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 & J# {- a, L$ c5 L8 D$ r. S, v
0 \* j1 U/ q# _# `8 v0 q F解释的不错' J1 ~2 c7 k X( f6 i
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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。4 R; e2 a5 i; Q- C: F
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关键要素 & E1 g( Z- e$ V1. **基线条件(Base Case)**) \4 I' f v2 E) H; r
- 递归终止的条件,防止无限循环" d( m0 O; C6 G! k+ S
- 例如:计算阶乘时 n == 0 或 n == 1 时返回 15 Q& E% k2 C5 P6 @# N3 {+ W! [( P
- ^0 j8 G9 V& u7 F% _2. **递归条件(Recursive Case)**+ h4 N3 J( x, J0 j- K. Z9 F
- 将原问题分解为更小的子问题1 N! B( ~( Y5 }9 a: r' b* K
- 例如:n! = n × (n-1)!" I' I- l7 w; T8 X% R
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经典示例:计算阶乘 6 V. a. k y# \2 }& u# _: [python 4 }+ W/ l7 R* B# E$ l4 k" Ndef factorial(n):9 r& I. N& t u+ P3 K/ D8 V
if n == 0: # 基线条件8 ^1 e9 ~# Z1 B2 V( f5 n y
return 1# @( e1 h/ a2 @5 b! j$ ~9 i/ M$ w1 Q
else: # 递归条件, E) G( M# q- }4 u
return n * factorial(n-1); H5 ?& k$ l C' A
执行过程(以计算 3! 为例): ; T/ k3 y" ^6 h1 x9 ufactorial(3)9 p6 C6 ^# M4 {
3 * factorial(2)' R7 e6 a( w0 M0 v+ @
3 * (2 * factorial(1)) e7 W: v ?5 z- ~3 * (2 * (1 * factorial(0)))0 f/ t6 O+ h9 d2 k; D5 X& t
3 * (2 * (1 * 1)) = 66 f& N- H1 B) S0 o: A3 O% K: t' S
, ]2 g% _ k* t& H 递归思维要点8 ]' z) E! X6 Q5 W
1. **信任递归**:假设子问题已经解决,专注当前层逻辑8 L7 C% ?8 M' U3 n; n
2. **栈结构**:每次调用都会创建新的栈帧(内存空间) : n+ U+ w+ c, D$ \6 Z* x! R3. **递推过程**:不断向下分解问题(递)5 I/ l+ x4 `# b& ?7 L
4. **回溯过程**:组合子问题结果返回(归)6 B8 U" T$ U6 N
/ o. F; s1 F8 I1 r7 k( r8 v3 F 注意事项) W- \7 x. m6 P1 O+ _- l4 w: s
必须要有终止条件 0 |* G: ^* u- I7 d递归深度过大可能导致栈溢出(Python默认递归深度约1000层)% s7 c% G; k- a/ l0 W$ R
某些问题用递归更直观(如树遍历),但效率可能不如迭代 # U5 ^+ z X2 q尾递归优化可以提升效率(但Python不支持) ! Y1 b; _6 A( t + l2 q# `% K/ F& \ 递归 vs 迭代; p Y9 w) a' W% Z' p
| | 递归 | 迭代 |0 G7 L* t3 `+ f- G8 C- O
|----------|-----------------------------|------------------|8 e3 Y$ T. k# @- X
| 实现方式 | 函数自调用 | 循环结构 |! B; ~6 d6 r) p6 ?
| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 | 5 s: P( s: x9 Z0 |( d0 _| 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 | 5 c, O. R( N7 E! m| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 |- o3 b- F2 _8 w! A* t
# W g9 s" k4 d" u8 Z 经典递归应用场景 8 g; `2 c% n0 H# K0 j) @1. 文件系统遍历(目录树结构) % K9 M2 W- G; h4 p) q2. 快速排序/归并排序算法 + J( k E Q, g/ u- s/ d; p3. 汉诺塔问题( R/ o; B1 k6 ], {: r
4. 二叉树遍历(前序/中序/后序)/ |# D, |, X7 M+ i& E `
5. 生成所有可能的组合(回溯算法)4 ^9 z/ P4 b9 \$ K- O! V
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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, / o3 }4 N# f3 j- O1 \) R* C我推理机的核心算法应该是二叉树遍历的变种。. p( J1 z) \* Z- B% z+ d
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: , c% y5 `. ?6 ~: d8 S) Z3 }8 \' uKey Idea of Recursion 0 }: z/ m( A$ X1 p8 } 4 h& V- K; h2 G; N p6 QA recursive function solves a problem by:! l! J8 v( s5 D
( F- T4 h* h- R* [+ A7 `1 J a Breaking the problem into smaller instances of the same problem. 9 H# X# b; q) k* J5 \1 e3 A# S4 ~* J4 e# e- { e
Solving the smallest instance directly (base case). 0 p8 M3 E. U6 i- T# Q. n) H* [0 J) E9 O
Combining the results of smaller instances to solve the larger problem. 9 x0 D: t8 V7 c# a9 s3 | 1 q4 F |5 R0 H& ~# Z1 J7 Q# kComponents of a Recursive Function' [ H, Q% F& E' Z$ r0 m; c
; n4 [8 q0 ]$ e7 ]: l Base Case: % }* U! Z7 n6 O8 m% S+ Q; n' `( C / ~; p% m( D9 x7 t' y8 B This is the simplest, smallest instance of the problem that can be solved directly without further recursion.& _, H9 H3 T& T: T% s
; V- V- ^* m5 P z7 E! g! O It acts as the stopping condition to prevent infinite recursion. * @7 |9 u4 N9 Y$ N" t& R; ]0 z- h3 x9 k
Example: In calculating the factorial of a number, the base case is factorial(0) = 1.: }( e! E9 Y6 K0 |/ `+ ]
1 b' I5 h' R9 d& k5 s) S4 j C Recursive Case:" j0 ?5 t/ ]8 V4 ~4 j8 n; z# b+ K
; H4 d: o6 v4 K9 r This is where the function calls itself with a smaller or simpler version of the problem.2 I: h) L. v) {$ | R% h( D
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Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1). $ w# [! t8 w9 k W0 R0 X. p% ?9 VExample: Factorial Calculation; J+ U* Y! W! b. @7 P0 N
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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:6 ^- B# T. I. I! H6 y7 E
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Base case: 0! = 1* @- R9 F# r4 K* h! W, N3 ~
! E, O% P2 z7 k Recursive case: n! = n * (n-1)! ) {, t( s" f' M" j' H$ P " o* f8 n7 r. B3 H$ M# Z& eHere’s how it looks in code (Python):1 J2 j% \9 Q4 O" Z3 C6 {+ \
python 0 C. p" S" p8 X4 j ) N9 k* s# ]; o2 ~8 C/ @- v: A5 P/ j: M0 I& A
def factorial(n):/ }6 _% ?# m9 S1 x
# Base case / j! R- s- W9 Z6 t; P/ ~ if n == 0:# q' v0 E% w9 i: k( H+ n
return 1 7 `/ p% X$ b9 t3 c # Recursive case $ l! R& x- S! [$ P; z; O else:9 g- V% ]$ G' ]" E$ Y
return n * factorial(n - 1)) U/ R: E7 w2 L- d+ v
2 M% o0 D. ^% k# w# Example usage5 H2 W" @4 w* `; y3 p( r% [9 y* `9 v
print(factorial(5)) # Output: 120 0 Q. r0 d0 L1 K9 d) V8 A , [8 ~ I% t4 b4 L- ?2 v; B2 ?How Recursion Works # y+ J9 q* ?( ?- @' C3 r. U, q 8 h/ w" m' }0 l2 Y. Z2 u1 s$ H The function keeps calling itself with smaller inputs until it reaches the base case. 5 c# e1 g) l9 G$ T9 \9 ]0 A4 R; ~4 }+ W& H' @2 E
Once the base case is reached, the function starts returning values back up the call stack.) w4 f. M8 F# ^) C% X8 b
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These returned values are combined to produce the final result. 4 n& v+ j: G" ]5 \ * B9 S4 R9 U* j/ d7 |For factorial(5):6 X" T; z: u P7 l1 P: T
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factorial(5) = 5 * factorial(4) + \$ w+ n7 a; ^* Z* Nfactorial(4) = 4 * factorial(3) 7 z7 i! e- N+ G1 V$ F% Qfactorial(3) = 3 * factorial(2) + h4 R# P( O! w/ q- ofactorial(2) = 2 * factorial(1)0 e$ `; {* G$ `; V( O1 L
factorial(1) = 1 * factorial(0)8 |2 D _9 S! C$ m
factorial(0) = 1 # Base case * o; r$ s# I7 L& q& M* o 0 T* V4 S, G9 c V4 n/ OThen, the results are combined:# \1 u- U8 O/ M3 I
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3 d6 H: Y. Y P: |factorial(1) = 1 * 1 = 14 W# z) M( c( _! x& p6 k. Q* [
factorial(2) = 2 * 1 = 2* v7 B& e' }8 }/ e2 m& ?
factorial(3) = 3 * 2 = 69 r+ L" M3 B/ B, a$ `5 U1 ? O% L
factorial(4) = 4 * 6 = 244 r+ b& m9 u Z7 U7 h! m
factorial(5) = 5 * 24 = 120 b& D% G+ e8 X. g3 G1 A& _1 Q3 T
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Advantages of Recursion . {/ j6 p. X; M4 h8 l5 K& j! U6 y! q( L& f. l( D0 A$ R/ R! S
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). : U2 O+ U2 ?- s w% M/ L 9 K G2 k6 a, d! | w& ]! w Readability: Recursive code can be more readable and concise compared to iterative solutions.. Z, U8 \ ?+ S1 a K8 M
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Disadvantages of Recursion : o" V4 n! o0 T7 ] T7 u& B- w/ D; a5 e/ m; U; E' V0 M4 y, _/ 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.: e. s/ Y1 ?" a5 Z6 |0 q7 c
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Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).& R: d C+ W) }% q ]1 w
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When to Use Recursion 9 E! m q% W9 `# B$ \0 V 2 t6 ]+ n8 ], m* z3 W/ d Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort). 8 _4 T# d Q0 ^8 ?- t, @; x + \( y% C# @8 V" _* g: e Problems with a clear base case and recursive case. 7 A* [4 s8 f/ ?4 F+ s N0 Z8 v ) e+ D7 R$ M% ~" [Example: Fibonacci Sequence 2 ~ T7 H) }3 a* l* l5 `9 K0 P % Z/ r" M6 I _& Y( ]- I* k8 oThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones: ) }1 ]! C, _; [6 G" E. S6 O * h8 p2 A x) a4 U: J. K- F Base case: fib(0) = 0, fib(1) = 1' I- J1 b/ M! B) Q& A
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Recursive case: fib(n) = fib(n-1) + fib(n-2) 5 x# F# _ l' R) ^ # i: W. {8 `3 ^4 A- [3 Kpython+ B+ H5 r/ w) r" w# V; _( H
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& w8 T4 t4 J7 d5 X* y. k7 z: |def fibonacci(n): , s( e3 I; ]* l+ \9 P: r # Base cases/ M9 P$ ?9 h% ~' X3 t& s
if n == 0: 6 i4 M- g; N# Y. C& X! j1 P+ T return 09 P, r5 |4 Q2 i8 l7 {8 l. p
elif n == 1: 9 k- {6 }; x: z3 L2 e5 a return 15 u% s, N- K; o( I- {7 o3 ~4 B
# Recursive case 6 Z+ @3 r" N. {+ a9 ` else: + C# r% o! Q8 S8 W return fibonacci(n - 1) + fibonacci(n - 2)$ K0 K6 C/ d# Q( O
+ l5 v$ l5 j" ?" C. u( \# Example usage 0 [* r4 P, ]' p0 {1 Gprint(fibonacci(6)) # Output: 8 , r X1 O9 |' o# A& |: H! P " B! N' i' a4 s5 a& vTail Recursion% H6 V" |* ^0 B8 u& ]9 X+ S1 H
. r, n. s% `% n( pTail 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 M z) P4 X ~. A" K/ ~3 O1 _
9 q1 K3 {8 m0 J9 i8 ~, M8 hIn 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。