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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
    ) }* F) C8 e2 E$ w9 X* F) q7 g' e) ?8 @8 g) h
    解释的不错8 D" a0 p' n; n6 Z4 e9 f
    / N8 Q9 [; q8 M6 C* Q8 V) ?
    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。
    & N8 `# A% @9 R5 I& T/ ^! w/ v' a( H: q# Z: i  f
    关键要素
    ! R! o. h$ M+ @1. **基线条件(Base Case)**; r8 j2 y8 f9 @4 D( R# L' I+ _- u
       - 递归终止的条件,防止无限循环7 R- l" v& N; k$ |. c
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    * {" k" G$ b0 n; h7 l) y- ]# ~( x* y  X/ s, _# l! t3 ~1 s
    2. **递归条件(Recursive Case)**# [5 i" |% b8 w$ N, M
       - 将原问题分解为更小的子问题
    ! e+ N/ [! t3 x3 f7 O0 o$ F" v1 s6 w6 ]   - 例如:n! = n × (n-1)!
    $ m; R$ C8 c* g- g1 y1 G  k
    , B. J! D8 `, m8 c  e! {3 M' `5 W 经典示例:计算阶乘5 [! i6 Y0 Y5 Z( T; ~1 Y, Z
    python) o( g3 I, S* f9 m8 w+ K" x  k
    def factorial(n):
    % a& F0 F9 z5 m) Y( N5 J3 y    if n == 0:        # 基线条件- Q# \3 O+ P9 Z; Y0 f: W- i
            return 1$ B& O) r0 I; g, Q
        else:             # 递归条件
    ' V2 a- H1 j+ @6 \  z- z* W/ f        return n * factorial(n-1)5 e8 @( H5 R5 e1 r
    执行过程(以计算 3! 为例):
    * I9 u( T2 J3 efactorial(3)
    3 B( z) ~: i( A: r3 * factorial(2)$ y4 o- w3 }7 M) Y# G
    3 * (2 * factorial(1))
    ) [1 K4 n) L; s* r- r0 L0 G3 * (2 * (1 * factorial(0)))  M5 x. C, H5 V1 Q+ M" z3 v
    3 * (2 * (1 * 1)) = 64 Z; |$ Q! k- E1 u4 W, r
    3 x+ x; y2 ]- V% k& I- F% M+ R
    递归思维要点" L$ g0 b* Q- b/ v2 Q) F
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑- a2 z1 h! c: T6 G/ J' {, E
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    * a9 p1 P: ?4 w3. **递推过程**:不断向下分解问题(递), X: m1 [) a9 |
    4. **回溯过程**:组合子问题结果返回(归): j7 P& a8 |6 }1 P, L

    0 [! h$ d0 Y- k6 T; R 注意事项3 t, ?$ A% b& x- l5 W" t! h9 n& M# K9 X
    必须要有终止条件& m* A; J& [; p8 W) c1 b% h
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)$ X* i2 {& f5 L% [6 G* E3 s0 e* w
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    2 b4 U& e& X3 [2 f! N; Q: {0 s) m尾递归优化可以提升效率(但Python不支持)) p+ L" u7 \: P" d
    + a2 F% Y" j9 M4 R% r; Q; Y. f
    递归 vs 迭代& F: S: D$ a/ a8 P% p
    |          | 递归                          | 迭代               |6 `0 C1 s) q9 z0 k+ O. h
    |----------|-----------------------------|------------------|
    5 E2 \$ `6 I9 V3 [9 v. {, a2 m& P| 实现方式    | 函数自调用                        | 循环结构            |  a- e/ b8 [2 l7 a) ?3 B
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |  @+ r' i& Y+ E! r8 P4 i
    | 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         |. S: C+ h8 [8 m. q3 Q# @
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |, H! \7 m+ P; n2 }( n4 ^" G4 ]9 J& C$ c

    ! f0 B; o' m3 G 经典递归应用场景6 u6 I! }1 C0 a3 Z0 l5 \
    1. 文件系统遍历(目录树结构)4 r( P. {; D# q3 k4 C! J5 G& E
    2. 快速排序/归并排序算法
    * l. f  U& s) `0 p* n% K7 U3. 汉诺塔问题
    6 f) x1 G2 }1 {! y7 g4. 二叉树遍历(前序/中序/后序)1 l+ ]* h5 L9 x
    5. 生成所有可能的组合(回溯算法)
    4 G0 a+ `2 v8 o
    + ]0 h: i3 n* G) G- B8 c% G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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    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,+ e! O$ n' a! v! Q: M
    我推理机的核心算法应该是二叉树遍历的变种。8 b$ X/ U/ s0 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:
    7 ~' d8 @6 {. ?# `! eKey Idea of Recursion
    " a3 Y1 n$ s+ a* l$ M: ?
    # v# R+ e' L3 v2 uA recursive function solves a problem by:
    " ?- w( w* J9 d7 h
    3 i  g5 m- X# y4 d    Breaking the problem into smaller instances of the same problem.
    / M* P8 R6 I/ I0 D# K& p* W* s6 W% h. x8 P- _+ g
        Solving the smallest instance directly (base case).
    # O1 A+ [8 j0 n& L# _  h# e: c8 X8 ]' `- g+ n
        Combining the results of smaller instances to solve the larger problem.% j9 q. a5 k. K& z7 P2 ~

    , f/ W0 ]  V6 RComponents of a Recursive Function
    % O# Y6 J' Z7 }2 h- _/ C4 K4 c# r9 n0 s- G+ n
        Base Case:
    8 f( Y8 E! n8 F6 p* r7 i
    , O; k$ O5 N, T$ H3 t+ }2 A- i        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.6 h# }; h" @# w& ~2 a; `# w
    " m3 y! r; p% I- @9 G
            It acts as the stopping condition to prevent infinite recursion.' V9 G3 i  h; [, K  u9 Z
    3 b' H$ e; @3 B+ _
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    7 g* N  L/ C: \/ u7 j0 D
    + |/ e! M# ^2 I" b8 p* i" Y    Recursive Case:8 E: z5 F1 o% H) a1 ], d& a8 _
    9 ]4 Z3 G* f  \9 b3 N9 h3 ]
            This is where the function calls itself with a smaller or simpler version of the problem.
    + D4 Z" A# P; x. S$ `0 ^
    # s# T/ T% J- }* H+ k        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    $ B' l5 h  B) ^" Q! ~$ t& _: I( A( T
    & ]( E, g! z! P; L! m" ~Example: Factorial Calculation" p- e& p7 z/ o

    ) [5 l! j5 Q* z1 N$ M  }) I% vThe 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:
    ) c( D2 b1 o2 D" A6 u
    2 U. e( M* T  |1 R* Y    Base case: 0! = 1
    . U0 |- @/ `' j$ C
    " f7 O3 V2 ]! C6 c0 o* I  l$ h3 P- w    Recursive case: n! = n * (n-1)!
    7 x( o* y" K/ o/ X6 G5 r* M9 L1 Z  T$ \. |( w! x( Y" s
    Here’s how it looks in code (Python):  F! k' N, V% o& z6 ?
    python
    2 t) s! E' h' t, q- `/ U; x. M4 R; U0 v% T. u
    ) A6 F7 r. D, n5 M, d$ x, U
    def factorial(n):
    ( F, \/ f- m. _3 Q7 F7 b. R, c    # Base case
    " j" {) g; z4 H4 H: L    if n == 0:  C( s. U( _# |2 T8 n9 O
            return 1$ Y8 ?$ P' B8 k% v1 Y
        # Recursive case( G' w! v: b: g6 k
        else:4 Y7 {" {1 u! {3 S5 K
            return n * factorial(n - 1): C5 l( E( }. ^% {& }* R. F
    - B5 c/ ~" T: \$ m) A& Y
    # Example usage$ j5 u5 {8 b) ~% H
    print(factorial(5))  # Output: 120
    6 ~2 W' b! G1 j2 v' ^) k
    " u+ k/ V8 m, z6 x, ]1 eHow Recursion Works
    . L& M  W; C5 c% F0 k6 F% k! |
    " I5 E+ R% B  O( h2 d    The function keeps calling itself with smaller inputs until it reaches the base case.
    7 Y* d" V! j* O$ Z5 i) W; f! F
    2 T0 x& T9 j/ ]0 l$ G    Once the base case is reached, the function starts returning values back up the call stack.8 M8 K( T$ m% ~/ w

    ' u1 h5 g$ O6 p% l    These returned values are combined to produce the final result.
    9 x8 y- H  ]4 `8 f! E$ m9 g) I; y. O0 ~& `6 r2 @
    For factorial(5):
    ! P/ E  L. m7 J5 j4 M# d( }9 W) D; m) ^- m

    0 H2 c) d8 _/ l$ X: ~; e+ {6 k3 Vfactorial(5) = 5 * factorial(4)
    6 l- C8 H' f1 r$ c9 b4 Sfactorial(4) = 4 * factorial(3)( ?4 S7 Q& A/ G' |6 _* _9 d: d
    factorial(3) = 3 * factorial(2)- x3 y7 a5 @6 [4 `2 E; w
    factorial(2) = 2 * factorial(1): w4 z6 f% K, @
    factorial(1) = 1 * factorial(0)  d- q2 i2 \- H( Y
    factorial(0) = 1  # Base case3 k% i4 e- G9 A$ h2 d
    $ W& F) W" }1 }" w# R1 B
    Then, the results are combined:
    . g+ l5 U' p/ q' @
    + m% V3 r8 R+ G8 Y3 ?
    ! Q7 Y: B! F2 @4 d/ sfactorial(1) = 1 * 1 = 1
      Z  u- h) P; m) [; F$ Afactorial(2) = 2 * 1 = 2# }3 H5 d5 A$ j$ ^% E
    factorial(3) = 3 * 2 = 6$ H  c* k) R. `) h3 L/ V* J
    factorial(4) = 4 * 6 = 24. ^8 S* g. ]: T9 X& ^* E' c
    factorial(5) = 5 * 24 = 120" U# ?4 X4 B1 a+ S4 V& z8 N) I

    * v7 @& a% ^+ Y" [& G7 XAdvantages of Recursion/ {7 G% i1 J: t9 d
    3 P4 }7 J& Z$ H. N+ [
        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).
    0 c8 P& b& M- a( g5 {! D
    ( q+ I% I- ?9 O" X) v    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    7 M* a3 _* u2 C7 Q; }/ Z9 I8 @, h. b$ f# I# l- E( z6 k/ F
    Disadvantages of Recursion. Y  ^* ]3 \% d2 H" [
    3 }# o& Q/ s6 y! y  ~1 Q% ]! u  b
        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." I2 `6 |+ c1 \9 U
    " z4 Y$ V/ J8 O# p% R2 }' ^- i: S
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ) U: q5 L5 Y5 e1 \* I9 L( n- T+ d$ p; z' k, x
    When to Use Recursion- l) L( a9 J4 I7 C5 ^- K; U

    $ w- s9 J, r% N/ \) r    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).5 t' u# H  O$ l4 h9 U

    , r- T- A8 Z: V8 l" S    Problems with a clear base case and recursive case.
    + W( E$ B) p; v# h
    , v- _7 s- g8 S3 j! G* {5 JExample: Fibonacci Sequence
    1 z1 R. ]' A5 Q' ~
    0 ?- d0 `4 o* i; B9 }The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    7 B1 v3 |2 `" O3 P# q- J2 p- R; C5 e$ N1 h3 x
        Base case: fib(0) = 0, fib(1) = 1) x1 |! S" p, k' W. l/ g
    9 g6 T. ^5 a+ d8 w1 Y; F
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    & R, Y4 ~4 ~7 G0 X5 n% j3 z" A: C2 q2 b+ ]
    : g/ r% b3 ?/ n9 G, |4 z# @python
    . _8 B- J" ]* t) r4 v$ D  D7 [  o' W& x1 ~5 |; n) l

    & m" A0 Y9 K; o1 @def fibonacci(n):3 X7 R* Q. |1 k2 q8 j) ^. A4 I
        # Base cases
    ! Q+ ?- |. E  H1 f9 y$ w    if n == 0:
    5 s# |6 l& T. x+ @% ]0 k: b* f        return 0
    8 i4 ^( b3 K9 S9 o2 w9 r: o) K  z    elif n == 1:
    7 L, R7 [1 [; k- z( S        return 12 \9 S, f: x. t8 v9 o$ \5 r0 ]8 Q
        # Recursive case6 C$ M1 J, _: t3 Q) U, o
        else:
    ) T3 o6 \) r6 p. j; R9 t+ H        return fibonacci(n - 1) + fibonacci(n - 2)' G( @0 c) p0 s, B9 Q

    5 P% `7 p; @# b" U7 @# Example usage0 q( v  F7 w  q5 w+ y- {
    print(fibonacci(6))  # Output: 8" `; q; i: I$ f! r" F, ^

    * A% ?0 v" z9 c: C5 KTail Recursion
    ( o1 X9 E8 C$ V
    3 p/ C$ e( a+ 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).& r, v( B: q# T5 x% @9 M6 h
    % Y+ s) w3 k7 o* }3 d" y
    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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