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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# T7 b" z0 A1 `; c; m4 L/ t
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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。
    + E. a* R! E5 }  a) s7 U0 C
    8 T3 m( X8 t; f) B1 r; j6 S, q4 {- _6 L 关键要素8 U& L8 a' L/ o4 G
    1. **基线条件(Base Case)**7 c* w' U$ @) K2 Q" d" X; r% i& T
       - 递归终止的条件,防止无限循环
    $ y) q/ }8 w- ^, X   - 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)**. _5 |1 K# P0 m& |" h0 I
       - 将原问题分解为更小的子问题+ ^/ q* \3 G5 ^/ E
       - 例如:n! = n × (n-1)!
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    经典示例:计算阶乘$ H; W6 M" I: X- Z7 z
    python8 N) y- g! ?- V' i) t: \4 Y: P+ s
    def factorial(n):
    - l( Q  Y# E8 `1 U: ^4 L    if n == 0:        # 基线条件
    5 Q0 e- C' j: m9 j7 n        return 1% d# \' U! ]$ B, f
        else:             # 递归条件
    5 Z  q7 E  Y4 `1 S        return n * factorial(n-1)  N, p5 r/ N+ o# C, R: _
    执行过程(以计算 3! 为例):: N7 B4 T/ r# [7 r1 O$ l$ T
    factorial(3)
    " H0 m' n  [. b4 Y3 F" @  R3 * factorial(2)
    0 P5 b0 S- P2 G1 \( q8 O3 * (2 * factorial(1))( ]- n* Z/ G, a: O( R; J' @9 \
    3 * (2 * (1 * factorial(0))); B* h$ T& D. T5 B7 W
    3 * (2 * (1 * 1)) = 6& p* O# C7 k, @2 X7 X* }

    : h* J+ R; b) |  B2 _5 Y 递归思维要点
    5 |. y. W; _; s1 v% i* K1 R1. **信任递归**:假设子问题已经解决,专注当前层逻辑5 r( P; I6 d& E) J% C/ j
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ( Q: B! n  ~' ?4 u4 c3. **递推过程**:不断向下分解问题(递)
    0 U: m! H0 k2 V3 f/ c/ L4. **回溯过程**:组合子问题结果返回(归)5 H$ {5 H) t# D5 @1 k
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    注意事项* S. N2 p+ ?; v: _0 r+ G
    必须要有终止条件7 B" P, |  D+ r" j* k
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    4 g  U$ X  P% w) h某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ) X9 a8 y+ W9 i7 {尾递归优化可以提升效率(但Python不支持)
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    递归 vs 迭代1 H+ }) p4 z/ M# v
    |          | 递归                          | 迭代               |% v/ f3 B; o! x: k
    |----------|-----------------------------|------------------|
    + P& \7 m+ ~9 I" [/ F- m3 o, e| 实现方式    | 函数自调用                        | 循环结构            |
    , t) `, ?4 q1 `  f/ e| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    " N0 u* U0 l3 ?- @7 r| 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         |
    . W4 J* Y# d) O7 s  [& ]| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    8 j  j* C* [8 Y  u& F
    / {. O9 C. k) L- p6 L' O3 T$ m 经典递归应用场景. V: P+ ~, [7 W0 t; Z3 o& _2 x
    1. 文件系统遍历(目录树结构)
    , x$ M8 {% l" S! J2. 快速排序/归并排序算法  J6 e/ U' R0 ]( O+ A6 |
    3. 汉诺塔问题
    ; B3 N) K  j' u4. 二叉树遍历(前序/中序/后序)
    & q/ {6 H' M  i( i5 H3 p9 p5. 生成所有可能的组合(回溯算法)8 n, ~: d* E4 I1 q0 |+ K0 t0 L
    + }: b1 e9 ~/ c0 s- b$ b: B# Y
    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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    昨天 06:39
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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," @7 X' x3 [( M; N1 ]# D
    我推理机的核心算法应该是二叉树遍历的变种。
    ' A! z. i3 G  }% c- _5 G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:) k7 H) B7 f- `) ?* X% M5 A/ H
    Key Idea of Recursion& R7 C+ c6 m0 _0 l9 h  B" J) z

    4 X* K  Y8 R* W! k5 h0 A2 t& A" cA recursive function solves a problem by:1 S7 ~9 w3 s6 V" |8 Q
    # a6 i8 @" Q3 [) x: H
        Breaking the problem into smaller instances of the same problem.
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    * a" S0 J% [* A! q: Y5 V, L! o$ A    Solving the smallest instance directly (base case).
    4 P0 }3 p+ _4 ~8 q" d2 ~8 p% V3 B4 t+ @/ W& E/ n3 S! y' O" Y
        Combining the results of smaller instances to solve the larger problem.0 o/ E; B, p, H. [% a8 D3 [
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    Components of a Recursive Function
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        Base Case:
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            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." k7 _! x( X* R0 V# U9 Z

    * ~6 h6 p6 a$ R9 W" J. }8 W, l        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    $ Z; Y9 V5 j% b% [$ q+ p; h% R
    $ r5 ~( |% l7 S$ e2 e, b4 ?2 ~; P    Recursive Case:0 N0 I/ H: p) r" y, K
    0 |# @5 G' ], h3 l# N  M) b' j& B
            This is where the function calls itself with a smaller or simpler version of the problem.
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    8 O% K* j! s+ R2 U3 e, k        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).6 N* @& }! `: P6 S* U7 d+ y: G5 I
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    Example: Factorial Calculation6 x3 `5 Z3 V) e

    # T' V, ]6 Y3 h: b- ?% O7 kThe 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:
    - m9 N% \! R; I! q; T1 D8 M  _, }0 L/ T) C7 b8 Q' @0 p  W  g
        Base case: 0! = 1- l1 [; N/ Q% c$ [9 o8 E% A
    3 b# t  B% O" ]* o: ^: W
        Recursive case: n! = n * (n-1)!
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    Here’s how it looks in code (Python):2 g& d, T5 w, \+ p
    python, O) I0 L2 |# m3 t# |
    + |6 ^/ T# g# L1 w5 F  Q7 q

    % F& C1 l8 l! d7 u' ndef factorial(n):- a. G# B9 _( B5 j$ |' t( {2 p0 W
        # Base case
    1 X$ E# o- U% S' F3 W; T    if n == 0:
    & t6 O0 R3 {! `7 v2 x8 M        return 1& C/ C7 W; e" f! z8 u' Y+ Z2 t
        # Recursive case
    5 {( _" Q# R, ]" |) w' u    else:
    ; G  ^& c6 p5 x# }1 W* H9 Z7 x3 P        return n * factorial(n - 1)9 M9 l6 N. }- g' @; h
    2 |4 L4 Z; d- D- F
    # Example usage1 ]( |6 o0 Q  b! @' Z! ]8 d
    print(factorial(5))  # Output: 120
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    How Recursion Works
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        The function keeps calling itself with smaller inputs until it reaches the base case.
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    3 k& \9 S# E* m$ j, r/ t    Once the base case is reached, the function starts returning values back up the call stack.
    3 O' {* l2 G' ?9 |
    - P* w2 B% `2 e8 V    These returned values are combined to produce the final result., b3 t$ d4 Z4 e  g# W- O) ]
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    For factorial(5):
    ; j0 c7 L' j/ T: M" h, [; I7 T1 }6 B7 S1 j# F: ]
    ) ^, ~: |8 Q! D& G  _# |
    factorial(5) = 5 * factorial(4)4 w$ p: `4 b1 x: N4 L
    factorial(4) = 4 * factorial(3)* c' h* N: X- \: h, M& C! P+ ?
    factorial(3) = 3 * factorial(2)
    & N: O; m; @8 x5 g1 H4 Kfactorial(2) = 2 * factorial(1)
    : }( v8 ?# V! P# `: a& K( J8 @$ jfactorial(1) = 1 * factorial(0)
    & e* W, ]1 J% ~# Kfactorial(0) = 1  # Base case- E. v; D% s1 D7 t
    . u/ z3 {  c% K8 |4 I1 t. `+ u
    Then, the results are combined:
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    8 D( U3 ^3 g0 X: P  z( H
    factorial(1) = 1 * 1 = 1
    6 K( o0 E" t+ C+ ~# W& F! efactorial(2) = 2 * 1 = 2/ w' k0 L% D" c: m
    factorial(3) = 3 * 2 = 6" x& f0 u0 @2 w
    factorial(4) = 4 * 6 = 24- e5 Q  p; K3 w6 q: r5 n
    factorial(5) = 5 * 24 = 1201 y5 S6 {. a8 c, D2 H8 t# O: A* S
    & K* h0 y. q" a) c
    Advantages of Recursion& P' u/ ?1 s! @. ~+ q6 {) ]: A$ a
    8 k3 l! I3 @4 G
        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.9 }& m) }0 t5 O  i, ?! J

    6 T8 @/ J* ~. K) _' o  CDisadvantages of Recursion
    - O4 O9 T$ @5 D) y
    % e$ g% j1 ?6 z( r& D- j- U* G, |    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.
    3 q) Y  M9 y& D" Y6 L
    1 x5 o" x. o1 |# B$ R$ ?    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    2 _0 J' @% \2 [. P# G
    6 n* |- Q. u; n4 m, cWhen to Use Recursion
    ; M# m1 o* I  A4 x
    ; Y2 h* k0 V9 i6 m5 }/ K5 b    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    6 e6 F  `  Q) G' |7 i* b
    / }4 r3 ?8 S0 {9 j/ N& i4 P    Problems with a clear base case and recursive case.8 T% y% H) Y( B1 H/ M6 w( A: j( p

    5 f# f0 W2 _6 ]) kExample: Fibonacci Sequence
    , A* D  f) ^8 j. j: h3 r
    - M! h; s% Z7 x. d+ c( e" T3 d  yThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    - ?" t# g* M) L0 y9 \$ m0 d+ i& s( m  G  |8 p6 ]" |$ s* q
        Base case: fib(0) = 0, fib(1) = 1: G  h$ e0 y$ W& g9 L1 r1 d7 R

    ) n7 I2 z0 ?$ r% b/ f3 Y$ w    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    python4 r+ V6 y2 B& v: {' g# K
    # s) F8 C  T( F5 o3 g1 }
    % a0 O' }1 \3 o. g/ E
    def fibonacci(n):
    4 b4 m$ V. q  Z) L, w. u    # Base cases' u+ e) D# c) {2 \  C' Q
        if n == 0:7 R! n1 E- ?* W9 g
            return 01 v( B+ l* U1 ~9 f2 W' `7 c; v
        elif n == 1:" ]5 x. J( |5 y: A
            return 1- x. Z, K# A) ?+ G) I% M# R
        # Recursive case' i  n' b, m6 c  X
        else:
    7 Z) v. e$ u- q0 m, o        return fibonacci(n - 1) + fibonacci(n - 2)! G- w3 g& o; e+ G3 T

    $ w! I6 z! f) Z; y# Example usage
    / @1 t9 ?9 v0 s) Y' S: }# kprint(fibonacci(6))  # Output: 8
    5 ]. i, {8 j# V
    9 Y  X3 {+ k% W' c" j8 x) i5 OTail Recursion
    * ]4 j; e5 @  x5 v- ?7 a3 I* ^3 z( Y) W
    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).
    : l6 U9 ?% ~( a5 c6 i' }# Q" W5 I7 O  @+ c- ^  D
    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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