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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 ' O/ J" J" ?2 o9 ^' F1 F+ q$ |) H

    3 \3 K: s% v- p+ {, p. I解释的不错
    . `. O, m: U5 o  \5 x! {/ D  F& ?9 c$ _# z& l/ F2 L
    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。# u* L: N3 N7 }+ M' C+ [' P9 b. j
    3 K4 ?$ O  G9 [6 {
    关键要素3 U& x9 v+ g  |7 k
    1. **基线条件(Base Case)**
    % d5 }; f) u2 x7 V1 b# t; k3 c" F   - 递归终止的条件,防止无限循环# X! F: A' s% K# y5 X9 G4 R  q0 w
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    8 q" I: \  x% w6 ]( l0 A
    $ c( B+ j0 H; i. L% n" Y2. **递归条件(Recursive Case)**
    # C2 f( b) J' a: O   - 将原问题分解为更小的子问题
    0 j: S3 f6 j, ~" @) S! J   - 例如:n! = n × (n-1)!
    9 _. f! j7 \- E# P" F7 b% m9 h$ U* b2 q( B# B( R
    经典示例:计算阶乘6 t% f' ^9 h/ T& a8 M) |
    python, r) m( |3 w" ?$ D
    def factorial(n):
    + v) ?! K% j( H: p& {+ R8 d8 ]    if n == 0:        # 基线条件& U6 {" ~7 I0 l; d
            return 1
    2 x' b2 p9 |. R8 w2 `6 [7 P3 D    else:             # 递归条件, e& C8 A/ p. F
            return n * factorial(n-1)
    3 B1 [" @# u+ w0 ^6 @执行过程(以计算 3! 为例):) [, k2 N: r9 E% N
    factorial(3)
      L2 q! s) P, d- N" P$ s3 * factorial(2)+ U% K0 V# l8 F' Y; m2 N
    3 * (2 * factorial(1))/ B4 ~6 p/ ]8 _6 w' m5 u
    3 * (2 * (1 * factorial(0)))
      V  Y2 c! u7 V3 * (2 * (1 * 1)) = 6. y& D1 {0 Y& }8 c
    + S$ A+ v' S: v! z. w. i& T) s6 D7 ~
    递归思维要点7 i0 J- @- U- G  E  j8 c+ i  T; ?
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ! ^3 r$ L+ a7 r2 o  _. _2. **栈结构**:每次调用都会创建新的栈帧(内存空间)* N- x/ L. c, P/ {" P
    3. **递推过程**:不断向下分解问题(递), X1 N9 W7 m- G) B& P
    4. **回溯过程**:组合子问题结果返回(归)% \" g. N1 A( o: k) S5 J

    8 O, D2 l9 N8 I. L# r 注意事项
    2 i1 ^' k, u& i; d$ P2 q4 W必须要有终止条件! a1 s( d" t7 \, F! i& O
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)3 p( ^+ F5 ~$ }
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    - d3 k7 P9 e' Z$ E. p" ^尾递归优化可以提升效率(但Python不支持)
    , T* z7 \6 E" N0 ^; }' `- v/ l  c7 T7 r: {
    递归 vs 迭代6 \" t9 F4 g* y  [9 p
    |          | 递归                          | 迭代               |
    * q4 H# `. R" G* w5 {|----------|-----------------------------|------------------|* ~$ ^. F2 O1 J! j: H7 A
    | 实现方式    | 函数自调用                        | 循环结构            |7 Q) X  k( _. N* z
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    0 }1 y8 ?3 @$ r4 J" k9 D1 T' d8 @| 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         |
    8 }; j" M0 h/ R& z| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |6 v+ n+ G( `: ^6 j+ e% x
    " ]5 N: \. k+ D
    经典递归应用场景
    * K+ b# [. @/ B1. 文件系统遍历(目录树结构)
      g0 C8 t. C/ s! D$ w2. 快速排序/归并排序算法
    $ }: \- V8 k, J2 O. [3. 汉诺塔问题
    5 u  H$ L6 x! U5 i4. 二叉树遍历(前序/中序/后序)4 n3 K3 }# p! g, O& @
    5. 生成所有可能的组合(回溯算法)+ O- |% E, G$ k4 P0 I% {' ~

    - W9 v' U- |4 N& s) l5 [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 09:58
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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,# M1 `3 C; I4 {$ W! S, b
    我推理机的核心算法应该是二叉树遍历的变种。2 t0 n9 P$ P# E& L+ \; 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:
    4 O" W/ `4 c* [. Z# U1 D) d- qKey Idea of Recursion
    . c( ^( \0 ~' E* W1 p$ y! }1 S( r0 R& h
    A recursive function solves a problem by:3 X4 \! x2 V, x; F7 T; n& o2 [* f

    % x  x8 L/ W1 V- H6 ?- V    Breaking the problem into smaller instances of the same problem.1 V; W" T- s) n) ]' W( a# }1 f

    ' `* i( p" l9 ?    Solving the smallest instance directly (base case).
    9 s- L( u- c$ K2 @' M/ Y" i2 W; U) a4 M+ L
        Combining the results of smaller instances to solve the larger problem.: m, ?% ^6 Q1 ~. C

    7 r+ s  g; W# d+ s* p/ H& ]  [Components of a Recursive Function
    8 t# _: i# F( W+ P0 }
    $ W, \+ [- l3 e0 z. t    Base Case:% e  ?+ d0 A4 ?: I" G
    % P0 V% t% E4 @
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.8 x  X, J$ o% V, ]
    3 z! L: w: [) }% N  i8 `" g$ P
            It acts as the stopping condition to prevent infinite recursion.9 \" W; ~: y- R* {

    ; R0 L" M- U8 ?0 `. w        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.( u- W: s7 X+ ^4 _" q( B2 y+ q/ Z

      j3 L$ ^$ X& [3 j3 k4 w% G$ i    Recursive Case:& D4 W5 d( ~4 U$ K" Q6 I) A: l
    . q# Y3 r2 ?0 R% Z" H
            This is where the function calls itself with a smaller or simpler version of the problem.5 U9 b0 D$ o  i

    : {: {$ Y1 n4 W: K, F. b3 ^        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    , c0 A5 V# A! H2 G, ]/ I) c/ w1 k4 b$ P  l# D: i. r
    Example: Factorial Calculation
    3 ~4 b% [2 i2 Z) [& `9 A2 s2 v" }, R& X0 C& n4 m" N: I& D
    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:  i* |, }5 c0 b# A4 P; ?9 Y

    ; P* F3 K' Z9 t8 H! n& q  S7 L    Base case: 0! = 1* k3 Z. x1 F2 Y2 ]! O9 H

    4 ~% o; G' l& t7 L$ W! Q    Recursive case: n! = n * (n-1)!
    3 |& v6 W8 Y' f- Z
    7 w4 H- ]0 \, D1 C8 n9 FHere’s how it looks in code (Python):
    " D& C6 T5 t+ e& f- w+ @0 A+ Y% Npython5 r# a6 N! c% u% Z$ \5 q( ^% v

    1 F0 t3 B3 k" W5 [' f
    2 g# D4 t8 ?; `) udef factorial(n):
    ' b+ x9 }# N+ @4 T- [) A    # Base case0 L1 @9 i3 x3 w3 T# n/ |. h0 B  V
        if n == 0:
    4 @- R" b: e2 b+ G6 i2 L        return 1
    2 {0 n) o+ C- C4 X  _0 m    # Recursive case: r# ?' V' Z3 A3 q5 ?, [8 @
        else:- l% F4 T& K/ m" C* z( K
            return n * factorial(n - 1)
    7 p% H. o$ t# Z& c* S
    * z2 d3 [. M/ B- A& E7 I" R# Example usage; \: e* d' z  c. J1 A9 R4 ?
    print(factorial(5))  # Output: 120
    2 k& b. l2 \0 g0 |9 H2 X' f- h. s- `& S# y# K9 i
    How Recursion Works
    9 P) n8 c2 Q0 |: r2 @+ y" F
    . O. s) s- Z$ N! m- d0 _    The function keeps calling itself with smaller inputs until it reaches the base case.
    ) J0 c) J# ^& v& B( I! r( y' G/ {& v" Y# G! n4 ]+ w& `8 X, z# q
        Once the base case is reached, the function starts returning values back up the call stack.
    % |+ _! l) c( c& k
    1 r: z4 b2 a: o, G$ x    These returned values are combined to produce the final result.
    # Y( |1 p. I. }8 K6 T+ L8 g3 G. U
    7 C. \; p4 ^1 V: L' u: a5 {1 BFor factorial(5):+ y3 V1 z. B6 I9 E7 E

    3 r& n* t$ ^; W2 h0 d# q
    0 K5 |8 X1 i# W1 o! E4 H2 a' J2 ~factorial(5) = 5 * factorial(4), k1 l" Q5 I! k4 P$ ^' |; i6 O
    factorial(4) = 4 * factorial(3)5 [8 R2 y! |1 C$ V' f- l7 K4 z
    factorial(3) = 3 * factorial(2)
    ' q2 N1 ^/ ?; s: e6 E, Pfactorial(2) = 2 * factorial(1)
    3 V  a. I; Q9 a( afactorial(1) = 1 * factorial(0)
    ) _7 @# J; {9 C+ }/ ^/ dfactorial(0) = 1  # Base case
    * F  ?/ Z& W1 S: W; g7 y( h: i( }; \: H8 U% f# ^. `! s# Y
    Then, the results are combined:
    $ N/ N9 s% ], ^; A, C$ X3 y1 B+ L: D2 B1 t1 _# `. p: Z6 n  h
    * q# G/ ]9 N0 v7 S* E
    factorial(1) = 1 * 1 = 11 v3 o4 G2 h9 ?  g# ^
    factorial(2) = 2 * 1 = 2; B9 c2 K8 @7 ?0 @9 i" M
    factorial(3) = 3 * 2 = 6
    4 T% W( O) ~; d2 a, ~; }' xfactorial(4) = 4 * 6 = 243 J( P1 i4 j) l/ @8 j* o$ ]
    factorial(5) = 5 * 24 = 120
      }8 X1 e; J, P- e; E# s; f$ M' ?0 r: R# c# Z9 W3 g1 V; \# j
    Advantages of Recursion$ {6 r6 B  r% U0 g: v8 Q0 ]
    + J# d5 T/ _+ |5 M
        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).7 h1 G8 s! Y$ T1 [/ a% E: j

    - b  W1 w) s/ y6 H1 s    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    : e& h$ x+ ^2 _& z# ~' d2 y$ r( q# N- T, `) I
    Disadvantages of Recursion
    % q0 l7 A  f$ \! M; J* f  R! Z" t( d5 O/ `
        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: D$ s8 n" O& \5 M

    3 w* v% B6 {; k    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).. O" k$ j8 `9 ~4 i" y4 @7 A

    $ c+ k' I: |( }9 v- a) |When to Use Recursion6 K: U( t! Y' F6 F% P" Z8 \
    ; i) `8 ^! w" ^7 C, s+ q
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).. W& T1 J1 U# d% u
    % q3 g# Q! t- r& P! h
        Problems with a clear base case and recursive case.1 `& j4 `9 i/ N$ C0 t6 h& O( T6 U& W

    2 @  s+ c7 E1 I0 ]Example: Fibonacci Sequence  h) a, J3 n$ r* l' S

    % @8 N. ]5 e: M# V6 r, B) JThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    9 j8 D' S! d4 H6 n+ K" ?6 C1 T/ j
    + X# D+ {* |' K( W% m    Base case: fib(0) = 0, fib(1) = 1, _& ]% u" X/ E) [" e2 d% \

    0 }3 {$ h  D1 J) r2 m$ t    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    4 P/ P6 G  u( k9 K4 b3 @8 E0 u) e/ Z5 l5 V
    python3 o4 A0 O6 b0 }- g* q7 O) Q
    6 ]7 ?; q' q8 g# D- l, M* o

    & J) c6 K) H# W5 P1 Y  X1 Fdef fibonacci(n):
    : l; ^7 ?) S/ Y2 N5 \3 v  O    # Base cases6 @% w$ `9 L( g
        if n == 0:9 S7 W8 _: c5 a
            return 0
    % O/ O) p6 k- J0 B: l    elif n == 1:/ g: v5 r# B; K* y6 a
            return 1
    ' G5 v) T3 C5 X! g  C    # Recursive case
    3 k4 _; x, p) s! y. z$ E5 k  l% _    else:/ ]4 x/ X/ E8 ~
            return fibonacci(n - 1) + fibonacci(n - 2)1 D2 G$ E, y. ?. F: Z" b' R
    . q, w9 l: o; E
    # Example usage
    % W) |0 e& _7 `, N* Lprint(fibonacci(6))  # Output: 8
    9 K: t9 [+ p& I; `  t8 u3 P9 Q& P
    Tail Recursion
    + y* j" k3 C0 \
    : n, Z8 a1 G1 k% [0 DTail 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).
    9 q8 T% |9 S$ N& ~" {
    ! r5 I( c! y4 w7 c4 p/ W8 HIn 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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