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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
    ( I2 N0 b4 M6 K( v# ~. t0 ?. c
    8 }& S1 o& H4 {. D. _0 v- d  V9 j解释的不错
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    8 u, b4 |" i7 u: `7 q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 _3 O1 i3 N% N# Z
    & W- y: m8 L  K/ D& L7 E$ Z 关键要素6 I6 H* \5 Q! i. p& x( Q& |
    1. **基线条件(Base Case)**& N. ?. \) H- r) E4 ]
       - 递归终止的条件,防止无限循环
    8 ^6 _$ I1 A- P& P$ c; Z   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    & G% l7 a9 d. e8 u6 K( v
    ! g( r7 M2 _- Q7 I3 w2 G2. **递归条件(Recursive Case)**
    ; K: D* Z. W9 L, |9 L5 D5 i& G- S   - 将原问题分解为更小的子问题
    ) P+ j4 f* m* l/ e2 ]0 j1 r   - 例如:n! = n × (n-1)!
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    3 x+ t! c- M* a# F 经典示例:计算阶乘
      z$ A: D- `" {( Bpython& W- ~1 r# I  {/ r7 b8 Y  ?/ Z. f
    def factorial(n):) t) s3 r: R/ ?# |3 S
        if n == 0:        # 基线条件4 }/ o4 Q" J4 c& x' e
            return 1
    9 u, [, z$ H3 _5 K+ ?    else:             # 递归条件3 u$ O1 Z% v, {: ]
            return n * factorial(n-1)
    ! u. f0 q0 c, t6 ?) B1 H执行过程(以计算 3! 为例):
    & R. n' _+ N3 i9 C1 a7 `1 s- |factorial(3)5 J* A& k% U) d# f
    3 * factorial(2)
    & N6 B# C6 K  S3 * (2 * factorial(1))4 E, C1 U$ b) {8 y, Z( D3 a: ^
    3 * (2 * (1 * factorial(0)))& x/ K9 d, f% M! p
    3 * (2 * (1 * 1)) = 6
    ! M3 Y4 b/ r  D% V
    4 y! E* K: l- x: g$ ]# ]9 r 递归思维要点
    / C2 P! q3 s: T: P% s1. **信任递归**:假设子问题已经解决,专注当前层逻辑6 `9 a: e& C* n$ H, V! }8 n+ a+ O
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ' t9 _% Q- t7 w+ l  U  B1 ~6 i7 U- I6 Y3. **递推过程**:不断向下分解问题(递)
    & n* h6 @8 f; x) {" e4. **回溯过程**:组合子问题结果返回(归)' w' g% ^+ a& G" Y4 Q# q4 H
    7 e/ \$ S% B4 n
    注意事项" O# L( q' B  j- J$ w8 J' h
    必须要有终止条件
    : V8 l- a4 p- l2 l2 W% R递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ) I$ M  ^- k  U& _5 M% C, Q某些问题用递归更直观(如树遍历),但效率可能不如迭代
    : \; I9 q" \7 y+ q! r: Y$ M9 z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
    3 t: E2 D5 c& ]( K|          | 递归                          | 迭代               |! x8 [+ S* W5 F5 T$ ]) e7 z# A
    |----------|-----------------------------|------------------|
    ; W% S  }& k" g. W! Z| 实现方式    | 函数自调用                        | 循环结构            |
    5 h$ r: u$ B; R2 [3 o* I7 T6 \/ j| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |9 D" \& B5 Q4 _5 o( 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         |0 l3 U; v, o  S8 g+ _
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    " _6 b$ X( b& }
    5 p2 ~9 M2 H. j 经典递归应用场景7 U8 o( ]' _, t3 p- h7 N" ^" h
    1. 文件系统遍历(目录树结构)  |( B: v% ]; S8 s, Z
    2. 快速排序/归并排序算法" O4 ], s' ?% ]5 V
    3. 汉诺塔问题
      {; V* s) l" I# R7 N' f6 i9 Y; Q4. 二叉树遍历(前序/中序/后序)
    $ b# Z. F" I9 b% h* N5. 生成所有可能的组合(回溯算法)& B2 O/ s+ |: g, f
    . R$ Q* o  J+ L( R- u* }
    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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    昨天 06:39
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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,
    3 g+ P0 J8 i" D0 p% d4 X我推理机的核心算法应该是二叉树遍历的变种。
    2 R8 p( P: W/ x7 Y7 r3 S8 A: k2 I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:
    2 K5 G. f* D& L: [( CKey Idea of Recursion+ G- {( D8 j, a/ G8 z3 Q

    $ U/ \$ s- L) G8 I; L1 _A recursive function solves a problem by:
    + F! r- C5 e0 F7 U5 V, E
    $ G3 P6 x. [) I    Breaking the problem into smaller instances of the same problem.! w. T4 Z1 h" I! n8 q* V
    # T% r" h2 z( w' F- j& c, ?
        Solving the smallest instance directly (base case).6 x* m/ r3 F9 Y; }: a" n( s

    % C/ G4 Y' W8 K    Combining the results of smaller instances to solve the larger problem.
    # ~! ~6 M4 e' B3 @5 v5 {# L1 F: z* }. h8 Q* B" F! ^5 m% y
    Components of a Recursive Function
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        Base Case:
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.4 D1 Z3 P( ]1 V" K4 G. I
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            It acts as the stopping condition to prevent infinite recursion.' K* C9 l% p! D, ^' i0 t
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.3 \2 [- P) J0 ?8 o: k" A& M

    $ k: r, @0 p  t& b( x( w1 H    Recursive Case:2 x+ \( g* A* L, }, W2 i

    2 X# G) j- h5 g1 a; o" Z        This is where the function calls itself with a smaller or simpler version of the problem.
    1 B1 o; i, |1 w" n% E0 O7 D. u6 {  O; l9 a; x6 c, K$ H4 o2 }
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    Example: Factorial Calculation- h) u4 J4 ~) ~
    6 O. Z( Z2 I# B5 u7 ?2 G+ }% E# h
    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:. O1 Q. ^& U" [+ K' j) S' G

    ' y, H& T# c. N8 S) S    Base case: 0! = 1* N4 g3 J$ I( w7 o1 T0 G" r

    - l( `2 j. f5 E/ m    Recursive case: n! = n * (n-1)!9 w& W8 U( _, l+ r9 g+ E6 `
    3 p, Q6 w4 l5 U
    Here’s how it looks in code (Python):1 b1 I' r- S! p/ O) }4 ?  x
    python' O1 v: |5 m9 N/ J" m. x8 T

    % W7 D, V8 V: W2 K' A6 V0 c; Z# X$ R' T
    def factorial(n):- c+ |5 f- J& V! Q  w0 k
        # Base case% w) H# c- d9 B& j8 K* w* E( x
        if n == 0:' {, M1 i; @3 y6 j/ N- Y
            return 1/ b+ M+ U: N4 _4 `% G/ T
        # Recursive case! M0 C) x  q% |! J7 u, o( r& t# ]0 e; D
        else:7 j. Q/ |: I: q" a
            return n * factorial(n - 1)
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    # Example usage
    3 ~" T7 }3 P5 X) ?. E3 a! D& Gprint(factorial(5))  # Output: 1208 Z. I2 Q* ^3 q9 l
    % Q: K& j& x' R+ a# b
    How Recursion Works
    # {5 b& t" ^& x/ L' Y1 H7 |+ A: w3 a* Z) g% o
        The function keeps calling itself with smaller inputs until it reaches the base case.
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    7 j' o9 H4 {2 }$ M2 ^    Once the base case is reached, the function starts returning values back up the call stack.3 V8 K. n) s! L3 A& Y* h
    ) W" V. ]: ~7 P* W- `
        These returned values are combined to produce the final result.
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    . [; e* P6 r% j0 T" W' |: oFor factorial(5):' P; L: Z. z7 `0 \" L5 h
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    $ ^- e9 k2 l; [! s  C
    factorial(5) = 5 * factorial(4)
    0 o  o5 o( B2 wfactorial(4) = 4 * factorial(3)
    8 g' ?$ ?  }0 I! p) hfactorial(3) = 3 * factorial(2)
    9 k; M/ h# D, xfactorial(2) = 2 * factorial(1)
    1 {- P9 O3 s# sfactorial(1) = 1 * factorial(0)
    4 Z/ S9 K! s- T, P' L5 D1 Yfactorial(0) = 1  # Base case
    & ?! y; M5 T6 i+ C7 K6 B; r% X) |2 |. C" J+ m& W+ k6 M7 b
    Then, the results are combined:! R5 @/ e( \8 Z7 K: X& r

    # f) Z8 c+ t/ L# o4 r2 i1 S4 s' s( ~/ R9 I# W/ u
    factorial(1) = 1 * 1 = 1
    / W! r5 i' M3 B* c9 s8 L' A. hfactorial(2) = 2 * 1 = 2
    1 T; i+ l  g+ L7 I. n, R2 _3 mfactorial(3) = 3 * 2 = 61 z" g3 i8 m/ w  X+ t
    factorial(4) = 4 * 6 = 24
    . x* @8 b$ B* ]2 zfactorial(5) = 5 * 24 = 120
    * X* x5 \( |0 a/ d% o5 R# ^" L
    - R0 }3 e0 ]$ ]) F% z) f0 JAdvantages of Recursion
    9 B% y- L+ g% H" ^+ z. Q0 q3 i/ Q. i  u
        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 s4 M* t% u! L7 e1 v
    * A' e6 V% N/ ^' _! k3 U; T    Readability: Recursive code can be more readable and concise compared to iterative solutions.% z6 I+ D% ~* C: i% |

    + }1 \  Z$ ?# Y: i+ W8 QDisadvantages of Recursion8 T6 v$ a2 ]! ^% V3 z+ r, o* L
    * |+ r+ R: @( M# a* O7 x
        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.
    ; _% d& q7 E& y5 d$ y3 {* E7 X$ ?" V" M* E1 f- O
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ) B* y4 s# B9 o" q  @8 i: k
    2 z8 J4 F0 F; y6 f1 k- ~' }4 MWhen to Use Recursion
      g1 C$ C3 c5 _% o
    + m' B% \, i5 K, D( W    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    : Z+ s* |$ ?) W1 M: Q2 I. @9 D$ W  I& E5 [! [- Z
        Problems with a clear base case and recursive case.  E3 {. S' T; Z+ b
    ' ~& @7 A& a$ ]$ I$ R6 T
    Example: Fibonacci Sequence
    9 K& z; N5 J+ f$ D4 X' u- w5 Z- V  d2 G! @# E
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    6 S: w6 E) Z+ [+ n$ b( M: M) b8 X( z6 p4 i4 Q3 A4 i) [
        Base case: fib(0) = 0, fib(1) = 1! U  Q) K7 D6 S9 O/ x% R1 j& Y" |

    ! u0 a" ~, n- L: O    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    python$ A, J$ F+ h. T' ~

    4 J$ J: V5 @; q% ]
    9 e3 N7 Z$ j6 j. c( _1 M) R( Udef fibonacci(n):
    6 D9 B$ k. |) q8 H+ A  C" |! K    # Base cases
    / S0 }3 P! u5 k3 o2 X    if n == 0:3 B- i6 ]: @" I( Q: S# r7 A9 a
            return 02 F& k3 N. U! T& ]0 I
        elif n == 1:0 G+ ?4 M, X& A
            return 1
    , a; k1 X. C- G    # Recursive case( `. j" N1 q$ Z8 M
        else:
    1 W; q9 [3 d0 w/ j9 ?8 L  u        return fibonacci(n - 1) + fibonacci(n - 2)# Y! |" [9 a  ^' ]4 b: l
    - A/ a! @$ j* K4 Y) K# `
    # Example usage
    & K2 @% g* i2 e: b9 m, z6 Fprint(fibonacci(6))  # Output: 8- Q# Y4 T% f8 [. [7 T7 E/ v

    # R4 T8 r& r4 g1 k* ^! w4 l- s. h8 QTail Recursion
    7 v  b. e3 }+ M! Z
    0 u' y( Q  T" h+ R# f% MTail 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).
    / A- l+ F: t# r5 u7 k% v- {4 i" f/ f2 ?1 D- E- S
    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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