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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
     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   h0 G7 j* h- \0 A0 [4 g9 ~
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    解释的不错
    6 ?0 ]* S! P7 C( Y! H( |$ u9 @
    3 `3 \  }& a- [# [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。
    9 j; k4 i: }' Y# _3 _) z+ f5 g( i  X. ?% I6 {8 `& u1 G6 J& P+ m
    关键要素
    7 P$ f8 W* l/ b: }( {1. **基线条件(Base Case)**
    * u2 u& K7 k7 c. A6 f7 T* z- c   - 递归终止的条件,防止无限循环, y+ [( I5 G  ~' ]
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    % Z, J( z, L  q% A6 v  [( N9 d& g6 {9 [6 Y6 i
    2. **递归条件(Recursive Case)**
    % f+ q: g$ j9 C9 S; |   - 将原问题分解为更小的子问题9 ]0 e" W8 l$ s$ `% C
       - 例如:n! = n × (n-1)!
    1 W5 @% m( Z6 w7 M1 N' ]' G5 Y; n, v2 g$ C
    经典示例:计算阶乘) K* g( q3 p" G
    python
    + d, u9 `# z# `' h6 K/ jdef factorial(n):
    ) U; b% g0 w8 q/ b* `8 h    if n == 0:        # 基线条件+ }; N! v) }  P5 N6 M
            return 1
    : J0 A  @0 U/ G! j" H$ _1 i    else:             # 递归条件
    ) k& M: U# _# g, C3 E        return n * factorial(n-1)
    " E$ ^  ^# P5 n% t2 C执行过程(以计算 3! 为例):
    : Q+ Q; h1 Y- tfactorial(3)
    / g; k6 O, Y) i. @3 * factorial(2)# R( n, L" @7 Y& n
    3 * (2 * factorial(1))
    - r6 H# s* i+ l( w: n# `  O" S3 * (2 * (1 * factorial(0)))
    6 k2 w8 ^, G' i3 F3 * (2 * (1 * 1)) = 6
    . L  A) @$ A% B) o& ^. m. e
    ; U) S" @! d( ~2 S 递归思维要点
    - Y4 b% I9 R+ d5 ^6 U1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    , `. \4 ^, h1 [. E2. **栈结构**:每次调用都会创建新的栈帧(内存空间)6 D- @7 U' `2 O! Y3 |/ p
    3. **递推过程**:不断向下分解问题(递)
    ) S7 Y( ]+ y9 X# M7 F' @3 c' I4. **回溯过程**:组合子问题结果返回(归)( J: P+ s& s+ a! M4 ~5 u
    # M- k7 E4 d6 e; N8 Z9 N
    注意事项, h0 l8 A5 d. k7 |# }) l
    必须要有终止条件7 C# F: u: e* R* @- r9 J
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    . G) [& i) P$ X! N# @) Q某些问题用递归更直观(如树遍历),但效率可能不如迭代! G& R2 q% N  d
    尾递归优化可以提升效率(但Python不支持)% U) k: p2 g3 h, B6 z  ?

    - ?" f5 c# n/ j' C 递归 vs 迭代
    : _, E$ g' u" k0 n' J) ~! b|          | 递归                          | 迭代               |
    / k8 O7 h2 q" l) Y6 G|----------|-----------------------------|------------------|
    5 C% n" p6 d; h* Z| 实现方式    | 函数自调用                        | 循环结构            |
    6 A" ^2 g& `1 H4 e| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    & ]' ^  ]. x; J* T; c- C2 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         |
    $ m# Z% w% N' S4 e# N) n| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    + r7 c3 Z/ N: P  z8 r, @" f" w( y$ W4 }
    经典递归应用场景5 M& q6 i& `, Y2 V+ y' f* @% r
    1. 文件系统遍历(目录树结构)
    7 D! |, x5 [8 m& R4 @* }8 l2. 快速排序/归并排序算法2 ^7 ?. K7 I  Y' f( H
    3. 汉诺塔问题; i. C) q4 N6 M- w
    4. 二叉树遍历(前序/中序/后序)
    ) I  U/ a: |- r" c7 Q% K$ j5 H0 o  L4 X5. 生成所有可能的组合(回溯算法): O, v1 k0 f# Q# D* x+ V
    % m. C! h( d( F" v) t
    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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  • TA的每日心情
    郁闷
    昨天 06:50
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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,- |$ S. s  @% d
    我推理机的核心算法应该是二叉树遍历的变种。
    - L0 l! t* x7 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:' d* z# l' i  N0 F6 Z' g
    Key Idea of Recursion
    ( _9 \  D9 b8 o& N; ]; s+ \3 W, J7 [8 Z% \
    A recursive function solves a problem by:
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    # o# L9 B! |+ m1 ~( ?$ r    Breaking the problem into smaller instances of the same problem.: E  S9 ]! Y- j: |& Y
    5 H) ?  o* i4 R# |1 G+ g
        Solving the smallest instance directly (base case).
    - b( {* V/ Z0 W6 `
    ' E/ v0 w& q1 I# ~    Combining the results of smaller instances to solve the larger problem.
    # f. |3 ^' a- ~/ K* x% z  c7 n) R, e6 ?) j
    Components of a Recursive Function
    ! A$ o1 ?; ^% c6 K4 u- q$ X
    7 y( h4 C' ]: W) d6 d    Base Case:9 r# L0 o, R; T9 i2 V* z

    8 c  Q4 U6 `4 n: n* U9 l% g/ \9 r/ u        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.: A5 p& o. D" M1 v

    5 j: J/ V+ k2 ?+ j        It acts as the stopping condition to prevent infinite recursion.
    0 T& Y: i6 @9 O0 T4 B  m
    , Z1 }1 I( o2 ^% d* \+ L& L3 h. z        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.4 H4 d/ S" h1 I9 o; J$ T
    . Z, y. W% t( F
        Recursive Case:
    / t" |, \4 k0 m3 G" l% M9 D( b2 x7 Y. @8 N* p' \5 I
            This is where the function calls itself with a smaller or simpler version of the problem.
    . j. S9 c( C: Y5 s  m) L5 S0 z& c1 E. u1 K, a, ?5 l8 v! g
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).% o' A& o$ }6 c# V: o3 Q) \9 ^
    ( }1 W* |, A5 _* \* `6 L/ z
    Example: Factorial Calculation9 [. X8 n6 B8 {, D5 p4 R
    6 j8 f  P$ n, S7 u
    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:
    ; Y2 M. J5 @7 ?' q" K- N  D
    & L# }: }5 O  [$ L, \! ]    Base case: 0! = 1% S7 Z0 |6 @6 _7 [! h
    $ f1 f1 Y  h2 ]! y, a
        Recursive case: n! = n * (n-1)!
    ; K* }4 o+ X! O& A' V
    " I/ z: J& q! n2 e) n: g2 P  RHere’s how it looks in code (Python):
    ; e; n* ]2 ?4 _0 ~$ e" T- c$ qpython5 g  `4 x! b$ U) S: }: r5 `/ J

    3 B: w, b0 ?6 G: m) s+ C; K2 Q! w9 r" M" A1 y) O/ B
    def factorial(n):
    , {8 a/ }7 X! N, r0 e# m1 E; T    # Base case
    + H% l. C  j/ n/ z+ {# |' j' k6 W    if n == 0:. F+ K4 q9 u. |2 B: M7 v! K. D
            return 1
    : I; U' Y4 i0 z1 b) y    # Recursive case; Q. _8 y; o5 n6 K' Y/ q7 x" |
        else:
    3 `+ `0 b$ C1 k! x1 y, w        return n * factorial(n - 1)* Z7 `% @+ y/ f/ `7 ]
    4 e: v9 C& `; W! t& q" G
    # Example usage' a9 J# x8 L3 N9 B
    print(factorial(5))  # Output: 120
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    : W) J! z6 h: a* g+ wHow Recursion Works) M) _7 w! d, s  ?

    ! t) |% F! [7 _$ Z* t    The function keeps calling itself with smaller inputs until it reaches the base case.! r4 ?$ c; S" H$ r% w- Q
    $ h) ^" l' N$ k* N7 g
        Once the base case is reached, the function starts returning values back up the call stack., k# y5 `# v9 p

    7 `# W5 T2 B# N& i+ c( @    These returned values are combined to produce the final result.( \9 v$ U6 s9 ?3 J# m+ S: Q% j2 h

    , J4 e% _1 f8 v8 o# SFor factorial(5):
    2 V0 Q( V* H# G! @- x) R4 p4 E$ `$ @/ m: |, D7 f4 G
    5 r5 u# [3 @! ?& C3 o) h0 D
    factorial(5) = 5 * factorial(4); w( Z& @8 D, r# Y2 a! k
    factorial(4) = 4 * factorial(3)' s1 l' u' b2 f/ c/ Y# E
    factorial(3) = 3 * factorial(2)
    $ ?% `0 m+ l) efactorial(2) = 2 * factorial(1)6 y% e! C; e) P8 r% P, {
    factorial(1) = 1 * factorial(0). Z3 }. A7 Z% Y
    factorial(0) = 1  # Base case
    # b( [6 k5 Q& @, F  f& P
    - W- L' K: S6 F! s  t. e2 @Then, the results are combined:
    6 X: C6 u4 L% @' S5 ^4 C. }2 i2 d4 b1 Z, C: M

    ; E0 N7 Z  U# `7 e; @- q3 T+ {0 hfactorial(1) = 1 * 1 = 1, d# Z9 d0 r1 e
    factorial(2) = 2 * 1 = 2
    , M& N8 Z. ^0 Y$ g# T3 Hfactorial(3) = 3 * 2 = 6
    ( g, n; a  C7 J/ ?$ qfactorial(4) = 4 * 6 = 24
    . |$ M" k' c; ~# v, u+ d. Mfactorial(5) = 5 * 24 = 120( J: P. Q0 b( D* L+ S) b

    7 Y0 V' o) p) O6 t+ z1 WAdvantages of Recursion5 |8 E4 }: b+ w4 H# f8 w) ]0 P/ ^

    3 }2 M8 C/ E& {, u. k" n3 S5 H    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).
      D4 K5 @! l6 B9 f% q- Y" w
    ; ?* K) V9 U6 P% }& }' W    Readability: Recursive code can be more readable and concise compared to iterative solutions.9 t, Q! Q) b3 z% u) F; j  U

    + Y2 D' E! u9 r9 M  z$ K- J, GDisadvantages of Recursion
    5 X( v! y: g- s# w2 o  Q3 k" L8 ~4 x- N3 x- s% h
        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.( U& |( u9 o" m. _- u' I
    ' D. |( `+ g8 [, H- _
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ( B9 i% y8 C7 `$ F$ |$ ^% i: b; S4 ~% w. t6 ~+ z" s  H; m  h
    When to Use Recursion
      B0 ^2 I# w, C7 i8 Q9 }! |2 |& o
    + Q! C0 N1 _% M- n) R; J* u: w$ O    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).5 h2 ~5 J1 w2 m/ j* [

    1 }% a+ l8 x* t    Problems with a clear base case and recursive case.
    + d7 g4 X5 j! g) V, s5 t( m6 t: _
    " K% r1 d" ]( B. xExample: Fibonacci Sequence  W/ o8 u# K3 w3 p% b
    - E$ ?% e: X1 M. g% s6 L' O& ~
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:  D! ~1 m3 n+ A( {, f
    - U1 F9 K4 E! F0 \* F8 j
        Base case: fib(0) = 0, fib(1) = 1. H9 {, l: v! o; ^' J: R
    : r( X7 N7 O" H* f% ^
        Recursive case: fib(n) = fib(n-1) + fib(n-2)) f& T8 w4 f- c4 q. g7 d

    3 z8 E' Z' z2 |: Upython6 ]/ g& A4 D0 ]
    . r- `( T; Q- w/ x- R3 J3 f

    5 q( L) M9 U/ d6 |/ a+ @: Ddef fibonacci(n):
    6 q( B6 ~! s. Z5 b! p- t7 [/ Z    # Base cases
    ' E) r' S2 H+ w  H2 B; b6 F; p    if n == 0:
    " \9 e0 [4 A9 c8 }: U  X5 ~$ E        return 07 i$ g1 R2 k( h3 u  X5 d* n
        elif n == 1:' t* G( R# H% L
            return 1
    8 f1 q2 C  L$ B$ W    # Recursive case
    1 u" z- U" j2 W    else:
    7 f% q5 N& g' Y5 e  Z  j; e: H9 k        return fibonacci(n - 1) + fibonacci(n - 2)
    8 V0 r4 {' j! Q: r$ ~) X3 [8 M5 s; e, Y  }' h3 A
    # Example usage
    0 P6 D5 I% E: e( x; l8 Rprint(fibonacci(6))  # Output: 8
    5 x8 y  P0 Q3 S& E9 q- U( `! ^( @& K* q) i7 i) G
    Tail Recursion
    , T5 P; K) |. F
    5 C& F! s8 S* x6 mTail 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).
    # t  ^% k# z2 I( |5 E0 G. G- m* q  X$ }& R  G$ h+ W/ A6 J, M
    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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