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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
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    ; n! v# }8 X5 |& P) ]# X* Y- {) S解释的不错6 Q& e5 }- x# ~4 `( x) O; S0 V

    , V: r+ {4 D) E* I1 [1 U2 r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。
    6 S4 b5 V6 T' W7 Y, B! H! B
    6 u% j' `, d  x% |$ \4 [9 E" ~ 关键要素, u- @# L7 L5 D& L7 I! l
    1. **基线条件(Base Case)**1 s" x/ m2 T* b3 a: C% K
       - 递归终止的条件,防止无限循环  f# L* d0 H9 B6 i8 J
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ' R. T1 \  K9 O) \- `" ~7 L/ E7 @" j* X+ l: b; D
    2. **递归条件(Recursive Case)**
    * }) P' O  S! w: @$ A   - 将原问题分解为更小的子问题
    2 M+ y" `/ g/ T   - 例如:n! = n × (n-1)!
    $ a# t" D  O- _& g" e" d4 O6 J
    , B1 U) D* C$ n9 t 经典示例:计算阶乘- j, P$ X/ f8 x& h0 h: i. j
    python
    + P) I) M; [! J- Xdef factorial(n):
    2 U8 d# ?" A; r9 q3 L5 s8 ?$ j- y    if n == 0:        # 基线条件) O  L+ a8 t1 f4 ]5 B7 H
            return 1
    8 q, \) [" [/ x; ^) T    else:             # 递归条件
    ' T$ K  V2 z  [0 |% ]  P2 R: K" ]        return n * factorial(n-1)5 B- y7 x9 R& {
    执行过程(以计算 3! 为例):
    & J) ~% O' \$ L5 N, a8 nfactorial(3)
    * B9 W2 ~+ W4 |; z8 ~5 K3 * factorial(2)
    2 w; h$ l( m9 ~$ M4 H& D4 R3 * (2 * factorial(1))2 B7 y9 Y' t8 V$ _/ T7 N7 C
    3 * (2 * (1 * factorial(0)))
    / v# F* P( B. q6 A2 T3 * (2 * (1 * 1)) = 6
    4 u3 ^5 b; ~/ c/ b2 k" A" A  ~* o3 d" a/ q. L
    递归思维要点
    4 `% C6 ~& Q4 a; ?1. **信任递归**:假设子问题已经解决,专注当前层逻辑  j- o- n0 N% h- h4 Z
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间), ]( M. ^5 L' m4 t; m
    3. **递推过程**:不断向下分解问题(递)
    ) ]# x0 [2 s# U$ b4. **回溯过程**:组合子问题结果返回(归)
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    5 |. w) ]  d0 |. ~% L 注意事项( [6 |4 K2 V' Y( F, y: w
    必须要有终止条件
    $ N0 x9 f' Y; ~+ }递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ( a; l' B3 x& h( m( i. W! \某些问题用递归更直观(如树遍历),但效率可能不如迭代. u  _- q; i. Z- ]1 E1 z3 c
    尾递归优化可以提升效率(但Python不支持)
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    $ t3 c( v* d% s4 V! { 递归 vs 迭代
      E+ l# P9 ?/ \% x|          | 递归                          | 迭代               |, n: l' H/ f0 s) I! a" _7 Q
    |----------|-----------------------------|------------------|; v4 h7 r9 L. |  U/ N
    | 实现方式    | 函数自调用                        | 循环结构            |& t6 C  F+ L) e# A
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |  d- {5 `  s  n. B
    | 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         |
    1 O) J1 \. U' B& O; D+ x| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |2 b  L- z' M- @1 Q" k
    8 |; E' ]  C, u% }0 V$ `
    经典递归应用场景
    : o% {% X) n0 I1. 文件系统遍历(目录树结构)
    6 H  z7 C. r* G2. 快速排序/归并排序算法
    / `9 J( F) M- A" {2 x6 M3. 汉诺塔问题( A. @" g5 x# p6 B/ ^! ^9 {! l5 P
    4. 二叉树遍历(前序/中序/后序)
    8 k; {% @0 }* F+ m, P, \9 o  E( d5. 生成所有可能的组合(回溯算法)
    * ~; x2 `! Y  |% _! ~) U% P6 D
    8 [; W1 V: ^6 l1 U- z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,. {$ y6 c, a( B' h6 W6 v
    我推理机的核心算法应该是二叉树遍历的变种。+ w' U) H" }  L" b1 F
    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:* `# u. D! V1 n1 K
    Key Idea of Recursion7 A; p% `, I$ ?* n9 L) J

    " Z- k4 t/ o+ Z4 t8 }/ r  XA recursive function solves a problem by:
    & t! q. W4 M  d6 i! @- j7 O! \% q. J1 C7 b5 Y1 [# b9 [
        Breaking the problem into smaller instances of the same problem.
    * B. Y' n  I2 F( C
    7 o. o' H# y: ?' w    Solving the smallest instance directly (base case).  Z7 T  U8 F+ l- K/ e( A1 _

    $ l/ l6 c, L: O" W9 ~, U4 q4 y    Combining the results of smaller instances to solve the larger problem.
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    Components of a Recursive Function" p$ i5 l+ X" n! Y7 S, I0 B  l
    1 u  J! [+ c+ g9 y0 a
        Base Case:: `& t- ~' U2 ?7 ]4 F' g
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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            It acts as the stopping condition to prevent infinite recursion.. N4 I* t! h/ H8 k
    , k( ]( w* l9 Y9 b7 l1 z. D
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    8 T! k$ {  u% |, r( Y1 j    Recursive Case:
    : w5 q4 B; \- F( B, |' C
    8 c" g1 h. q1 M, D        This is where the function calls itself with a smaller or simpler version of the problem.' L. u! `, F4 o7 {/ F/ z7 v
    # B2 [# M4 Y6 ~1 f6 R9 ]8 S8 R- l
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).* S0 X" \6 N- L: o1 \: Y
    * W. H- I- U! @  Y3 T
    Example: Factorial Calculation% x, t# ^  Q$ m+ R) S" L

    & z) J/ {6 G/ z3 E- m7 ^# e7 kThe 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:
    + V8 r7 l' a( z2 v$ I" }7 ~8 m! t: q- ]' L$ K8 j8 N
        Base case: 0! = 1
    1 H& @" N1 H/ h$ z5 D, {- x. G2 I/ j! l8 H
        Recursive case: n! = n * (n-1)!
    2 U/ v/ S& M* f" [0 ~# \) w9 x2 U! p6 n5 ?$ X( g+ V4 R$ G
    Here’s how it looks in code (Python):. Q# y9 l& ?$ o8 c. S2 B- J
    python
    # E* s, w. G7 Q, @
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    def factorial(n):9 H; G1 }6 R$ A$ M8 w" `) o6 o
        # Base case
    * ~2 b& u& }* E7 {    if n == 0:
    0 k3 U+ I7 [7 P  ~) g        return 1
    2 ^6 e9 I5 ^- |8 W+ X- |    # Recursive case( o0 A6 F& D* j
        else:" }5 B( c* Z7 }# s# c. q8 B$ f
            return n * factorial(n - 1)
    3 [+ ?( D  H0 O4 u* e: ^+ H1 ?# S$ d8 |9 R- N% z
    # Example usage. J  a4 ^. q8 I. s6 S( w0 O4 j
    print(factorial(5))  # Output: 120
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    How Recursion Works
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    ; u" ?6 o! D7 K( m    The function keeps calling itself with smaller inputs until it reaches the base case.
    1 h8 H) J, ?7 w+ s
    8 C0 y4 U- M, }6 n    Once the base case is reached, the function starts returning values back up the call stack.
    3 ~6 c! Q) q' Z1 I5 `8 k/ o. d, N& N* y  {: l: @
        These returned values are combined to produce the final result./ q# K' B3 Q8 c+ c$ u3 {2 B

    6 j" L: l4 c8 y2 ~& F7 y; r3 qFor factorial(5):
    3 j! k" x7 R- j: u. y6 n
    / z. y3 H: V- K9 x" s0 U. t0 U0 m* I/ ~9 Z; X
    factorial(5) = 5 * factorial(4)' K7 j1 w# v4 R0 G9 v; V
    factorial(4) = 4 * factorial(3), L- f, L5 h1 n+ m1 x
    factorial(3) = 3 * factorial(2)
    $ r" N( `/ L" y2 y0 W- \9 B6 _) c+ wfactorial(2) = 2 * factorial(1)2 F, b$ h( S+ A; e
    factorial(1) = 1 * factorial(0)
    1 v/ e2 q) J  `% ]factorial(0) = 1  # Base case
    2 d" P) b9 q: s  \+ f  v
    # u) M4 E  c/ o: Z$ ?Then, the results are combined:: N3 a8 \% n( y# g/ H; r

    , V: \% |& h+ t2 Y; O0 T
    & l9 i! [* C" c6 g5 X6 a; e) z: ofactorial(1) = 1 * 1 = 1
    * T7 X( R% o. f9 qfactorial(2) = 2 * 1 = 2. M# W" N4 R, w/ Y; J4 ?! {/ ?
    factorial(3) = 3 * 2 = 6
    # ]8 f# Q9 M9 {+ ~: d/ hfactorial(4) = 4 * 6 = 24
    , v6 V  W" E! gfactorial(5) = 5 * 24 = 120
    / \7 i1 M  H# j# B2 W3 A+ ?/ I) s2 B6 c# n% |- O
    Advantages of Recursion& L4 w9 ]. ~# [+ O0 e

    " D% Q  ?5 J# R  ^, l    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).- g% b8 }: K& b6 O9 Q2 B

    % @4 C9 d) u) f5 f2 {    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    % n- C/ z' u4 P5 a9 f( E4 I- z3 t* _& y6 e* g3 e
    Disadvantages of Recursion8 |  A" o2 {! X: R* Y) p# V0 d

    ' e, `$ v9 X: T% v    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.
    ( u' @% ^5 H8 x+ C  G4 C
    ! A  B0 L8 J6 u8 Z  ]$ x* l    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    9 I% Y, Q: z3 l6 M7 p6 ]! X1 R  C' S# T; S# ~: _: Z) L0 i+ m
    When to Use Recursion& k. O7 q/ M% Z* S
      R! Y" w+ O* a: s  W! w& V' }
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    1 t  l6 K7 Q0 e+ t+ q
    # u" t  C. O; ^& t7 |/ `$ P    Problems with a clear base case and recursive case.
    8 {5 m) ]8 e' y
    6 d7 Q& e' H9 ?( YExample: Fibonacci Sequence" u9 c/ y; s: i6 B8 {( \. M
    ' Z+ t1 h/ G- [! H) G. m& \
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:" ^- _. A0 F  q# h- k: T' f
    $ Y4 \; X- ?! K' k+ A! t1 o! p( A
        Base case: fib(0) = 0, fib(1) = 1
    8 w/ I; }/ [( [8 P: K: |  N" G) O! d9 ?
        Recursive case: fib(n) = fib(n-1) + fib(n-2), p/ S! t4 g, d: g3 ]% {" B

    & o( v7 [) W( @0 A9 P! Z  N$ Epython( C4 u1 E3 N' o/ o
    4 {# i9 c; l  f) i- F& H! M1 W) g+ R
    2 ]0 o+ V. F" N4 Y$ N' M6 R
    def fibonacci(n):0 I6 y) N8 G% z& |
        # Base cases0 J, O: Q9 V" n) L* J& R
        if n == 0:
    . H7 g# R, z) j! Y: v5 \        return 0
    " \& @! f( W4 R! {    elif n == 1:: x  ?% M! h- F& y& h
            return 1
    & J4 P! h  G: j9 p6 I. d4 K    # Recursive case
    2 [) V' Q+ e, s0 a4 E8 @' h" ]    else:( A2 N( ]8 y. B: W- n& {1 C" }
            return fibonacci(n - 1) + fibonacci(n - 2)' c6 j* C3 y) m$ h1 f
      R) ~$ f, w# L$ E% I
    # Example usage1 Y. J0 {* e( _; w  ~
    print(fibonacci(6))  # Output: 8) ^: _7 H& z  i, J1 A
    7 l/ J5 ~8 y  O  F% h5 p( t, B
    Tail Recursion
    % {. y; {$ M. P2 V
    + z. Y& g6 ?4 J( nTail 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).0 }" z' E9 h9 O, |" Q- M

    : e9 s! k/ `3 e; V. jIn 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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