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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
  • 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
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    解释的不错
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    # P% l. X8 J& c& 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。" w+ @# r& Y; d% w

    0 w7 b) h: O$ n9 ]3 [ 关键要素
    ' n. L( I4 }5 i9 B1. **基线条件(Base Case)**8 R4 ~: q# _" H3 e  b
       - 递归终止的条件,防止无限循环% x3 ^4 M3 o7 ~0 u$ W
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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    2. **递归条件(Recursive Case)**8 p6 z1 b6 A4 w' f
       - 将原问题分解为更小的子问题) c. q' r* Y' H7 f+ y& o
       - 例如:n! = n × (n-1)!
    ) a* x% G4 C6 z" d/ E
    5 {7 m4 w# r" _' h7 d% H 经典示例:计算阶乘* M) V% G& G, }4 c3 c3 Y% w
    python
    , J3 B& p* ~) p( zdef factorial(n):0 w( c, {! r4 z* D5 ^
        if n == 0:        # 基线条件
    + k; W) w9 }6 ?/ `" }7 o& O        return 1
    2 W$ C* i$ F* v    else:             # 递归条件
    0 N& P* K/ n! u7 I0 {1 G        return n * factorial(n-1)# k, o8 R' y4 z2 D' U( ?; D
    执行过程(以计算 3! 为例):8 N: }& y9 r% H; n
    factorial(3)/ K: Q5 r9 F) N, E2 V9 ^
    3 * factorial(2)! z3 b& `% B8 w7 b
    3 * (2 * factorial(1))
      m7 L# o) d8 }/ k8 E: Q0 e9 Q3 * (2 * (1 * factorial(0)))
    , H% @7 e: n& l$ R. L4 w3 * (2 * (1 * 1)) = 6
    ' S* A- |* b- B( ?# F2 G+ b  t( l! X5 B: u; c3 K5 h
    递归思维要点5 N( L# u1 K. |# ~' ~
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    7 ^& o% ~5 P/ K$ n2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ' v5 C0 ~( d  t/ E4 w8 S3. **递推过程**:不断向下分解问题(递)$ ~* P( |( e% A! E3 G3 x' G! u
    4. **回溯过程**:组合子问题结果返回(归)
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    6 A( O% p* a. u3 X7 a 注意事项
    " [9 S3 a% g3 ^" C0 S必须要有终止条件% l/ k; x% q$ P2 Q+ o$ |
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)$ q1 }( p* _- i
    某些问题用递归更直观(如树遍历),但效率可能不如迭代  v+ w+ s0 |+ D' Y+ T. l# _
    尾递归优化可以提升效率(但Python不支持)
    4 x6 K- }0 Y6 B8 H! u# S; o/ h5 f0 @& R/ H5 Z! v8 N7 u
    递归 vs 迭代2 V( ^: b, h/ |3 u' ^
    |          | 递归                          | 迭代               |  x4 K: T3 E. {  R
    |----------|-----------------------------|------------------|
    5 S  y$ l  F4 @0 H| 实现方式    | 函数自调用                        | 循环结构            |5 [' O% y" {! ]0 a  p4 m4 N
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |0 M& F/ a8 l8 i3 g6 r/ _# u
    | 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         |+ N3 ^$ W3 D- Q* I: Y% A% i, w
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    # s% ]" u( ~1 U6 S3 W+ d- B
    + ?) ]8 P6 H+ E) L5 I/ _ 经典递归应用场景
    ! v& f7 G& j3 P  Q; A! a1. 文件系统遍历(目录树结构). w* o" s# h( w8 Q+ t" B3 M
    2. 快速排序/归并排序算法
    1 D" E1 u2 f: h. z* {3. 汉诺塔问题7 q$ T: ]+ s: k& n
    4. 二叉树遍历(前序/中序/后序)
    8 x' d  A" ^4 |- e  C5. 生成所有可能的组合(回溯算法)
    ; u6 T& I- z7 [% o! g+ o& f8 t, }! t! g: X
    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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    郁闷
    昨天 12:19
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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,
    : p2 _6 a5 M% }9 T4 o' ~5 R我推理机的核心算法应该是二叉树遍历的变种。
    . u7 T* ]* T, ?& d& J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:
    ( D7 H; }0 O* d' Q4 s" i' b" [Key Idea of Recursion
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    9 b, a, T6 T) \* }; bA recursive function solves a problem by:3 p4 t  Z! \% e, W+ f1 @

    6 R, G+ L5 \% g( n# p- m+ V8 c9 q    Breaking the problem into smaller instances of the same problem.; k. f+ y2 W0 l7 m3 ]
    + ~4 z- X5 k9 ~* w/ S1 R! x
        Solving the smallest instance directly (base case).) S4 i; `, s* N. o( D2 f3 y
    1 _: y( n: r* w) R/ T
        Combining the results of smaller instances to solve the larger problem.
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    Components of a Recursive Function1 |4 K2 ~. W! Q  e! y; X$ W

    # |3 U+ e9 c- M0 v    Base Case:. D: n' [" L$ i
    1 j. \* ^/ x) X; u$ w* ]. B( I
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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            It acts as the stopping condition to prevent infinite recursion.# ?2 m. E/ [. j0 a
    ( q6 ]7 U1 Z+ a, {- u
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    7 I) ]! R+ q: t8 j    Recursive Case:0 w. `  ]  t& T; o
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            This is where the function calls itself with a smaller or simpler version of the problem.1 _0 s" A3 z: M7 e

    + n9 F2 G) X) Z7 x# X! [8 d        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).* N7 t3 [9 J# p, w' P3 e3 b
    9 U. Y3 D3 _) P" N5 W
    Example: Factorial Calculation, h* w4 X! Z6 L% `

    / M/ Z' M8 M/ C& @' h+ e9 [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:0 P) [# J0 H6 M  t

    ( n+ d2 c* z$ p7 R; I    Base case: 0! = 1# b% Q  d3 [+ |3 W1 i% t* y

    2 X, ~" i! J8 R( ~    Recursive case: n! = n * (n-1)!4 h9 O" P" N/ I" O3 Z! b  n+ c8 ^
    3 M, n) Y/ ~/ D9 n9 Z) a: R
    Here’s how it looks in code (Python):
    ' |) L/ {* x7 Y$ u6 i- kpython
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    def factorial(n):. F+ A$ x& _- p1 k
        # Base case+ ^2 C% B* X. N: Y. J& _  o
        if n == 0:
    , S1 w$ F( r* T5 z4 \, k0 q& k        return 17 L. f: Q+ }1 N7 x; W
        # Recursive case- n% F  j" H9 B7 m+ o( q' r& w' W
        else:1 G7 l' g6 o1 K7 w. X8 X, Q
            return n * factorial(n - 1)
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    # Example usage. P8 O. v& M+ _; F* S
    print(factorial(5))  # Output: 120
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    How Recursion Works! |" n4 {4 o0 X

    7 ~( R" e. x' f8 h: h    The function keeps calling itself with smaller inputs until it reaches the base case." q9 y8 K3 @4 e- I5 q$ u1 {

    / n) ~) i# J% P7 c0 y2 p9 N- l    Once the base case is reached, the function starts returning values back up the call stack.
    ! B+ C. ?0 T* n0 ]  X& {' g6 _# x! K; w& Q: w- j4 k
        These returned values are combined to produce the final result.# Q$ l$ b5 J+ J$ J: }# a' b) [9 }6 z' {

    # Y$ r# m+ |2 e' Z# }9 S! c% b7 IFor factorial(5):& G& k! m8 R5 \

    . M% B( ?7 }2 z9 E6 O, J4 Y( i$ y4 \3 \
    factorial(5) = 5 * factorial(4)
    5 c2 I% w- z6 t; V6 D  o4 _4 Dfactorial(4) = 4 * factorial(3)( I# q$ F$ J4 S% O
    factorial(3) = 3 * factorial(2)6 ~9 j  V! _0 L+ C5 B
    factorial(2) = 2 * factorial(1)0 p3 M4 Q3 a: B/ `2 i
    factorial(1) = 1 * factorial(0)$ ?, V# e" {2 M6 w$ a
    factorial(0) = 1  # Base case% ], j* O, l' z- s8 R/ n3 P9 x
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    Then, the results are combined:" K* ^' T6 g6 Z1 M' l
    / a- ?; c' {8 A7 {

    6 @* c' E9 C7 Z! t! i! Sfactorial(1) = 1 * 1 = 1! E6 D. `! y- E8 a' @5 F) T" |
    factorial(2) = 2 * 1 = 2
    ! u& W! J6 s9 H( B# R! T/ l! Q+ gfactorial(3) = 3 * 2 = 6
    : z- p7 S/ M6 x" |factorial(4) = 4 * 6 = 24
    3 [: K' u1 v7 q$ |# @; A7 V3 cfactorial(5) = 5 * 24 = 120
    * v0 ~- b' B$ U6 [* u# K
    # L' z! R  W! T. P( AAdvantages of Recursion
    3 W5 l$ A9 Q2 Q' D" @+ Z4 M( z- N7 B. |7 o3 L! }
        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).
    & u$ n8 x# }8 H. \' D* l6 L% v1 M; g( _' p- u
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
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    0 `! o9 k; D. }" ]Disadvantages of Recursion
    1 l: `- P$ i' Y2 p. V. f2 ^3 l3 V9 `
    % q1 ]! g! f* X/ I    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.
    ) Z! T- Q# ?4 D% e- b; }" H* U
    9 }* X& E( t6 F1 z) S" y$ M    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).# M7 Y8 Z$ Y7 p) }0 Y1 m
    * v1 y0 ^4 b2 J1 L! K9 X
    When to Use Recursion+ P+ o4 h0 j7 A8 y
    7 B! r& s+ a7 @- d# n& ]' Y4 H
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    - P* m/ D' J5 o2 a
    # ^0 A: `- i6 \/ _$ V    Problems with a clear base case and recursive case.
    7 \7 Q/ p; U2 {) s: e8 O) ~! v0 l# `
    Example: Fibonacci Sequence
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    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:" ~/ i3 y: _! O
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        Base case: fib(0) = 0, fib(1) = 1
    ! v8 T7 Q& S- P7 b5 N: C/ ^$ y3 Z3 d2 ?4 K4 w! ?4 G1 X
        Recursive case: fib(n) = fib(n-1) + fib(n-2)) W1 @6 ]! u8 j5 X3 g" S
    + I3 x$ \: G* W( w
    python
    % h, o2 U! e! v+ l- c+ e2 }6 K6 P5 l; i% q' b- I- X3 E1 T7 ?

    ) o. V& {+ i9 @9 v& Adef fibonacci(n):- D+ k6 I' p' u) @2 n) @
        # Base cases
    5 [+ B, }4 c6 D9 H  f    if n == 0:4 h2 R( p/ [- S
            return 0
    / F' q) v7 b& B; z! f. `    elif n == 1:
    8 X# \0 h5 j2 [# l        return 1
    ) t0 N+ K: t" b& [    # Recursive case2 L! y& o& O+ Y! \+ l
        else:3 d5 ]  w" ]0 F
            return fibonacci(n - 1) + fibonacci(n - 2)8 u( I. B& N1 r( S) k  ^
    ( j4 b$ o# E, F) R+ E# s& M
    # Example usage
    ; j$ X1 j& D* V2 fprint(fibonacci(6))  # Output: 8) d2 D9 f7 O2 i1 F  ?5 Y

    3 C1 \. p4 N& A" UTail Recursion
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    , c/ _0 G* w8 W! Q& H9 X# JTail 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 ^; j7 Q- Z, n/ z& x$ e% N7 e& _
    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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