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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
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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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    解释的不错
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    " L' f2 ~, U$ _1 U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 Z2 P; `4 ~, l) R# |+ l
    7 j( s0 ]" ~% O2 t* A
    关键要素
    6 n& w! q. @/ ~/ S1. **基线条件(Base Case)**
    , j* l4 o9 y" d% L& r' h, {7 ]   - 递归终止的条件,防止无限循环, x3 G2 b. T: t* F7 M% |% V6 C
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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    & X3 F2 ]9 `! L; }2. **递归条件(Recursive Case)**8 x  ?# O  U5 A# u( Y7 G
       - 将原问题分解为更小的子问题
    ' I5 P0 x: H, \# N   - 例如:n! = n × (n-1)!; w2 o5 q! m4 t- ]

    2 j5 f9 ?) O' ]  w 经典示例:计算阶乘
    # g3 \* P& v6 cpython
    ( P8 A& e0 e8 w2 w6 H4 r& c  Ddef factorial(n):/ C: p+ o7 W, A& L" g
        if n == 0:        # 基线条件" a$ g6 W. P0 {- `/ o' B) T5 a( w
            return 1* i9 q* Q0 q/ ~% c+ x
        else:             # 递归条件9 G; A" Z4 i" N% O6 x$ B+ ~6 O
            return n * factorial(n-1)7 v0 j, E: v6 b
    执行过程(以计算 3! 为例):
      A  h! u7 G3 a! h  z+ lfactorial(3)
    : q* C, \1 S5 `9 L, O3 * factorial(2)$ g# P/ l0 M- D9 [4 ?
    3 * (2 * factorial(1))
    0 G1 a( {- Q; J1 M$ l" X4 V3 * (2 * (1 * factorial(0)))
    2 M2 j  l, ?9 u& f1 O( O$ u5 f' J3 * (2 * (1 * 1)) = 6
    * @' u, Z3 U: `5 Q$ S
    + _5 O: D3 S& m7 g/ n# x) u; [2 h 递归思维要点( V5 e/ N- o( J0 U
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑% z# @6 K; E. r/ ?% N4 y
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    8 `( Z5 p, \$ x8 ?( K3. **递推过程**:不断向下分解问题(递)
    8 l. S' y1 U. ^4 |9 M  H8 I+ K3 z2 K4. **回溯过程**:组合子问题结果返回(归)+ d3 U! r9 ~8 V' i

    ' q: I- L) M! s% W- c% f2 V1 y4 Q 注意事项
    ' ^  B1 L$ W$ D必须要有终止条件
    " X" F' V) L6 |8 w( N递归深度过大可能导致栈溢出(Python默认递归深度约1000层)+ l9 L& `3 R/ e) p& M0 q
    某些问题用递归更直观(如树遍历),但效率可能不如迭代; L  S4 q* q& b0 i( I
    尾递归优化可以提升效率(但Python不支持)
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    4 S; j! N9 n/ _  |7 x 递归 vs 迭代9 e# w7 x! b% q/ {: b3 d8 N
    |          | 递归                          | 迭代               |
    ( ^! c1 Z( V+ Q|----------|-----------------------------|------------------|
    ) F  p/ ~7 b# r1 R2 \" L| 实现方式    | 函数自调用                        | 循环结构            |
    * i( A" T  x+ n; ]7 l3 M| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |9 g- z: A- \9 a+ r8 p
    | 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         |! R( v# l3 ^4 b! a% e
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    # @5 _# Z1 V+ e3 ?: ~/ N  f
    . e. W. K  P( i- P" L! ~/ Y 经典递归应用场景
    $ C) |* A, m  T5 M0 j/ B% ?1. 文件系统遍历(目录树结构)
    6 x. M, m' n" t6 {1 W' d: ]+ Q" J+ ~2. 快速排序/归并排序算法
    % V' B$ x% l5 k4 m/ k+ P3. 汉诺塔问题: x. x. t1 u2 }9 q9 n# l& N
    4. 二叉树遍历(前序/中序/后序)
    + @$ g% l6 P: U( I7 f! F: }* P5. 生成所有可能的组合(回溯算法)
    # _4 m4 K" I# t9 I0 }0 x9 W/ l. n7 Y) w2 w. D) n7 K
    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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    17 小时前
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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# x1 @  k3 f! e( D我推理机的核心算法应该是二叉树遍历的变种。1 Z  Z4 h5 X+ E2 @2 r: d! k
    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:
    % n4 D  y- v: [4 pKey Idea of Recursion
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    6 c; s2 T: C6 M' ~( u: hA recursive function solves a problem by:
    + o: A9 b, S5 ?  T# X- J1 v- T$ G1 K; f: w9 C
        Breaking the problem into smaller instances of the same problem.. x+ V- N. d& M& o
    % X) N' s* ~& z8 j, _7 Y! N
        Solving the smallest instance directly (base case).
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        Combining the results of smaller instances to solve the larger problem.0 w2 X3 V: Y5 @1 s" B/ N

    " d. B3 s- ^0 a" lComponents of a Recursive Function
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    1 n7 k; O; P1 H+ p7 _    Base Case:
    2 u$ z1 p4 {9 `2 t) [( c7 O. V$ u0 h) U# U8 L
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    0 U' U! J% @* U" V. O. C8 v* U* z& u6 P( i6 E
            It acts as the stopping condition to prevent infinite recursion.5 ?* X4 {2 U0 u& }' c8 h

    , n- |9 W( k! G+ D        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.4 e' W/ {+ [2 t: }0 ?8 c

    1 ?' O' E. ~' l    Recursive Case:5 e, y: M% ~! J2 T+ N5 y

    4 Y8 V0 H5 E% P; R        This is where the function calls itself with a smaller or simpler version of the problem.  j" I, J; G/ R' I

    . S6 V, `3 i# ?  s& h5 b8 _7 m        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).; \- \9 |4 J, X3 I! T

    - _; ?3 s; h* ~# J, t4 wExample: Factorial Calculation; e$ v/ [6 u" j/ q7 {# e! q! C

    ) {  D' T; i3 m1 [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:
    9 m: `, E# h; E' A9 D+ c; z% Y: |
    ) p. }+ V) Q) ?& X* v0 {2 J& v    Base case: 0! = 1
    1 e: T) z& w' K4 ?4 p
    1 j. I1 |  r4 a% K5 p    Recursive case: n! = n * (n-1)!$ b8 Q/ @: l& x0 e+ X) n3 G# a
    2 i; E5 q2 H" W/ `  @5 K, R
    Here’s how it looks in code (Python):+ G( ?, D2 P) Z
    python4 V' |) ]% z1 }( m" u/ E

    ; ^: ^8 k9 f/ D% Q0 H
    6 \7 v  W: `; t6 Idef factorial(n):
    ' v. |0 @0 r, R+ x7 c    # Base case
    + I( u) ]/ _& Y: P, Q  d    if n == 0:' Z' o0 h# k9 y/ u! C7 ~3 R3 `4 U
            return 1
    & N0 K: l3 o7 m    # Recursive case% S/ W& c! X3 J2 i8 W7 P
        else:- B3 B9 E2 \1 r, Q* ]; d
            return n * factorial(n - 1)1 d; ]- \+ F& |* t

    ; b( U" z+ Z0 T# Example usage' \+ o* y+ q8 [2 R2 w1 {3 v( J
    print(factorial(5))  # Output: 120* ]0 c1 h+ c( }3 A3 Z8 b5 n

    / D/ b) z: N1 z# j! k/ ]) m. V: C. ?How Recursion Works& i- e9 G/ f3 i; M# Y/ l

    * T" V/ K, m" W3 X- E    The function keeps calling itself with smaller inputs until it reaches the base case.
    8 U7 k5 \* X9 r/ v2 q6 @' G6 s+ L) d2 m) V( P. v
        Once the base case is reached, the function starts returning values back up the call stack.
    6 l6 r& G0 Z7 d+ r. s. H- J1 C
    5 N2 ], n4 @' ~  x1 }    These returned values are combined to produce the final result.
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    % V% Q- `5 z8 K( ^: Q8 AFor factorial(5):
    5 _. D4 {: U9 S  _
    % G9 ~* T: J: L. r; ~6 l1 j' {6 ~+ {5 h
    factorial(5) = 5 * factorial(4)# a0 g1 _! W& D) [1 W& w( I+ _& M
    factorial(4) = 4 * factorial(3)
    $ V) }0 d+ d8 X* A7 jfactorial(3) = 3 * factorial(2)2 @6 T" _8 x. t# [2 t* D
    factorial(2) = 2 * factorial(1)+ I9 B  g0 s. W* ~' {$ i
    factorial(1) = 1 * factorial(0)
    ( ]6 E* m! s* U! X* s, rfactorial(0) = 1  # Base case
    / W; [: B$ d5 U/ |2 ]: Y# ~4 ~" t: k! z
    Then, the results are combined:2 P; ]" f; C  `$ t( E( K

    " G7 f/ ]2 k3 ^6 O0 C
    0 ?9 P0 Y& P+ `' T4 Efactorial(1) = 1 * 1 = 1) Z- O% S" }# {4 J- i6 s5 q
    factorial(2) = 2 * 1 = 2, R# u  c. n4 E/ ~$ W. \
    factorial(3) = 3 * 2 = 6% h& V4 M$ ~/ T
    factorial(4) = 4 * 6 = 24
    ; V% @4 t# A- G+ b% z" o; r' \factorial(5) = 5 * 24 = 120# @" }( p6 \) @# y* }% v* W* B
    ; J; U4 e' j, h. U
    Advantages of Recursion9 c- D( F0 U1 l$ |- s8 Z/ c

    " q& U6 u6 e5 G# `" R1 [    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).
    0 h% A4 e7 Z! g; z! A3 W* C$ i3 _" }0 g% \
        Readability: Recursive code can be more readable and concise compared to iterative solutions.7 t* N7 [5 e6 w8 J

    ) p: N) G8 Q0 _  F; ]Disadvantages of Recursion+ W/ N. F. r6 N0 T" Q

    3 g/ R0 N6 ~4 j/ x9 C. 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.
    * `) t$ c5 l6 }* B7 R& E8 f$ f
    ( R. u8 r+ D0 y" T    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    $ L9 m; }: W" E. T$ U$ E% }6 l) N/ C! p/ e" k, v0 e5 `
    When to Use Recursion
    ( Q6 M/ l  j2 R7 `) E: w5 b% ^' ~7 r/ O
    8 }% c. i+ t+ N3 b; j, ~1 a    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).$ Y+ G9 M  j$ C2 z. l

    - ?! r. b3 q- e7 d2 {2 S9 h" y2 t    Problems with a clear base case and recursive case.9 t4 Q: q7 r/ Y2 q
    / _: s$ ~( {. o
    Example: Fibonacci Sequence" ~6 i# ?: i% R' w" @: C

    ' l% g7 {& x$ A1 l5 W( c* `5 hThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    , B: k5 g* X3 X5 M' M1 \8 Y" |4 |8 V6 V3 K; i
        Base case: fib(0) = 0, fib(1) = 1
    / k6 N$ C5 H% q
    ) G+ {9 u2 G& i6 Z1 c! Z5 d  T    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ! a, K4 y3 P3 T# u, |/ Z% B% d
      m6 I# I. e/ z4 B: k  Wpython
    ; @  m' H. D- O; |. ^
    : T% x0 m3 J# [+ Z' R; G0 B  [7 F4 Y9 z0 j& Q$ s; K
    def fibonacci(n):/ T" p3 h. }+ H1 Q% t. e8 e
        # Base cases5 \9 E! C+ O$ n. s: K' a
        if n == 0:8 n4 p( k3 }( r$ h6 h9 P& c
            return 0, {( Z  T! l2 {  m2 i2 n3 v) s' q
        elif n == 1:8 h; C8 |8 S3 c4 b
            return 1
    ( O! c. ^. k9 O0 |( e8 @    # Recursive case
    6 a$ h# s! [& A5 r% a" P    else:" T' \  v$ ?8 t. o& t
            return fibonacci(n - 1) + fibonacci(n - 2)/ n: t, ^; n/ m& V

    " E( u3 T8 m2 f# Example usage7 V. K* S$ C- t0 V0 N7 k' n
    print(fibonacci(6))  # Output: 81 c# {: A9 |# G9 T3 T

    * u; {# j# [( \9 N5 cTail Recursion
    , p% S7 T* `0 S) B; h9 Q& k( i/ H* v' n2 z/ A) e5 a4 c
    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)., a6 j# Z: z5 {" R/ g2 P

    $ ~& ?1 \8 E, O, [. E( v8 D* ]9 nIn 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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