 6/ C$ O9 F8 X* b' l3 ]# C6 S+ G4 g

    5 d/ `' j2 x% L  v% K- u$ I6 k( r 递归思维要点
    ; S# g* }2 @. s6 u- S: e1. **信任递归**:假设子问题已经解决,专注当前层逻辑/ r! t7 S5 e* e
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    " Y2 j; m: p: Q3. **递推过程**:不断向下分解问题(递)
    8 F: z4 ?" G9 r6 S2 J" ?4. **回溯过程**:组合子问题结果返回(归)6 x. L4 @) }- ?/ k5 s

    9 ~; L: v4 ?; b 注意事项
    6 {' i3 u* O" G& ~9 {必须要有终止条件
    5 g0 u/ w$ E4 }1 {+ c+ i) I递归深度过大可能导致栈溢出(Python默认递归深度约1000层)3 M, z3 s, _2 y. p- T
    某些问题用递归更直观(如树遍历),但效率可能不如迭代4 d6 i) o* r7 Q& y. l
    尾递归优化可以提升效率(但Python不支持)
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    递归 vs 迭代8 u4 Y5 h: _& U, |4 Z
    |          | 递归                          | 迭代               |* u4 E4 w% ^* b5 _+ D# ^
    |----------|-----------------------------|------------------|: k: g) x( S7 k6 P
    | 实现方式    | 函数自调用                        | 循环结构            |; Q* S/ `1 s4 b/ u; q* M1 m/ z# A
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |9 y! `  T" |& @
    | 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         |0 r7 y8 {$ Q+ R. ?# U2 K6 i) m9 l( v
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |2 H) A. N) q: l- |8 K
    ) x  a" a; `6 n6 S) h) j
    经典递归应用场景* _1 z' I- L& F' G9 I- }
    1. 文件系统遍历(目录树结构)
    ' X' h4 R2 [$ w+ b4 {! s2. 快速排序/归并排序算法
    8 w( ?: n8 e6 e4 k2 G% {3. 汉诺塔问题
    ( A# B) ~& _$ k; T- @$ q+ u3 g4. 二叉树遍历(前序/中序/后序)( C6 _# v; c. Z
    5. 生成所有可能的组合(回溯算法)" j8 y" l4 s+ M/ ~
    ! f2 |) w. ]9 O3 B
    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," h; u9 R& z: H% O' A6 [2 y
    我推理机的核心算法应该是二叉树遍历的变种。; x/ Q0 P3 N8 A% y9 v
    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 f. @2 f! S) K# S. \
    Key Idea of Recursion& I1 t  ]9 u- ^0 j4 d
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    A recursive function solves a problem by:
    ' H- S) e0 b9 O! F/ A+ x
    : V# M8 C, B4 g3 l' f, j: D% h/ H3 G1 f$ n    Breaking the problem into smaller instances of the same problem.
    ) F, q, x$ @4 Q0 \( R8 T
    $ U% e' B. `* Z  K! {+ A    Solving the smallest instance directly (base case).
    + i; L# ^7 y* n3 g6 j: w( o1 v$ H+ ]5 j& ]
        Combining the results of smaller instances to solve the larger problem.
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    - U4 Q9 c' b0 O( K5 |  O" wComponents of a Recursive Function
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        Base Case:
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    ! O  v7 M$ ]- a* G        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.- n1 E; |* s, x9 N6 B
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            It acts as the stopping condition to prevent infinite recursion.
      [- V. l/ Q- I/ x0 Y7 ?0 O1 g# {. M& S! {7 C# P
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.2 a1 e' Q$ A3 P
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        Recursive Case:
    & r7 {: ]7 x4 q# J( `
    + ^2 e5 Q* `2 V$ S! q        This is where the function calls itself with a smaller or simpler version of the problem.
    ! r4 c9 a/ J! ^
    / M+ B% a* w" P) k        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ; G( \: _. `1 P9 a$ V1 {! o# k4 n7 d3 S7 Q. j1 s* z- S1 k, R  x: T
    Example: Factorial Calculation
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    8 ?, Z! U0 w! y0 w+ cThe 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:7 y# O( F9 P. x- G8 F' t
    8 t$ ?( ~' E& e& [4 H) I
        Base case: 0! = 1$ g7 ~6 b9 B" n) V7 p3 _6 U3 g

    ( F, n; L2 A4 J2 W    Recursive case: n! = n * (n-1)!4 Y; Z* P4 c8 u

    5 ?; j9 S0 b2 w& NHere’s how it looks in code (Python):
    ' B3 E' h( H; M  R& Qpython( n+ m6 J  N  c& D, d

    % v; @. Q4 m5 i! @; H9 Z( O5 G7 b# \# E* ?& |
    def factorial(n):1 z- r8 ]. ]7 t2 N
        # Base case) u0 ~9 h! q$ ]! S0 Q
        if n == 0:
    2 l; _( h- u$ t" p        return 1
    9 O! I* l- H8 Y, O    # Recursive case
      I1 V0 O6 }" u6 e' a7 z! \    else:! ?3 b" S5 j! F9 J
            return n * factorial(n - 1)" j+ `0 h0 b0 m( X
    ! I& N( M) s* p4 b9 C
    # Example usage1 e3 m, g) V0 c
    print(factorial(5))  # Output: 1201 l9 J+ U0 H- t4 T, b% o& A3 E% N
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    How Recursion Works& b! r$ ?9 M) o+ q2 J

    $ X! x+ M9 S% [! K    The function keeps calling itself with smaller inputs until it reaches the base case.
    9 u: B' G1 j. ~! H" h- D/ z  Q; K3 S4 f# g6 i+ o% x
        Once the base case is reached, the function starts returning values back up the call stack.
    : b* G& E4 D% K: M! l$ W$ \- O- \5 r! F) R- w4 J% J9 M1 R0 I5 }" Q
        These returned values are combined to produce the final result.* V! @5 X5 y6 K. m9 C5 U8 i
    . \) z! R7 d/ y- ^- D1 b
    For factorial(5):4 U' \& b3 r$ L# s$ W7 z6 n: [- w
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    factorial(5) = 5 * factorial(4)
    & ^8 h: B- H: O. ~factorial(4) = 4 * factorial(3)
    ; l# `/ k) S9 a5 k" Qfactorial(3) = 3 * factorial(2)4 B' v5 N& U7 Z9 S4 U: u
    factorial(2) = 2 * factorial(1)
    . Z2 [2 Y% @4 Y% h2 ^; pfactorial(1) = 1 * factorial(0)
    + P) A# M) A7 y2 K% ]7 dfactorial(0) = 1  # Base case- R" }/ O2 z" o$ `% E
    : |. T1 S+ X% Z. N9 R, t
    Then, the results are combined:
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    ! s9 l+ [' x& f7 U9 i+ t' @3 V' z# n/ h
    factorial(1) = 1 * 1 = 1
    ! Q$ H# j) b( A/ P* t) Xfactorial(2) = 2 * 1 = 2
    3 n5 R2 `9 A8 ]) sfactorial(3) = 3 * 2 = 6
    4 A! Q- w  ?4 t9 N* }% c4 vfactorial(4) = 4 * 6 = 24
    0 e/ h  W/ t( B7 [- X: z/ Tfactorial(5) = 5 * 24 = 120
    + [5 G: _+ G. }! x0 L' F* n3 s3 \0 |  l0 r; z/ J
    Advantages of Recursion3 _8 m/ E4 ]! g. ^

    , S  L- F6 \: P/ 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).
    9 s; G$ l) D! e- J* b
    7 G7 i3 U# _3 c1 C+ n, Z- l: K9 h# c) d    Readability: Recursive code can be more readable and concise compared to iterative solutions.2 O. ~3 G4 o, o6 x! h/ v
    : }$ W- Y9 e8 q4 }  n
    Disadvantages of Recursion
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    " _! J3 y& v: v& I1 ?+ X: 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.+ q9 O* }' o$ _3 g2 a% n% l# D

    ; t1 \( X; h  W  W3 g; F    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
      A8 W# ^( t& B1 a, L- h! b) _& S  C4 Q7 _# a
    When to Use Recursion; H5 U& X7 L/ E9 A+ \7 X* f  |

    % ]% P- {- l3 a  t. x    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).: g4 _) s  E( Y- T* g

    2 }% b0 d, n, _' M; Z9 k" c    Problems with a clear base case and recursive case.4 M# S. \0 y  s* @: Y/ m* T

    . F- M3 O5 S" ]4 Y: kExample: Fibonacci Sequence7 y7 I/ c4 G! X$ x/ U7 x) S# _* z

    - r* r# w% I) QThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    $ @: @/ I: H% ~& j: Z& l8 K" f8 F. z1 L: A: c1 l
        Base case: fib(0) = 0, fib(1) = 1
    2 c( X  T) z3 I: E& q6 t" E
      R2 _; T4 R( _    Recursive case: fib(n) = fib(n-1) + fib(n-2)' R* @/ S6 @. K& o  U; |

    . n2 D& }- W6 p. z5 I1 `8 fpython
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    + c, [, x; a& l( j$ Y5 Y) X
    4 G! A, i7 u! hdef fibonacci(n):
    : t" ]% I+ X' r7 a' H: Y' d    # Base cases& {0 |6 R! x( S  I1 b, [$ O; ~
        if n == 0:) I: N; o4 k3 F5 ~3 b# B
            return 0
    5 G" ^9 k' o. d  }* S    elif n == 1:. S0 V% K7 o, g& ]' C' D3 I
            return 1, m7 h6 S4 V5 ]1 E
        # Recursive case
    % N. ?% D% M+ O6 E    else:; B) _" f: S% N: n" {7 L$ g* b+ ]
            return fibonacci(n - 1) + fibonacci(n - 2)
    # j7 ~  p  K5 H/ l+ M3 b5 Z6 j, Y: I4 u( }! `4 ?, |! T
    # Example usage# x. \2 D8 l% q, d, o8 P5 |0 \! x
    print(fibonacci(6))  # Output: 8  y$ G& C$ w3 G% x4 h/ ]7 X
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    Tail Recursion. T! }- l" l' K+ A9 s
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    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).
    6 O- a$ D, m3 q$ u- M, I& H: c4 ~1 s$ P' s% S5 H
    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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