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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
  • 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
    1 E7 i' [- n# O& u* h
    0 ?  l7 }$ A2 \8 `/ X  z. t解释的不错- U( o/ t. O0 Y8 R4 K& R( u6 c
    8 P5 H- \6 s0 [
    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。# r7 }# q$ C( ?- {% U
    . x; J# p! q& m, X- t
    关键要素9 G8 }6 s" q. t% N* c3 T
    1. **基线条件(Base Case)**/ t! Z3 Y7 |) ~1 t3 G
       - 递归终止的条件,防止无限循环
    5 `% V$ P) f/ Q% E8 s+ W   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    5 f# m; \7 G- O- `2 n
    ) R- P' v1 o8 v( B2. **递归条件(Recursive Case)**
    1 \0 x% B. f! h% Y, t% G; x1 f) \! A   - 将原问题分解为更小的子问题
    0 c* j$ _+ S; |1 }   - 例如:n! = n × (n-1)!: ^" y3 T. i0 R& H8 n2 A
    9 R  l' \6 r. e' S# v" x0 A% W/ t
    经典示例:计算阶乘6 F% x( w8 Z/ ^3 Z2 s0 _- D9 M; O5 Q1 g% i
    python2 ^# y5 V0 a* M+ b9 j
    def factorial(n):- p0 s/ P: T0 D. W, @% ]
        if n == 0:        # 基线条件
    % W/ E5 x5 M0 b" ^, W- T        return 1
    ) ~# f4 j2 J( ]6 V2 j9 j    else:             # 递归条件  `/ l. v3 [3 N3 C$ O4 E# v
            return n * factorial(n-1)4 \) e; S8 o" A7 L$ {
    执行过程(以计算 3! 为例):
    3 @; ^7 _- i/ O3 a# pfactorial(3)7 Q6 _7 W. ~/ ^3 n
    3 * factorial(2)+ h: Z& o0 k& c# P/ ]; q; L/ y2 o
    3 * (2 * factorial(1))
    0 l& j) |. v3 w+ r7 ~4 T3 * (2 * (1 * factorial(0)))
    . `, I( D) M  {! u3 d. ^3 * (2 * (1 * 1)) = 6
    7 r  {! _2 r$ \( _9 F$ g1 B% K
    1 C7 ]- K) x0 O, M' v 递归思维要点+ A" M  T6 ?3 O: |; R/ F# l9 V
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑; O5 j- b  ]" U4 v) |3 g) Y
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)$ u! P( f. w/ o0 D; h6 w' F5 k. e
    3. **递推过程**:不断向下分解问题(递)4 S/ X9 E2 {" I4 q& m
    4. **回溯过程**:组合子问题结果返回(归)# B2 l" G* P3 P- n& W/ r% ?
    ; T" l. t5 t& N: `
    注意事项
    , |% u0 J4 P' {& o! {必须要有终止条件
    % d) f! \9 u5 s% I3 S递归深度过大可能导致栈溢出(Python默认递归深度约1000层)5 {0 c% [, c1 U' F- b: q
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    - Q9 |, [( b# {1 p/ ]8 y0 s; z尾递归优化可以提升效率(但Python不支持); K' J, j6 ], R; e; e/ ^

    ( g+ j. Q  ^* n) [ 递归 vs 迭代
    4 I5 U* O. m; h|          | 递归                          | 迭代               |
    $ x! o5 e4 ^3 m5 P( ?|----------|-----------------------------|------------------|% p4 A" v) @) E3 D8 K  y& a
    | 实现方式    | 函数自调用                        | 循环结构            |: C( `; @+ \' i2 a. e& k5 R' N
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |8 Y% B4 L2 Q" k& K  ?- V5 K' {
    | 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         |" b7 @! N6 c% ^8 n9 K3 [1 H
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |/ o  Q/ T9 O! e0 W% q+ M) l  ?0 x

    " I7 v0 y+ ^7 j6 w 经典递归应用场景
    8 C% e; n8 Y' k4 T! [1. 文件系统遍历(目录树结构)
    . [$ Y: e6 ^, M- d  e2. 快速排序/归并排序算法$ c* u8 M8 }( g/ e
    3. 汉诺塔问题
    ' X. V4 I* |/ D9 L0 {4. 二叉树遍历(前序/中序/后序)
    , i" e3 Q$ j1 c: d; e* o5. 生成所有可能的组合(回溯算法)8 w$ E) D5 i8 h

    , y6 h) Z3 N$ h9 X: ]1 B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,
    ) h( a# f+ M: p我推理机的核心算法应该是二叉树遍历的变种。
    # P# u9 z* E" P0 E4 J8 i1 E# c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:5 R: P. y1 A: R' H
    Key Idea of Recursion
      z3 {. t- l3 \* E8 o9 v$ N0 l6 a7 |% q% t  u: i6 b' H: K
    A recursive function solves a problem by:
    1 C# x  ^1 P# b# G* T8 R
    8 |5 U) D1 D3 a) g- t" @    Breaking the problem into smaller instances of the same problem.; |0 K/ A6 i3 K( ~1 d
    9 A/ S' t) a- U2 f- [5 g: l
        Solving the smallest instance directly (base case).
    7 ]5 S+ S: ^) X6 K! ~- u7 [) Q% A/ Z' h
        Combining the results of smaller instances to solve the larger problem.
    6 A# X( w2 x+ Y+ _1 n
    " I0 Y6 n; C) E! C" xComponents of a Recursive Function& M$ h  g! U% b! b0 }6 W& w/ q6 |

    1 _  t2 h: p5 P+ L" y; f' U    Base Case:
    7 `; F( ?7 F) P1 A0 U) j2 p' g3 X. Z; K* |
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    : @( z4 W2 J- ^8 O2 w/ x
    9 k. |5 ]) W0 B1 ~# Q        It acts as the stopping condition to prevent infinite recursion.
    , ^. _' s. p) g8 a
    5 D1 G  [" L" w, S( @( D( X, m7 X        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.: ~) @  `% s5 `2 S+ ]

    7 G# t9 N$ e: Q, Y# W, s    Recursive Case:
    / }' D: r. |7 }, b+ r+ I( q5 V8 |
    ; c7 d2 I1 a- s        This is where the function calls itself with a smaller or simpler version of the problem.
    # ^6 p1 K" K; }8 Z8 Y9 ^5 r% E5 d2 w2 }9 o
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    . H( `3 d  \. a" n# ^2 L# h- y2 }( ?% ]7 T& p
    Example: Factorial Calculation6 k7 [7 U' P2 U  z
    $ u; \. ~% U7 j& `: R- G
    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:7 g8 [8 B2 y7 D1 k! S, t3 _7 U

    - i, F$ S( X. G' s$ Y1 d5 B; j    Base case: 0! = 1
    $ _6 i( l1 ~! r. N2 q; b6 F* L- d2 [# G8 X* K+ j  n
        Recursive case: n! = n * (n-1)!
    5 C+ {. p, M/ a8 w2 m# n6 C4 E7 Z
    , L& I. a, a& }' n' J  n, c* VHere’s how it looks in code (Python):
    0 `: G- v6 `, G5 l9 Q8 K8 xpython
    : K9 U/ C% e! G& x; l/ d9 ~" }
    - v& P1 A! _3 `4 i
    ; A8 `7 ^& S7 n, v3 g/ [# y- D2 pdef factorial(n):7 l1 \4 ]! V/ t* P( D
        # Base case
    8 Q, f& H4 Q( z1 N5 A# |6 I    if n == 0:
    # z0 o0 S' X9 w  G2 W" h        return 1' }9 Q7 F/ X; b. A; X
        # Recursive case
    " e2 V; [1 L" _7 Y3 G    else:) J( M* f+ \7 }0 g! Y5 m* C2 j
            return n * factorial(n - 1)" Z, z( v9 l) R% P7 ^

    1 K/ h! S$ y  r2 t$ |  d- U0 f# Example usage
    1 h% ]% z3 x% uprint(factorial(5))  # Output: 120
    5 T4 {& E8 n, c( ]' a! R) N/ E5 F! R+ P
    How Recursion Works: E& C, D* C" p8 `6 H) R

    1 e! ^" F5 g( ^! n7 _# d    The function keeps calling itself with smaller inputs until it reaches the base case.
    ' N7 ?# b$ A! I8 {" O+ r  f5 @; L1 R4 a4 I
        Once the base case is reached, the function starts returning values back up the call stack.
    ) q1 e# k7 f2 `- W- V; C; I! P" g& ~! o* T6 f
        These returned values are combined to produce the final result.
    ! P/ S5 W8 }; D) @* n4 g, Q2 k# b- @3 O' K7 i
    For factorial(5):$ C, B* I8 e1 p9 H$ C8 y/ E9 [& Z: W
    4 a4 D7 |, d4 _* U; M

    : V! Q8 i; a, pfactorial(5) = 5 * factorial(4)
    # o0 o1 N/ O2 A7 x9 b' ~  ~factorial(4) = 4 * factorial(3)
    ) Y' T5 M9 o; j6 ffactorial(3) = 3 * factorial(2), T* X$ ?, x. g" l( O; |! X, i
    factorial(2) = 2 * factorial(1)* {; b3 M. l$ U- X2 d
    factorial(1) = 1 * factorial(0)3 B* W3 s  `) y  ~
    factorial(0) = 1  # Base case
    " @+ d* p# k  N1 p! c& L3 f. j; b) ^
    Then, the results are combined:
    ( o: k1 \3 W1 |& n* l) ]
    , v" `- ]7 J6 \+ R+ V2 J/ v" q* w6 x+ u2 b% K
    factorial(1) = 1 * 1 = 1
    # b! p& G1 D) p# k, m2 }# cfactorial(2) = 2 * 1 = 2/ _. j% E! i/ q5 l
    factorial(3) = 3 * 2 = 6# H. B1 D6 |( e3 Z# ?8 W
    factorial(4) = 4 * 6 = 24
    , E! P# U' e, @factorial(5) = 5 * 24 = 120! \/ o& F; K: g5 @# b
      z# S% [% `, r
    Advantages of Recursion
    ! i1 X8 h5 M& z- [$ O
    8 s. i# I  {% x. R$ a5 e    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).% m/ d3 R$ ?' c+ J; ]- |
    9 g9 P7 u! g! }1 V' \
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    8 q$ V; U$ j( ]; C# ~, K# l) w$ C" l. m+ @$ e
    Disadvantages of Recursion
    $ W* h, r7 c4 {$ u7 W* z8 B4 R3 A: l; [6 \9 X1 ?
        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.
    7 b% L* n! ]( }1 L! F% {( D! V/ F" R/ k3 p
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).7 _! o' n' _$ \1 C

    8 F% Z* g: r5 C/ x' F3 d7 }When to Use Recursion, B% \, Y: h. x5 o0 d, d

    , x# O% y; j5 k4 N/ {6 u) Y6 t    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    + d6 t( e6 K. N) |2 A/ J; W. N5 |0 P+ D6 A: f
        Problems with a clear base case and recursive case.  t9 x. v9 @% o$ r* j4 r( J* g

    ; _# Y5 T- o0 c5 t; k2 w8 \Example: Fibonacci Sequence
    " ?, X) Y: Z+ r1 q2 {
    . a' t+ R0 F1 k! T, AThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    2 c+ @7 @. e/ Q" e  J; |- T0 n1 m- d' C6 a0 {! o
        Base case: fib(0) = 0, fib(1) = 16 V" _; B' U. }% b+ v3 c

      c! T/ `4 U$ y, q    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    # I4 I0 g  `( J( r% S0 H" y( Z6 ^7 `) j4 @; M6 w7 J5 c' S; @9 b% R
    python
    * L. `2 P4 @" V! j; D  u% L
    * k8 E# X4 `/ d) v8 O, T, C; t; s. c/ l# O' Q5 O
    def fibonacci(n):
    9 S# S9 _: t0 }& e! G! x# u    # Base cases
    2 M1 k& a+ b" T: \! u7 m    if n == 0:
    ) I: y7 C1 [1 L" _        return 0
    / Z  _& @* b8 o6 S* T2 T6 s& |    elif n == 1:) J/ f+ _' {8 N" ?
            return 1
      ^% b- L. J1 R    # Recursive case
    $ I. V1 f0 q1 x5 z5 j8 [6 F    else:
    : y. W) M8 i& A  ^- v  x        return fibonacci(n - 1) + fibonacci(n - 2)6 n, Y+ L( y# l9 ?' x: k
    6 V" K  b. g9 x( h
    # Example usage3 ?' c9 m& K0 Q3 _) D
    print(fibonacci(6))  # Output: 8
    8 R. v- s+ c7 Q- l0 _! Z% I- O$ U4 _
    Tail Recursion$ u9 A" [& }" e: e/ k
    + Z& i, \+ H$ M, y8 k1 S
    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).- O4 c, a% q( p' ~
    8 L- y# r' I% m- Y6 q9 _
    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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