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 5 x# x+ t* d( @, c V5 p! B
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解释的不错$ i+ b( m+ q- f/ A
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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。 " e O0 k2 M- b% | Z0 U0 O9 C. n- I7 j9 w0 b- b" q
关键要素 a$ `0 N5 Z0 k, q! ~1 C( Z
1. **基线条件(Base Case)**2 s3 l* c' g2 D2 I9 ]
- 递归终止的条件,防止无限循环 ; ?/ V1 V9 y. t' O' P - 例如:计算阶乘时 n == 0 或 n == 1 时返回 17 W& ^, r' b5 \- B/ C) o( k$ }
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2. **递归条件(Recursive Case)**. O5 m+ y7 Q; r
- 将原问题分解为更小的子问题 : w" i" C5 ^7 ~5 k( L - 例如:n! = n × (n-1)!* V3 I# I; u" `1 K9 O% n3 s
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经典示例:计算阶乘. x3 h; ^4 ]+ h' g* S& L$ F
python & u- K* |* o1 n+ z! J/ S6 G6 Qdef factorial(n): C3 ]- `3 a* j. H2 L6 R0 r9 { q- { if n == 0: # 基线条件 6 ?% }8 L# o+ q1 |# j return 1 9 J2 ^# i' B, `' H4 A else: # 递归条件 . i8 i5 @" {5 D; T a) g' e return n * factorial(n-1) / ^2 y8 a' [; G R* V" \1 q执行过程(以计算 3! 为例):' v/ H2 b$ O' P) m4 K
factorial(3) 9 l5 a3 T: p, |/ S8 K3 * factorial(2)$ {) O7 k7 U! d$ V, }2 S. i* J1 C
3 * (2 * factorial(1)) * }5 \% v, u" R" [# D* Z3 * (2 * (1 * factorial(0)))9 m( Y# k. {; {5 @: A& S
3 * (2 * (1 * 1)) = 6 2 N) Z! C) O/ I. |1 X1 B+ I2 I3 z `4 Z9 I; R- a
递归思维要点7 U& \2 ` H( X2 \8 ?, p8 e
1. **信任递归**:假设子问题已经解决,专注当前层逻辑 $ b6 J* \" ` }2. **栈结构**:每次调用都会创建新的栈帧(内存空间) 2 t% `1 @, r( w5 m' {: [- L9 b% }3. **递推过程**:不断向下分解问题(递) 5 k1 r* a: M" k4. **回溯过程**:组合子问题结果返回(归) ) g# }* k( g" F3 @0 A8 C( @" D* i+ R6 q) n. w' _+ l. F+ I
注意事项 ) Z( B, h; X4 {; c必须要有终止条件4 Y& ]) P+ F% H9 @1 Q/ S
递归深度过大可能导致栈溢出(Python默认递归深度约1000层) : N) ~* S! v0 d, {. I& g+ o; Z5 [某些问题用递归更直观(如树遍历),但效率可能不如迭代 ; Z8 R; A' D; q0 E尾递归优化可以提升效率(但Python不支持) + E" v! A9 A# H) k$ l2 Z+ J( R1 Y2 s0 w( [7 l3 d, W
递归 vs 迭代 6 G1 I Q: L% I& C; d! h5 T| | 递归 | 迭代 | ) o& j& X1 K- K- m' i|----------|-----------------------------|------------------|7 x' N) C) w1 O5 I4 F# s! r
| 实现方式 | 函数自调用 | 循环结构 |8 S3 b" g1 z7 z# e8 f4 F" _: ^
| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 | ! F: K: D; h6 m$ k2 p4 z+ e/ d+ 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 | ) }) g H G5 g' A! k9 W| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 | 7 J' h# u- G* ^6 k e4 ~" y( r. Z% g H6 }2 d5 W
经典递归应用场景 / j# B* {; R% r" h8 \1 _) x+ R1. 文件系统遍历(目录树结构) 8 t. H m+ s" `$ c, k4 Z2. 快速排序/归并排序算法 " Z8 k5 |$ n6 L9 v3. 汉诺塔问题 % l/ \( J/ d2 K5 ~4 E0 Y4. 二叉树遍历(前序/中序/后序). n7 @; w2 u8 m8 r3 y
5. 生成所有可能的组合(回溯算法) 3 ^0 Z+ O. c6 n, v, n4 D2 _' C 4 ^3 q5 C m0 X: y. V0 ^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,+ E3 s5 d3 h" r" @& ?1 D
我推理机的核心算法应该是二叉树遍历的变种。, I5 d6 r# Y6 h) f+ {
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: : j! _% l$ h! b9 x( L. `Key Idea of Recursion ) V2 y3 n3 V' c4 l& |6 ~; ~3 j/ K8 C4 @1 c% ?5 K
A recursive function solves a problem by: ) R! ?6 H1 M2 p/ Y' v ! @% u( D; s; i1 g- S& ]) K Breaking the problem into smaller instances of the same problem. \( g! f7 I; R4 p" ~9 }
, D2 G" h2 z' J# }( E Solving the smallest instance directly (base case).& Y5 t' G5 N$ c( E8 V
k% S, Q p* ~; H4 f9 I4 S& N* e Combining the results of smaller instances to solve the larger problem. * V$ h2 \$ U2 l% O4 ]5 ~" Z0 V* Q/ g# T1 }# t# d0 n7 N0 N
Components of a Recursive Function ; ^! [; K9 w( `; z) A% d/ ? ' q, m- x3 \/ g( i# M Base Case: 5 W& X+ |- |& z R0 Y# a: M1 S ) a' ?% m2 y& G j This is the simplest, smallest instance of the problem that can be solved directly without further recursion.6 g) ]2 _6 C X# a2 ?
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It acts as the stopping condition to prevent infinite recursion. , U; F# A: J( D# `; [" s6 i% i' S0 `/ g/ G3 R; z
Example: In calculating the factorial of a number, the base case is factorial(0) = 1. / ]& j! W/ W. r; O: Y8 x; P i& |- ?" _" b' l3 B, ~" ~/ B
Recursive Case:1 ~) w, c' ~6 @. W. l! {! r! {
! A6 t2 U* D# R This is where the function calls itself with a smaller or simpler version of the problem. ( I; E8 g6 g! z& W : K1 h4 j# N# x. I( o. t+ L Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1). * E2 S2 Y6 ^* j. Y 4 P2 }5 e0 C/ m( R3 p0 y# j6 CExample: Factorial Calculation 4 {6 H5 L' G/ }8 J& u$ P& T& @; Q
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:6 p/ B9 C& R2 J$ V, U- |
8 c9 z& L/ X( W, V2 S2 m Base case: 0! = 1/ w' n: o6 Z+ j7 e4 E0 C# k
3 R6 f: P. t4 m! u! ^ Recursive case: n! = n * (n-1)!) E. A" Q4 S) o& ^ i
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Here’s how it looks in code (Python): # P' U9 f* Q$ Y) ^) f" jpython. H9 p6 l; F: P( s2 f, M) \
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def factorial(n): 8 j* F# d* I4 g# W # Base case7 |9 i* t! z1 c, k) N
if n == 0:* q4 F8 O& ], {" x* U; a
return 1: {0 F: S# ~2 L. q5 [. h/ j
# Recursive case" `9 C5 u9 S7 l. q. ]. x, @
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return n * factorial(n - 1): ?6 Q8 `3 x/ z- _5 [0 G; s! j
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# Example usage! |3 [2 U$ w" d& [8 }" Y
print(factorial(5)) # Output: 120) a7 ?4 |0 [! B$ ?2 s
" i7 \: s) _ A6 MHow Recursion Works % k9 M, d7 z+ r- V5 j* W% [% F5 t3 n. R: k& {
The function keeps calling itself with smaller inputs until it reaches the base case.) B. V, ?" Y! u' U7 ?. a
' E# T$ }# w1 N Once the base case is reached, the function starts returning values back up the call stack., L/ M5 `& s0 s3 N
[& \4 v* N/ P l% j# I7 ?/ @' F These returned values are combined to produce the final result.! |: l/ l3 v8 g9 K
8 ]/ h9 ~( e# e1 n! X 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)./ T* a+ k' l# F
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Readability: Recursive code can be more readable and concise compared to iterative solutions.$ l- k- @, w+ [
9 X n* f2 V7 @7 ^ lDisadvantages of Recursion+ i% r: Z3 q- R8 ?
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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.5 R2 M; x2 Z' E+ e ?
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Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization). # c0 Z {7 {4 [* z9 P# q& L , b3 F! U# {. Q7 F/ \7 [When to Use Recursion . h0 x E: r! b. [9 ] p$ L* l 3 |0 o p) k0 k3 T" x; M Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).: Q# W. @' j: z
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Problems with a clear base case and recursive case. * s& s( m1 L& p) r6 J3 `1 d& d5 \! }$ G0 P
Example: Fibonacci Sequence : m6 i' w/ Q) g8 M5 y1 i' ] e, \/ W% c) r9 e, H* DThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:: T: [8 e2 E# X. S" c0 y
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Base case: fib(0) = 0, fib(1) = 1$ ~. t# X3 M+ p! ~2 u7 [$ s
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Recursive case: fib(n) = fib(n-1) + fib(n-2)5 m4 ]' X/ O, ?
. c8 g7 \% C. g/ _3 G 3 i* U+ u# l1 A2 vdef fibonacci(n):' Y- S1 v0 y7 y0 B6 u) I& R$ N
# Base cases , @- U n: D0 ~( D if n == 0:" X1 `7 g6 [+ t6 P# e
return 0+ W1 m p( }- M
elif n == 1:5 a: a( A- L% x2 {
return 1, X! U$ y( i, A' P9 P* O4 P- U
# Recursive case 2 G* e- ]: {1 Z else:1 i. \# A. S( t6 ?6 O( r
return fibonacci(n - 1) + fibonacci(n - 2)# w& A4 h) ?; Y X
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# Example usage3 c* H7 l5 R" e h* v! w
print(fibonacci(6)) # Output: 8 D- x$ } N3 Q! e0 J
1 I! G* d4 z: hTail Recursion 9 t, X P' S9 S, A4 N2 T0 F- I8 n( y' I. N; a) w$ A# f
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).% K' x K. J: B" Q
' V0 y [3 P; y3 YIn 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。