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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
    ( ]4 y* W/ @/ z( q9 ?' C& d7 D! T4 [2 o
    解释的不错9 a" S, V4 |7 X; [# {) n

    0 r, V8 X0 o; [/ R+ o" V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。
    3 R/ d( k/ J6 I+ m( _" Q* V' y* [1 @
    关键要素
    9 L9 t$ g3 @3 U( S. B1. **基线条件(Base Case)**
    / `; I4 K' I6 i- n, d, E- G# }+ w& o   - 递归终止的条件,防止无限循环7 e) T9 E! j+ T: g2 [7 G# v
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1& H3 Z7 ?; F* _0 t

    0 `7 ?" l* u( _8 p* W2. **递归条件(Recursive Case)**
    * i; F$ v+ H0 X# T7 T! B1 v: j   - 将原问题分解为更小的子问题
    + \- n" E3 C7 l8 N* Y: a& O   - 例如:n! = n × (n-1)!/ P4 K# |! g# ~% h1 K
    2 Z4 d( ?+ i# v  n
    经典示例:计算阶乘
    ' ^# r. m) H! b8 q0 ^9 hpython
    " k+ Q9 v5 ~4 q; c) \. jdef factorial(n):
    ) m. p+ s: @4 J    if n == 0:        # 基线条件1 o( o# H6 O$ l! V! ~- i! H' o
            return 1! Y' k8 t4 n2 G
        else:             # 递归条件
    $ \) W5 _1 s/ t, N7 c        return n * factorial(n-1)1 N& L+ T% b+ U0 V  H
    执行过程(以计算 3! 为例):
    ( [! ?" c7 P/ s; c0 h' ofactorial(3)9 U1 V: [; a6 p9 l9 ]0 M  a
    3 * factorial(2)
    & _' V) G; y! m0 U' O3 * (2 * factorial(1))
    ; M0 d0 F" Z% ~+ Z/ c3 * (2 * (1 * factorial(0)))
    4 M  k; ?% @# E5 D  A: ^3 * (2 * (1 * 1)) = 6* `1 O% ~& a: ?) v, S  ]

    # W4 O9 S$ T- j+ N- } 递归思维要点+ K) l" P  |- f
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑/ j" L' b, L1 y8 O3 _" {
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)& P4 |$ i8 M! U7 j$ n; w
    3. **递推过程**:不断向下分解问题(递)
    ( u3 N5 }! t8 a- b$ h6 d4. **回溯过程**:组合子问题结果返回(归)
    & i9 v; w( l# j' C. S+ e4 @6 [7 a; I% A
    注意事项) n  v1 M, x0 E; a
    必须要有终止条件2 D$ c. s4 b) h- F
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    # Y  _; O- ]( \/ i某些问题用递归更直观(如树遍历),但效率可能不如迭代* P9 d- x  z1 f/ D& j/ q- B  D0 o
    尾递归优化可以提升效率(但Python不支持)
    & D3 C( \, K& p3 f6 B9 m: j
    9 [, U" k! J8 z: Q% A 递归 vs 迭代4 ^5 u; h3 ^# V0 V* i1 n
    |          | 递归                          | 迭代               |& P% R* w  F2 ?
    |----------|-----------------------------|------------------|
    + i  e+ k" H/ m' Z. N2 j| 实现方式    | 函数自调用                        | 循环结构            |' f( P0 [5 V+ u" D& x
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    6 q! d2 b+ Q3 U  F8 [% J| 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         |4 z. N7 ?$ I$ Q" a$ V" m  G& A
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |- x2 P* V) P2 m1 l* r) a

    ; r, V8 s  p7 |* q 经典递归应用场景( {4 g) g& m* u; k0 J- q- \0 G
    1. 文件系统遍历(目录树结构)& H6 B/ a* F8 ?) [8 b$ ?
    2. 快速排序/归并排序算法, i* B; i0 D* |- z: p- C
    3. 汉诺塔问题
    ! ]2 ]# }3 P1 F* Y% ~& W2 y! B" [4. 二叉树遍历(前序/中序/后序)
    + j% Y4 {% e$ s3 |; l+ {# \5. 生成所有可能的组合(回溯算法)
    + q$ |6 ~7 Y, p) b  R, J2 C0 w# h# V6 c3 E( L' b: w1 w
    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,% Q. ^: [$ O  I
    我推理机的核心算法应该是二叉树遍历的变种。! Z% k& r( j. ~. W
    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:" H" V4 y& O8 h' p
    Key Idea of Recursion
    , ~+ m& d. T9 `% s' d- |; m8 d2 R. i7 M0 U- [
    A recursive function solves a problem by:
    8 K1 X; f8 t) t9 a5 K
    7 F, T1 j3 y1 d. p    Breaking the problem into smaller instances of the same problem.
    , ?+ O8 h# n/ \# ^; o' S8 X$ Q; m
    1 p7 T2 B4 g( I  D; b    Solving the smallest instance directly (base case).
    + X" U0 H7 i: ]! ]8 L# m% _3 U" j& H+ B! ^& \
        Combining the results of smaller instances to solve the larger problem.9 ~, U" v5 M  o  b4 A4 F5 _0 [" U- V/ j& ~

    + a9 H' v7 N3 [# v' SComponents of a Recursive Function9 `: y7 u6 w. B) O9 E* p

    # z) w1 c; `9 t  c& C1 l; ~    Base Case:
    * G. B0 e- f1 A8 D0 L. I2 G) Y! M5 J7 T; x6 V& ~
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    7 X; |0 b! \; x! [1 y+ E+ Y7 _2 l
            It acts as the stopping condition to prevent infinite recursion.
    0 B2 \' X# F8 U1 r0 S9 a
    8 O+ p; {& a4 E        Example: In calculating the factorial of a number, the base case is factorial(0) = 1./ p$ b) H8 N$ q( N% y
    2 F8 ?+ u7 x; b. `' C/ H0 B
        Recursive Case:
    $ I$ U. L2 p+ A3 v+ N8 y8 a- p" T
    . x; Q) s1 x- ^$ A0 ^5 N% O        This is where the function calls itself with a smaller or simpler version of the problem.4 T& X8 c& t3 h1 E1 L! q
    1 D' I* C5 ]1 Y* ]1 ]
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).3 ?% V' |5 D- Y
    " e) V- I) O8 ^9 T$ ]' Q8 \+ _8 H$ P
    Example: Factorial Calculation
    6 J8 F9 X& b$ v! J' n4 C1 i2 t% C8 W* z$ r9 I
    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 Q- ~2 c, G! I7 t1 d$ T& Q) ?# r# X/ p: i
        Base case: 0! = 1
    ) j9 M0 V. N" P4 J9 G
    , q8 d& j0 j/ |8 h& s/ m    Recursive case: n! = n * (n-1)!/ x% ^6 ]& K/ d( q3 Z0 D( a" O
    % w% a9 J! |" s( a& j" U% c
    Here’s how it looks in code (Python):7 `) }5 H, D# o, e+ p6 E- e
    python
    / s2 K# x% s) L& [# L' }; h
    . x; c& A  d- T& q3 e5 ~8 W
    # i  F! j( ~; [; U: o9 rdef factorial(n):
    3 ?9 N8 f* r& ^" Y' |" |$ J! S; W    # Base case
    ; W3 a2 D- |0 z! I' d5 \    if n == 0:
    4 u+ O: v- \7 P* q$ _9 t6 n) t        return 1. a# K* V' L5 S& J" F; F, @
        # Recursive case
    6 o) J5 N  o! Z( l3 u1 T0 \    else:9 r2 d8 }, Z  I. J+ ]5 \. w/ |0 F
            return n * factorial(n - 1)4 S+ _2 [$ I" T2 d+ \6 ?+ `
    * |$ _- |$ Z: a
    # Example usage
    . `4 \: f; S2 f2 v' eprint(factorial(5))  # Output: 120. _3 n- u% u7 K$ X3 I* r5 f  I
    7 i  R4 o7 K. o  {9 ?9 ?' h6 [! H
    How Recursion Works% Q; K; k# t* N, z1 _
    ; Z& w% H1 O+ V" y% c: m5 G6 u
        The function keeps calling itself with smaller inputs until it reaches the base case.
    ) t5 U; k$ X. G; _; A5 y& Q5 N
      W0 F+ `% G7 _7 j    Once the base case is reached, the function starts returning values back up the call stack.
    6 S+ _5 r* O- o  x
    8 Z  Q3 q( A+ `& u" n* M    These returned values are combined to produce the final result.
    % w; U1 E. \  ]1 t
    1 r' L4 X6 g) n. }! d& w/ z1 ?For factorial(5):( e3 T# M* K4 _3 c. f
    . Y4 b- P: C' H. _. V' G
    0 b. ^  A2 n3 s" T% M9 d
    factorial(5) = 5 * factorial(4)
    . F$ R0 w9 g0 W. Q2 f1 _0 J( Y! Lfactorial(4) = 4 * factorial(3)
    & X5 L5 R7 X4 B( |, y( gfactorial(3) = 3 * factorial(2)! u  [- L. G8 m. C
    factorial(2) = 2 * factorial(1)2 R9 O' Z# d# l, c; b/ c& ^. l0 q% Z
    factorial(1) = 1 * factorial(0)( ?1 X: N* q$ P9 {8 K
    factorial(0) = 1  # Base case
    " z6 }0 ^, D! r6 w' X( M  X, Y; c3 l5 N( r* p( o: ?
    Then, the results are combined:6 z, `6 M% q3 Y

    - R& ?/ P6 ]- |/ u. ]0 A( J* ~7 v8 \9 z, L) a  I/ w8 E
    factorial(1) = 1 * 1 = 1/ m* N0 O- M2 K+ B
    factorial(2) = 2 * 1 = 2
      Q' \* F' E0 [factorial(3) = 3 * 2 = 6
    . X9 f0 Q1 S  |factorial(4) = 4 * 6 = 24; M& ^. i$ p2 k) S) {; x4 E- n
    factorial(5) = 5 * 24 = 1209 M: m; w2 Z& a+ k, r, R

    . v1 J  y! P" h( |Advantages of Recursion/ s4 t' t6 ]( M) G" j( V
    ( I' C2 B5 `2 U$ D# c
        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).
    6 X! S$ I" A% l/ N9 z
    5 C: F0 j& h5 k) G1 P" z5 q    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ) g0 @! T; e7 C4 [9 f
    ; _' ]8 `3 s  t. _, oDisadvantages of Recursion
    1 W' s) w- Y) V7 G3 Q4 W
    ( k7 |% U& U: a/ j1 ~    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.# `2 `( {) l% B+ g: O
    / f, ]* w" z) N' P
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    1 E' l( d' V9 ?# Q0 p! V6 Z4 V5 ?& W3 ]! e
    When to Use Recursion
    8 m8 S7 ^0 E! c1 S) u% j# Y
    9 p% \# A1 a; X& q: Z5 ~. s* k$ g    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).- C) x2 a: F( u4 R

    # }) w0 `0 S7 c; v) q    Problems with a clear base case and recursive case.0 d) `. f! C0 S% m0 D! C

    ; c+ U6 @% Y5 ?0 `3 S: X: xExample: Fibonacci Sequence; Y$ e; q* Z2 U* P5 T0 m& w' g. Z

    / s, N. H3 |  G6 i9 xThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:" h0 T. `" }; N! D0 h3 t9 {1 j+ s

    + i+ H3 Q3 Z1 L) U. C2 e6 p9 o5 _5 S    Base case: fib(0) = 0, fib(1) = 1) G9 V1 B9 w# w

    0 K' n% _2 m9 e2 ?    Recursive case: fib(n) = fib(n-1) + fib(n-2)$ X6 A" R8 c- Z0 U$ H8 y( p0 V- V

    - `1 x* s* [: U% |+ Tpython
    3 c! F# b! U! C! w3 ^& Y+ p: W4 s* A5 M

    8 Y" H% D% X. N: a6 b1 Jdef fibonacci(n):
    5 e6 O& @0 @0 i7 d# P; j5 X    # Base cases
    ' d0 j1 i% A; \8 C& T  t2 {    if n == 0:
    % W4 y7 z* q8 \        return 0
    - z7 Z" ~- A/ d- Q1 i3 }' L3 |! e    elif n == 1:, K7 C: A  z2 Z" e
            return 1' i' ~- i$ ]$ |8 _
        # Recursive case
    . K; V! C* \' e4 \7 Y3 W5 S. x    else:
    3 |& M, K# f' I4 Y! G8 z; x% I        return fibonacci(n - 1) + fibonacci(n - 2). ?  G. T6 h& I* }
    5 W0 w% q% p- u3 X  U/ ]  F
    # Example usage
    ( H3 L+ L- c8 {2 @+ Wprint(fibonacci(6))  # Output: 8. L' z- q' V! v/ i
    % O- F" H4 [5 V1 E  S1 ]/ M- t* X# E
    Tail Recursion
    , G4 J  G% p7 \8 Y( T0 Y/ W; V# [& T# F% x! S0 w
    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).
    4 a( [6 z. W0 U. U1 p" R" }: Y$ m5 Z9 @. j
    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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