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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
    ' R; d& E5 ?$ B5 I
    2 m6 G8 @$ s$ C3 O' w; P* `. q解释的不错
    + b2 E3 I# r9 D; ]. }  s
      ~/ T3 J! q- `) n9 [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。
    ! ~, R$ [# }$ A" E' s, m& ]0 m$ v3 ?" ?
    关键要素
    & w# R- x' T1 A6 B1. **基线条件(Base Case)**+ Y  I' ?3 f5 k+ T1 f' k6 L
       - 递归终止的条件,防止无限循环' o6 G8 d0 d. ]+ r5 |; w( N
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1! z( j4 @/ U. }- Z9 {) m! U
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    2. **递归条件(Recursive Case)**- G" @2 j( c  H/ p+ U
       - 将原问题分解为更小的子问题
    5 H* H1 k$ z+ A   - 例如:n! = n × (n-1)!
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    经典示例:计算阶乘
    8 n8 `) O& [: Z7 r' A3 [9 b- Fpython
    3 ?5 D  W  I$ A5 u% y1 H; r, \def factorial(n):0 F5 t+ o- X. R3 x% D9 u' ]
        if n == 0:        # 基线条件2 |2 }4 y4 C1 a) F9 c* n  y
            return 12 M0 T9 @- J4 |
        else:             # 递归条件) j) B3 P3 d( @8 q7 ^, i
            return n * factorial(n-1)" X- a% o- J5 ~  p
    执行过程(以计算 3! 为例):$ i* ~, M7 M- Q) S8 \* u
    factorial(3)
    ' w5 O, Q) ^9 ^* U; a! ^3 * factorial(2)
    ) b1 b2 h! k( o! s/ k) e3 * (2 * factorial(1))% ^8 a( p8 d, A
    3 * (2 * (1 * factorial(0)))
    . _! R+ \6 n. Z8 M3 * (2 * (1 * 1)) = 6
    - K+ a: T! v5 W7 _# z, w+ A! R5 S" H% Z2 I
    递归思维要点
    - P/ K+ b( c) u! J4 u1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    8 x) A, r. u# r/ a. v! l$ D* k& Z2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ) V" R; v+ W# o/ l! ~3 Z/ M3. **递推过程**:不断向下分解问题(递)
    , {( z0 E* G6 N0 a" A: C4. **回溯过程**:组合子问题结果返回(归)
    1 n* r& v; W: b! _5 {8 T: o/ Z- \2 e1 @* }2 `  N+ Y
    注意事项
    3 E5 a, j% e2 J4 p1 F/ {必须要有终止条件) O* ^/ N% H( Z6 A
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    , ]/ p4 V0 W- J% p) p某些问题用递归更直观(如树遍历),但效率可能不如迭代
    * f4 L1 q- i( z9 M7 N1 z* [! e+ b尾递归优化可以提升效率(但Python不支持)  C/ h5 U5 {9 W3 v# U1 r# y3 }2 S
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    递归 vs 迭代
    * i+ m$ q) S2 i* l0 i2 C|          | 递归                          | 迭代               |7 p3 i$ h+ K% B% y  v
    |----------|-----------------------------|------------------|1 E" b& e) [8 A  w  ]1 r
    | 实现方式    | 函数自调用                        | 循环结构            |
    ! D+ K+ C6 G; L7 _8 Y$ {9 U| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    : s+ v$ d# S" Z, || 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         |
    " o/ I& Z) M" q8 j! q| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    2 o9 N9 q+ c) r- h4 u* Y1 x; s% [+ m; P) F+ h0 ~1 a
    经典递归应用场景
    $ d6 Q* K7 R. F2 E& v1. 文件系统遍历(目录树结构)
    " v& b2 B1 F5 s5 r) k7 ?2. 快速排序/归并排序算法9 J2 [2 L* s) P0 K4 i3 V
    3. 汉诺塔问题
    % Q5 p4 P+ b' A, a" l4. 二叉树遍历(前序/中序/后序)
    & A2 K% i, p, v8 _$ b* s; I5. 生成所有可能的组合(回溯算法)' y, K( B9 f1 F+ X, ~) n1 t- X

    / X" `0 D, p" x  g! A/ 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,7 u9 S7 |& H. l+ I7 P; v
    我推理机的核心算法应该是二叉树遍历的变种。$ N0 E: ^$ B% z: B. n: 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:: w' x" [4 H* E  ?  M) K& S
    Key Idea of Recursion% o/ ]2 Y4 R* N8 y' V9 D

    7 J" Q4 M( O; R, kA recursive function solves a problem by:8 ~' L6 @  T; o2 f
    : J, Z; X* `$ X
        Breaking the problem into smaller instances of the same problem.
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    - H" q3 N4 [3 s' }    Solving the smallest instance directly (base case).
    5 c: n0 g5 f- w* [
    ; `6 u' t; |( S( z    Combining the results of smaller instances to solve the larger problem.
      o* U# {! J) D" z! g" d: W% s' K+ Q5 }0 I8 N$ v% P* M3 ^
    Components of a Recursive Function
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        Base Case:
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    ! b. _$ e6 X/ }        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    / \4 v: F9 z6 B( O! L
    3 `/ N: `' K4 D        It acts as the stopping condition to prevent infinite recursion.
    7 [1 Q+ ~- K/ Y% g0 z4 e' D+ ^' Q9 M* h+ w8 b
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
      b3 @' m3 V3 J6 n) Z' Y# B5 k
    ' i/ ]2 {0 [% B7 T    Recursive Case:$ K- L; t! V) o* N

    ! c1 D$ T$ ]3 V! I: Y+ F& A( a        This is where the function calls itself with a smaller or simpler version of the problem.
    + s4 P4 Y2 j% N( f0 ]6 u! k  h5 ]5 ~6 h& F$ S0 I8 V
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ! i/ R1 ?" m) b: ?7 f$ [8 N) Z& P% Q( A' K. \& ?
    Example: Factorial Calculation
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    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:
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    ' \7 u- o8 s  y5 q5 ]- e+ n# ^    Base case: 0! = 1
    0 r3 d' x( O8 w! v' m( \% T- ?4 d
        Recursive case: n! = n * (n-1)!2 y1 g6 O. C9 M& {( `5 C
    $ t! T0 B9 T+ @+ W3 k
    Here’s how it looks in code (Python):
      U5 k2 L8 A# H, @1 x+ f# Hpython& X1 d' V$ _7 m  s5 E  J* W+ f
    : x% i- t  X7 ~* Q% j" U& U
    0 `/ T; H: h, `- O/ n3 B) U" C
    def factorial(n):$ p, s6 V* W% [$ ]' I/ s& T
        # Base case
      Z! g( O. K2 ^3 g9 n    if n == 0:
    4 n( c7 d, Q4 m; R) D+ }        return 1
    + u' K% F9 |! v$ I0 u3 t) B    # Recursive case! c/ c* b% u5 l+ s' S; M6 ?' S
        else:
    " p' G% F" h) f2 }        return n * factorial(n - 1)# Y) P* B1 Z% v. J8 G
    ( }  z' m% p4 w8 C! `
    # Example usage
    ' M1 |6 I" @- Iprint(factorial(5))  # Output: 1202 g. p, [) U! T

    ( ^) p( i4 `- q4 [! mHow Recursion Works
    & x+ _& d& ~8 [/ `* D8 s( X! C. t" o) f+ p- K8 x6 B2 r
        The function keeps calling itself with smaller inputs until it reaches the base case.
    7 z! Q/ j5 x: x% s7 p
    2 c6 a8 E. g$ p$ j0 e. c    Once the base case is reached, the function starts returning values back up the call stack.. L8 ~; V9 F1 m7 d* \6 r+ \& K4 R* c

    - {; _& i4 I9 R) l- _; o    These returned values are combined to produce the final result.
    / M) \' J+ n, q! R
    7 G; d- q6 a& m! R* VFor factorial(5):
    ' X& B, a5 p2 X" b* \
    " I) F1 V4 b" H$ c' ?# V, H+ P9 Q0 c
    factorial(5) = 5 * factorial(4)6 Q2 i8 w" T7 O  a7 l
    factorial(4) = 4 * factorial(3)
    " r4 G) O# S) s; X) Z8 o! i) L9 yfactorial(3) = 3 * factorial(2)
    3 @7 m: G& N. Ufactorial(2) = 2 * factorial(1)9 i8 g( n; o4 s- x+ ~# Q
    factorial(1) = 1 * factorial(0)% h1 s# n- r) S8 _
    factorial(0) = 1  # Base case
    3 V6 G1 m" S9 s( }4 \* r- Z% A$ n
    Then, the results are combined:
    % q5 ]  C8 n# w, k+ N0 q0 r6 Z9 y
    ; r; u2 P1 ~4 R9 L. i4 j  y$ e$ g8 J- R
    factorial(1) = 1 * 1 = 1
    ! K8 @; i! t7 E1 x3 ufactorial(2) = 2 * 1 = 2% A( n' U4 K: u+ ]- v7 v0 B0 \" C
    factorial(3) = 3 * 2 = 6
    4 n5 _4 w1 I7 R. c9 tfactorial(4) = 4 * 6 = 24, S5 d/ s' w9 v3 S  }
    factorial(5) = 5 * 24 = 120
    ! B( w+ C) N* a, O. r$ `# ]! W7 [- w( y- v2 V7 T4 i8 \
    Advantages of Recursion& O4 T0 ]4 q" t0 m6 `+ e& r6 k$ Y! l

    : x0 ~4 Q6 i: D7 B8 h3 ~; d9 q    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).. L8 \1 ^  |2 f. C
    3 ?7 \8 o, o, ^2 _
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
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    Disadvantages of Recursion' ?; h; I4 G- m0 P" B" s* N0 l
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        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.
    , \$ _5 m/ o( K6 r/ m$ [: l
    % z5 [! q0 H. _# q0 i    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).  q. b' k8 `& [. N( @+ p

    1 E* B8 C1 W8 x$ O6 g8 k: i* qWhen to Use Recursion
    ; Q5 H- P; X$ s, c$ S2 g; r6 f7 p3 `# D' b# T
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).! v1 e  H# k  m& M+ J$ }

    : y$ j) j- P; O" f4 s! `- o. T    Problems with a clear base case and recursive case.- `. h$ x$ y3 \: l% _* f& Y9 N

    & W2 G8 g: h! v/ L/ a1 h+ s+ aExample: Fibonacci Sequence) J' y5 \% B, ^+ e
    - z: k: n3 S& v
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    0 Q3 v+ |# C" s. L
    + N6 s( p# O6 M8 r    Base case: fib(0) = 0, fib(1) = 1; t  @: K; h7 G( C! q) ?
    / F% O5 r* T" ^: ~6 O" M6 u
        Recursive case: fib(n) = fib(n-1) + fib(n-2); V& N, K/ ]" D

    + \' H6 X" z# P7 m, r$ _8 ~- ^* @python# \$ v) d( o4 x1 C

    - h* k7 q! r5 ?) H2 W7 j$ u4 o
    1 c, e. \( h$ H: q+ G# {def fibonacci(n):7 m( g, Z0 y/ N0 l7 ^/ W
        # Base cases0 |- o7 u9 p/ s  P$ D
        if n == 0:, f8 b! n) I0 G6 S& S8 D
            return 0% Z  H) H8 Z/ S+ P! S# T( B# D, u
        elif n == 1:+ M8 k. L3 @' S4 N
            return 17 `" o. A: J) H- t  p3 u
        # Recursive case
    - }1 c/ W* c' @    else:6 t: R: d: `5 p. }- |' _
            return fibonacci(n - 1) + fibonacci(n - 2)! r8 Y) @( N$ r. s/ L8 b
    ' }: q' C( C& Q7 h
    # Example usage4 G+ I1 x) B' c: O5 n0 q4 ?% J; J
    print(fibonacci(6))  # Output: 8
    " u! k, ^4 Z: \& r7 U
    + W$ e& o$ _$ |Tail Recursion
    0 X4 N* R8 }. P+ @3 q2 J: I1 I, n, c9 J1 i+ ^
    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).
    + f# g3 z0 K: J: ]! [8 u  F) Z* B( N0 ]4 O% R! k
    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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