设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1267|回复: 3
打印 上一主题 下一主题

[科技前沿] 突然想到让deepseek来解释一下递归

[复制链接]
  • TA的每日心情
    开心
    2025-9-8 05:08
  • 签到天数: 3 天

    [LV.2]筑基

    跳转到指定楼层
    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 ) L0 T2 F) s4 O; Q( q* \# I( ?& Y
    ) N' b8 t8 a0 }1 M$ x# Q
    解释的不错
    " g& l3 V" S, l( W/ H
    ( J8 t1 L" f- o( 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。
    7 J  L5 J+ L8 }' J2 L) r+ K6 X' N9 x& S
    关键要素* z0 ^" S$ `7 ?# `  k# Y6 g% z  a! [2 K
    1. **基线条件(Base Case)**
    9 S" ?; a$ w$ n' w) ~) d! j   - 递归终止的条件,防止无限循环
    % i1 r4 X5 T. s% I9 b   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1+ m' s3 K( {8 q- w8 l0 ^
    ' N4 z4 \' C$ u& M0 f# k, C
    2. **递归条件(Recursive Case)**
    6 P! w# O4 U0 p   - 将原问题分解为更小的子问题
    + Z1 p' A5 x3 {7 W   - 例如:n! = n × (n-1)!* v+ j& W5 Q. \" h3 z
    5 U" k9 l# B' w
    经典示例:计算阶乘
    ) d) _8 O4 O5 P, Z' A6 `) kpython
    . M8 j2 }2 y1 o% ydef factorial(n):# ~) X* F/ R& N' M) }" |! J! r
        if n == 0:        # 基线条件0 ~# k6 E' e1 a1 \) l" t) F$ o
            return 1
    0 y# s7 q9 F0 X% j    else:             # 递归条件
    $ V2 w8 G/ |) ?- k        return n * factorial(n-1)
    * T# Y; K# Y, X! H9 `9 ~/ F执行过程(以计算 3! 为例):
    - h% T8 K# c& ]8 h& afactorial(3). @: ]2 ^0 P! c! v& `! s2 D
    3 * factorial(2)
    , S9 P$ P1 y3 e+ X# C( S/ \0 n3 I. i6 \3 * (2 * factorial(1))
    ) K6 V# `1 b5 l7 ?5 _; j3 * (2 * (1 * factorial(0)))2 I( |' I: K, `/ H# e
    3 * (2 * (1 * 1)) = 6
    + P& X/ [" g3 A; [
    ' D, r: _  ]# x8 r: [ 递归思维要点
    # t0 q/ g2 [% m* ^! D1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ; c8 ]( }! P: X2 I" z4 Q2 W2. **栈结构**:每次调用都会创建新的栈帧(内存空间), g. H# N1 c  d/ `7 a
    3. **递推过程**:不断向下分解问题(递)
    4 t2 `7 R3 \3 g* E! a" B4. **回溯过程**:组合子问题结果返回(归)& f) u* ^3 t2 J2 F5 s; C5 O& Y; w
    # N' n4 g  y3 s2 ^4 {  N
    注意事项& a# P$ p3 h5 B
    必须要有终止条件
    0 b6 f) ]3 w0 P  B递归深度过大可能导致栈溢出(Python默认递归深度约1000层)3 A* k7 D& K2 V( m# k
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    * p$ E* ^. Q6 _0 k# M9 k. D# Y尾递归优化可以提升效率(但Python不支持)
    ; D! i: S3 w7 W' I% }  {0 g' q: ^& G0 m6 n
    递归 vs 迭代
    2 b- p9 S, F( K' a) s% Y|          | 递归                          | 迭代               |
    : A; T" }8 ?6 l/ g6 o|----------|-----------------------------|------------------|
    ! N( [; m5 `) n2 N4 t  x7 Y2 T| 实现方式    | 函数自调用                        | 循环结构            |
    ! c: S' P5 V  w+ D# U) B| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |. k5 A7 b; p' {# 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         |
    ' p" k, E) a4 D1 L5 C| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    * o% L+ Q, g( v( W& e* Y" j
    ; q, Q5 q* z" u! h; j# O7 r) g 经典递归应用场景
    9 c% O/ i3 M) V# p5 }1. 文件系统遍历(目录树结构)
      B. [% p" Y) m- K5 Z" q2. 快速排序/归并排序算法
    6 X- c9 ^' b& V" D  t5 w/ B" b3. 汉诺塔问题
    1 m  v6 }) P% r5 t8 T; Z4 j4. 二叉树遍历(前序/中序/后序)
      L% c3 ?- k, q  Y6 s3 l. {$ I5. 生成所有可能的组合(回溯算法). |3 a$ n1 N7 o- u1 f# S
    : z1 V7 i  w0 ]7 T1 K" A0 K0 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。

    评分

    参与人数 3爱元 +26 收起 理由
    pcb + 4
    老票 + 16 涨姿势
    住在乡下 + 6 给力

    查看全部评分

  • TA的每日心情
    奋斗
    昨天 06:39
  • 签到天数: 3105 天

    [LV.Master]无

    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,
    # v. @& r; J. ?, G3 C我推理机的核心算法应该是二叉树遍历的变种。
    $ g# Y1 E  |8 v/ T. M+ 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。
    回复 支持 反对

    使用道具 举报

    该用户从未签到

    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:
    , i' ]( L7 ?! Q! oKey Idea of Recursion
    " R2 X1 K3 w/ C. P1 S( {, M1 }8 r4 |; r$ Q- i$ H
    A recursive function solves a problem by:
    " O8 T6 k; m6 |# Z) n" S
    0 q0 x  ^4 ~+ T2 ?$ n( M0 ^6 h1 a    Breaking the problem into smaller instances of the same problem.- o" L. i& t% h9 U; n3 q

    4 `8 R9 O( e4 j- z/ [; J% k    Solving the smallest instance directly (base case).
    ) D$ _8 ^3 z, \* I  b! e: f& B6 s0 t( H% d( X& S: c% E! E" a
        Combining the results of smaller instances to solve the larger problem.
    ! |+ `6 Y# `0 A& n0 {* Z6 ]1 K3 a
    9 `7 e# ?. a( D8 NComponents of a Recursive Function
    . a7 ]8 v0 z3 F; @
    7 h* M( V7 N6 f4 d! \    Base Case:6 \4 r7 J( N- k: _
    ! n8 w% ^4 A& o/ b
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.5 z* l' P* m8 p9 d6 K+ }
    ) c7 z6 o8 @2 ~: ^3 m# s
            It acts as the stopping condition to prevent infinite recursion.
    4 H. G  d% r" G4 l8 i
    $ n+ [9 w) d3 L* `5 Y+ W$ x8 U        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    + X8 [8 ~% i7 t* g) Q  g- R& J2 @  o6 X5 S
        Recursive Case:
    - u+ O% w! C- _
    ) n. y6 {3 F. H7 Z' v" i- }        This is where the function calls itself with a smaller or simpler version of the problem.
    3 @( U2 a) A% e  {9 ~8 ]
    6 G1 L* K9 }, q# ^; a        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    0 R% v5 U) W+ t7 W+ u) Y$ ?; c. u5 U) J& d% ?  z
    Example: Factorial Calculation. N% ~) p6 r( |- m
    : ]1 W8 K/ o$ n  C2 w2 ~9 }
    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:
    % Z) q- ~) k0 w' z+ F& x  t/ q, a+ B& H9 [5 L- {7 N
        Base case: 0! = 1
    , ~1 w+ p8 l# l5 c0 g
    ) v6 G3 j' T1 J6 K' R6 Z2 N) }    Recursive case: n! = n * (n-1)!$ c( _4 Z* v7 y" f0 e( c7 G
    7 S, E0 ?3 M5 b' G
    Here’s how it looks in code (Python):1 V8 f! I" h8 X1 O4 g
    python/ M6 z: r, M/ ~+ x
    ( [! T& h$ t: e

    2 R/ g% B# Z4 k0 X- [9 u8 Y3 idef factorial(n):
    , N: H1 U7 A. X5 `$ F  L' x    # Base case
    , a; z' i8 Z! {9 N% Q! C    if n == 0:
    + P5 c2 \0 n1 x$ U1 u8 y        return 18 ]' f5 e/ n/ C! y0 g1 i
        # Recursive case8 o5 c6 ?8 ~3 S1 F! @" d" a3 a
        else:! e' n: K. k- D$ _
            return n * factorial(n - 1)
    # t# [) N# T  p. j+ i. a" X. x% |. ], c8 Q6 A% q; r- u. O
    # Example usage% S; J; a9 F, ~$ V
    print(factorial(5))  # Output: 120
    $ ?# W7 L8 T* |
    4 y, @2 _6 G) dHow Recursion Works
    2 o) }, r. f8 f% `( S3 y# X4 S; U! {' P$ o3 }
        The function keeps calling itself with smaller inputs until it reaches the base case.
    # E$ Z. O  b5 ^' _* y. [, w  F+ ]1 i; ]" f: }: D' L7 d4 a% {
        Once the base case is reached, the function starts returning values back up the call stack.' B5 s# @" T0 O
    1 h. P! \6 k' A* J4 T
        These returned values are combined to produce the final result.
    2 E$ D: M; c1 `4 l! X9 p* t- [
    4 G4 s. |* H2 F, @( B7 OFor factorial(5):8 Z8 C3 S% e0 c# ^

    1 ^4 A0 O8 w: w8 ]* {* b
    / m: S4 m' ~+ @: j5 Cfactorial(5) = 5 * factorial(4)
    & X9 M1 u5 V, q3 W3 a' |factorial(4) = 4 * factorial(3)9 ]# I9 r3 N8 I6 X* q
    factorial(3) = 3 * factorial(2)$ k. Y: I6 ~+ ]" e9 B: m+ B
    factorial(2) = 2 * factorial(1)  N9 V/ `, }2 n8 p+ A
    factorial(1) = 1 * factorial(0)
    6 k" f. M6 o' _( Y0 H. sfactorial(0) = 1  # Base case7 B7 I+ X* U  V6 _/ L  u
    7 r+ O* R- U" Y  M4 |% f
    Then, the results are combined:4 G1 O7 x% U% R# W
    6 A* ^% p3 B" v! c* c/ u. D

    ( x/ a( D& }( k& e0 d1 v) Tfactorial(1) = 1 * 1 = 1' U( ^5 [- s- ?# f- f. w5 q
    factorial(2) = 2 * 1 = 2' b4 E, @3 }2 L" W4 z
    factorial(3) = 3 * 2 = 6
    5 d0 L' }0 G: a: \) lfactorial(4) = 4 * 6 = 24
    , Z- f% F. v& o2 m# c( mfactorial(5) = 5 * 24 = 120
    " V  E3 T3 o" B/ S4 k* W
    + \- U$ j/ N1 @& T) r6 m$ BAdvantages of Recursion4 C" u: c8 J1 ], z

    9 `6 R, F7 I  g( i7 L    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).9 e8 m# N: _( O5 R  l) g
    2 K+ ~( M# A3 p; V- y1 C
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
      i; y6 Q" F. n5 Y2 p, X: P$ D2 Y, n9 q: ], D. \: k
    Disadvantages of Recursion' P. U& }1 N4 N/ o0 @
    ( A7 o; j/ H! V
        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.
      M0 ~0 e5 K# o$ q, }0 ^; S1 [% M: N8 n( {% i
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    3 |1 K; r! y) _8 t; |+ T5 |! ^  e; c0 D7 H$ ~; M3 v
    When to Use Recursion( V) L% m2 M" `
    ! O' j! f4 ~7 o# G) }) b
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    0 r0 \( o' w4 c+ l7 q$ E" K, [- i% B+ M& j
        Problems with a clear base case and recursive case.! Y$ J$ j/ I* R

      U2 H6 m  y. W9 l" a* t9 [  ^Example: Fibonacci Sequence
    ! I1 R4 C/ e5 L2 ?) [0 p# S& B6 _# d$ Y5 S
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:% y/ J* a& R! H
    / m2 e1 e2 L$ H
        Base case: fib(0) = 0, fib(1) = 1
    ' g* G' G6 f& p* P0 z. B
    % [: R0 ^/ y" _( r    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ; u  c0 C2 _0 a7 O' ~1 U) t! i, m8 P, ~
    python
    $ H) h: Y5 H+ |- e0 `/ L* H! F& x% L9 G( k% ~

    / q5 t6 i+ S6 L; r9 l& _def fibonacci(n):
    , Q- _2 `; t7 g    # Base cases' `7 [5 F5 W9 T
        if n == 0:9 ^: ]. S2 y: J
            return 03 D; F; F  k$ ^0 n9 C0 U7 o
        elif n == 1:
    $ ]* g* A/ r0 [. D& R        return 1
    $ n& ]# x$ X& d% g" T    # Recursive case
    8 x- @; R# }1 C- Q4 T3 @    else:7 k7 c7 s& Z+ X' x2 l6 I' X* `
            return fibonacci(n - 1) + fibonacci(n - 2)
    / J- r0 a) J2 M0 k& `
    + W+ Y  k1 [$ _# u  G# A' Q( ]$ c# Example usage1 ~* f) `. d/ f% F3 k
    print(fibonacci(6))  # Output: 8& A  s5 p* o- G

    ' A7 X( ]* [2 GTail Recursion
    ) y& [" _: v: z" z& X+ y
    $ @, |  V8 M& Z  H/ g: ]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).
    7 h$ K; l* F2 u2 Z+ ~2 n% C8 g
    * |. r/ a/ z; y& B+ eIn 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.
    回复 支持 反对

    使用道具 举报

    该用户从未签到

    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。
    回复 支持 反对

    使用道具 举报

    手机版|小黑屋|Archiver|网站错误报告|爱吱声   

    GMT+8, 2025-12-1 11:39 , Processed in 0.031448 second(s), 19 queries , Gzip On.

    Powered by Discuz! X3.2

    © 2001-2013 Comsenz Inc.

    快速回复 返回顶部 返回列表