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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
     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 ! K% l! m7 k+ |  _  n, ?

    2 ]( n. ^0 O$ y- o9 o8 i解释的不错
    + t0 \4 y# T3 G' o0 b) e3 F: `6 o1 l9 I+ c5 d# S7 C- S; A6 A& 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。/ s; t9 d1 O6 C% p5 J

    $ t8 k% [1 x3 l' J5 r9 K 关键要素$ u) g$ X7 v. B) N: a& y6 u
    1. **基线条件(Base Case)**( |0 d. k/ f5 q
       - 递归终止的条件,防止无限循环
      m0 O: O$ [" s3 M6 n5 f" C) C   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    % ~" A) }6 Y7 r' S/ T8 |1 j% t4 V. x) }8 i; U$ O- r/ S3 L
    2. **递归条件(Recursive Case)**) S+ T# N8 G4 r, h* s2 B
       - 将原问题分解为更小的子问题
    ( B, \$ D4 D7 u" x( `2 ~9 R' S6 N: r   - 例如:n! = n × (n-1)!
    + I3 [: T7 ]! L, T& V4 w1 s, Z
    经典示例:计算阶乘
    9 Z  \, i3 m! M/ L$ Xpython
    / U7 F6 h9 f) H7 q7 T- q4 R, Udef factorial(n):/ |. E( Y" @1 i) }
        if n == 0:        # 基线条件
    2 g: S) p1 @/ [        return 1
    % w3 ^/ a8 |! d" g& l+ {- }. [    else:             # 递归条件8 ]# a: t" Y+ H( Q/ n9 @' j" A! |% |$ C
            return n * factorial(n-1)7 q3 ^  a. H4 B
    执行过程(以计算 3! 为例):
    6 ~$ Y- O: ^8 @factorial(3)1 e1 J; o0 ~. O/ h( q& L
    3 * factorial(2)" D: m' ~5 [: e4 n
    3 * (2 * factorial(1))+ t0 K+ z+ A. D! a; x
    3 * (2 * (1 * factorial(0)))
    " V4 E& w% X0 y# ?7 d  M. f, |3 * (2 * (1 * 1)) = 60 k( B0 R3 g$ d5 ^$ q5 O
    + p$ w9 [/ B* s" e2 ~  Z
    递归思维要点
    0 d; G5 E1 a6 z" s2 ~  m1. **信任递归**:假设子问题已经解决,专注当前层逻辑  c* B; M, q) M3 a* V$ T
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)0 Q  w  b. |; |2 ]  b) ^0 t0 Q
    3. **递推过程**:不断向下分解问题(递)! J0 N% i9 [) B6 a. n! r  G
    4. **回溯过程**:组合子问题结果返回(归)
    - E3 S  E$ T" X7 _2 Q
    " ?) _5 Y  B2 t4 y; m' e 注意事项
    0 ]& [" Q$ E- K, ~6 g+ S必须要有终止条件
    - r+ u; b; N, l7 |! ^# L递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    / e& i) Y. `3 T某些问题用递归更直观(如树遍历),但效率可能不如迭代# s/ l6 y" T/ v) m
    尾递归优化可以提升效率(但Python不支持)
    5 _# T& S5 U' G  w; ?* `
      J. t2 T- e6 Y" Z5 f 递归 vs 迭代# ~. s8 m3 ~" |2 Q5 l' g
    |          | 递归                          | 迭代               |
    & x; n6 u% p$ ~) C! `$ T7 ^|----------|-----------------------------|------------------|
    2 n: u9 L5 i' w; @| 实现方式    | 函数自调用                        | 循环结构            |
    & f+ e( E: W: Q; O6 L| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |5 X% e/ c3 b& B4 ^/ A
    | 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         |7 y2 e8 W7 L5 V% H( D8 e1 Y
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    " V/ P; V, j3 `, d3 w& e( q/ y& ~0 ?/ J2 W  ~# H+ h
    经典递归应用场景* `& I* _4 J/ o7 o2 c! g9 B; E
    1. 文件系统遍历(目录树结构)# `% d* n' \) m& C7 v5 n
    2. 快速排序/归并排序算法
    ! q$ _- N! k( o3. 汉诺塔问题% j! N, a& d9 D3 ~( \
    4. 二叉树遍历(前序/中序/后序): B# `2 S% O" @9 y
    5. 生成所有可能的组合(回溯算法)5 a8 O/ J4 T/ ?6 c2 ^" v

    5 ~7 p" q( I; `4 l: t4 X, l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
    郁闷
    昨天 06:31
  • 签到天数: 3239 天

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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,- Y0 M) W4 i6 p3 b
    我推理机的核心算法应该是二叉树遍历的变种。& `  c3 g; b% V% Y
    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:% X7 t' h7 l* Z" A
    Key Idea of Recursion7 [  \6 D$ T; \! d' J" r0 G1 M

    ; A9 s% e4 T3 d, E$ x9 M. G0 mA recursive function solves a problem by:9 e; v6 e3 A: b
    3 }$ W8 l0 i  ]( N2 \/ v8 z- M
        Breaking the problem into smaller instances of the same problem.% }- E* o) x, D- V
    * `- X+ Z) k$ O; t/ w8 C
        Solving the smallest instance directly (base case).1 N; b' z9 t1 _) }+ s" S- m" X5 ]
    5 H& E9 m* T( v, t9 H
        Combining the results of smaller instances to solve the larger problem.
    ; u  ?1 Z& |. \  |! V, J0 N+ x1 M4 x4 ?! P3 m
    Components of a Recursive Function
    7 P6 o# I# t8 G3 I( W/ [; F2 Y- X$ P0 K+ Z2 s
        Base Case:
      J# k8 k! s( S3 C% A' @% A" I7 U/ v, G0 S% D
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.9 v/ f6 z8 G! C$ s0 s! r

    % A* e( A' d" S6 y+ n7 y" C        It acts as the stopping condition to prevent infinite recursion.
    " ^* b) B! }  J4 G1 w. ]/ A+ z! z  o. t0 X2 ^6 m* @: e2 U6 b
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.8 h0 F8 b% S* ]" e8 B7 k3 b4 Z

    2 n+ w3 U& r  B4 x' ^8 C4 M    Recursive Case:
    2 ]8 f# y/ L1 y4 V' V
    4 ~, y4 V& V- E1 @0 q- j' |; M        This is where the function calls itself with a smaller or simpler version of the problem." ?1 `( Z0 C! w3 r; p# h9 W
    9 u- t" I8 ^; r1 `1 }" f* @& F
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    0 B5 t' \/ s$ F( n/ g2 U1 b# W; Q, [- f+ o- y
    Example: Factorial Calculation
    ; j3 ]9 h3 @7 y8 }
    ) \- q5 k* _4 z' W2 r7 @4 HThe 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:
    8 {$ z8 h5 ~7 [% b
    2 F0 |* n+ e; |; E; [( Q8 T    Base case: 0! = 1
    + F5 W1 L! r! A+ U0 d+ l
    - R; K3 L2 \7 B# _7 [    Recursive case: n! = n * (n-1)!
    2 _: c, V0 J+ O' E( X: X8 {
    " F/ \+ m! b% G$ J# RHere’s how it looks in code (Python):
    $ h3 r. j$ i, e! upython7 K5 v& e5 d( y' X9 l
    - ]+ r3 b8 ~/ G  ?

    4 T( Q. U; i# E3 x9 Ndef factorial(n):
    ) d1 _4 m7 a- A! E: `    # Base case
      S' F- x4 k  O- B5 ?6 g) l0 J4 C    if n == 0:
    & T; r6 Y' L, i3 x3 T+ n        return 1
    4 R& V# T" X/ l' @7 q7 W9 {- o; |    # Recursive case
    : F2 [# P) S& z; W( B$ y    else:
    : s1 z' [4 K+ a$ i: c/ |; h        return n * factorial(n - 1), z  G* E/ \2 ^
    2 D. t/ O2 \1 [
    # Example usage2 L5 T* Y* H9 {9 B$ x
    print(factorial(5))  # Output: 120
    4 d1 T3 J: F* L- e" [5 O/ k0 L% T# e7 m# j" {. w
    How Recursion Works
    % p' t+ R4 P6 B2 c# ~0 s0 {" I; v* L
        The function keeps calling itself with smaller inputs until it reaches the base case.
    " v  n7 X- g9 D* h, Y) W" a( ]; D  d3 A% N( |1 @  P
        Once the base case is reached, the function starts returning values back up the call stack.
    ; h: [. E- z; j4 b2 L7 ]1 Q% \* \; G; G6 X
        These returned values are combined to produce the final result.
    , [7 ?' M% p' P
    , m) M7 |* [1 ~% J& _! F- g. DFor factorial(5):3 B7 Y% U3 Z* W" t
    5 G: k- X$ @, Y) o7 D& K- ^; z2 h

    9 n1 C* [6 _' s. s6 Kfactorial(5) = 5 * factorial(4)8 E3 ~5 m1 g8 O! b$ G
    factorial(4) = 4 * factorial(3)
    ' R% H' R: h+ \2 Q2 Yfactorial(3) = 3 * factorial(2)8 H, H2 }5 d( e* m+ F$ g$ d
    factorial(2) = 2 * factorial(1)6 ~" C4 `' L* f- S6 ]7 Q
    factorial(1) = 1 * factorial(0)
      C9 k; t( D) O) `+ v, Gfactorial(0) = 1  # Base case
    ; E$ T" n5 m7 q! j% u
    : V! F+ L, l0 l1 j$ oThen, the results are combined:; k; L/ w3 [% ]5 ]# |3 ^

    ' p: ~; w% r, Z2 s5 z: `+ o7 ]& Z- X! D
    factorial(1) = 1 * 1 = 13 V8 V( |! U, G6 ?
    factorial(2) = 2 * 1 = 27 ^1 \' K; A: L3 E
    factorial(3) = 3 * 2 = 6
    7 x- P( t/ E' `8 Ufactorial(4) = 4 * 6 = 24& ~1 R9 m0 i: p# k
    factorial(5) = 5 * 24 = 120' Z/ d6 g$ x7 |; y7 w, P# q7 x

    - h& n7 a% x& N9 yAdvantages of Recursion+ L! d7 O# \) j- t
    ; Z: k3 {1 j7 q6 D) T3 w4 B
        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).5 A0 h) u) s' i# l' e. X

    - E  `- _& X$ f2 P6 J/ o: k9 n! T    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    & q) T& K; k0 Y: t( Y/ f. j7 ^9 g& A: B
    Disadvantages of Recursion8 ?" B& o% c8 [9 [1 @' V* v$ [$ `

    - Y+ C/ I9 d& k+ V+ x5 }( R    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.
    . s4 y( R9 _4 y$ S5 i
    + G4 N7 H3 p& O" ~4 M; k6 D$ \    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).: \  X# x' H/ o8 A: _

      r$ g6 i- O0 \( A5 zWhen to Use Recursion
    / u" @4 O) E3 n6 M: }
    ! B0 ?; D( f. m2 b8 b" M/ B    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    . E9 k& S$ `6 P+ w$ R& w  e3 a- P& [' c7 W: B, ?( I' A- X" T3 C
        Problems with a clear base case and recursive case.2 d2 R6 e0 ^! i4 x( D+ [; B1 M2 O% Z2 Z

    & y$ V0 Z9 Z. [Example: Fibonacci Sequence" _4 f3 h2 P& y' x$ V$ O

    ( Z9 B3 Y% F' `0 K; e, C1 xThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ; O9 h  w8 p* H' r! h5 [
    : \; K$ Q6 Y& p( X) W5 }) m    Base case: fib(0) = 0, fib(1) = 10 I6 }* I; L6 z: V% S
    : l: v$ s$ p' d4 Q6 q
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    $ e5 r6 s+ d7 m( y! B. i+ }0 A4 Y9 {0 G0 l. ~) ]3 H
    python* f1 ^" R4 l9 B

    ) l) g- Y6 C5 s* P# S9 Q* v+ ]# Y1 [6 \. Y  X. M1 G3 c
    def fibonacci(n):
    7 F8 m6 A2 _; l5 l% V' |+ T    # Base cases
    4 g# Y3 J3 w( v$ Z3 w    if n == 0:
    . o& b; L( M; k        return 0
    . {4 W( Z/ s& n  U* O2 V8 I% u    elif n == 1:
    " q* B4 I$ j1 e( K        return 1! ?% Z3 c+ t; w! j- Z
        # Recursive case
    5 i) }! r5 }4 C. {    else:8 x' H- G9 ?; `0 g, k  h# `
            return fibonacci(n - 1) + fibonacci(n - 2)
    8 f9 E9 G5 z. J: W* k
    ; j3 G9 a4 ~" F9 n5 _6 @# Example usage/ w8 K0 N0 f; w% m5 F' |5 [
    print(fibonacci(6))  # Output: 87 ]4 E. y- T7 ^4 @

    0 v. _! Q3 n  ], G$ W" c5 HTail Recursion1 u% O5 _4 Z' c  u9 B4 H

    ! a; i, k: ^& m4 f# aTail 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)." B6 l4 R1 _# y
    1 ~1 b. Z/ y8 c$ T6 J
    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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