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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) d6 ~( d3 z2 K3 a

    / |$ u2 A% K% A4 S) ]$ {: ^解释的不错
    % X" J0 F0 P) z
    4 {. f3 w) R: T6 F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。
    8 `( K5 `, S8 f% N3 E! M( x
    7 v* @( E2 ~8 Z) _8 e& ~ 关键要素* m# a- O8 g  S" a5 l: i3 _
    1. **基线条件(Base Case)**
    3 N* S, ?: k+ B, ~   - 递归终止的条件,防止无限循环
    ' g; V5 E$ y& C% Z; c# d* V2 e   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    . p5 }0 e9 G0 X9 T* X: ]3 j- K0 F/ i, u
    2. **递归条件(Recursive Case)**
    1 B5 ^# ?: k/ R- t   - 将原问题分解为更小的子问题
    7 f' A" t% u" V% B   - 例如:n! = n × (n-1)!7 G8 w7 G; \% u) s* Q

    , q! k; n, Z7 G, m  V% F% i& g+ r 经典示例:计算阶乘) W6 T3 k8 p/ k
    python
    - C; `! e7 f% l" s* O% i: Q+ xdef factorial(n):
    % o9 N; Z; v! c    if n == 0:        # 基线条件
    % }5 L# {* P- [0 ]        return 16 J3 B/ ?! @1 h) `! F) }
        else:             # 递归条件
    & |" ^/ W( ^2 r1 S3 j        return n * factorial(n-1)
    * Z+ h% l9 |% X执行过程(以计算 3! 为例):& M9 P/ S* ~, q* F1 D: R
    factorial(3)5 w' W" t, ]7 G$ i
    3 * factorial(2)6 f# Q- O$ @, x2 Y
    3 * (2 * factorial(1))
    + Q/ u: d, m& B$ A6 s3 * (2 * (1 * factorial(0)))
    9 V. k, G1 h- @$ l  W; m/ P: m3 * (2 * (1 * 1)) = 61 e" {" t9 x- {) p& f- I
    + j/ E0 Q) z+ t% a! c
    递归思维要点
    : j! Z/ u: U4 I1. **信任递归**:假设子问题已经解决,专注当前层逻辑( v1 G0 g2 F5 D5 i& Y+ b9 V1 \# V
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ' a3 `, O. m4 |( Y  ^3. **递推过程**:不断向下分解问题(递)
    ) r7 V2 K" t2 I5 v4. **回溯过程**:组合子问题结果返回(归)
    , ~$ }' p3 a6 C/ a2 i: u
    . ?# a# @: _8 N( ]; o0 K7 V 注意事项
    2 A1 f7 M! n/ W6 X3 ]$ K4 _! Y必须要有终止条件, i& M6 ^" ]& H( |. E+ r  _+ n( \
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    + z8 P6 G  M: u某些问题用递归更直观(如树遍历),但效率可能不如迭代
    9 h) A) c3 z6 ?& L) [# m尾递归优化可以提升效率(但Python不支持)
    5 e5 |, b' Y: S3 w: P* F4 u" ^# \6 v3 C& m  t: N$ @
    递归 vs 迭代' h0 \6 U5 @' _7 L9 A$ E
    |          | 递归                          | 迭代               |
    7 K/ u+ r" w4 t, Q|----------|-----------------------------|------------------|
    4 A% |/ C8 V* D) M( K" A& E9 g! m, k| 实现方式    | 函数自调用                        | 循环结构            |/ Z8 l2 d3 y% a3 }) ~/ e
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |. H5 ~# z0 C( I9 W* c0 d
    | 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         |; E8 W% W5 Z6 S! w
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
      n, I# s# B% n2 @- \
    8 H5 m8 a8 z0 P9 k) D0 c+ l0 p7 J4 S# n' s 经典递归应用场景
    4 {2 L5 Y: X% ~: S3 ~# X+ K1. 文件系统遍历(目录树结构)
    6 R8 P5 J0 a8 s; I/ `2. 快速排序/归并排序算法
    7 q7 {: G7 V) Y1 ?3. 汉诺塔问题- x% f  r# Z+ W2 \
    4. 二叉树遍历(前序/中序/后序)% L2 Y2 A8 X1 x- @2 _% G
    5. 生成所有可能的组合(回溯算法)
    8 N  ]# a) w. p; f4 w/ y1 a! I' |5 v; W3 {6 h/ ~: }' {# c
    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,
    + p( v7 B  T8 V. E我推理机的核心算法应该是二叉树遍历的变种。( o1 e8 {2 [" w& c5 v* `: p
    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:
    # S/ w7 }2 M" C7 ?- D; `Key Idea of Recursion) v3 ]0 Q, L& U; p" Z" O
    . \; |, T# c" ?- H5 _$ p1 U. Q
    A recursive function solves a problem by:
    0 x. i" }( ]) t. e0 N2 K, [
    5 [# D. `' c+ K1 e5 B, _: @: P    Breaking the problem into smaller instances of the same problem.
    8 ]2 u5 C+ V& J1 i; `# a3 a. F9 G+ [& e, r( w; D
        Solving the smallest instance directly (base case).
    . T8 E! B( T$ ~& X' b: g) E' F- W3 ^3 d+ T
        Combining the results of smaller instances to solve the larger problem.
    1 S1 D8 z' K: t* L# y! m* h( J5 a+ X& S: B6 T9 N( o4 T
    Components of a Recursive Function
    # Z0 `5 v4 q* Z4 F8 j  j7 c) i/ h' o. A; C
        Base Case:
    8 c' `$ |) |; k( r1 _+ d: @$ h4 T( {5 a* X; J5 E
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ) x7 d+ V' g5 R9 a  v
    ! C" @- l4 q% F' U3 j' @        It acts as the stopping condition to prevent infinite recursion.& j( |: m: M0 i' S" d
    8 k+ ^- z# }  y2 {2 F8 }
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.  V" W) n- u, q! t- y+ p

    1 J, [8 W+ H, C* U/ M0 u    Recursive Case:
    2 C3 d: b% K/ E, p( s+ T' Y3 Y% c; T* b$ D2 k
            This is where the function calls itself with a smaller or simpler version of the problem.
    3 [6 {$ O( o6 D, _. i6 q
    2 P% m4 c  t5 X8 Z! e        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    # @% J& K8 u% b- C, j$ {9 |3 z# t/ h" ~+ U5 q4 [9 Y9 }7 x
    Example: Factorial Calculation8 Z  \% J. M& m+ t, ^
      x* E5 K0 i7 d3 s
    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:& j( e3 B, E4 p7 n2 R0 M5 H
    ( _3 O; l* u  I5 W  N: T  V
        Base case: 0! = 1
    0 H% Z2 T! |. v
    * u, i/ _0 F6 K7 C) O* l    Recursive case: n! = n * (n-1)!
    % D3 F+ e- M# b1 n& b' t
    : ^: n* d6 {" jHere’s how it looks in code (Python):
    * ?' K* y& t0 s. e+ u' zpython
    . H2 b/ v# ?$ S7 r7 X9 M4 G  P4 C, z% f6 m, y: T" x4 f4 k
    # o8 L1 c4 Z0 h0 D: m3 a* s; z
    def factorial(n):
    9 p" s* b2 T7 A7 M" F    # Base case
      k! w" c" A0 a    if n == 0:
    / O; B% H4 W6 a: e2 q$ o5 A        return 1+ [/ f" v) P4 W1 c  p( `( K
        # Recursive case
    . f& U# t; b6 z0 e3 N    else:
    : B$ H4 i6 p/ D4 @1 ]% p3 t! \        return n * factorial(n - 1)
    $ Q; R: ^1 V; R, [. c
    8 `! s% c9 S. r, O# Example usage
    ! l2 c  _( j0 c$ ?print(factorial(5))  # Output: 120
    2 Q3 w7 N7 Z6 q# D. ?/ R/ l. M( {6 T" g, E. u
    How Recursion Works6 g* z- F1 M) \( r9 g0 m) Z, e
    ) o, R7 G9 r# [! U( b
        The function keeps calling itself with smaller inputs until it reaches the base case.4 X- K( B7 F4 Q
    ) k+ R5 O. j8 K3 {: J6 o* s
        Once the base case is reached, the function starts returning values back up the call stack.% |! W) T7 b1 R
    5 d5 m+ T0 H8 O9 Z8 c
        These returned values are combined to produce the final result.
    3 O$ l1 w# r9 w/ X: a1 ]7 f  G
    2 C  |! t1 ~, D9 }# Q$ oFor factorial(5):2 Z( W: k7 D7 ]+ M/ T5 `0 B2 b
      r' O+ S+ E! T1 n) j" j

    , \: u) m' _+ u& x6 \factorial(5) = 5 * factorial(4)
    0 L) q9 _# h# ^% c3 i+ Yfactorial(4) = 4 * factorial(3)
    # c' Z% d3 x* u1 j% |2 F" hfactorial(3) = 3 * factorial(2)& R: A; Z& @" G/ T1 Z
    factorial(2) = 2 * factorial(1)
    9 b% J% h$ ?( B) F0 O+ o1 _+ Tfactorial(1) = 1 * factorial(0); o' Z6 J3 u) q
    factorial(0) = 1  # Base case' w7 t" k: {: y/ E' U

    ! Y. A/ T( c* M! @Then, the results are combined:2 N- \0 X- I' o% e0 D& d
    - T8 N# a* L4 r4 P- H

    2 D( `6 `/ W4 O4 ^factorial(1) = 1 * 1 = 1
    7 J2 ^. }4 R+ v" sfactorial(2) = 2 * 1 = 2
    . h% v- c3 [2 @factorial(3) = 3 * 2 = 6
    & {# X' @' w* m5 o- S  k) @! ]factorial(4) = 4 * 6 = 24% h3 q6 @5 r1 q
    factorial(5) = 5 * 24 = 120  t- }0 ~% d. r5 i$ u

    8 e& q  W8 e. F0 }% Z9 z' @8 T0 D  ?Advantages of Recursion( v% `& Z  X2 h
    # \+ r4 m8 ~9 U$ z# V5 A' S
        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).* q, D6 \' E0 }5 q) v* _

    5 J* H) S' |& v: A" q; D% Q. n9 N    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    7 j. g# V0 S9 m' Q+ w+ t* g  w, y5 ?% p2 ?: |5 e0 o1 |
    Disadvantages of Recursion9 C; t! l) _( X3 v& c8 a  k
    . O" Z& S/ w- d8 a1 r( G
        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.; @1 N! x1 M( s0 d4 D' `
    ) ^3 m* D/ t/ ^6 S8 i. A, u
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)." y) A. [9 ^6 n( X

    5 c4 J9 H% z" eWhen to Use Recursion
    4 ^) a8 F' i) i5 {- \' s( d+ v- E. k5 d8 h
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).+ Z$ c  e' k$ \. C5 _, ]0 L

    # U% P. u& V7 c    Problems with a clear base case and recursive case.
    ' z+ l$ {% C  K5 a& i% M4 K4 t% Q# o' m: m5 g, \1 L. Z
    Example: Fibonacci Sequence, d5 V. U( q  d8 l3 W# k/ U3 R

    ; H5 t6 g1 V% C) K& z5 n0 wThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:4 |3 K% i/ n( Y8 E  }* ?' B
    ) Q! r# O+ F. v* F! Y
        Base case: fib(0) = 0, fib(1) = 13 A' o1 d: p* I9 u, f" R

    * g$ g* X2 C! Q9 t0 |; z7 U    Recursive case: fib(n) = fib(n-1) + fib(n-2)2 ]1 ?+ ~( \, B& P; y" g/ h( I

    2 d, _! Z" f, U1 Xpython9 ^0 V; T" `) Y; b3 B
    7 S" A! t) l& k
    + y/ u% H. _; g0 s" L0 w
    def fibonacci(n):
    0 O, i+ ~% Z* `* C& L( B' @0 v    # Base cases; s1 k- Z) r, M; c) {
        if n == 0:- x1 ^$ F& S# r7 C
            return 0) w" w& l: V5 Y( c2 `% L
        elif n == 1:8 G0 Z# {( m/ F/ }. ~; K
            return 1
    8 o: O# Y9 x1 u    # Recursive case
    - g$ ~! M! h# q    else:
    1 B0 J1 @5 }, }; X" U6 N1 r# r        return fibonacci(n - 1) + fibonacci(n - 2)
    8 ]/ B1 [. ]7 v, r( z8 R% C) O" T2 H3 x$ N8 a' j& r* }
    # Example usage. ~2 m1 f5 o5 v; D% p( G) A
    print(fibonacci(6))  # Output: 8
    ( |* P( C  R" d
    1 L! L: Y: l$ F1 q$ G6 eTail Recursion
    : a  W# V) ~; H; u& i% P+ ?! I; E& d! t* n. ~& J% B
    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).
      R! v! `. v$ b+ {  c4 U) E" t; w4 e: ~) I3 z
    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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