爱吱声

标题: 突然想到让deepseek来解释一下递归 [打印本页]

作者: 密银    时间: 2025-1-29 14:16
标题: 突然想到让deepseek来解释一下递归
本帖最后由 密银 于 2025-1-29 14:19 编辑 " U- ?( ~. k8 @/ d5 v

6 B$ y8 G6 w) m' v& I解释的不错
+ z% F) n1 y/ a6 }% D) k
. O/ L, Z2 i) |3 |& j% Y6 @. i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 f/ T1 X7 }# F6 l6 P7 L

: c. ]. e( T& b$ S/ V 关键要素
7 c, C; z9 g. d  d1. **基线条件(Base Case)**4 B: [/ ^9 j+ u/ v5 G8 f; ~
   - 递归终止的条件,防止无限循环, [( P9 a& k, k7 E- A5 A4 r
   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
# c% G, j! }+ }' O  k; I1 Q; o2 Y
2. **递归条件(Recursive Case)**
- `5 t6 W- w1 ]: ~: U5 D, [( c0 @. h   - 将原问题分解为更小的子问题
( F$ d$ b! N- l4 g% p   - 例如:n! = n × (n-1)!, L* U6 c* W) F) T6 E' s
# p$ l1 G3 j8 Q, c
经典示例:计算阶乘
( a" D  Y5 G$ m0 m- J+ c) R, Q- Vpython
4 i+ c0 x9 z0 j$ E6 xdef factorial(n):
& k" b& H  n0 ~: O( P9 C5 A2 ^    if n == 0:        # 基线条件$ m! W7 @. ^% r0 r0 L2 d
        return 1
" [) e) \( _7 W- E$ H4 q9 j    else:             # 递归条件4 e) ]# X7 j3 j1 ]8 l
        return n * factorial(n-1)
8 D: x2 K1 m3 g3 |4 Q; X执行过程(以计算 3! 为例):* V4 `9 C" q$ a
factorial(3)
% s4 u% Y6 e# |8 ~" J' \3 * factorial(2)9 k. G7 o1 f2 ~  J& d
3 * (2 * factorial(1))6 Y% Z- H$ F5 U2 n* X! g
3 * (2 * (1 * factorial(0)))
# R6 b) k3 O3 Q! ?. E4 ~3 * (2 * (1 * 1)) = 6
( z' ~# \, _: H7 ?1 w5 M) K, `: o) X" r% G8 r
递归思维要点7 J8 l6 ]" ~4 I4 @5 ^
1. **信任递归**:假设子问题已经解决,专注当前层逻辑
5 ^& S; c. S! T$ y2 k2. **栈结构**:每次调用都会创建新的栈帧(内存空间)# b. P) M/ M! X6 s' U
3. **递推过程**:不断向下分解问题(递)
" Z/ W! J2 ]$ @' _* U  K4. **回溯过程**:组合子问题结果返回(归)
5 r0 H3 E6 B+ n
  F" l7 r( P  [- K( g0 u5 a 注意事项
6 X9 M% N- V8 p# Q$ S必须要有终止条件* n5 a1 T+ z& |4 U7 ^' N. T
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)9 |  H( D+ V7 T% k6 o& {$ Q
某些问题用递归更直观(如树遍历),但效率可能不如迭代# _1 N8 P9 _3 w$ w0 J+ M* @! m
尾递归优化可以提升效率(但Python不支持)
2 {& ?! O! U3 E& S; W
3 u2 W! j$ W1 h/ K% ?# S- x) G  U 递归 vs 迭代8 c! c& {! }! Y* d$ D
|          | 递归                          | 迭代               |
  g; d7 Z5 w9 K|----------|-----------------------------|------------------|7 _  u% e5 C8 B3 `- v6 c3 c
| 实现方式    | 函数自调用                        | 循环结构            |
! b: w+ @2 G+ q  F4 \: O( Z| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
, L6 J9 b  D! G# ~| 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         |% e7 o6 R; x) ]8 B) S- L* r
| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
2 }- S" c6 l3 J- p  G' g4 d1 B+ ^9 \
经典递归应用场景$ ?8 d6 R) ~( Q+ n( I
1. 文件系统遍历(目录树结构)6 u. E. Y8 F: j/ i7 {2 R" z3 h
2. 快速排序/归并排序算法
" a7 A$ g( K& U/ u3. 汉诺塔问题! V, E0 Z, m( G
4. 二叉树遍历(前序/中序/后序)" o: h: q$ }7 H7 W% F( \
5. 生成所有可能的组合(回溯算法)
* k. t' V2 d( O7 C# g# l7 u2 M" U0 M; T
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。
作者: testjhy    时间: 2025-1-30 00:07
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,
9 E0 u- n1 z: R我推理机的核心算法应该是二叉树遍历的变种。
9 W' E$ V6 l$ l4 i$ G9 `0 R( Y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。
作者: nanimarcus    时间: 2025-2-2 00:45
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:
4 l' `. e  \+ F9 m' rKey Idea of Recursion
7 V) a0 Z1 P% e0 r! k; E# P' E0 k
+ {5 W" g& D% `% kA recursive function solves a problem by:3 O. W! z: [$ \" B/ ?2 K0 k

/ T1 O8 L$ G+ W3 j5 @! m# w    Breaking the problem into smaller instances of the same problem.
0 k+ x7 ]6 m. C3 |
* @$ m. A$ B& b' ^$ S7 J    Solving the smallest instance directly (base case).
6 f7 v: \5 r( L* n0 ^9 m. E0 S' i- @8 B1 i
    Combining the results of smaller instances to solve the larger problem.
5 Y3 x2 g5 m4 D  p: j2 y3 E
3 v" A0 ?2 |# t6 ~4 oComponents of a Recursive Function
8 |' W% S! c+ r0 W; q7 [! Y2 L& X" {9 y# i
    Base Case:5 z( `% [. c- L, z, y
% b  a0 Q9 Z7 i) z7 y0 `+ w4 d
        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
8 Q% I2 q' E7 Z9 B. O6 S& r% b' Q
        It acts as the stopping condition to prevent infinite recursion.
* M3 p6 \% T* m9 l1 d# W5 n0 T
6 ?/ @! ~% S; D- }0 L1 l        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.$ A2 Q/ b5 z1 j5 L9 d! T! \
% M3 z! h3 @( f) l; I. D4 l
    Recursive Case:0 q' `% @0 B6 m9 J5 ^
7 T, X5 w6 B/ s" y, h0 C
        This is where the function calls itself with a smaller or simpler version of the problem.
* l* N3 N  }3 Y, ?" r1 g# b& S' k: F! l7 s1 v
        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
0 X2 x; W' y" y+ `6 D
. Z1 k7 X; u2 [( q& q6 jExample: Factorial Calculation: D' T9 c! l# J( X( M' S

# ]: G; p" T2 Y) c3 eThe 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:4 q, r4 o9 h+ b2 Z* ^
" Q" F3 s: l5 N* q( v, B
    Base case: 0! = 1
  W+ i* e# _4 \1 x8 q& H2 [/ z4 u0 H7 W9 N0 Z8 s  q
    Recursive case: n! = n * (n-1)!
( f5 q5 n5 S: A1 C
* X! I2 ?, j6 k/ NHere’s how it looks in code (Python):
' D9 A( r5 o+ c; O: Gpython' Y: h/ m( X& A* u" f& u
* J; e, S! c. n7 F' Y+ O& X
" D! F6 k+ \; v3 i0 \
def factorial(n):
2 |/ P* U, x& C" k% A8 C    # Base case
2 y* w1 A6 A( S2 T; U    if n == 0:2 G4 c5 U( u; n1 n
        return 1
9 a: P. f; e8 ^$ A# W# K    # Recursive case6 w. ?" v1 ^" O: c$ M" A6 X
    else:
+ a% I5 s5 N4 U        return n * factorial(n - 1)! t# j2 ^; A0 @2 j+ |! }

& m5 k. a" P& i/ ~: G# Example usage% f+ U! i5 W$ Y1 e( s
print(factorial(5))  # Output: 120
% Y" x/ p2 V$ Y) W1 O
% y5 X, |5 F4 p2 ~How Recursion Works- i) a% x1 X% v1 _0 Z: b+ |# T: `( _

* e: ^- R( E; d2 }/ n  i4 |    The function keeps calling itself with smaller inputs until it reaches the base case.9 o+ p9 U7 P6 F* l9 s, R6 t

2 {. [" C5 s. y3 k    Once the base case is reached, the function starts returning values back up the call stack.7 a+ U5 X# T! @/ \
+ n! D" f" Y5 K8 I' X; _
    These returned values are combined to produce the final result.$ Y- Q4 k% t- V+ U+ V7 x3 Z, b1 U

3 u5 r0 m$ Z! _! S; o) [, IFor factorial(5):% B- m# y( l. r, l! P  V

. \& f: s0 G) {" B2 y$ V3 ^# F
* i$ i7 b5 m3 B- ], Rfactorial(5) = 5 * factorial(4)  k4 R" z7 _8 l' f" r, h
factorial(4) = 4 * factorial(3)) V& X* i" g( _" |  o6 K$ u
factorial(3) = 3 * factorial(2)
7 b* M. K7 |% N& U9 A8 Jfactorial(2) = 2 * factorial(1)
, x& Z; {! w) ~% vfactorial(1) = 1 * factorial(0)
( ]. c! Y8 x1 K) tfactorial(0) = 1  # Base case
: v, ?! ^. }8 ?% q: _' U! f* h1 U
Then, the results are combined:2 T4 X8 R% L" n9 }" S

# D' U/ Y/ h6 U. x5 d( Q
0 b& V+ {# M0 K- T* k# pfactorial(1) = 1 * 1 = 1
8 p* `7 @, I0 i, s$ f" i7 T* n0 x# M% ^factorial(2) = 2 * 1 = 2/ \- Y# {4 B8 |2 e# m# L
factorial(3) = 3 * 2 = 6( u2 c; i, l2 b+ B: z, b( ^
factorial(4) = 4 * 6 = 24
1 L* @: b' V9 }2 rfactorial(5) = 5 * 24 = 120
+ X- U0 J# k' k5 K5 j
) n* V/ r! |! J9 GAdvantages of Recursion
5 _6 |3 P: _2 s) E, a1 x1 t1 o% V* y" w7 G6 c: 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).
5 r2 K& e4 r' f! d- \
% o# f& v8 [0 l4 K    Readability: Recursive code can be more readable and concise compared to iterative solutions.
8 V) G  R3 x# B) q3 N
# r+ u1 r$ |0 |1 R* ?3 {) |Disadvantages of Recursion/ i; M1 C. u- y# ?/ R7 M. k
" y" ?7 G! W& F/ E: k- k+ K# S
    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.$ e* x) x: m0 U. n
- k$ A) ]. F/ o* y0 A& p$ ^2 C
    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
6 a, P* Q. Y2 n. h8 s) r4 [; k' b* [! F' y* X7 b' y8 g' o
When to Use Recursion6 @8 n0 W. W9 ]0 g8 A3 j
. W( b6 Z! K! W$ J1 f( ~. c8 _
    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
# U% Z  w/ n7 u( x6 N
! Q( p- M. `4 {1 }' {    Problems with a clear base case and recursive case.& K& V3 c8 P8 ^9 F

" T  [" D  ]* c; ~: m$ V7 yExample: Fibonacci Sequence' t% Y% I. J; D2 m8 Z# a

3 j' ^7 @/ E" ?$ f6 @. E. V* J3 ~The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
, f' E0 P0 ~' h9 P( n! X9 N1 {, f4 c! o3 J! D- k
    Base case: fib(0) = 0, fib(1) = 1# \1 \4 B3 G; j+ B8 q1 d

* {) a5 q' K+ C' W1 o    Recursive case: fib(n) = fib(n-1) + fib(n-2)7 ^9 |! H( Y. m) {

5 x# c* e" A! f% f3 Fpython0 Y% ^# e* B1 i( O+ B- R
5 }! X  p6 L8 ?% F9 J
% T  q3 t9 \8 @& d; m
def fibonacci(n):& ?  O8 N% r% k& j6 G' \
    # Base cases/ x7 ]" W" U6 D7 m
    if n == 0:) O) o5 P2 W5 ?4 J" X* m1 U
        return 04 C) ~; F7 P' x$ v" L) N1 A( r) |: Y$ I7 C
    elif n == 1:1 T* Q: @  U: I
        return 1# W4 |. o4 y7 C' ?4 }2 L! i4 E
    # Recursive case
4 y* i3 ?/ |( v7 Q' {    else:* J; H6 d: c* z
        return fibonacci(n - 1) + fibonacci(n - 2)
9 F0 ~$ Y5 b2 @) H& s$ O) e3 R( N2 a# I4 O8 l& v/ }
# Example usage
7 z5 x- S( X  Iprint(fibonacci(6))  # Output: 8
( ?: H# A% K' ~' A7 t3 [
5 H' E  E; F* W$ dTail Recursion
* w' N. @- _# j0 y( J0 O6 M( m- a% y, _! v
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).
+ W% _/ Z  ^( P5 h& C2 S/ t& U. a! Z& }3 M; S6 c+ f; `
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.
作者: nanimarcus    时间: 2025-2-2 00:47
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。




欢迎光临 爱吱声 (http://aswetalk.net/bbs/) Powered by Discuz! X3.2