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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
    . @: I# \. ]+ Q" ~5 j1 A% B  @+ x3 `- ?5 v8 ]6 I; \5 p0 P
    解释的不错1 o2 u+ m4 Y, ^4 \; K, @

    9 h9 o1 R/ c' o+ r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。& p7 s1 q1 ]  g* c7 u
    7 A  R( T* t. ], Q
    关键要素
    . O  V6 O4 X0 i& K# P" F: ?1. **基线条件(Base Case)**
    : J, ?6 b- N7 `+ l/ c6 h3 v   - 递归终止的条件,防止无限循环/ Z1 H) }; u5 N- A
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    8 n) y' C, g' Q% V( T! F+ l4 s7 ~; K' ?4 }' D
    2. **递归条件(Recursive Case)**9 m8 r! x! U7 C
       - 将原问题分解为更小的子问题8 c- y0 m# z6 `; [! \
       - 例如:n! = n × (n-1)!
    ( c- x5 o# I( K; R8 _- O" F8 \
    8 I4 `  B' \0 b6 c5 c 经典示例:计算阶乘
    # L# z' X& S$ Mpython
    5 \2 q0 Q2 {, m; Q  S7 K  Ydef factorial(n):/ W* F8 F( P# X9 T6 x
        if n == 0:        # 基线条件
    # N" g! N5 t2 K. k8 k        return 17 j3 `# i5 }5 \" S1 a( J
        else:             # 递归条件
    # W" }, q; Q( [4 y6 f$ U        return n * factorial(n-1), F+ e9 H! ^) f* U
    执行过程(以计算 3! 为例):9 v" ?+ J9 n9 j, v# o2 a
    factorial(3)
    7 P& [' s0 e/ C  v; c: r3 * factorial(2)
    % H& s! v$ N7 F# C$ _+ x3 * (2 * factorial(1))9 R: ^0 L) `' E8 |" x) J" p
    3 * (2 * (1 * factorial(0)))
      W( b" o, g& C3 * (2 * (1 * 1)) = 6% f( ~$ K2 Z4 T" r' }

    9 @/ \, ^4 P; y. p3 s 递归思维要点" ]9 {: Y5 |( J
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    + {( l7 E0 k7 ^3 B0 X$ z8 ~2. **栈结构**:每次调用都会创建新的栈帧(内存空间)0 c/ {$ f  j/ v, @3 x
    3. **递推过程**:不断向下分解问题(递)9 n4 ^/ t0 m% d! G$ s
    4. **回溯过程**:组合子问题结果返回(归)5 W2 b+ x& b# g$ k

    8 \0 C: Z# }8 ^/ y0 r, c 注意事项
    ( B( g, B: g, {* ~* F$ t! l6 B! O7 N" T# {必须要有终止条件
    3 |; }- P+ v  c" T: m2 v递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    " j5 R5 T4 Z+ V5 h* e8 {  ?某些问题用递归更直观(如树遍历),但效率可能不如迭代7 h/ m$ P+ q* l5 K6 b
    尾递归优化可以提升效率(但Python不支持)
    . t5 H& Y: I+ ~+ c8 V: p& r
    & ^; v0 ?  a/ k: @3 M2 i 递归 vs 迭代% _- f" r# y8 Z1 I' p
    |          | 递归                          | 迭代               |
    " D) B+ i& c6 s/ _& ^6 y|----------|-----------------------------|------------------|
    2 X) U3 P8 e- @2 C% e: h0 L( z& Q| 实现方式    | 函数自调用                        | 循环结构            |  R5 n8 x& K1 V' |: Y, B
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |4 u. K& Y1 a; e) ]( [( F1 e
    | 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         |
    ; G! B2 B' J) n  p! h| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |! ?% ^9 N: ]% S8 k* m

    ( m7 p; l9 g3 k$ L1 t 经典递归应用场景( @) E2 o+ M* T4 @1 D5 Z
    1. 文件系统遍历(目录树结构)9 Q/ Q- G& B( n! [& X
    2. 快速排序/归并排序算法8 b8 e) y8 H+ r. o3 E2 l
    3. 汉诺塔问题
    1 e/ c- |$ c1 A' [5 R2 |: s4. 二叉树遍历(前序/中序/后序)" }! m! p$ {/ I. K; @
    5. 生成所有可能的组合(回溯算法); ]3 {4 j; L" e
    : @6 y' h& E- u& Y4 o" U7 ]+ l" J" M# v
    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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    17 小时前
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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,/ O3 I2 i# a/ F
    我推理机的核心算法应该是二叉树遍历的变种。
    " \% C) K) f! ?% ]$ i3 M7 |$ ~  P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:+ J! d/ [" D* z, m
    Key Idea of Recursion2 D% g7 |0 E, R$ V

    ( x# ^* i+ Q6 h+ G, ?0 bA recursive function solves a problem by:
    + P, O5 U* j! H% v* I) O6 N% T/ J( b+ b3 a
        Breaking the problem into smaller instances of the same problem.
    0 R+ b, n7 C% I& J& C! }
      n" v8 R8 k  y    Solving the smallest instance directly (base case).
    ( f6 J, j2 E* e7 B6 v" t& e+ e3 [& u1 P" `( D. J/ M
        Combining the results of smaller instances to solve the larger problem.5 k6 j9 o3 A, Z+ z4 W3 z
    4 \" S  N3 ^2 H# I) f; N
    Components of a Recursive Function" B# v( K! Y+ C
    , f! {$ `( K5 o: h
        Base Case:
    2 |6 a: O, R: `( F
    3 T  E0 O' {" H1 P  V        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    8 L, v8 ]9 L7 d; T/ I! B3 `# M, g
    # G4 l$ f0 j, K6 H, |/ F        It acts as the stopping condition to prevent infinite recursion.
    1 H& \" u1 E- u% U6 d8 J6 J5 u$ R: |4 ^( v' u, P5 V
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    : r' {* j$ [. w2 c2 x* @. ^$ t! T9 c3 a8 r2 W1 N
        Recursive Case:/ l# I4 G7 c: h) @/ D
    % H+ `2 I. E% u! c& V! J/ U1 o
            This is where the function calls itself with a smaller or simpler version of the problem.
    ) R, B8 |( u! k" d
    2 C# J- F, C0 d! Z. d3 V        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ! ?, c( W# e* M( n8 q: g  }) U' |8 a$ q6 y# F
    Example: Factorial Calculation
    ! u  E& V  p$ {
    3 _+ r1 \" D, n' B7 x. p$ CThe 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:
    & |/ S# i* S; Q) W; k
    , _& C% b! e9 z4 x( N1 r    Base case: 0! = 1' d, F* d+ [+ t# X, a
    " M* e! R4 S5 |8 W) x  ~& O3 {. T
        Recursive case: n! = n * (n-1)!
    9 I0 q" g$ y7 ^1 E" j! \$ r+ A) V$ m
    Here’s how it looks in code (Python):
    ! Z1 G" M6 X. ?" q: v- ipython
    3 T2 ?0 B* H  M3 X- W0 Z: V( E
    , u8 q% J/ s& W* Y5 @$ D( g# [  ~7 W
    def factorial(n):# {4 {- r! T; y9 h7 {
        # Base case
    , _  A$ A6 }# T' B  O9 C    if n == 0:
    - O1 C# c4 t% Q) q/ N. a        return 1
    ) |5 P% J, `  y7 N' r    # Recursive case
    ' D! U  a6 z2 n# U# t) ]    else:
    2 T3 _3 K8 g" A  C% u% d: I        return n * factorial(n - 1). d# D9 Q- Z& i, ^" V: m) }
    + j  A7 a' D9 X/ o
    # Example usage
    , ?/ `2 d0 i9 l1 x3 X& @print(factorial(5))  # Output: 120- I- ?7 r) s; Z; p  O3 X4 [/ v6 r9 f
    & c; \- O) u/ W$ d, v
    How Recursion Works
    " K% C4 o5 B( |
    1 U3 s  {- z2 H% d6 d) z+ i: G    The function keeps calling itself with smaller inputs until it reaches the base case.
    0 @& C; D4 ^: L8 P" l* o7 c( W, m) \3 O" k
        Once the base case is reached, the function starts returning values back up the call stack.9 y3 P. G3 H; z$ `

    $ n$ v' Q8 Y/ k    These returned values are combined to produce the final result." ]3 j% D+ X9 O2 V8 Y- ?2 p
    7 T+ j4 c$ i9 k# n& z
    For factorial(5):: i8 o# c) x0 J

    7 k4 B# x5 w5 ^' _% A* j5 }$ r" x# B" }
    factorial(5) = 5 * factorial(4)- t1 m" B( Q/ n+ u  v
    factorial(4) = 4 * factorial(3)
    2 g: M. r3 r8 gfactorial(3) = 3 * factorial(2); o& U# j2 T" M' L: v7 k/ t
    factorial(2) = 2 * factorial(1)( S9 }" j' ?6 Z
    factorial(1) = 1 * factorial(0)4 I; Z/ v4 v1 N9 W
    factorial(0) = 1  # Base case
    2 n5 L  E/ N1 E) B0 ~/ n
    , K( m$ M& I- qThen, the results are combined:& R) v3 c/ [9 {
    / }0 u" G3 b' Z2 e" ?

    0 i$ H- `% g* u  J4 I% xfactorial(1) = 1 * 1 = 1
    & r2 D# g' j/ [3 f  [factorial(2) = 2 * 1 = 2
    ; R/ \- B( r0 d9 p: w2 g4 d' ~9 J9 bfactorial(3) = 3 * 2 = 6
    # E! w7 ^; k8 J  \& U, p+ afactorial(4) = 4 * 6 = 24/ q; c. p# _! r% m6 O; M* C9 \
    factorial(5) = 5 * 24 = 120
    $ b: N- w. }8 j( E
    5 H, ~7 N2 r8 N, BAdvantages of Recursion+ f- ?: g5 y' T  ]

    : u' h* d& z) W    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).7 Q! v* |; r  V5 O: z
    ' o9 n5 k9 V! b0 k' v+ N# o2 K% o
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    + m! M3 j& ^8 f
    & g& f/ B% x' n4 E* @- HDisadvantages of Recursion0 M* A& }2 w$ O) l' M, r

    2 \/ q! z4 k+ j" i6 _/ m    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.
    % m2 X/ s) L- ?4 D% ]
    . F3 Q" h2 k3 p& L    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).' z- M6 X( L8 D& h
    9 |' Q! S! Y4 w/ X6 b3 [
    When to Use Recursion% u# s, Y% k5 |+ p0 \, p, O+ k7 {

    2 t# m' s( W4 ?    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)./ |8 w+ i  K: a8 c+ D

    + ~: R) Y: Y$ E( m    Problems with a clear base case and recursive case.7 a6 f4 U* _! ^# p/ h

    0 p6 j* k% J+ W1 \9 d% PExample: Fibonacci Sequence1 ], A( M: }" Q$ m; f3 c4 t5 U: U
    * m( z# W0 N" N4 W0 T2 Y7 d9 n; T
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:0 m( {1 o0 a( O2 A

    - I3 r' _7 e- U! t( g; `. T' A& f  A    Base case: fib(0) = 0, fib(1) = 1! y( X& b6 B! J4 X0 y* {
    ( o& S1 Y' J2 a* X3 P+ V
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    , s7 r3 ?. c4 f( h( Y/ H, I- z& q2 c. T& i
    python
    % F$ u" Z( V& L) |& ]
    ! z1 g2 h0 U4 k: P- r  {. u' @" N7 ?; l: K* ~: Q
    def fibonacci(n):
    , i8 B, S  w$ y. I- |    # Base cases
    : e5 X% j8 Q8 y+ {1 @7 r* o8 p4 G- \    if n == 0:/ l  \: O4 w8 y
            return 0
    . I- H  {) O+ _; V    elif n == 1:, w2 u- e% Z4 W4 r7 z
            return 1
    ; y3 o/ Y, a/ v    # Recursive case- h3 a$ o* Y0 |; R/ h4 i3 P0 I
        else:2 u9 G. Q% J1 F5 d4 x" a
            return fibonacci(n - 1) + fibonacci(n - 2)7 t1 C/ M# ], v
    , d3 N* N, M4 q7 k: R0 |
    # Example usage
    $ q; k$ Z$ Y& `print(fibonacci(6))  # Output: 89 O3 \0 _' E. `( g1 c- g
    9 n# ?4 c8 }( V; ~2 r* s, ]9 a
    Tail Recursion
    ( H2 Q1 J( U# R2 P( E' h* {& I# x& z  y7 Z1 a: U- p- a
    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).) h  t8 `# }# \. T* \' \6 ~6 I

    % ]5 S+ m. D/ u1 V' _2 _7 DIn 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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