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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 6 A0 w+ k9 [) g  E) s
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    解释的不错
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    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。
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    2 V& i( Q& @% r9 C 关键要素- V- L! \2 R& G7 R
    1. **基线条件(Base Case)**: y  \9 h7 q* {6 S+ g& ~
       - 递归终止的条件,防止无限循环
    3 u# R" |7 ^/ G* k. y9 r' I   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    # @$ c7 v$ v: `' N9 p- z6 ~4 m4 H! }. I# V7 @8 ]; c, F
    2. **递归条件(Recursive Case)**
    4 S+ E' S2 ~6 d8 U9 i   - 将原问题分解为更小的子问题
    * C" n1 r! C! i0 U! D* a  d   - 例如:n! = n × (n-1)!
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    经典示例:计算阶乘) l; p1 K: q: U( [& Z
    python" [! I, _0 `  D1 {
    def factorial(n):' v5 C. `2 n! d2 Z7 K
        if n == 0:        # 基线条件& q. @+ H9 n5 s' U5 u
            return 12 p4 M4 m  v4 o+ U
        else:             # 递归条件. Z: M, p( S" c: T! C; f( v# b
            return n * factorial(n-1)( J$ R+ e! z# ]( k0 K% b
    执行过程(以计算 3! 为例):8 l+ q6 e5 t& ]4 \2 e5 R% A
    factorial(3)
    ( t9 ~) u6 {+ N3 * factorial(2)
    " p5 M4 d, w3 o6 E) U3 x3 * (2 * factorial(1))( ?# j0 r$ u: t, C0 l7 _, w
    3 * (2 * (1 * factorial(0)))
    9 ?3 m9 P! R# y# g/ s3 * (2 * (1 * 1)) = 6
    9 u' R9 p; n5 I4 ]6 ]! @% z" N" M4 l3 h* o
    递归思维要点
    4 S( b# a1 V! V- ?$ N8 Z) o1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ( |( ^; o: F3 \. ]2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    / D3 F0 r+ v9 w3. **递推过程**:不断向下分解问题(递)' `' `9 z' g/ y" y3 a& v! ]
    4. **回溯过程**:组合子问题结果返回(归)& I- X& O% ?' f( b4 |6 W

    # g: h; z% I8 h9 }5 n3 D* r! U0 s& D 注意事项9 ~. u/ ~* x% S4 a7 r
    必须要有终止条件
    6 ?' w( U3 C+ @# X递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ) I; l. c  J" p, y) t* y' M某些问题用递归更直观(如树遍历),但效率可能不如迭代
    + _1 Z& `8 d- h" n2 o" C; F尾递归优化可以提升效率(但Python不支持)! c. O5 a! g6 B/ h* W% N7 ]. J1 l

    + X  f, `9 |9 p  y& R* N1 V7 ] 递归 vs 迭代
    4 g7 |* w, J8 ]|          | 递归                          | 迭代               |
    / g0 }: B- o; x6 c( Y' [6 A|----------|-----------------------------|------------------|
    6 D2 ]6 G; p; Y6 `3 M| 实现方式    | 函数自调用                        | 循环结构            |- g+ Z8 C/ B0 B: n( W0 ?
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    * D& j; r5 f$ _* v- o$ }0 _| 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         |/ S4 N' T3 |; a& ]( A5 F
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |5 Q! v. D" I: O) `* T& A
    , r9 m* z! }4 i7 t2 T: E
    经典递归应用场景
    7 g% b- Y! P6 l1. 文件系统遍历(目录树结构)! f6 }/ _% K6 x3 u
    2. 快速排序/归并排序算法
    * L, u6 G3 v' F) P3. 汉诺塔问题4 b; l( z8 G- B$ E4 M* `. B
    4. 二叉树遍历(前序/中序/后序)$ q- ]( U6 ^1 Q
    5. 生成所有可能的组合(回溯算法)2 `6 C) x) k4 B$ |# E# M

    2 b. P; Q2 G7 O$ x7 N; k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 07:28
  • 签到天数: 3152 天

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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,
    2 v7 X3 \" r% y1 e我推理机的核心算法应该是二叉树遍历的变种。' H- m5 K2 s) T+ `4 T5 Q3 |& D. x  S
    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:7 a8 G1 i5 X( r" I, N. U
    Key Idea of Recursion7 Y/ |+ s; r+ Y1 @  K

    9 x/ }0 |' Y4 r# i9 r2 UA recursive function solves a problem by:  r. S$ A; i* F# w9 J+ `% P) B
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        Breaking the problem into smaller instances of the same problem.
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        Solving the smallest instance directly (base case)." R+ r! U* k3 y5 M
      C# }# K1 m# R, F
        Combining the results of smaller instances to solve the larger problem.
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    - T. e' F; ]# iComponents of a Recursive Function/ K2 V6 N1 O+ p' c

    4 O4 H9 a+ ~' k) Z$ k  j. D& {    Base Case:
    $ b4 X+ R6 }' K7 ^- N' T1 _3 B3 j" x! ?
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.. v  p: g2 x& N0 `% T

    ! {+ @* i! k6 m0 X  }        It acts as the stopping condition to prevent infinite recursion.. F. _/ C# m3 o/ b, s' B% p
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.% E$ |% e) _3 ^/ O7 g
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        Recursive Case:
    . `/ l3 p& g% w# B. x
    # E  t7 t4 L! t        This is where the function calls itself with a smaller or simpler version of the problem.$ D: Z  V% W2 G1 e4 F

    4 \' Z3 l8 L) S1 U( z1 N        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    Example: Factorial Calculation( F/ t& S+ u/ V* J! G
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    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:7 E2 `) l! P- L3 }5 O

    " S3 N# S% i1 ~    Base case: 0! = 1$ Z3 \" X; q6 @; k
    8 Y/ z1 ^7 s% A* n+ I
        Recursive case: n! = n * (n-1)!
    2 c" A9 N& j4 |: @5 _( ?; j1 W, v1 b- i" h6 e# D  s7 d
    Here’s how it looks in code (Python):
    8 B+ J% k6 S0 npython3 K% g7 I, Q. Y; K* m

    ' m& d$ r" G; F" ^0 \) G0 L2 ?$ c- Y. ?& |
    def factorial(n):, i$ m# N6 q9 ^. o* Y
        # Base case
    4 s8 ]3 ~4 s, a    if n == 0:( t+ z. Q" a  o0 G; R% j+ X0 H, l
            return 1" q* G+ `3 \* G& r( \
        # Recursive case0 ?1 U* }6 }# Y' Y. A- L
        else:
    5 @' {+ F. ?3 p$ L" d: e, e        return n * factorial(n - 1)1 S& o3 [3 _: z8 t! [' N, E4 k+ d
    * L/ G) G, x$ s6 `4 g, Z$ E
    # Example usage+ m1 D! A% D" ~! G. o
    print(factorial(5))  # Output: 120
    ! C4 w- n. i6 e3 W
    % _' A. t/ _) O, H2 EHow Recursion Works
    : ^: I) l) m% e3 b" N* K
    2 {  h* ~8 T5 x7 c6 }    The function keeps calling itself with smaller inputs until it reaches the base case.: \) D- D7 C; b6 F: ~
    ' v1 v- Y7 @2 L+ F5 x) ^2 D
        Once the base case is reached, the function starts returning values back up the call stack.
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        These returned values are combined to produce the final result.# G4 K' J4 t+ ?: X
    $ [/ _' @7 i' I
    For factorial(5):
    * [1 X2 ~0 ~) N- j( v
    ' W: v6 Q; {0 ~$ ]6 i# d2 g8 t* g, Y! v/ x
    factorial(5) = 5 * factorial(4); K2 W, b" X' a. F- ~
    factorial(4) = 4 * factorial(3)
    6 e5 h) ?6 o$ M# K& t" G7 Efactorial(3) = 3 * factorial(2)
    * T2 `2 M9 Z! `0 |" S0 A/ Gfactorial(2) = 2 * factorial(1)
    . Y$ {0 A, X( Ufactorial(1) = 1 * factorial(0)7 C0 U) |9 e: [8 s3 V+ d( Z
    factorial(0) = 1  # Base case
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    6 P$ M  K( Q, }' }% D. A. tThen, the results are combined:5 t" J2 G: W$ v9 `7 X5 n) A
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    ( f; U$ B! v' K3 x8 W1 c
    factorial(1) = 1 * 1 = 1% i3 f2 S. Y4 k, j, o" d. V
    factorial(2) = 2 * 1 = 2
    8 F4 |6 r8 H/ G7 E# ]# Kfactorial(3) = 3 * 2 = 6# O0 d. ^. K9 k% q( B1 @
    factorial(4) = 4 * 6 = 24
    ; L$ }% N& ?% a2 F9 X+ H8 _factorial(5) = 5 * 24 = 120
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    Advantages of Recursion+ a$ d. ~* E5 n
    " J: m$ W* h0 D( ]& ^
        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).
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      O' @/ n- G1 f2 @2 E    Readability: Recursive code can be more readable and concise compared to iterative solutions.& ?0 M& E3 R1 m0 @9 l, ]

    8 o( R- q$ ]- cDisadvantages of Recursion8 W9 h" p- |) W8 h( U

    * S1 P+ K; G, s, V' z) K4 \% i7 C    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.4 k" j8 _4 O. c3 t
    - b' n4 z' l& D  Y) H, R3 H2 a
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).& l; b& @; F# M: s3 O& t  `
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    When to Use Recursion( J/ g3 N5 T- n% \# z6 ]1 S
    " w' E+ ~2 N9 R7 L0 R' k+ Z' i
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).8 G; U9 l( _, U0 q
    ; ~! j2 U; E4 I, q4 D& ^
        Problems with a clear base case and recursive case.
    6 v0 E  A7 u; |# l* W3 v" t8 E* E# @1 N! \8 k& u
    Example: Fibonacci Sequence  j+ b0 E/ Z& u: _8 m
    5 {- C7 u$ d" S0 K! Q6 C$ i1 r7 ^
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:0 @  j7 f3 O* A$ `& v/ i
    3 W6 v- c1 z6 ^' o
        Base case: fib(0) = 0, fib(1) = 1
    & a! \' O; n: f$ B
    ) C" v8 v- i' f; |    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    python7 _4 z, \3 J& ^6 J
    3 c; C  M: O  R; c
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    def fibonacci(n):0 {/ z3 f4 S% d1 R
        # Base cases
    2 i$ @& v0 C- b* Z4 [, I) H    if n == 0:
    3 z2 |+ o5 f( H) i        return 0( E0 r- j2 T, ^0 I5 a' A8 C
        elif n == 1:. j" j0 U% D& N4 Y- e
            return 19 p1 h+ M' H& e* U
        # Recursive case7 v; U5 j+ A, \% Y0 j1 w7 x
        else:9 k: a% c! S3 t1 W. ]
            return fibonacci(n - 1) + fibonacci(n - 2)
    3 n; j1 G  e4 ~) K( f
    5 D8 S4 \+ n9 Y+ ?9 J+ @( W# Example usage
    " g" _6 O; ~( E1 e* S- O8 vprint(fibonacci(6))  # Output: 8
    ) F* w4 L: Y) M: |/ m' ?6 n0 A' T, T" I! I; n
    Tail Recursion
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    2 t5 F& z4 t7 R5 \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).3 [9 _/ `) u( i

    0 E$ P5 ^8 I% r1 H' H2 A% QIn 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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