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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 # Q8 f+ H! I: Y; t# d7 g; Y- J0 k

    ' e) ?% H/ D; f% D' q解释的不错2 L. k3 w& \) N, S% D, s
      W; c2 b% R$ W; V5 J& f
    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。4 B$ M5 v# I( Y, ~) o6 ~. ?
    ' n+ g- q* z; M8 V7 I
    关键要素
    - H9 R/ e/ R! K1 D0 l$ {! q/ L1. **基线条件(Base Case)**3 O( }/ ^# \; Q8 R
       - 递归终止的条件,防止无限循环1 B! T% m+ b5 J( {& C
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
      W4 j: f- @9 H* q2 {* x& c' j3 x" ]) P7 u
    2. **递归条件(Recursive Case)**! p: ]% u1 P' v( Q
       - 将原问题分解为更小的子问题! u* h1 U( Q: |& u5 k& R
       - 例如:n! = n × (n-1)!3 Q* d: i0 B3 j* z+ r

    7 Y" a# K  U. r( K, z" R 经典示例:计算阶乘2 w* }2 b0 g' L/ }; v
    python' d+ j% ]( t$ `$ U, l) D
    def factorial(n):) X( V! W) `8 c2 o2 W( g6 g; x5 @6 N
        if n == 0:        # 基线条件
    5 ^5 U" |& Q9 b% E& @        return 1% ~9 R- g2 K( r( ?) m( e& C4 L
        else:             # 递归条件
    + ~; O- Q5 }8 f- f6 C& A        return n * factorial(n-1)4 L7 R  q- G: o) ~. ~1 e& [
    执行过程(以计算 3! 为例):
    2 S+ g- N7 j' D! e( pfactorial(3)
    * @" E# \$ C) n1 t2 |+ T  ~3 * factorial(2)
    - v" d+ c3 F) X3 * (2 * factorial(1))
    1 R4 W; \$ u- W* u5 I3 * (2 * (1 * factorial(0)))' p2 J& G7 B4 l- b$ j- B$ v# V
    3 * (2 * (1 * 1)) = 6* t* W+ m& r* m: K" c* q  A
    0 L6 |3 b4 a# ^& ]* r
    递归思维要点
    , y/ `. G7 W* V5 l  g, z" `1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    0 d: D3 _) M/ t$ P. A; T2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    : ?+ T# Z8 J, B1 `8 y- S1 Y3. **递推过程**:不断向下分解问题(递)
    ; u$ r  Z0 m& w, ^- L) I9 {4. **回溯过程**:组合子问题结果返回(归)
    2 G* e+ G- h# `- J6 F5 T
    4 k7 V  n$ C0 Q6 v/ C; P& r( n 注意事项) V* U: y7 G' @2 L( ]* x4 P0 Z
    必须要有终止条件0 k! e+ |% `- J/ z, v0 O, u5 e* M  M
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)5 f0 O+ a" B- a' p: z$ z& f
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    . W2 g$ Y2 G) B  O尾递归优化可以提升效率(但Python不支持)  |7 Z& s: ^. a
    7 s/ Y% ~+ B: {+ P7 P( t% s
    递归 vs 迭代1 u3 N2 D% i/ c9 S% a
    |          | 递归                          | 迭代               |
    1 o6 b' U  [, W$ h+ Z- ~3 V|----------|-----------------------------|------------------|
    0 K3 a6 ^( E1 N, T| 实现方式    | 函数自调用                        | 循环结构            |
    % v9 H" d% T0 h+ p8 O| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    " M) m/ d( m6 s8 c) R| 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         |
    ) ~1 F+ b$ ^, j3 q7 ^| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |( L! ~8 M8 ~. |+ E

    2 I5 w9 X3 I- d5 U, S& U: V 经典递归应用场景
    " \  Z9 t' f# H7 m1. 文件系统遍历(目录树结构)
    6 l% ]8 @5 i0 K: n2. 快速排序/归并排序算法
    ; O! C+ C8 p3 H, i! s0 S4 W3. 汉诺塔问题  E; e# w- h0 j( a; f
    4. 二叉树遍历(前序/中序/后序)
    - u( t: h$ |, |7 Q- B% ?5. 生成所有可能的组合(回溯算法)! q& E* n1 y' B, k( A+ r1 `9 f
    6 W- R& D6 O. G, s: f" h
    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 06:41
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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) s# a/ ?! z
    我推理机的核心算法应该是二叉树遍历的变种。) Y# |# X. A9 @& y
    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 }* k' z* l9 o0 v
    Key Idea of Recursion
    & A' a9 F, l4 P9 P5 \! d0 q7 h
    & y3 K# c# I1 P$ X& Y% xA recursive function solves a problem by:
    / l! q) F6 ]' y3 f
    4 y4 i! B8 R- ?1 X) _5 h3 l9 p% I    Breaking the problem into smaller instances of the same problem., f% D/ G" I+ E, P) b( i' \0 e

    / H* |, B1 d0 z# |# |  N    Solving the smallest instance directly (base case).: a, E/ ]' {* U5 G

    9 {/ T. A9 Y4 m6 H    Combining the results of smaller instances to solve the larger problem.
    ; _) M  k' E# j. y' U$ v- h( P9 z0 B( j8 W. i
    Components of a Recursive Function0 X0 v) S) d* M4 o3 ~

    : h" a+ N2 [: y    Base Case:
    7 x! X! x2 r, r5 D) ^6 o" O
    1 l4 @" e9 V* Q$ C+ D        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    & e, H1 M4 S8 ^! W
    ) T* r) u; }' n% f+ Q/ d$ p, N$ W        It acts as the stopping condition to prevent infinite recursion.; }5 u6 z/ ?3 n7 _

    , F! ~5 v4 b, p$ Y9 |& E        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    0 m/ N4 c1 D" V5 I
    ) X- `" M8 D% w, C* r1 C    Recursive Case:
    $ R& l9 }- F# D
    * x: C' |1 k$ M2 L        This is where the function calls itself with a smaller or simpler version of the problem.7 K- i3 ]4 p! p: o+ j) K

    : I# g# t, I3 v7 s$ V        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    5 `5 s, ~9 V3 A7 A
    ! V0 t3 {7 S$ @" l" eExample: Factorial Calculation
    % z- M! I- M: n& D6 @
    0 o) E9 e' Z/ \  L, @% z1 Y$ VThe 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:
    & V' d" j$ F0 r$ N! \* b0 A8 y
        Base case: 0! = 1
    1 S& v7 p; J3 R9 B
    9 y: M& M; a: {( E    Recursive case: n! = n * (n-1)!
    # k, [) ^+ ~+ J: u$ E' k5 q! F" e; r4 f8 r* I
    Here’s how it looks in code (Python):4 N# I4 d. o. T8 ^. _4 L/ \6 Y
    python
    & j8 A: U. i  l! n# @0 U2 o, S( u- ^% I$ A

    ( J8 m4 e5 [% ?! r6 _4 U- p4 odef factorial(n):
    2 {1 D/ R) E$ R! P. i& D" p' Q  i    # Base case5 g1 G" X- f7 E& o5 V- A9 H
        if n == 0:
    . ~1 v% L+ [% M, a        return 10 p( B0 l( F; B) r$ E) I# X$ w: L
        # Recursive case
    2 B' ?6 K3 |; ~- b# Z! l    else:% [7 s  ]1 \5 k3 ]6 C' o9 `  p3 F
            return n * factorial(n - 1)
    3 i9 W; T0 `; V9 |' U; i" A6 \9 S; T! b# |% g
    ' X  O. v% m( O2 U# Example usage
    # g# X2 C7 s: T. u( I) |4 zprint(factorial(5))  # Output: 120; n" J% T- `, v5 u# O' M

      M- |# X* _8 j4 hHow Recursion Works
    * e: ~" O' m$ H: C0 \% r% L
    0 a9 c' E# e  N3 g: ]    The function keeps calling itself with smaller inputs until it reaches the base case.9 I! x- O) n7 D3 J' p4 m
    " J# s$ X: {: Q, p
        Once the base case is reached, the function starts returning values back up the call stack.
    + u( N' K, w) d) I/ ]9 ?) T/ l; n
    8 B* j4 w/ n7 Q/ O8 }* P. M    These returned values are combined to produce the final result.
      {; p6 u- w/ w2 p7 J5 d9 K. l% M4 H- y# \  G, h) q6 c
    For factorial(5):
    $ @' w2 f- L# }+ S: b3 K& h$ a! ?  A4 }% [3 O% m

    + @: P$ ^  ~" G; L" y, ofactorial(5) = 5 * factorial(4)
    & y. H$ q) u& m  Q0 T/ mfactorial(4) = 4 * factorial(3)
    ) {. `* Q4 _% P& e% v" c6 jfactorial(3) = 3 * factorial(2)
    ) m2 O2 E- ^8 b/ U' a4 ^# C* ?factorial(2) = 2 * factorial(1)+ l: v9 [/ c( h* P) d5 C, G
    factorial(1) = 1 * factorial(0)# g% U, K4 i1 s3 P
    factorial(0) = 1  # Base case! K( q  k0 x, n' L7 z, f
    9 P- Y" z3 ~0 P" w
    Then, the results are combined:( P3 S5 x2 _( |3 R6 b) Z8 @( D8 J
    2 h6 `, U% d' _6 N

    - ?5 i# @/ _. n5 }/ R% N5 o* n8 pfactorial(1) = 1 * 1 = 1
    4 }9 Z$ |' L3 hfactorial(2) = 2 * 1 = 25 ~% Y* s" H. ?" H" T! }
    factorial(3) = 3 * 2 = 62 B( Z) k, H) t
    factorial(4) = 4 * 6 = 245 P: [, t( [( h
    factorial(5) = 5 * 24 = 120
    9 v) ^7 E) z6 V5 ^/ r' q0 d) `2 W; E7 U" s" @! m4 m
    Advantages of Recursion0 u  \- |. k, W# d1 k0 z8 \
    ; H  U! r3 M: f
        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).2 x- M, ]/ o2 V8 D; L. R  c" V

    7 n" L- D3 [; j: a* G    Readability: Recursive code can be more readable and concise compared to iterative solutions.3 K  l8 r; F/ m- `' ^
    , k1 t7 |1 U: r, ]" N
    Disadvantages of Recursion# ]% L5 V5 H. y7 h
    4 w! L  L3 p4 y9 |% u: e
        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.
    3 h6 X# p8 O$ \5 Y. t* a- t  N/ y( z; N! X8 u
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    " i* s2 [; \3 E; v, r6 @
    5 E" p9 |6 r  }, H9 y/ _4 VWhen to Use Recursion
    ( E! K- z5 y% F1 ?8 \  |. I8 N1 P+ M$ k1 D& T5 [4 ^' B3 f" K' }
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).# {6 ~! P( |% Z% f4 i
    ; F0 P( i4 O! x, I
        Problems with a clear base case and recursive case.: P) D, ?" g" D
    / m* X7 K+ i4 H) Z5 j& j
    Example: Fibonacci Sequence, M7 l6 J- ?# ?( l) r
    3 C3 _* ~  f; E) V; u
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    * K  `% c. j+ ]7 o$ E) L9 @
    ' K3 ^" k8 C! S- a  R; Q+ z    Base case: fib(0) = 0, fib(1) = 1
    " v! W. f$ K+ Z& Z3 H+ p. a5 t
    + c. X" z4 h- V7 d$ A4 o    Recursive case: fib(n) = fib(n-1) + fib(n-2)6 H: R6 }: L; U  m, ]
    ( v2 E& k2 T$ ~3 {' E- E" `4 D
    python
    ' d6 R/ @% |2 D) N6 s2 }: i! j" Q. b- K3 c

    9 ~+ X! A5 S  C% F8 y+ q5 cdef fibonacci(n):
    / J. H4 z. j( ]3 w- G: Q    # Base cases+ a. ?5 e& F4 l& N& a
        if n == 0:6 k8 x) @3 J  r) j  A( ^# D5 k
            return 0& w) t+ M; w) m/ G& q0 ^; s2 x! i3 k8 `
        elif n == 1:
    ; P0 p: Y0 e3 @& k; _) _        return 1( e: z( u5 z# O  T: R' v
        # Recursive case
    9 w8 o  @& }1 ^, ~2 t8 y    else:
    ) u- l  k" l9 k: \        return fibonacci(n - 1) + fibonacci(n - 2)2 m: m+ m) K( l- z3 Y5 f/ _
    8 k/ @/ Y# d4 j8 Q
    # Example usage6 @+ E0 _! z4 B/ b
    print(fibonacci(6))  # Output: 8
    $ r, E  m( S, q; i, a! _) L8 B, ^  C0 X$ J/ R* J4 B
    Tail Recursion% J7 K: B+ r% I' O' E

    3 [& C! o0 Q5 J7 {) R+ Q3 m- V- fTail 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).
    ; l+ W9 a9 [8 d0 a# b# A) r; A9 i; Z- T9 _
    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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