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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 ; C( Q$ x7 u7 p
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    解释的不错  k: Z) V2 k9 v" m0 Z
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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。
    / d( k$ F3 w( W2 ?8 R& p
    ; ~% W2 z# x$ M9 u 关键要素
    - W1 m/ e0 v9 J( x: r1. **基线条件(Base Case)**. z0 }7 U5 l3 e8 m
       - 递归终止的条件,防止无限循环. F8 _; ~# N$ t* ?; }& [8 J) I
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ) T' r- J9 @" A
    : u& D$ @" C9 K! n" x2. **递归条件(Recursive Case)**( G' }8 [9 o+ A+ |0 Q, B4 B
       - 将原问题分解为更小的子问题
    3 v6 |, L; B, w$ I5 F2 i   - 例如:n! = n × (n-1)!
    ) r9 n' {+ M( D; R# S
    4 A2 C  c& d* P4 \ 经典示例:计算阶乘; b" L: M( V8 s! ?
    python
    % T( c. r: d2 s+ Z0 [) \def factorial(n):% d( ^( ?/ p2 c& |2 _% w
        if n == 0:        # 基线条件
    % @$ c& h0 f- q7 E. h        return 1# i4 ]7 V! C% s  n: M
        else:             # 递归条件2 r1 r: x0 A! v1 i% I
            return n * factorial(n-1)
    / y% n+ l* d! h执行过程(以计算 3! 为例):
    ) T; m9 t8 [& g3 q) c5 {factorial(3)0 A6 H: A8 N0 Z
    3 * factorial(2)
    , |' l/ i2 U1 ^9 |/ t) H3 * (2 * factorial(1))& c- S' F. K6 ?; N
    3 * (2 * (1 * factorial(0)))2 B% O, b2 a: r$ m
    3 * (2 * (1 * 1)) = 6
    3 j! P% l- j  l  C% {0 [
    / |1 e. b( U# O) h, Q 递归思维要点
    1 Z0 X$ J9 p) {: L, @/ f1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ! Z; {  ^, m/ n2. **栈结构**:每次调用都会创建新的栈帧(内存空间)6 Z/ E! B1 J) P* a" Q3 b
    3. **递推过程**:不断向下分解问题(递)& i! m$ u& ]5 ^: E- v! Z
    4. **回溯过程**:组合子问题结果返回(归)
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    $ ^8 Z# A. p) B& ^- T' y3 k 注意事项: k" \- ^+ A$ {+ @% {5 f
    必须要有终止条件
    # d# [8 _1 F6 _: H* ~7 M' {3 \1 N递归深度过大可能导致栈溢出(Python默认递归深度约1000层)5 \8 c8 P+ R- G, d9 e+ b& Y
    某些问题用递归更直观(如树遍历),但效率可能不如迭代  S5 z! I3 e" L% ?6 {0 ~
    尾递归优化可以提升效率(但Python不支持)  E0 z; a& k  {3 L

    + \  v$ c* W  h' F; N: a 递归 vs 迭代; S2 |+ K/ B1 g5 D$ {0 U* K4 c8 \' v+ K& C
    |          | 递归                          | 迭代               |
    9 k: @' s+ R+ R|----------|-----------------------------|------------------|* P4 ?8 d' y8 ^1 y6 P
    | 实现方式    | 函数自调用                        | 循环结构            |- {5 P3 R$ [; P; l& ^
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    . F+ \; m8 v6 F9 d& B& L| 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         |
    0 ]# v4 d8 x; g. c( ^' x0 C# U  Y| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    4 {0 J) O: p. i5 S
    ( V% d0 O7 l, z0 t+ J 经典递归应用场景, P) S% e9 `5 P" f
    1. 文件系统遍历(目录树结构)2 b( h) W/ N2 \; ^2 l
    2. 快速排序/归并排序算法2 j0 b: [2 Z* T# Y# }: B: d* }
    3. 汉诺塔问题8 X% ?5 \* Y4 a( L
    4. 二叉树遍历(前序/中序/后序), f" k, u  X3 v4 E* P
    5. 生成所有可能的组合(回溯算法)
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    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

    3 天前
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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,1 k% o7 s! c2 o; ?# I
    我推理机的核心算法应该是二叉树遍历的变种。
    ( H$ h. j8 l1 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:  c3 [: O; }% C4 r5 B5 j
    Key Idea of Recursion/ m- a/ a8 Q3 t8 c; E7 d# ~9 ~

    0 `, G! n, S- O8 |4 b! b1 O8 N. |A recursive function solves a problem by:
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        Breaking the problem into smaller instances of the same problem.8 |+ L! Z7 {& _+ l
    + i+ I3 o9 k# k
        Solving the smallest instance directly (base case).
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        Combining the results of smaller instances to solve the larger problem./ a. m' _3 y9 C8 \7 Y
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    Components of a Recursive Function/ X5 `3 x. d. M
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        Base Case:
    5 K' n. k$ k+ u& s+ Q; f/ M$ y9 Z
    % A8 N, j7 l' N: A' W  L        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.0 E5 m  w, }5 b* q- h& v) D1 D

    + Z" d& j4 t4 a; `; o        It acts as the stopping condition to prevent infinite recursion.- b6 _+ H3 @  `$ b( ^
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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        Recursive Case:8 [: L! R4 V' Z" T2 g

    3 z3 w$ Z1 r( z" f        This is where the function calls itself with a smaller or simpler version of the problem.! i! Z  b6 F7 M, X3 u5 G2 ~

    8 E3 u6 V3 M: P( F" q3 `4 p$ z        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)." d: S+ `0 `- G+ B
    3 `8 e0 y* U2 p! h& W* E) T- g9 Y
    Example: Factorial Calculation( D0 e+ `+ E7 _9 X5 I1 N- S& z) d

    $ E9 }5 O% ^; L1 ?) q( b+ gThe 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:
    ; T% T- [, N6 f( ?) }0 U( B. |# j# W: J' w
        Base case: 0! = 1
    # V; ~! m- x, v5 e8 g8 _
    / I. V, ^. B$ w    Recursive case: n! = n * (n-1)!
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    : i/ B7 h! f+ Q- iHere’s how it looks in code (Python):4 d) \; o% H3 l
    python8 Z: \: K2 ~9 w
    & O5 `7 ~4 t6 N& p  d

    $ u+ j  z# L" ^6 W7 @( wdef factorial(n):' A8 {5 R2 k0 f2 J# [
        # Base case
    9 ?: K4 D4 g/ h: b& O    if n == 0:
    ! B; W+ J% V5 Z5 V- J% {0 C        return 1# y' v! v5 B8 A) x+ _3 T
        # Recursive case
    / l4 B2 d3 G+ ^9 N) `' X    else:
    * N( c% I6 ]0 V- j- n& p        return n * factorial(n - 1)' @) ~8 u/ |# |' r

    5 k: |: n& P  ^5 Y# Example usage
    6 I' q; G& \( O4 X9 D% I/ X: oprint(factorial(5))  # Output: 120' u+ S' e4 A( n( J, w2 y
    6 T* |$ D: f' S0 p+ f( O9 t
    How Recursion Works6 L! h  d2 j& @2 H' Z5 i

    * t. b8 Z6 z4 a& h    The function keeps calling itself with smaller inputs until it reaches the base case.
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        Once the base case is reached, the function starts returning values back up the call stack.
    1 [) ~- M; b) J. S9 \* R# V* k5 ]5 ~) ?, O  G- i8 D
        These returned values are combined to produce the final result.
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    ) o3 M# `2 c, T: N" R4 QFor factorial(5):* G; l, |* A& P: _
    0 y* i$ D( j  K1 @/ M* f8 o

    7 X5 T' i5 [% z: F, jfactorial(5) = 5 * factorial(4)  t! U/ k& n: F5 }6 I
    factorial(4) = 4 * factorial(3)
    - U8 |1 J$ d9 ]# q7 L- \; Xfactorial(3) = 3 * factorial(2)
    ( h. A& Z( k7 y2 S; ^5 Q2 pfactorial(2) = 2 * factorial(1)
    . W3 f, O0 B- [1 f0 V. Pfactorial(1) = 1 * factorial(0)+ [( p0 V6 f; L0 z& `% v1 u
    factorial(0) = 1  # Base case
    6 B/ I! ^( R- h) c1 q. ^
    # H4 U( G1 L  A; oThen, the results are combined:# H+ B8 b+ n9 y- N

    - b& S. M" s; s0 u( k! O2 U4 ~2 S3 e+ {& M0 j- T
    factorial(1) = 1 * 1 = 1+ D) r( E9 K' a5 c% r5 t0 m  A/ B1 q
    factorial(2) = 2 * 1 = 2: T5 h: Y; A8 ^( z; _' g. X
    factorial(3) = 3 * 2 = 6
      `; |7 i9 j/ z) Q7 L& |3 Kfactorial(4) = 4 * 6 = 24
    : s4 [2 g2 J! k0 V* Y& _factorial(5) = 5 * 24 = 1201 `0 ~- C- ?0 k* s7 u. }

    8 f$ T# L& v& k8 iAdvantages of Recursion
    # b- o+ f$ c8 p# P: U8 N; i4 M: E! B; v3 P/ a6 _* P3 ]8 J
        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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        Readability: Recursive code can be more readable and concise compared to iterative solutions.
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    Disadvantages of Recursion$ w. L% A: v: h, k
    ) i3 C$ G5 o+ R% T( Y0 y8 M! n
        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 U0 S. r1 W' D+ d
    9 z; e. Q( o' b" l1 k
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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    When to Use Recursion' v9 L- n. ^# C! H% K7 L1 c
    & B1 ~8 [. R% ]" n% r
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    7 s# R% @6 K2 f. e! R* |" r0 P
    7 s1 |' U$ Z: z1 p# }, n" o, Q1 g    Problems with a clear base case and recursive case.
      {1 D5 a0 |; z8 F( e" S& c! {
    / e6 M; N- y+ W' W8 r" I$ _. Q3 dExample: Fibonacci Sequence: j2 |2 v5 {5 \2 Y
    $ t( F: p" t9 q- ^
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:7 B$ r% a4 W3 G* G0 Z! x; c' l3 y* K

    7 h( \- o7 E, b4 |    Base case: fib(0) = 0, fib(1) = 16 M4 g% w" q8 C+ K- ~

    4 a. N, q2 {' Q    Recursive case: fib(n) = fib(n-1) + fib(n-2). U/ S% b' `4 {( R5 T

    # ]" y; N" a6 B' u/ [python+ J# C; G& k* R& c

    ; L$ [0 ^: t1 w9 n  d- ?  l, ]3 R$ I/ X1 p3 A; s
    def fibonacci(n):
    ; T' v" H" J% s  K1 k9 g. k    # Base cases
    & H2 }6 n7 |9 S) N; O    if n == 0:
    1 m( B. e9 i5 Y5 A1 `% H        return 0
    0 f; D* o% E& V( g7 K    elif n == 1:
    4 U, U' F' P, G% S" v! z        return 1
    7 e" x* q! S3 v3 w  v: O    # Recursive case
    ) h9 C0 n5 ?# X5 c& v# B2 j, V    else:$ G6 E$ ^: q- v  \: p) P6 r( U2 P+ s
            return fibonacci(n - 1) + fibonacci(n - 2)
    : M4 K1 O6 i- m2 J, U; T8 f1 l$ o9 u" M: o# I' W7 s$ T& J
    # Example usage
    $ T, S# J9 M. Y+ g9 p" yprint(fibonacci(6))  # Output: 89 Z: ~6 d/ T, M
    1 t# K0 g: X: E  @/ a# p
    Tail Recursion
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    + W, [: W' A* t, D* 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).! K4 z7 f$ o9 E# |0 i

    $ d5 p7 Z0 I; C( i6 L+ d6 TIn 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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