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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 6 p% q; {( U  h" o
    9 I" q) n9 Z6 K+ L! o
    解释的不错+ |4 ]/ Q$ j, c4 B4 ~5 P3 ?

    , @; Z  g( j% T9 F9 ?: S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。) x+ s! O! S2 j& }! a! v2 E
    " m# c/ j2 O/ X# R7 @2 L7 j3 t( J# e1 {
    关键要素% S6 U. V' i. k2 F- F8 o1 g3 b
    1. **基线条件(Base Case)**8 Y, W9 Q7 b+ J
       - 递归终止的条件,防止无限循环
    / g: z" w, _: p0 b# n( I. R$ _   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    0 G4 T# G* n4 E+ G* L3 r7 s* F5 U6 R( V+ ~' u1 A- f
    2. **递归条件(Recursive Case)**
    ; b- `% O" }& R7 E+ |: X; D5 m+ f  c   - 将原问题分解为更小的子问题; d7 g6 d2 \! b+ O  L- f" b
       - 例如:n! = n × (n-1)!, _" G# r! z: c1 w4 X! v

    / h+ a3 H* Q8 ] 经典示例:计算阶乘
    1 I# R, C) @8 g. s, Q2 Q7 bpython
    ) _6 z0 f2 j, u- Tdef factorial(n):# `( R# Y' D/ t& \/ S' q
        if n == 0:        # 基线条件& a6 {) y' {3 r4 K( i" R
            return 1& R7 r3 z% _: o% Y: h# k! [( B
        else:             # 递归条件/ x# G5 `( ~2 g" Y1 H7 v5 E6 z: w
            return n * factorial(n-1)
    4 n0 f- u3 o& A9 v: V# u* _% `执行过程(以计算 3! 为例):4 p: a; i( }3 c1 M4 v# y
    factorial(3)  f; e' x1 i  M0 q+ T! m* L
    3 * factorial(2)+ U+ A, f; n( t6 @0 C5 f
    3 * (2 * factorial(1))
    : a- u4 O" m  S* Y/ u/ l. V3 * (2 * (1 * factorial(0)))( ~/ e+ R& _% ^7 y
    3 * (2 * (1 * 1)) = 60 I8 a, A' X/ J8 o7 C
    " i* F; l% }) r. W6 o3 i: P
    递归思维要点9 z- V8 p( q/ t3 P: J/ J
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    1 e# H- Y# \3 P$ R. q( D, T4 n; f2. **栈结构**:每次调用都会创建新的栈帧(内存空间)* H) R2 p/ E  o2 B  S" A5 }1 {
    3. **递推过程**:不断向下分解问题(递)3 V3 _. l7 _9 w, U- W5 l4 e& C
    4. **回溯过程**:组合子问题结果返回(归)+ q* L4 y6 j9 S
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    注意事项
    % R4 b- _0 p+ ]! _必须要有终止条件$ V1 q. ?# ~) t) j3 E; a& F3 S
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层), K: O" P6 V, t. D
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    / U7 P0 @6 ^7 o# Y, I尾递归优化可以提升效率(但Python不支持)
    5 Y* ?! y/ c3 ]  A$ s- D. O
    4 l7 @/ F0 b+ A+ e, Y' S 递归 vs 迭代
    2 G/ f7 G7 X4 \|          | 递归                          | 迭代               |
    $ `! [+ p. d) k& X! J- ]/ p9 Z|----------|-----------------------------|------------------|
    0 z* I, I  n! \5 D6 s  n| 实现方式    | 函数自调用                        | 循环结构            |
    " ^( m) Q6 ^9 D% Z| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |7 y: t1 ^. l1 C. \; B1 f
    | 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         |
    # {2 V& z5 p* V4 ?- z| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |+ J1 o0 E3 B6 \* m

    2 [' `' E: }- F+ e' Y7 ? 经典递归应用场景
    3 d5 O, r  z( Z7 r) f$ g1. 文件系统遍历(目录树结构)0 z7 q, @/ d' Z& n' U4 k
    2. 快速排序/归并排序算法
    * i( w( f$ g; a, r4 A0 e. [3. 汉诺塔问题
    $ Q5 n( ~5 ]4 N! V4 R4. 二叉树遍历(前序/中序/后序)2 q3 \$ |3 o5 I( E
    5. 生成所有可能的组合(回溯算法)# l' B1 T4 [; G8 l2 x' y7 Y
    8 M2 V1 S' p" D2 W
    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,
    " J/ A, G7 R% \; R7 c我推理机的核心算法应该是二叉树遍历的变种。+ I  A' O8 _9 d6 l/ u& y3 m! W! 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:7 n5 r2 r3 m- A* E8 W
    Key Idea of Recursion
    3 E% h$ @% m, V+ a& ]- T/ w' n3 z. [: {
    A recursive function solves a problem by:
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    % X  w9 l7 O3 o0 U: B) M    Breaking the problem into smaller instances of the same problem.
    $ j9 r5 x1 L; }$ Z
      W/ j& W) @  r* y: j    Solving the smallest instance directly (base case).
    6 ^0 o$ m4 l& k2 V+ ]/ i: w7 Q$ b
    ( P; e! @( |) Q  o, `    Combining the results of smaller instances to solve the larger problem./ R5 T) o: q! V$ L  C2 Y
    ) H& _1 K* Z( k) l! `
    Components of a Recursive Function
    . S  Y! W$ y8 [
    ) J6 f9 h9 B/ t4 i8 I    Base Case:
    . `# P$ L3 s% g  T" E" U' c3 W* a: t' C0 e! b3 g' l
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    # W5 k! ?4 R: C  E7 A( Y7 Y  b4 v# `; t
            It acts as the stopping condition to prevent infinite recursion." C8 H5 z/ p. v- Y
    ; c3 m2 s, z+ t: c7 L  _
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    , A4 X" [. P) N4 f. D0 C  D
    5 `& F) c/ C5 d    Recursive Case:# R% u2 Y* W! J: {: o
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            This is where the function calls itself with a smaller or simpler version of the problem.- p( q+ B1 [3 D* n" A

    9 y1 }, D; T# A0 Q# l, [        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ' {0 ^1 c! x+ g' c, B% ^" _: N6 t4 h1 @- t. ?; A
    Example: Factorial Calculation% A+ B$ X" [4 N# T9 \4 X% k  W
    # d: x+ q: c6 W+ M
    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:9 T7 b$ d# `5 R5 L6 h4 J# w

    - u3 e# \" B  q, ^: `& U+ p; ?    Base case: 0! = 1
    7 Z2 k9 h9 w1 _- K7 Q# l( ?: p. x
    + J. D) C5 ]* Y' C% X# a    Recursive case: n! = n * (n-1)!6 b' P0 ], D+ C- D

    1 C7 s0 U# p2 o0 v5 q" @Here’s how it looks in code (Python):& e: h) @: Y/ G/ y% x
    python8 {- b; C) q2 b9 q. F4 G

    ; O: R* \) v8 W+ _% E6 Y" K- H
    / d8 a. A4 V  s9 {8 Adef factorial(n):) A% M% n6 y$ M* i1 ?
        # Base case0 \0 s, a/ ^, ^9 {3 ?, t
        if n == 0:
    , Q9 x6 Z; a# B3 M6 j( m        return 1
    ! f4 b5 a) F: W- V    # Recursive case1 G, j+ H4 W* [/ U
        else:
    ( R( |: r% q# O9 L3 D        return n * factorial(n - 1)4 r& o, A  [* L7 \4 q: e3 V6 Z

    9 s5 z7 r, I/ i5 ~, p" O# Example usage+ M- j4 n: o- m8 q, |
    print(factorial(5))  # Output: 120
    ! d6 _) a( `0 f3 Y) N
    3 L. Y! E' _% O. [4 M1 k- r: {1 THow Recursion Works
    9 ^, D6 o4 E' F: I! S' R8 Z. W# B( A1 y1 |3 i4 N) d' D- X
        The function keeps calling itself with smaller inputs until it reaches the base case.! f! x: a+ {! R
    % ~2 l" I* [, m# P- U
        Once the base case is reached, the function starts returning values back up the call stack.- y1 q$ b' S6 g. h' L
    9 c! w! d/ v8 _, H2 c
        These returned values are combined to produce the final result.3 e3 w+ Y* R; X1 w2 B! W

    & C; P- d7 Q' C! `+ \! lFor factorial(5):
    ; L* m) f6 Y0 n; M- ?/ u
    5 i  C# d5 O/ V2 P. H( \( \% H; O( \
    & ^! [9 c. t! hfactorial(5) = 5 * factorial(4)5 D7 h* L- |& X8 b/ C; M1 w: `2 \
    factorial(4) = 4 * factorial(3)
    # x. z9 n/ l9 w/ `# E, c8 }factorial(3) = 3 * factorial(2)4 n7 p* K9 v) T3 e- a
    factorial(2) = 2 * factorial(1)# ]7 `0 ?5 m, |  P  c
    factorial(1) = 1 * factorial(0)0 \1 M4 Y. h4 w8 }6 d0 @
    factorial(0) = 1  # Base case
    / o3 D7 {: w& x) r7 \9 j* A0 F% l1 J1 B/ `1 e) l4 q
    Then, the results are combined:8 O4 k7 b0 g! m; Z- t
    ) q* f/ j* o; [8 D4 ]1 H) j

    - G/ H+ i) ^* Y9 a' m3 {factorial(1) = 1 * 1 = 1* q+ a' k4 |( S% G
    factorial(2) = 2 * 1 = 27 F6 m' z3 f9 Y7 W
    factorial(3) = 3 * 2 = 67 [0 {, d: K. ~6 Y$ M- U0 |- W. e
    factorial(4) = 4 * 6 = 24
    & l: @0 B# C3 s! U& t) Ifactorial(5) = 5 * 24 = 120
    8 v5 Y2 @  |1 N2 ~
    : k6 |: G% {5 d2 cAdvantages of Recursion
    . M7 a) ~* }* j  y6 `  O- j7 h4 @: K6 r7 I/ o
        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).% j3 m0 x: n: V; }0 J6 U4 h- S, S
    ! o1 I& D6 F4 D6 P$ {" b
        Readability: Recursive code can be more readable and concise compared to iterative solutions.7 F% c# ]& |2 V
    6 k+ s2 j- g( H% u/ A( {8 K1 i
    Disadvantages of Recursion$ I- c- o' t, o0 R, D/ O# [

    8 Q. B' \; Y3 D. ]  w, S: R    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.: z# b+ u5 B2 F" Q- F6 Q9 \) M
    . @6 f6 V0 [5 Y* _% U7 w6 R
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
      Y$ L& a* N8 u$ L
    ' _% B$ a" J4 T7 J% w# {# rWhen to Use Recursion
    & b" N& R  M* o. ]" V& t5 t' f! g6 x
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).. V5 V% O8 [- D8 Y/ o  M

    ( o% y! U# f" x/ _6 u5 q7 `    Problems with a clear base case and recursive case.
    ; F* q. t$ e' H8 u6 y5 F2 q& G8 A3 c, f# `; U3 j
    Example: Fibonacci Sequence
    ; x7 E. E  ^. m: K6 h
    ( A( p: V8 M* ]' H( q+ L1 M, AThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:$ z* ]7 j$ i/ q$ s6 K0 ]

    0 v. Z- d3 |# p2 N2 b    Base case: fib(0) = 0, fib(1) = 1
    ' N( ~$ H: D( b! V' w* ]+ W& t9 f3 y
        Recursive case: fib(n) = fib(n-1) + fib(n-2)1 Z& R3 c- t7 _3 n
    6 d3 w& K. o. S# ^3 ^- @
    python
    / m* L! W% h5 J) r) K
    7 `8 Y, M: t, h$ b( \6 g
    " e! g2 k$ i5 {2 kdef fibonacci(n):  Y9 {: [! R& y1 A' q
        # Base cases' y& t7 k4 }' J3 n) W3 ~" \
        if n == 0:
    " r$ D8 G' F# D, ^' i1 r8 g        return 0
    . z4 V: `8 z# Z% I1 _4 e# h" u    elif n == 1:
    : Z: S4 i1 q$ T  R        return 1
    6 S2 t1 s6 z: J9 y) d/ j- B    # Recursive case
    . R: x6 ?* p  ^# T( a/ l( Q- v    else:
    % l# e9 ~7 M  g/ f0 t        return fibonacci(n - 1) + fibonacci(n - 2)
    $ }3 `' F# m4 Y+ y5 z  f. C1 A9 h4 W2 D( K1 D8 D) ^+ e1 S
    # Example usage# D. x% Z/ a1 a* D. n; `
    print(fibonacci(6))  # Output: 8
    , f4 j7 v7 v  G# h+ Q2 h" \. O1 [9 x6 ^* _$ \8 s* T+ M9 K( ~
    Tail Recursion
    8 f3 L& v# j4 b3 W
    2 ?5 ]- ]# G$ S* Z/ HTail 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).
    0 J* n; \* l/ C1 v" v/ c
    3 i  o5 k1 V3 _) V6 e. i+ Z3 KIn 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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