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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| 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 & }$ x2 A$ p5 `

    ; V) u/ _- i9 q1 ]) S9 U2 W解释的不错8 K' H6 A& r6 u& i4 ?% F
    " s7 S3 {4 t& N4 @. T
    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。2 s' {9 T6 D3 E2 A, f  T( p) f! ^$ n2 P
    . h1 R# h- e( l4 Z
    关键要素
    6 |" D# A& [' \8 x* g' \1. **基线条件(Base Case)**; J5 L0 G7 p9 l( J: \4 P0 Y
       - 递归终止的条件,防止无限循环3 p2 v9 y2 A, N! J9 n
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 19 r/ h0 e1 G3 I+ Y2 m

    : j& d: \$ n% u6 g2. **递归条件(Recursive Case)**
    7 a( K* x& v" n! M5 k   - 将原问题分解为更小的子问题
    ) ]! i1 M# m/ b0 k" S   - 例如:n! = n × (n-1)!8 p/ Y9 g( [# d
    3 V, @0 P$ i7 t7 L4 V4 A# t: T
    经典示例:计算阶乘
    + O2 r' l5 K1 i# A) lpython
    7 A; a3 C- d3 R" @def factorial(n):
    0 Y% W" |7 S2 p7 w+ U& H1 o    if n == 0:        # 基线条件
    0 `: H# U* _! o8 X: F# f9 R& f        return 1
    % q' `3 C; C( m) T" ~' d7 j    else:             # 递归条件7 V7 v' k* W  ^+ g3 O3 O
            return n * factorial(n-1)' o, `& [* W" h/ M+ S1 V( t/ [5 ^
    执行过程(以计算 3! 为例):; Z9 j7 L) n: P1 x8 U
    factorial(3)
    7 _* m+ Z9 V+ H; l+ A2 M( P& |3 * factorial(2)
      ~% w" Q) X3 D, r) F3 * (2 * factorial(1))$ w, q0 G8 L/ f4 |5 I
    3 * (2 * (1 * factorial(0)))
    ) }1 a* r8 z, V4 _3 * (2 * (1 * 1)) = 6
    ! W0 Z: N  V3 K* B: N( g
    ; u8 ^: V% O$ K) H# P) O7 {. g' q 递归思维要点; m9 J! r/ z4 ^  k4 G
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑4 P( a1 c5 Z8 j. S/ p
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    * }' y7 l: @* K- J9 j; ~$ F# P3. **递推过程**:不断向下分解问题(递)9 i2 Z9 Y" F, k: L* T
    4. **回溯过程**:组合子问题结果返回(归)/ m& g1 I; I# L+ ]9 s5 X) n
    * S$ @6 I1 R8 b
    注意事项( M3 _" C: T: E1 [" f4 q
    必须要有终止条件8 x- m4 T5 `+ f! `: b7 f) s/ {
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ) [) s3 _, o) ?8 p% D某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ' ~1 m, y4 d* T尾递归优化可以提升效率(但Python不支持)
    8 o0 _' R3 m, i, L/ o  q/ Z; f' Q3 o( q6 s4 m7 R* L
    递归 vs 迭代
    5 m( N. w3 V' M# x6 E9 q|          | 递归                          | 迭代               |) |$ l4 }6 w; _. e' }9 V
    |----------|-----------------------------|------------------|% Q4 J( t6 s% X8 {4 m6 c- a
    | 实现方式    | 函数自调用                        | 循环结构            |
    , w3 b4 p: @; W9 {| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    * l! D- d6 f4 i- I) N5 I0 G| 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         |
    & w) W- Q9 F. d* f| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ! z* S# q$ m0 K" b/ _" I
    5 `$ S- t- {' T% D) K 经典递归应用场景
      H$ n) U! z2 d- `, b: R2 A3 z4 V1. 文件系统遍历(目录树结构)- }- N" P& _) R/ c7 B
    2. 快速排序/归并排序算法
    - K7 I+ }8 q+ T  z5 s- j; Q3. 汉诺塔问题
    4 r$ w: J4 j- N7 n# O4. 二叉树遍历(前序/中序/后序)
    6 x6 n8 _6 M) ~% y* c5. 生成所有可能的组合(回溯算法)! x$ ^$ g1 ^: r4 A, H+ n: t( W

    - G  \5 B' [4 F# r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,
    : p" [3 y" h3 Y# }. t: V$ F我推理机的核心算法应该是二叉树遍历的变种。
    8 B2 Q- \0 J7 B8 M1 `6 o( N0 X9 Z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) k* D: G: Y" F- D9 h  oKey Idea of Recursion$ _, N: [. r% H4 p6 i3 n& g% W9 X
    8 J+ ]$ }3 U7 U7 D" C" D2 S
    A recursive function solves a problem by:1 k) N8 M& q# _7 ~* L1 n

    . U+ t+ j  d+ ?# z! J' @. ]% x    Breaking the problem into smaller instances of the same problem.1 v  }/ P, s5 @8 O0 I  M
    & j/ @2 \2 |6 V$ f( E+ k
        Solving the smallest instance directly (base case).
    * \. {! k6 L5 b0 [" p$ m/ V
    6 z! r6 z! K; a# U% W/ T8 O    Combining the results of smaller instances to solve the larger problem.
    $ E4 X6 G1 c% N. }3 g
    ) W; Z! I+ `% L- lComponents of a Recursive Function
    + t* q  I+ b" F8 ^" e. |. i* N. Y2 O' `) g* v! Y+ B0 x
        Base Case:
    ( [! Z4 V9 q4 _3 A# N0 R) y" k5 F% j" x
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.8 D# K, ]- U# l. O! o

    5 O. I; e: M; T$ N        It acts as the stopping condition to prevent infinite recursion.
    3 `: o" u0 Q# L1 [7 f
    / @/ V3 \! N9 I2 J. A+ J3 L4 K        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.! }! D6 w8 ]) R/ g/ ]6 \
    * D& m( b" G6 m( e4 n3 r8 X% `. M  ]
        Recursive Case:( d( E, s: G9 p. N" g6 {
    + m; q6 Y2 o# O, Y
            This is where the function calls itself with a smaller or simpler version of the problem.
      W$ q& ]' J1 ]+ [* l# k' H. \. A
    % R9 |: z6 V, Q  K' G        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ' a* U3 f) A% u$ N4 n# `2 c
    / }  p: C; k9 I( a- e; `Example: Factorial Calculation" S, K" z1 M0 O7 T

    . @* I# x" O& L  S& W" `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:  ?( P. z2 R$ ]/ H/ T* {) R

    * e1 i( K: \2 P+ k. }! h    Base case: 0! = 18 V7 U7 a8 k$ G4 t) p
    ( O0 u8 l# g% Q" T1 X! q- E2 u4 l$ P
        Recursive case: n! = n * (n-1)!
    8 {1 B2 f. R: }- T2 P1 G6 N
    4 l# c( s. _* [/ Z  ^" i7 VHere’s how it looks in code (Python):% E" |7 H5 e* [  ~5 u7 }
    python+ K) p" a/ s7 d- z

    1 e! J* W( p2 W" ?# E
    ( b% R: ]! @: ?# g+ Wdef factorial(n):3 ?- G6 `" ^7 b  d
        # Base case
      R, t  F" ?9 ]# C; \( D6 _4 @( I    if n == 0:. T/ ^7 k9 m) Z# q2 J0 V7 _$ u
            return 1+ n; i* j0 v2 c6 B6 x/ k+ V2 J& W. O
        # Recursive case# T& x! q# Y: \5 u8 r
        else:
    9 o. d& c: ]" C+ |        return n * factorial(n - 1)( p, y4 L- w% ?5 J% T- Q- ?

    ( i  S5 K9 f2 n8 m5 Y# |+ ?# Example usage
    + g6 t7 R3 a+ a7 H9 x2 nprint(factorial(5))  # Output: 120, A% R$ ?4 H3 F# ~. c

    , M: [/ ?$ L6 S/ lHow Recursion Works
    ; ?4 G  R6 R" }7 w9 f, B
    ) M: E; n% a6 W: `8 N    The function keeps calling itself with smaller inputs until it reaches the base case.
    3 v( e- j, i* S
    1 m+ l$ H  ]) P    Once the base case is reached, the function starts returning values back up the call stack.. S0 p( X  s1 B* K6 w/ u5 a  n

    : w7 h6 u% X* k/ @' D0 k    These returned values are combined to produce the final result.+ X, t3 i& J% m. l

    - I7 B: n8 c4 b, N% W4 t& KFor factorial(5):3 j/ f, t1 a; W% M' }! N

    ; c/ H( x* D2 }3 R" [+ h( A$ o( N6 P- X
    factorial(5) = 5 * factorial(4)  A* x, E8 I7 ~- E
    factorial(4) = 4 * factorial(3)' ^8 [+ i) V3 B; C7 y5 ?
    factorial(3) = 3 * factorial(2)
    4 W! L& X/ L  H- r" Afactorial(2) = 2 * factorial(1)
    : C+ j4 C# D8 u3 x7 O. Q7 `3 \factorial(1) = 1 * factorial(0)( \, j' e' F1 s, _3 J4 L7 ?% l  u
    factorial(0) = 1  # Base case
    2 Z$ j- r/ |' _" L: b7 X, D* E$ y0 A1 t3 S2 Y1 z7 b
    Then, the results are combined:
    / s% R- ~, y! @) G
    ' \. T) y# H, S/ ^6 b) h& w+ W) ]+ J) c$ ?
    factorial(1) = 1 * 1 = 15 Y3 {6 G. h3 k, a, j( a! v- Q: [. y
    factorial(2) = 2 * 1 = 2
    * `. n5 p5 f& i8 t% gfactorial(3) = 3 * 2 = 6
    + c2 @. C" O' U! ]factorial(4) = 4 * 6 = 24
    * `+ P. J- ~3 E( H8 Wfactorial(5) = 5 * 24 = 120
    6 f. z" Y% E( h
    4 @' ?7 c) ?1 w# KAdvantages of Recursion9 [7 l; B  ^. K9 H: g

    . N3 _* w% z4 {) Q- v" S    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)." N9 J% f9 |: |# D  D

      t4 W  w( i) [7 p' K; b- a2 _    Readability: Recursive code can be more readable and concise compared to iterative solutions.9 h9 \$ \% @4 k  k3 _; I1 g" x" R
    4 _7 h1 i' a' o- T
    Disadvantages of Recursion) d0 H4 x& W: ^: v8 t' n1 j0 q. u+ z
    : ?$ N5 h6 Q8 @, s/ o. P
        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.
    1 ?9 @. o' f5 j
    $ X; z9 Y0 `  X    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    , Y+ i2 f: p% h# d* L2 \% w3 h3 D: B5 D
    When to Use Recursion
    - K7 K' P! C4 w1 _, {) l& j/ D: E2 Q1 K  T
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    + M) \* ^( G$ J( h0 p7 C' H2 ?
    $ E  L4 G$ d9 I" b8 d6 ]9 t( m    Problems with a clear base case and recursive case.
      s& ~* U3 L- ^  C' W1 j* @
    1 c( B; [& {, ^. A7 t3 ^7 nExample: Fibonacci Sequence; s" a9 r2 M$ U1 p5 ?& m

    + V4 a/ j* \% T: e" t7 F# UThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:, q" \! y5 t2 E$ Z( R  A7 q& C

    $ h, o1 X; j" l' r    Base case: fib(0) = 0, fib(1) = 1+ s1 c+ \7 N8 i- u2 [: v' `; E

    , |* Y/ T  T" e$ v% U    Recursive case: fib(n) = fib(n-1) + fib(n-2)% E. M/ c+ U' c  U* u9 X
    ) \2 n+ `4 X. l. U7 C7 y$ w( \
    python
    3 I7 M: \* D' g, F& Z& k
    3 W8 u* A1 w' _: J
    ! ~5 ?1 s3 q  q) j6 wdef fibonacci(n):
    : P; V: ^; V$ u3 A3 [0 V    # Base cases( ], T5 V0 Q% V8 a0 b3 ?9 f
        if n == 0:
    , F- J. c2 y" q9 ?        return 0
    . ^7 }4 U6 Y; E! w  l    elif n == 1:6 j1 i4 y6 R4 G7 i6 i2 b  A
            return 1
    9 L1 O7 c1 w0 Z  `, |    # Recursive case
    # M& r3 k$ X/ {    else:
    7 y% K4 z& L5 L3 Q        return fibonacci(n - 1) + fibonacci(n - 2)
    6 l& E6 p* C/ U6 I8 y( A$ D. c( d6 _4 ?* M5 @' K$ w6 R+ t
    # Example usage
    . J: [9 S5 U! W8 o: g1 A. eprint(fibonacci(6))  # Output: 8
      n8 H, q" j8 j# P: y  K8 m
    / @% I/ A2 K4 j/ l' @0 qTail Recursion$ a( ~1 y) [' a; I
    ' }. Q# `. o- ^3 M' x4 u& l7 }; `
    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).
    0 g1 B4 Y4 K: O. N. q& y4 h/ g, D8 X/ l# _5 _5 A' b
    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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