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 + j1 W% p0 J. I0 ?

; m  w  V" @$ q5 N: q6 ^" K* T+ N/ g5 o解释的不错' D  ~  \2 N4 y' f7 n+ R/ t

$ a) b1 r, K/ a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。, B3 j8 C4 l/ a3 [  m

0 M6 U0 H8 R& m6 C. R+ u 关键要素' q( z/ |8 D% \
1. **基线条件(Base Case)**7 _$ M/ |- J- W* u# j/ O" G$ X$ ~4 h
   - 递归终止的条件,防止无限循环( q4 r. J* y3 ~9 w3 y  t
   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1! B% \) O, T; v; m

% f4 X  c3 ]) U8 p2. **递归条件(Recursive Case)**, K+ ^  F! l- u! V' _
   - 将原问题分解为更小的子问题* l1 z: O2 M# {# P
   - 例如:n! = n × (n-1)!  f0 }, Q+ c5 e3 U" P) x* W

4 {6 `* Z/ v& r* k( U/ W7 H 经典示例:计算阶乘
1 r# ^' F/ q* _5 j7 t  ]python
1 Y4 H: L- [- b  M6 xdef factorial(n):
) L# _/ o8 M, N- {: T3 I/ I' K! x( E    if n == 0:        # 基线条件
1 d; i2 {" S* @  E* X        return 1
& \  W+ c; L4 ]$ q* p6 E0 P    else:             # 递归条件
& q9 Q$ ^* Y. ^! n        return n * factorial(n-1)
' h* E2 G; K7 `0 i2 O5 z执行过程(以计算 3! 为例):, q8 B. a4 \( e3 ]  d
factorial(3)
" n8 s% l7 N7 F& B+ x3 * factorial(2)
# p; k4 ?% Q0 d7 f& |6 X6 L3 * (2 * factorial(1))
' \, Z" i) t# _& f3 * (2 * (1 * factorial(0)))
& O) H. B& P% D- i. S9 j7 O5 @1 _/ K3 * (2 * (1 * 1)) = 6
3 s" R1 Z0 o- g$ [' K) O+ h3 ?% ]& ^4 L) Z$ @2 \# y
递归思维要点8 s! v# N+ p8 b2 E* i
1. **信任递归**:假设子问题已经解决,专注当前层逻辑
6 B7 ]* Y, G( @- u- G; K1 H2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
" G/ Z$ f8 ~3 o! m. m3. **递推过程**:不断向下分解问题(递)
- j9 D" W9 Y  ]! k$ s9 q4. **回溯过程**:组合子问题结果返回(归)- T" r# e; f" p' N! _6 E
3 @6 P) m" ^& F0 i# z4 U
注意事项
8 T1 S. K  Z* H5 K1 _; g必须要有终止条件8 B4 u, j' A6 D% x; j
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)1 `) P" S" T/ z9 S2 |6 ~
某些问题用递归更直观(如树遍历),但效率可能不如迭代0 U3 F) W9 ?4 R6 k$ Y
尾递归优化可以提升效率(但Python不支持). b/ O5 H0 A7 g6 @" u2 E
2 [: U2 H3 Q; V3 X
递归 vs 迭代
1 R8 j! _) r$ k1 b|          | 递归                          | 迭代               |
0 x& r* g' u- |' }# C|----------|-----------------------------|------------------|( Q; `& _6 J  B4 [# L+ b9 W4 B
| 实现方式    | 函数自调用                        | 循环结构            |
8 v; b8 j8 x; I9 l* z6 s| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
8 T8 A& L$ L4 ~| 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         |
. ^& N) c6 Z! }  l% C3 o| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |& R- E% \, x# @9 B6 k* W  U) P6 z

9 W! }) V  m3 ^& z& K, \- w5 } 经典递归应用场景
3 s) A) P$ D- r4 m+ X" l1. 文件系统遍历(目录树结构)1 E1 l9 h3 R1 ]+ f
2. 快速排序/归并排序算法
( C7 {1 i% `0 c# @" ], V3. 汉诺塔问题& Q2 h' I) z0 X* \2 O5 s
4. 二叉树遍历(前序/中序/后序)- J* W' ~# A4 T6 Y7 s2 R
5. 生成所有可能的组合(回溯算法)
# ]! K& }7 @' @" f& r+ U; T4 d% P9 r8 h/ P8 A+ g4 Y# v! 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。
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,/ x* f& V1 P. o  S0 X* j
我推理机的核心算法应该是二叉树遍历的变种。% q7 E: g' o" }# s/ ?$ E6 z
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:: [; q4 T7 s2 E6 Q5 ]& n
Key Idea of Recursion
6 a% Q- j! m  m1 N( k8 b. C5 o5 H( k# _2 _
A recursive function solves a problem by:
# x% ?% n: w, u
5 P* g& l" {. m: `    Breaking the problem into smaller instances of the same problem.- k2 Y0 I+ j3 j6 u# D' x

7 t* K/ Y; v% V! V- e    Solving the smallest instance directly (base case).
+ t( w8 T" G+ W7 c0 z3 C8 g4 z8 i' H( m' J  j. x
    Combining the results of smaller instances to solve the larger problem.# r0 a& e$ i6 N$ Z  G7 e3 s4 _
1 k5 f2 }! N; E3 ]7 G% b
Components of a Recursive Function
5 J1 p4 G" w$ o0 |0 |* c4 v9 c% u2 w8 q6 e1 s
    Base Case:
+ Y8 j3 W: G0 i# @/ f' D: R- n/ |; t
        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.- Q* P4 b, N% E5 K% q6 ]

; V  z! @$ b% H        It acts as the stopping condition to prevent infinite recursion.
5 C" o& s2 w0 z; A% X2 i( W
& v  w8 V; `6 b        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
; ]1 r4 d. B* m2 s7 ~# b9 x# V& l3 U3 ]$ Q
    Recursive Case:
: U5 G4 {9 ]$ q$ e! S4 ]: |+ J  {5 k" a3 W+ _
        This is where the function calls itself with a smaller or simpler version of the problem.
5 J, o! O9 Q- x: L" n7 g+ I0 [6 h
8 T: K, N" H2 Y        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).. w( D. ]9 P  B) X7 j$ R! d- o

$ x& p* d. a- @. K- d7 R' vExample: Factorial Calculation# b% M/ l- m& w0 Z" I6 u+ u. e) I
, v. H* J, t4 P* z  h
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:  @- J0 B' g7 _; q  O
1 H) H4 y* V* s. S; w) Z: [# K
    Base case: 0! = 1! J) w  z; t8 d9 O

7 B8 W$ P; d7 }    Recursive case: n! = n * (n-1)!
; B2 w( s9 G1 e/ y/ A5 s& g* G
1 M' B/ j% ~5 _! S( rHere’s how it looks in code (Python):
# [5 W; y* l! g, U9 w& y+ Jpython7 ?& R" L0 E% Y$ K6 w: H

* J9 l$ P5 M! v. D. }" q! h! k8 M8 F6 ^
def factorial(n):
; L+ w$ o8 U2 v+ x. a    # Base case# g; g5 Q0 k  B2 |8 e
    if n == 0:- U0 j/ z* n; b; C" v+ d. c9 \
        return 16 h4 x! l0 ]# \/ Q. v, k! S$ v
    # Recursive case
  F0 R4 f! S$ S, I9 P# C! x    else:2 e& \, F6 f; |6 N. D0 t( A8 o
        return n * factorial(n - 1)6 d  u- t% P! A& I. R" X6 H- F2 L1 u
7 A) e# w; v+ Z$ G
# Example usage' F2 [5 n, S! c$ z
print(factorial(5))  # Output: 120
. F- y5 @1 Z5 u0 l6 K4 _
# G5 j( Y, B+ D2 C8 f- c$ LHow Recursion Works
8 Y( D$ g7 Q) `1 ~' C/ F
5 B+ l7 A- I' ~4 ?2 X6 [- x    The function keeps calling itself with smaller inputs until it reaches the base case.
; C) E, U7 O0 A" V4 N' }( k( ]9 K( X0 R+ m0 T+ h: C
    Once the base case is reached, the function starts returning values back up the call stack.
2 _; x1 m3 o: M0 g0 l. D- d+ q4 B) F, Z
    These returned values are combined to produce the final result.
$ H6 {% Z; r$ l( n$ O# R5 E) k
  @0 l5 }! h. r* L0 ^" [, dFor factorial(5):* N( p" D& g8 i
. w7 U8 ?4 d7 |. S2 F8 b% u
. M+ y. X' ~" |9 V& D
factorial(5) = 5 * factorial(4)
3 }# T) r# ]/ y1 P: ifactorial(4) = 4 * factorial(3). a" ?4 [; }: Q! F
factorial(3) = 3 * factorial(2)
, C! b/ v& O- o7 Qfactorial(2) = 2 * factorial(1)( I# D! R9 x) z
factorial(1) = 1 * factorial(0)# i+ v6 o+ \( _1 f
factorial(0) = 1  # Base case; o7 {  s5 }  O' g7 P& p: ]* o

* j3 u) v+ s+ g6 Q& D& k) y$ VThen, the results are combined:
9 n% r; ]& A. o8 N/ I0 |/ P# s: U" c4 U
( u8 g' ?! v- p+ d
3 `# t; K) `; G$ `; \& ffactorial(1) = 1 * 1 = 1
8 w+ P2 L3 L3 F1 Qfactorial(2) = 2 * 1 = 2
3 O) }+ f' ^" b( r0 Pfactorial(3) = 3 * 2 = 6. x7 D" Z' X/ w
factorial(4) = 4 * 6 = 24
2 y# R# [  i& i. s: S) mfactorial(5) = 5 * 24 = 120
: B: z9 R9 }1 x' P
5 i% C5 z# {! Q3 f# @' SAdvantages of Recursion# U" {6 Y3 U6 i6 w) C9 `6 j
0 U" s1 a% Y* C$ C
    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).- l) |! c* r* e2 z4 f& W

$ j/ E& F* Y6 E! y# @5 V+ n6 F    Readability: Recursive code can be more readable and concise compared to iterative solutions.
) C4 M2 N) n* U. O8 }& _, f+ Y2 b8 n, U4 K" b
Disadvantages of Recursion' F+ m# @4 N, O
; j% M5 l+ v. }& n* E- ]
    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.
( t: S- v4 F6 h
+ B( r& n8 ~, Q; n% R. Z7 r    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
8 o. T% g5 s& \3 K; ^2 i  f
6 _& A& V$ D" @! z0 X" n- iWhen to Use Recursion1 m! i% I7 ]% x9 F
1 Z& ~- K' E; N5 k: z5 \) {
    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
) V2 a: ], L% @, X  K5 O; E* y1 g) `: r, b* Q8 j
    Problems with a clear base case and recursive case.- G) q3 Y6 H$ Q# S6 ]$ l

- _1 y9 P/ E# G, {1 S) A1 DExample: Fibonacci Sequence
/ K' G7 t; v1 X- [9 L" A
9 O; |9 E5 T' y% e0 ^The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:8 V# k& u- M3 T
: [. ]  d7 g8 \8 [; _
    Base case: fib(0) = 0, fib(1) = 1
7 ]+ b% [( n: R7 U' S/ [
4 [; K* R$ S3 B" b! |: o9 T    Recursive case: fib(n) = fib(n-1) + fib(n-2)
0 L1 n, T; L8 z$ V4 A8 w) n! u
7 v! U& I- u+ p* [4 cpython
3 k( N+ |, r8 Q# ?
1 I5 u/ K3 o6 u* k0 @! c/ d* e* R7 F7 X: Q7 ^
def fibonacci(n):
" l5 }; k. S# |1 C8 o7 W$ j) a    # Base cases. Q! p! y$ ^/ h0 @3 g! ?
    if n == 0:* Q. M1 w; x# z0 ?5 ?
        return 02 W2 Y0 P; _1 y4 Z0 K
    elif n == 1:) h% C* r# |% F  }
        return 1
* Z' l  _1 s3 b" Z# p    # Recursive case
- r; S8 F2 Z/ l2 P    else:" ^$ A& ]4 t; W$ r* l
        return fibonacci(n - 1) + fibonacci(n - 2)
% R$ K- ^6 S5 w' K7 o6 U# Z; K) v0 Q& X+ k
# Example usage& \# ?' F7 A; G: j  T- V
print(fibonacci(6))  # Output: 8. ^& G8 k7 f- \( ~0 o
3 |1 p/ e$ z1 U% n) s
Tail Recursion
9 A5 E$ @' M1 B0 f* B0 n# c$ R+ J5 o  U) V1 H
Tail recursion is a special case of recursion where the recursive call is the last operation in the function. Some programming languages optimize tail-recursive functions to avoid stack overflow, but not all languages (e.g., Python does not optimize tail recursion).
7 \) u5 t8 V1 Y: g" C
3 u$ ^4 m7 _4 xIn 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://129.226.69.186/bbs/) Powered by Discuz! X3.2