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 # E2 M% e! T1 d7 q, F
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解释的不错 + m. s4 q7 x M" E3 t7 L7 z9 j2 P- x* p: C2 @
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。 % W# e4 e+ T9 m! V) k7 u2 H% [7 y5 B2 J1 `
关键要素2 t; R" J* `" I; C$ I, P# c
1. **基线条件(Base Case)**1 H) ]$ y w6 y# r G% r
- 递归终止的条件,防止无限循环( P+ g* T3 _) F; X
- 例如:计算阶乘时 n == 0 或 n == 1 时返回 1( e9 [. J# l: C8 z* v
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2. **递归条件(Recursive Case)** % h& D6 a: Q: |/ w# n! v% G) l5 ` - 将原问题分解为更小的子问题3 S9 C( p( M3 K8 s Z" t/ `
- 例如:n! = n × (n-1)!$ F5 G6 i9 p$ O0 Z
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经典示例:计算阶乘 & C% |, Z6 |+ B& i# U; G; kpython * O4 y0 `' A- \$ Y: T0 Wdef factorial(n):/ H" m. x& |# T. L s
if n == 0: # 基线条件+ h* ?* W5 C+ E% o0 I
return 13 u& ~/ p: u2 I/ l1 [
else: # 递归条件6 s0 q9 n8 H# d$ x$ k! o0 j
return n * factorial(n-1) + i! ]* v0 I, ^0 ]5 u# S9 ^执行过程(以计算 3! 为例):) ]. e1 t! m, H+ ]
factorial(3)3 i [8 n; ^. n: f$ j; u6 S
3 * factorial(2) 4 E% j2 A/ L" ~' D9 y+ I" T: q3 * (2 * factorial(1)) 3 \4 J9 s; R' b$ n' E3 * (2 * (1 * factorial(0))), ~" x0 \% c [* Z
3 * (2 * (1 * 1)) = 6' W* {; z" G- F- G# I4 P3 K
% Y: S m8 x! x2 \0 U 递归思维要点4 I" V* U8 N/ r0 x5 V4 W
1. **信任递归**:假设子问题已经解决,专注当前层逻辑! i' k. @- J1 v1 s. g8 Z4 m
2. **栈结构**:每次调用都会创建新的栈帧(内存空间); T3 e& d0 n a+ M0 T: H
3. **递推过程**:不断向下分解问题(递)7 p/ j3 z( R, z
4. **回溯过程**:组合子问题结果返回(归)3 a( z" C$ r, `1 ^* u& G% v2 {/ z( J
9 A9 w2 P* O0 e6 P1 L' s. r 注意事项' R: U; g7 M) H2 L( b4 ^0 k" [
必须要有终止条件& |1 [' d7 b& x7 U5 U+ ~
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)9 K: }9 y; {( f) ^ T. k2 {
某些问题用递归更直观(如树遍历),但效率可能不如迭代 ) }- W5 E5 m7 q4 t1 y& s- Q; L尾递归优化可以提升效率(但Python不支持) u$ \/ f x. L
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递归 vs 迭代& h4 l4 \. ^+ o4 @9 ^
| | 递归 | 迭代 |; ]3 Y K: \: @8 m
|----------|-----------------------------|------------------| ! g. C$ q) M& C0 V$ m3 s& \| 实现方式 | 函数自调用 | 循环结构 | : v& p& `& M" t7 c6 r& O| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 | * r7 g R$ J! z+ U8 B6 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 |$ ?$ f; @6 U3 g5 \2 D; {
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 |) O9 O; g O: W, T/ c4 A A
1 j+ N. j) {' h4 T7 u" v j 经典递归应用场景 6 q; S2 ~/ x' i" ]1. 文件系统遍历(目录树结构) * s" z+ a, n2 s! t2. 快速排序/归并排序算法 ( F6 A# h- r5 L3. 汉诺塔问题8 W! f( ~1 z2 ~ N+ ~
4. 二叉树遍历(前序/中序/后序); L G/ \, |' t, o7 A3 u
5. 生成所有可能的组合(回溯算法) ) q% Q3 a3 K3 i0 Q: r: ?* H) ~: C9 ?2 P7 q/ P# X
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,% `3 d7 Y8 x3 p# {, q
我推理机的核心算法应该是二叉树遍历的变种。 & U1 k" R* C+ P- F3 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:" W7 o9 }3 f5 W4 E8 A* j0 k, n) }
Key Idea of Recursion, z* I2 @! g* H8 C, p/ j
2 \7 q! L0 B7 u3 o1 z6 e6 fA recursive function solves a problem by:4 w. I& a `* q) V8 q
* f+ B6 z$ L3 s# w' d Breaking the problem into smaller instances of the same problem.8 D$ n. o- E& J4 f7 d& _* U8 M$ h
# |4 R. f* T6 B: s% l) V; S Solving the smallest instance directly (base case).! D: l1 c+ e& P, L. O* v
% T7 P6 i+ {$ f+ H( E6 Q8 [* C8 m Combining the results of smaller instances to solve the larger problem. 0 f2 Q: h8 O8 t& e* ]6 X2 k5 ^% M+ @. f4 H c5 P+ D
Components of a Recursive Function/ m% T6 q( F% J8 m
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Base Case: . x% n/ u6 g3 } E: |: y4 S) v" m: @8 Z. L" @" }& N/ B$ k6 F
This is the simplest, smallest instance of the problem that can be solved directly without further recursion.: p! H" ?4 z* v* Q- V
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It acts as the stopping condition to prevent infinite recursion. & z% u$ I" l5 r9 [: M' ?9 I. |$ S3 l 3 ?5 I5 r+ i) o6 @) K ` Example: In calculating the factorial of a number, the base case is factorial(0) = 1. ' W2 A( M+ w: v# A * e9 _3 t3 q% Z- Z& U( z Recursive Case: 7 V% e' D9 Z, O+ E! Q4 ~9 e4 r$ n0 D& g q/ g2 x8 b, ^5 g
This is where the function calls itself with a smaller or simpler version of the problem. ; h8 N+ V" u7 L8 v' W" j$ p! s6 k4 t; a % T( R3 O. d& ~, k Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).' R; w6 I! T. j$ z) i$ K' n/ Z
; p$ P: X- A# ~+ e/ NExample: Factorial Calculation- b6 E; r! c% a: T. L
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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: & }) P- e/ t% L4 c: r * V: V2 F; Z# u* }$ b( q Base case: 0! = 1 # }$ |# o4 I- g& T4 p3 H 3 K ~2 q2 ^2 e% f* U; R% H& d Recursive case: n! = n * (n-1)!/ K. y# m' A8 c9 X: P
( m/ Y: `$ D: U: q* THere’s how it looks in code (Python):5 y n2 r% A! o) N
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# S4 T% c/ r& e( s + ~! Y+ p! A+ D1 odef factorial(n):9 e# U. X+ _+ q3 ]
# Base case! {: W! b3 C/ E$ R- P. l& _% x j0 X' Z
if n == 0: ) w+ P- W4 b, R' n. _) z return 1 , z7 P6 z7 R- o r+ T) Q+ r+ w2 ~ # Recursive case + l8 l; f2 k, \( M' @; } else: $ d* F+ G* a [6 u' i1 [1 b return n * factorial(n - 1) + K/ I0 x% B b9 N+ E1 _ 5 p/ l/ A6 i1 o8 \8 @. P6 p1 I# Example usage ( J2 \9 V2 M5 \/ H1 Q' hprint(factorial(5)) # Output: 120 - A4 }, e! n- {2 p2 Q4 @1 Z - h8 u; ?* O; ~- O7 s$ fHow Recursion Works 2 y$ W) j, n9 h# a1 q, C1 d/ w ' P5 v/ C# z- \ The function keeps calling itself with smaller inputs until it reaches the base case.& L0 a; q7 I# L- h
4 i6 a* J' c' K; C- T7 n Once the base case is reached, the function starts returning values back up the call stack.; Q7 P, `: a" b/ F# c7 `4 |3 U
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These returned values are combined to produce the final result. 6 V3 N; H# B- E! G 1 }$ R' a$ Z* d) ]) eFor factorial(5):$ @9 T0 F$ Q6 g7 N8 d0 U
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$ n9 t* }; l3 C! L/ @factorial(5) = 5 * factorial(4) % h2 k3 Z7 v; `9 p4 ^factorial(4) = 4 * factorial(3)/ g8 A* `" g1 Z; s2 L8 x W7 S
factorial(3) = 3 * factorial(2) 0 }6 w" S) N6 |" J: ^9 |6 kfactorial(2) = 2 * factorial(1)& G# e. i8 n# y' C" [
factorial(1) = 1 * factorial(0)2 g+ D6 V" M5 X$ t4 r5 x
factorial(0) = 1 # Base case4 O7 [" S. j8 E$ `# q- y
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Then, the results are combined: 7 h. @# N+ T$ z2 X0 |% i$ o; O: d8 V2 c0 _0 |2 k. E
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factorial(1) = 1 * 1 = 1- {7 o* L, i, F$ V$ I' b6 A
factorial(2) = 2 * 1 = 2 3 m0 O1 ~$ `+ Afactorial(3) = 3 * 2 = 6 / m+ e2 J$ v% v' Gfactorial(4) = 4 * 6 = 24 * I" n+ ] L: d2 h* K9 P0 ffactorial(5) = 5 * 24 = 120 : ^: K' C$ C& a8 l: m* t 8 H+ _7 y/ C' |, T8 k7 S) q G7 ~Advantages of Recursion) H! F7 V. f, u5 |5 K
/ U" S2 L, y5 g! g7 `: Y9 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). 0 ~$ W$ u# t6 ^4 c- `+ r * e$ X7 Y' d2 N9 ~" ^3 z5 ~. m8 I Readability: Recursive code can be more readable and concise compared to iterative solutions. : t- F* A! p2 c2 e% A; o8 ^& S# w1 a2 ~( N! [
Disadvantages of Recursion 9 h+ _+ s) [, N K" o# Q6 ?* i+ U9 d% D3 @6 J B
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. , O* M$ z! B7 P; V8 l ; A- l4 Q/ g; T( K6 Z% O Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization). ) j1 _' u6 g. S/ L# U9 E5 e$ i, F9 c L- V' v) A
When to Use Recursion $ m' a' h2 X, G) E: N Y$ C% x# K- V
Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).: e! j$ N$ i4 ^7 h: H8 |2 m
7 D [4 m9 H T0 {$ X$ ?- X Problems with a clear base case and recursive case.7 U9 t9 d/ s$ t7 O2 r0 D/ {! ]
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Example: Fibonacci Sequence2 u* E8 f/ d; E" b S
* ^) J* l1 _9 u2 o1 eThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:/ q- \3 i5 B4 T8 |! A
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Base case: fib(0) = 0, fib(1) = 1) N* G: v# G4 V0 j9 y" j1 R
$ e2 p/ N& G) y' V8 i YTail 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).5 Z4 x- p/ W( {2 r3 Y
9 T9 y( A% g% V( m( ]' s! V, ?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。